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The roles of spatial heterogeneity and adaptive movement in stabilizing (or destabilizing)
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simple metacommunities
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Lasse Ruokolainen1,3, Peter A. Abrams1*, Kevin S. McCann2, Brian J. Shuter1
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ON, Canada M5S 3G5
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Canada N1G 2W1
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Department of Ecology and Evolutionary Biology, University of Toronto, Toronto,
Department of Integrative Biology, University of Guelph, Guelph, ON,
Current address: University of Helsinki, Viikinkaari 1, PO.Box 68,
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00014 University of Helsinki, Finland
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* corresponding author: peter.abrams@utoronto.ca
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Running head: Adaptively moving consumers
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Abstract
Adaptive consumer movement and between-patch heterogeneity have both been suggested to
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reduce population fluctuations in spatially subdivided systems. These conjectures are explored
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using models of two-patch consumer-resource system with fitness-dependent consumer movement
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in which at least one of the patches in isolation exhibits consumer-resource cycles. Neither
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conjecture applies generally. Under relatively low heterogeneity, highly accurate and rapid adaptive
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movement most often increases both the between-patch correlation of density and the variation in
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the total density of both species compared to a similar system having a low rate of random
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movement. However, such adaptive movement can decrease between-patch correlation and global
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population variability when: (1) the consumer's movement is moderately sensitive to fitness
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differences and heterogeneity is relatively low; or (2) one of the patches would be stable in
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isolation, and the stable patch supports a sufficiently large consumer population. In both cases, the
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dynamics are typically either a stable equilibrium or a simple anti-phase cycle with low variation in
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total population size. Under adaptive movement, population variability is often lowest for
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intermediate levels of heterogeneity, but monotonic increases or decreases with increasing spatial
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heterogeneity are possible, depending on the fitness sensitivity of movement and how the
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characteristic that differs between patches affects within-patch stability and population size. High
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rates of random movement can lead to greater stability than does adaptive movement when
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consumers are very efficient.
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Keywords: consumer-resource interaction; dispersal; foraging; metapopulation; patch dynamics;
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predator-prey system; stability; synchrony; variability
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Introduction
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Spatial heterogeneity is regarded as an important determinant of the dynamics of metacommunities,
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where habitat patches containing interacting species are connected by movement of one or more of
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those species (Holyoak et al. 2005; Goldwyn and Hastings 2009). The nature of movement is also
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thought to have an important effect on metacommunity dynamics (Rosenzweig 1991; Blasius et al.,
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1999; Leibold et al. 2004; Holyoak et al. 2005; Koella and Vandermeer 2005; Abrams 2000, 2007,
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2008; Valdovinos et al. 2010, and many others). These two factors, heterogeneity and movement,
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have long been known to interact in single-species metapopulations. For example, in heterogeneous
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environments with stable dynamics, random movement produces sources and sinks (Pulliam 1988)
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or pseudo-sinks (Watkinson and Sutherland 1995), where fitness differs spatially. However,
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adaptive movement (habitat selection) is thought to produce ideal free distributions, implying that
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fitness does not vary spatially (Holt 1985). In single-species metapopulations with fluctuating
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population sizes, random movement has been regarded as a synchronizing force, driving increased
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variation in population size (Hanski 1991; Ranta et al 1995). In contrast, heterogeneity is believed
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to reduce temporal variation of population sizes in systems with movement (Hoopes et al., 2005).
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At least one single-species model has suggested that adaptive movement away from patches with
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low fitness has a much greater stabilizing effect than random movement (Ruxton and Rohani 1998).
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The interaction of heterogeneity and movement of different types is understood far less well
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for metacommunities than for metapopulations. This is true even for the simplest community units,
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consisting of a consumer and a resource with simple diffusive (random) movement (Briggs and
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Hoopes, 2004). In spite of growing evidence for adaptive aspects of most animal movement
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(Bowler and Benton 2005), the vast majority of the literature assumes random movement. Those
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studies that have examined adaptive movement have generally either assumed that an ideal free
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distribution is attained instantaneously (reviewed in Morris 2003), or have modelled movement
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implicitly, using a switching function (Post et al 2000; McCann et al. 2005; Rooney et al. 2006),
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which also implies instantaneous responses of consumers to resources. Models with instantaneous
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behavioral responses often do not have the same dynamics as those where movement is represented
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explicitly as time-dependent flows of individuals (Abrams and Matsuda 2004; Abrams 2000, 2007;
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Schreiber et al. 2006; Amarasekare 2007; Rowell 2009). Thus, models that take the latter approach
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are needed to better understand metacommunity dynamics in nature, where instantaneous responses
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do not exist.
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This article investigates the interacting effects of spatial heterogeneity and adaptive movement
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on metacommunity dynamics using a model in which individuals of a single consumer species
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move between two non-equivalent patches with distinct resource populations. One or both patches
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are characterized by a predator-prey (consumer-resource) cycle. The resulting temporal variation in
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fitness favors consumer movement that increases fitness. Random movement represents a limiting
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case of adaptive movement, and we investigate how this special case differs. The two-patch
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consumer-resource systems considered here represent the simplest possible metacommunity
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(spatially extended food web), and this framework is also a good approximation of many biological
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communities. For example, aquatic systems can often be divided into pelagic and littoral or benthic
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and pelagic zones; terrestrial systems can often be divided into above- and below-ground
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components. Other systems that can be represented as two patches connected by movement by the
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top trophic level are described by Rooney et al. (2006, 2008). Temporal variation in such systems
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may also arise from purely environmental forcing rather than predator-prey cycles; this case will be
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considered in a future article (see also Abrams 2000). Abrams and Ruokolainen (2011) previously
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analyzed consumer movement in the homogeneous-patch version of the system considered here.
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Our analysis of system dynamics focuses on the related characteristics of between-patch
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synchrony and population variability. Synchrony (used here to mean a high positive temporal
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correlation of the densities of different populations) is important in part because it usually reduces
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the probability of persistence and increases the variability of population sizes (Heino et al. 1997;
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Holyoak 2000; McCann et al. 2005). In addition, synchrony allows the use of unstructured
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population models to describe metapopulations, and leads to responses of the global population
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densities to changes in environmental parameters that often differ from those of asynchronously
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fluctuating systems (e.g., Abrams 2007, 2008). Variability (or conversely, stability) affects almost
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all aspects of the dynamics and evolution of species. Between-patch heterogeneity has been
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regarded as a major determinant of both synchrony and stability (e.g. McCann 2000). However,
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different results have been obtained using models with different assumptions about movement, as
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noted above. Limited experimental evidence on synchronization in systems with adaptive top-
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consumer movement is available, but the degree of heterogeneity in the system is generally not
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known. An example is provided by Ims and Andreassen's (2000) study of vole population
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dynamics, which argues that the (presumably adaptive) movement of avian predators synchronizes
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spatially separated vole populations with cyclic dynamics. This and other observations of spatial
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synchrony over large areas (e.g. Ranta et al. 1995) probably involve patches with at least some
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degree of spatial heterogeneity, but spatial differences in interaction-related characteristics have
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generally not been measured.
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Our analysis of consumer-resource models represents movement as a dynamic process, and
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examines the impacts of different types of movement and different levels of heterogeneity on the
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dynamics of a simple system. Results suggest that the amount of heterogeneity, the rate of
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movement, and the degree of fitness-dependence of the movement have inter-dependent effects on
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system dynamics. Among our findings are: (i) adaptive movement can greatly reduce variation in
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systems with intermediate levels of heterogeneity, particularly when one of the two patches would
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be stable in isolation; (ii) a low rate of random movement produces the greatest stability and
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asynchrony when patches are relatively similar (low heterogeneity); (iii) high rates of random
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movement often produce the greatest stability when systems are very heterogeneous and consumers
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are very efficient. Our analysis shows that adaptive movement in slightly heterogeneous systems
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produces very different dynamics than in perfectly homogeneous systems (Abrams and
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Ruokolainen 2011; see also Goldwyn and Hastings 2009).
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Models and analyses
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Models
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The central component of the models presented below is the movement function.
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"Adaptive" movement implies that the per capita movement rate at all times is greater towards the
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patch currently characterized by a higher per capita birth minus death rate. The probability that an
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individual moves in a given (short) time interval is greater when the fitness gain from movement is
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greater. These characteristics have been incorporated into most previous models of adaptive
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movement in metacommunities (MacCall 1990; Schwinning and Rosenzweig 1990; Abrams 2000,
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2007; Abrams and Matsuda 2004; Schreiber et al. 2006; Abrams et al. 2007; Amarasekare 2007,
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2008, 2010). Here most of our analysis is based on a fitness-dependent movement rule used and
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discussed in detail in Abrams (2000, 2007); the model corresponds to a similar analysis of
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homogeneous systems (Abrams and Ruokolainen 2011).
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We assume that a single consumer species moves between two patches, each supporting a
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single, self-reproducing resource population which does not move between patches. This model
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could apply to a single or two spatially isolated resource species. The dynamics are defined by,
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dRi
CNR
 Ri  ri  ki Ri   i i i
dt
1  Ci hi Ri
(1a)
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dN i
 N i wi  mN i exp[ (w j  wi )]  mN j exp[ (wi  w j )] .
dt
(1b)
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wi 
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These equations describe resource (R) and consumer (N) dynamics (Eqs. 1a, b) in patch i, with
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movement depending on the between-patch difference in instantaneous consumer fitness. Equation
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(1c) gives wi, the fitness in patch i, which is the instantaneous per capita birth minus death rate. As
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in Abrams and Ruokolainen (2011), the form of the logistic resource growth in Eq. (1a) separates
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the per capita resource growth rate into a maximum rate ri in patch i, and a per capita density-
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dependent reduction in that rate of ki. The equilibrium resource density in the absence of
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consumption (its 'carrying capacity') is ri/ki. Resource consumption is described by Holling’s type
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II functional response with an attack rate Ci and handling time hi. Consumed resources are
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converted to consumer biomass with an efficiency of bi. The parameter di defines a constant per
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capita mortality (or biomass loss) rate of the consumer.
biCi Ri
 di
1 Ci hi Ri
(1c)
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The parameter m is the baseline per capita movement rate to the other patch, which applies
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when the difference between the consumer's per capita birth and death rates is identical in the two
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patches. This baseline rate is modified by an exponential function of the fitness difference between
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habitats, exp[λ(wj – wi)], with λ  0. If λ = 0, movement between patches is random with a constant
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per capita rate of m, while λ > 0 leads to an accelerating movement rate with an increasing fitness
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difference. Larger λ (the ‘fitness sensitivity’ of movement) increases the movement rate towards a
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better patch and decreases movement rate to a poorer patch, but has no effect when the patches
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confer equal fitness. Thus λ reflects both the accuracy of fitness discrimination and the rate of
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responding to perceived differences. Previous studies have shown that other movement functions in
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which the rate increases with the fitness difference produce dynamics that are qualitatively similar
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to those produced by the exponentially increasing function of Eq. (1b) (Abrams 2007; Abrams and
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Ruokolainen 2011). Some alternative movement rate functions as well as alternative population
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dynamics within a patch are discussed in the 'Alternative Parameters and Models' section below and
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in Online Appendices B and C.
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Analysis
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The main goals of the analysis are to determine how the level of system-wide population
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variation and the between-patch synchrony are related to heterogeneity, and how this relationship
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itself is changed by the two parameters of the movement function. Achieving these goals is
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complicated by the fact that there are many ways to be heterogeneous; patches may differ from each
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other in any or all of the six parameters that define population dynamics with a patch. Random
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movement with  = 0 is already known to produce different dynamics in predator-prey
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metacommunities given different rates, m (Wilson et al. 1993; Jansen and de Roos 2004).
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We make a number of assumptions to limit the range of systems (parameters) examined.
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First we assume that the conversion efficiency, b, and handling time, h, are independent of the
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patch; in systems with a single resource species these are less likely to be affected by spatial
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location than are the other parameters. This leaves four parameters whose values may differ
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between patches; r, k, C, and d. We focus on systems in which patches differ in only one of these
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parameters, although some additional cases are presented in Online Appendix C. Because we are
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interested in systems where the relative values of the two patches to the consumer species vary
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temporally, we restrict our consideration to parameters that produce consumer-resource (predator-
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prey) cycles in at least one patch when the patches are isolated from each other. The impact of
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heterogeneity may depend on the type of cycles that occur in one or both patches. The form of the
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cycles produced by this Rosenzweig-MacArthur (1963) system in a single patch depends on three
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quantities (Abrams et al., 1998): the distance of the parameters from their stability threshold value;
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the relative rates of consumer and resource dynamics (relative ‘demographic speed’); and the
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proportion of a consumer's time spent handling resource when the resource is at its carrying
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capacity. To measure the first determinant of cycle form, we use the proportional difference of the
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mean value of the spatially variable parameter from its value at the stability threshold (Hopf
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bifurcation) value in a single patch. For the second determinant we use maximum per capita growth
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rate as a measure of demographic speed in each species. For the third determinant, the expression
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Chr/k gives the ratio of time spent handling to time spent searching for resource when the resource
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is at its carrying capacity. A similar analysis of systems having equivalent ('homogeneous') patches
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has shown that the first of these three determinants has the largest effect on dynamics (Abrams and
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Ruokolainen 2011). That generalization also applies to the heterogeneous case that is explored here
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(see Online Appendix C). Thus, we concentrate on exploring the effects of the distance of the
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consumer death rate from its stability threshold value.
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Numerical methods are described in Online Appendix A. We determined the coefficient of
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variation (CV) of the global (2-patch) population of each species and calculated the Pearson
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correlation coefficient of the patch-densities for each species. In most cases, the CVs of the two
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species changed in the same direction with a change in parameter value, permitting an unambiguous
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determination of the qualitative effect on the variability of the populations (and the corresponding
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opposite effect on stability).
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Because random movement is a special case of adaptive movement, we begin with, and
devote most of the analysis to adaptive movement. The analysis of these systems addresses two
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types of questions; how the fitness sensitivity of movement affects dynamics, and how
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heterogeneity in different parameters affects dynamics. Most of the figures are based on a set of
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'reference values' of some of the population dynamical parameters in a homogeneous system. These
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are intermediate values that display most of the entire range of qualitative responses to changing
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patch heterogeneity and movement parameters that were observed in the entire set of numerical
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results. The population dynamic parameters used in the reference model are based on a
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homogeneous-patch system with r = 1; k = 1; C = 1; h = 3; b = 0.25; and d = 0.02 (analyzed in
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Abrams and Ruokolainen (2011)). The values of r and k can be removed from the homogeneous
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patch system by scaling, so their mean values across the two patches are set to unity. The impact of
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different mean C values is similar to changing combinations of the other consumer parameters, so
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the mean C is also fixed at unity. The value of h is intermediate between the minimum (h > 1)
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required to have cycles (given r = k = C = 1), and a value of h = 10, which is an order of magnitude
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larger than that minimum. The value of b is within the range of observed conversion efficiencies; it
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yields a maximum consumer per capita birth minus death rate that is more than an order of
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magnitude less than that of the resource in the homogeneous case (r = 1 for the resource and
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bC(r/k)/(1 + Ch(r/k))  d = 0.0425 for the consumer). Given these parameters, d = 0.02 is roughly
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one-half the maximum d that yields unstable dynamics. We explore the impact of different mean
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death rates, which results in larger or smaller amplitude cycles (for smaller and larger death rates
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respectively).
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We assume a baseline movement parameter m = 0.0005 for the 'reference parameters'; a
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small m is appropriate for modelling adaptive movement, because large values of m would imply
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significant rates of movement to the poorer quality patch when the fitness difference is small.
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Larger m values are treated below in the 'Alternative parameters and models' section. We are
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interested in behavioral movement, so it is also appropriate to restrict attention to cases where the
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maximum per capita movement rate is greater than the maximum per capita birth minus death rate
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within a patch; i.e.,  such that mexp[(bC(r/k)/(1 + hC(r/k)))] > bC(r/k)/(1 + hC(r/k)) – d. The
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minimum  we consider in any analysis of adaptive movement satisfies this inequality or is > 1,
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whichever value is larger. In general, we then examined larger values of  up to a point where the
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basic measures of system dynamics (correlations and coefficients of variation) ceased changing
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greatly with further increases in . In all cases, the maximum  allowed large changes in
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proportional patch occupancy during time intervals with very little change in total consumer
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population size. Most results below assume  = 500; for the reference parameters this often implies
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that the theoretical maximum per capita movement rates could be ten or more orders of magnitude
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greater than the maximum per capita growth rate. However, such rapid movements are never
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realized on the attractor because adaptive movement prevents large between-patch differences in
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fitness from developing. In general, the most rapid realized movement rates when  = 500 under
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the reference parameter set are between two and three orders of magnitude faster than the largest
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possible per capita growth rate. Thus, an alternative movement function (discussed in Online
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Appendix C), which imposes a cap on the maximum movement rate in Eq. (1b), results in limiting
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dynamics essentially identical to those based on Eq. (1b), provided the maximum movement rate is
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at least on the order of 100 times larger than the maximum per capita growth rate. Vertebrates that
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live many years may change their foraging location on a daily basis, so the realized movements
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implied by  = 500 are biologically plausible for many species.
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Our analysis begins by illustrating the qualitative types of population dynamics most
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frequently observed in our simulations of Eqs. (1). We next show how the fitness sensitivity
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parameter  influences patch synchrony and population variability for one (representative) type of
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parameter heterogeneity using the reference set. The bulk of our treatment of the reference
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parameter set then examines how each type of heterogeneity affects synchrony and variability when
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movement is accurate and rapid. The remainder of the analysis of heterogeneity seeks to determine
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whether these results are altered by different parameter values. This is followed by a comparison of
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random and adaptive movement. Many of the numerical results are relegated to the Online
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Appendices.
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Results for the reference parameter set
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General dynamical patterns
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The majority of the dynamics observed for various parameters can be classified as belonging
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to one of the four basic patterns shown in Figure 1. This is not an exhaustive or mathematically
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rigorous classification, but it does help to understand the results that follow. The first type is
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characterized by cycles in the two patches having identical periods, but significantly different forms
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and variances (Fig 1A). Here, the period and amplitude of cycles in the patch with the higher-
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amplitude intrinsic cycle are similar to what they would be in the absence of the second patch. A
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second common type of dynamics (Fig. 1B) is characterized by simple short-period cycles with
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consumer populations in the two patches fluctuating close to a half period out-of-phase. These
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approximately anti-phase cycles of consumers primarily reflect movement; the total consumer
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population exhibits little temporal variation. Near anti-phase dynamics are representative of most
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cases where adaptive movement has a large stabilizing effect. A third pattern (Fig. 1C) has small
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amplitude anti-phase cycles superimposed on a longer-period predator-prey cycle, which is nearly
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in-phase across the two patches. The fourth type of dynamics (Fig. 1D) exhibits fluctuations with
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high positive correlations between the patches for both species. However, the consumer population
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occasionally displays rapid movements to one or the other patch when the difference in fitness
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becomes large enough. (The sharp spikes in consumer numbers within one patch concurrent with
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spikes in the opposite direction for the other patch reflect rapid movement.) Dynamics like those in
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Fig. 1D are characterized by a between-patch resource correlation significantly more positive than
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those of the consumers. The dynamics in Fig. 1D are similar to those for high  in homogeneous
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systems (Abrams and Ruokolainen 2011); they are often complex cycles (as in Fig. 1D) or chaos. A
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fifth type of dynamics (not shown) consists of nearly in-phase simple cycles in the two patches that
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are each similar to the cycle that occurs in an isolated patch (Abrams and Ruokolainen 2011). This
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was only observed for very similar patches. Finally, stable equilibrium points are also possible for
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systems with significant heterogeneity and relatively large baseline movement rates (m).
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The rest of this section examines the effects of the fitness sensitivity parameter on
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population dynamics and compares the dynamics of systems with different amounts of
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heterogeneity in C or in r. It concludes with a more general summary of all of the results for the
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reference parameters.
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Results for reference parameters; the impact of fitness sensitivity on dynamics
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We begin by using systems based on the reference parameters with heterogeneity in the
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attack rate parameter (C) to investigate the influence of fitness sensitivity () on system dynamics.
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Figure 2 shows how between-patch synchrony and population variability change with  for different
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heterogeneities. At low heterogeneities (Fig. 2A), the dynamics are simple anti-phase cycles
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similar to Fig 1B for the lowest values of , and are positively correlated complex asynchronous
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cycles similar to Fig. 1D at high . Variability changes little with  over the upper half of the range
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shown. When heterogeneity is greater (C1 = 1.5, C2 = 0.5; Fig. 2B), variability (CVs) and
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synchrony (correlations) reflect the sequence of dynamics shown by figures 1A through 1C. In this
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case, consumer variability is maximal at low , where predator-prey cycles driven by the high-C
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patch dominate the dynamics. A relatively narrow range of low-intermediate values of  produces
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the type of low-variability, approximately anti-phase cycleS shown in Fig. 1B. Finally, when  is
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large, the resource cycles are positively correlated between patches and the consumer variability
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increases to an intermediate level with dynamics similar to Fig. 1C. The transition to anti-phase
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cycles causes the greater than 10-fold reduction in the CV of total consumer density when  goes
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from 100 to 180 (Fig. 2B). The within-patch variability of consumer populations increases in both
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patches over this range of . Higher heterogeneity (fig. 2C; C2 = 0.2) reduces global population
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variability compared to systems with less heterogeneity over almost all  values. At low ,
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consumer densities are close to anti-phase cycles (as in Fig. 1B) in the two patches; increased
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fitness sensitivity leads to more nearly anti-phase cycles having a shorter period, smaller amplitude,
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and a larger magnitude negative between-patch correlation of resource densities. Note that
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decreased variation with increased  also characterizes the low- end of Fig 2A, which also has
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anti-phase cycles.
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Graphs corresponding to those shown in Fig. 2 have been obtained for an evenly spaced set
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of values on the C2 = 2 – C1 trade-off curve (at intervals of 0.1 between values, as well as for C1 =
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1.95). A monotonic decrease in variation with increasing  (as in Fig. 2C) is first observed at a C2
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value between 0.5 and 0.475 (C1 between 1.5 and 1.525). Given our reference parameter set, the
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lower-C patch (patch 2) has stable dynamics in isolation when C2 < 0.54386. Thus, high fitness
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sensitivity produces the least variability and the strongest negative correlations of densities in
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different patches when one of the patches would be stable in isolation. However, at extreme
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heterogeneity (C1 = 1.95, C2 = 0.05; not shown here), patch 2 offers too little potential resource
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intake to attract consumers most of the time. Thus, short-period anti-phase cycles like those in Fig.
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1B are not observed for any . For some heterogeneities, alternative attractors exist over short
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intervals of  near the zone of transition between anti-phase and positively correlated cycles. In
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this set of reference system simulations, such alternatives were only observed when C1 < 1.6, and
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then always occurred for a limited range of  within the interval 115 <  < 165. This range is the
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zone of transition between negative and high positive between-patch correlation in resource
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abundances. Alternative attractors did not alter the basic patterns shown in fig. 2, and were not
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observed in any cases with  > 250. See Online Appendix A for more details and some illustrations
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of alternative attractors.
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The pattern shown in Fig. 2 changes significantly when the baseline homogeneous system is
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more stable; i.e., if the value of the potentially heterogeneous parameter is close to its stability
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threshold (Hopf bifurcation) value. Increasing the mean d or k and decreasing the mean r or C are
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all stabilizing. With a higher death rate, for example, graphs corresponding to those in Fig. 2 are
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similar to Fig. 2C for a wider range of heterogeneities; variability of both species is low and
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decreases as the fitness sensitivity  increases.
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Results for reference parameters; heterogeneity in attack rates
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Next we consider in more detail how heterogeneity in the attack rate affects variation and
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between-patch correlations when the fitness sensitivity of movement is high ( = 500). Global
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population variabilities and correlations become relatively insensitive to  at large values, so these
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results are representative of a wide range of large  values. (Some analogous results for a smaller 
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value are given in the Online Appendix B; smaller  can decrease both stabilizing and destabilizing
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effects of adaptive movement.) Figure 3 shows how variability and synchrony change with
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heterogeneity in C for both the reference mortality rate (d = 0.02; Fig. 3A), and for d = 0.04 (Fig.
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3B, representing a system closer to the stability threshold). For both death rates, there are U-shaped
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relationships between heterogeneity and variability; anti-phase cycles with low variability occur at
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intermediate heterogeneity. More complex cycles with positive between-patch correlations in
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resource abundance re-emerge at very high heterogeneity. The zone of nearly anti-phase cycles is
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narrower and occurs at higher heterogeneities, given inherently more variable patches (i.e., a lower
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death rate as in Fig. 3A).
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Figure 4A compares the boundaries of the parameter space of anti-phase dynamics to the
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boundaries of the parameters where the low-C patch is both stable and able to support a consumer
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population in isolation (i.e., is not a sink) for a continuous range of mortality rates. These two sets
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of boundaries correspond rather closely, although the lowest heterogeneity producing anti-phase
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dynamics is usually greater than the minimum heterogeneity required to stabilize the low-C patch in
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isolation.
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Results for reference parameters; other types of between-patch heterogeneity
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Patches may also differ in the resource growth parameters, r or k, or the consumer's death
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rate, d. This section presents results for the resource intrinsic growth rate, r. Corresponding results
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for heterogeneity in k, in d, and for combinations of both r and C, and r and k are presented in
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Online Appendix C and are summarized briefly at the end of this section.
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When r differs between patches (Fig. 5), there is a small decrease in both consumer and
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resource variability (CV) at intermediate heterogeneity. However, the largest magnitude change is
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the increase in variability at higher heterogeneity. Short-period anti-phase cycles do not occur for
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any r-heterogeneity assuming the reference parameter set with either of the two mortalities shown.
352
This is in spite of the fact that the low-r patch is stable in isolation when r < 0.737 in that patch for
353
d = 0.02 and when r < 0.948 for d = 0.04. Compared to systems with heterogeneity in C, the
354
variability of the less stable (high r) patch increases much more rapidly with heterogeneity in r. The
355
equilibrium consumer density in an isolated patch increases linearly with its r, so the ratio of
17
356
population densities of the smaller-r to the larger-r patch decreases rapidly with increasing
357
heterogeneity. Figure 4B compares how the relative consumer densities of the two patches in
358
isolation change as a function of heterogeneity for the cases of C-heterogeneity and r-heterogeneity.
359
The combination of relatively large mean density and large amplitude cycles means cycles driven
360
by the larger-r patch dominate the dynamics of the coupled system across all heterogeneities.
361
Stabilization is possible in systems where the mean r across the two patches is lower (e.g., if r1 + r2
362
= 1.2); cycles in the isolated-patch then have a smaller amplitude. If k is proportional to r, so that
363
the equilibrium population size is unaffected by r, then changing heterogeneity has only a very
364
small impact on consumer or resource variability (Online Appendix C; Fig C-3B).
365
Online Appendix C illustrates the impact on dynamics from four other types of
366
heterogeneity; heterogeneity that affects both resource growth and consumer attack rates (r and C)
367
similarly; heterogeneity in k, heterogeneity that affects r and k similarly, and heterogeneity in death
368
rates (d). Positive correlations between r and C could come about when the lack of physical
369
structure in one patch facilitates feeding by a resource species but also increases its vulnerability to
370
a consumer. This scenario can produce 'fast and slow food chains', which were considered to be
371
highly stabilizing by Rooney et al. (2006). Positive correlations between r and k are similar to pure
372
r-heterogeneity under the typical parameterization of the logistic model. As shown in Online
373
Appendix C, all of these examples also conform to the general rule that intermediate heterogeneity
374
greatly reduces population variability when it implies that one patch would be stable with a
375
relatively high consumer density if it were isolated.
376
Results for reference parameters; summary
377
378
Our extensive numerical analysis has uncovered two distinct mechanisms that can greatly
reduce the population variability of a 2-patch heterogeneous system relative to the variability of its
18
379
component patches. The first case is when the fitness sensitivity is low enough to produce a large
380
time lag in movement following a shift in relative rewards in the two patches. This in turn produces
381
short-period anti-phase cycles in systems with low heterogeneity. Similar anti-phase dynamics at
382
relatively low fitness sensitivities also occur when the two patches are identical (Abrams and
383
Ruokolainen 2011), and the result is again low variation of total population sizes. The second
384
mechanism that greatly reduces variation is based on the presence of a patch that would be stable in
385
isolation. In this case, a range of intermediate to large heterogeneities can produce anti-phase
386
dynamics when  is large. Having one stable (or nearly stable) patch appears to be a necessary, but
387
is definitely not a sufficient condition for system-wide stabilization, as shown by the lack of low
388
variability systems with r-heterogeneity (Fig. 5). When heterogeneity causes the less stable patch to
389
dominate system dynamics due to large population size or large amplitude fluctuations (or both),
390
reduced variance due to approximately anti-phase dynamics does not occur. Heterogeneity in the
391
attack rate C is particularly likely to reduce variation, because the density of an efficient (low d)
392
consumer in an isolated patch often increases as its attack rate C decreases until C is close to its
393
minimum possible value (Fig. 4B; see also Abrams (2002)). This increase is the result of more
394
'prudent' exploitation of a self-reproducing resource.
395
Table 1 provides a qualitative summary of these two main mechanisms that can produce
396
greatly reduced population variation in this model. The generalizations in Table 1 are based on both
397
simulations using the reference parameter set, and many additional simulations, some of which are
398
described in online Appendices B and C.
399
Comparison of adaptive and random movement
400
401
This section compares the preceding results to those for analogous systems having random
movement. It is known that perfect synchrony is by far the most common, but not the only outcome
19
402
of low random movement in the system considered here when patches are identical (Jansen 2001;
403
Goldwyn and Hastings 2008). However, there has been less exploration of the impact of
404
heterogeneity in systems with random movement (Jansen and deRoos 2004). Goldwyn and Hastings
405
(2009) showed that between-patch phase differences are common for the two-patch consumer-
406
resource system studied here with very low random movement and very slight heterogeneity. These
407
limiting cases usually involve 'phase drift', in which the phase angle between cycles in the two
408
patches changes over time.
409
Figure 6 illustrates the CVs of total consumer and resource densities as a function of
410
heterogeneity in the attack rate (C) for both adaptive and random movement, given two different
411
baseline movement rates (m = 0.05, 0.0005) and two different consumer mortalities (d = 0.02, 0.04).
412
Other parameters have the values given in Fig. 3, except that  was reduced from 500 to 250 in the
413
systems where m = 0.05 to make the overall movement rates more comparable. The type of
414
movement producing the least variable populations depends on the level of heterogeneity, the
415
consumer death rate, the baseline movement rate, and, in some cases, whether the consumer or
416
resource population is being considered. In general, when d = 0.02 (Fig. 6A), random movement
417
produced less variation of R or N than did adaptive movement at low to moderate heterogeneity, but
418
greater variation at high heterogeneities. Averaged over all heterogeneities, random movement
419
with a high baseline rate was most stabilizing, and this result was particularly strong for the
420
resource. The mechanism of stabilization was different for the two movement categories. Systems
421
with high random movement had point stability for a broad range of intermediate heterogeneities
422
(1.16  C1  1.72) because the resource in the higher-C patch goes extinct. Additional stabilization
423
then occurs because consumer individuals spend roughly half of their time in the no-resource patch,
424
reducing their effective capture rate in the patch that still contains resource. Stabilization in the
20
425
most stable systems with adaptive movement involved individuals moving from the more inherently
426
stable low-C patch into and out of the unstable patch to produce anti-phase cycles (low m) or a
427
stable equilibrium point (high m). Neither of these outcomes involves local resource extinction.
428
When heterogeneity was very slight (larger C < 1.1), a low rate of random movement produced the
429
greatest stability because it allowed phase drift, while the other varieties of movement did not.
430
This same comparison was repeated assuming a larger mortality (d = 0.04; Fig. 6B). The
431
higher death rate reduces the ability of the predator to bring about apparent competitive exclusion in
432
the random case and increases the occurrence of stability of one of the patches in the adaptive case.
433
This is point stability rather than anti-phase cycles for a broad range of intermediate heterogeneities
434
when m = 0.05. (These are cases with a zero CV of R in fig. 6B.) However, the most or least
435
variable type of system still depends on the amount of heterogeneity, and, in some cases, also
436
depends on whether consumer or resource variability is of interest. Under very high heterogeneity
437
(e.g., when the higher C value is 1.95), a high rate of random movement produces an asymptotically
438
stable equilibrium (due to resource exclusion), while adaptive movement or lower random
439
movement both result in highly variable populations. For a narrow range of very low
440
heterogeneities (larger C < 1.05) low random movement produces the least variation of both species
441
due to the occurrence of phase drift.
442
Alternative parameters and models
443
An evaluation of some alternative models and parameters is presented in Online Appendix
444
C, and is briefly summarized here. It considers: increasing the baseline movement rate; different
445
parameter values and models with different functional responses; movement decisions based only
446
on local fitness; movement with a sigmoid relationship to the fitness difference, and movement
447
modelled by instantaneous switching (following Post et al. (2000)). All of these deserve more
21
448
detailed treatment than is presented here. However, none of the results we obtained suggest
449
qualitative changes in the basic predictions outlined above.
450
A larger m does not greatly change the general effects of heterogeneity on variability and
451
between-patch correlations when  is high. As noted in the preceding section, the main impact of a
452
much larger m is to change anti-phase cycles to a stable point for a range of intermediate
453
heterogeneities. A large m does eliminate the approximate anti-phase cycles that occur when  is
454
relatively low. These cycles are based on a time lag, which arises because of low movement rates
455
when the fitness difference is small; a large enough baseline movement eliminates this lag.
456
Alternative sets of population dynamic parameters for Eqs. (1) generally produce impacts of
457
heterogeneity and fitness sensitivity similar to those described above. This holds for both larger
458
handling times and different relative speeds of predator and prey dynamics. Type-3 or predator-
459
dependent functional responses generally produce less variable dynamics within a patch, which
460
makes it more likely that adaptive movement will greatly reduce variation. A potential movement
461
rule based on local fitness only assumes that the per capita movement rate from patch i in Eq. (1c) is
462
mexp[–wi]. This produces quantitative differences and more often leads to alternative attractors
463
than does the function in Eq. (1b). However, strong stabilization via movement was often observed
464
for similar circumstances; either relatively low  with small heterogeneity, or when there was one
465
strongly stable patch. A sigmoid movement function produced little difference in the results,
466
provided the maximum movement rate was at least two orders of magnitude larger than the
467
maximum per capita growth rate. The (Post et al. 2000) switching model reproduced most of the
468
features of the variability vs. heterogeneity relationships in the cases of C- and r- heterogeneity with
469
high fitness sensitivity. However, correlations in resource density did not qualitatively match the
470
corresponding results under adaptive movement. The switching model also predicted no effect of
22
471
heterogeneity in d, and it did not predict the anti-phase cycles arising from a lag due to slow
472
movement when the fitness difference was small.
473
Discussion
474
Adaptive movement and stability
475
Our results do not support a uniformly stabilizing or destabilizing role of adaptive movement in the
476
context of cycling consumer-resource systems. Fitness-related movement by consumers reduces
477
population variation under two broad sets of conditions that were summarized in Table 1; relatively
478
low fitness sensitivity of the consumer, or parameters leading to one stable patch with significant
479
consumer density. When baseline movement (m) is small, low population variation is usually
480
associated with short-period, approximate anti-phase cycles in consumer populations; larger m in
481
otherwise similar systems often leads to a stable equilibrium point. When one of the patches is
482
inherently stable, global dynamics have low variation over a broad range of high fitness
483
sensitivities. This is likely to be the most common scenario under which adaptive movement
484
greatly reduces population variability in the type of systems considered here.
485
Given that adaptive movement occurs, our results also suggest that being better at assessing
486
or responding to fitness differences between patches (greater ) may either increase or decrease
487
population fluctuations, as shown in Figure 2. The largest increases occur when the initial fitness
488
sensitivity is relatively low, and are usually accompanied by greater resource synchrony between
489
patches. When one patch is intrinsically stable, system-wide variation of each species usually
490
decreases with increasing fitness sensitivity (as in Fig. 2C). Systems with little heterogeneity are
491
generally less variable when connected by a low rate of random consumer movement than by
492
adaptive movement; low random movement allows phase relationship between cycles in different
493
patches to change over time, while adaptive movement is more likely to produce phase-locking.
23
494
The question of the stabilizing effect of adaptive movement is related to the broader
495
questions of whether any type of switching by consumers or even more generally, adaptive
496
consumer foraging, stabilizes systems. These issues have a large literature, which has been
497
reviewed recently by Abrams (2010a, b) and Valdovinos et al. (2010) among others. As with other
498
questions about effects on stability, the conclusion depends on the definition of stability (Matsuda et
499
al. 1996; McCann 2000; Carpenter and Ives 2007), and the type of adaptive behavior (Abrams
500
2010b). Uchida et al. (2006) and Valdovinos et al. (2010) both attribute generally stabilizing effects
501
to adaptive behavior. It is certainly true that adaptive foraging for substitutable resources reduces
502
the possibility of resource extinction, as rare resources suffer less consumption; this result is shown
503
in a general context by Uchida et al. (2007). However, the present article, as well as many previous
504
ones (e.g., Abrams 1992, 1999; Abrams and Matsuda 2004; Abrams 2007), show that there are a
505
variety of mechanisms by which adaptive behavior ca increase population fluctuations.
506
How does spatial heterogeneity affect variation, given adaptive movement?
507
As noted before, increasing, decreasing, and U-shaped relationships between patch-heterogeneity
508
and population stability were observed, depending on consumer efficiency and the parameter
509
involved in the heterogeneity. The impact of heterogeneity is dependent on whether it implies
510
inherent stability of one of the two patches, and how it changes the relative population sizes of the
511
patches (Table 1). There is little empirical evidence for widespread anti-phase cycles in natural
512
systems. This may indicate that such systems are characterized by higher baseline movement rates,
513
m, which often produce point stability rather than anti-phase cycles. It may also reflect the paucity
514
of long-term monitoring of interacting populations in metacommunities, or a lack of monitoring on
515
the same spatial scale affected by adaptive movement.
24
516
Some previous work (McCann et al. 2005; Rooney et al 2006, 2008) has argued that point
517
stability or low variability is likely to arise in two-patch systems with an adaptively moving top
518
predator when those patches differ greatly in the speed of their dynamics (where they define 'fast'
519
patches as having high productivity, high biomass turnover, and high vulnerability of prey to
520
predators). Our results suggest that the key stabilizing feature in this scenario is the low capture rate
521
in the 'slow' patch. Differences in resource growth parameters did not have a general stabilizing
522
tendency in our models. However, the models considered here differ in several ways from those in
523
Rooney et al. (2006); our models have two trophic levels rather than three, and our basal resources
524
in different patches do not compete. The effects of these features need to be examined using the
525
present framework for representing movement. The system-stabilizing effect of having a stable
526
patch is analogous to the stability produced by weak interactions (low C) in McCann et al.'s models
527
(1998), which also incorporated a switching function.
528
How is synchrony related to variation?
529
Adaptive movement can produce different degrees of synchrony between patches. In most cases,
530
negative correlation between the densities in the two patches reduces variation in the total.
531
However, even this generalization must be qualified. The between-patch correlation of resource
532
populations is more closely related to variability (CV) of each species than is the correlation of
533
consumers. This is not surprising, given that the synchrony of consumers is largely related to
534
movement rather than changing population size. However, it is possible for significant decreases in
535
resource synchrony to be accompanied by increased variability in both species, as in Fig. 5A. This
536
is because the heterogeneous parameter (r) has its own effect on variability, independent of effects
537
on synchrony; here, variation of consumer density in the large-r patch increases more rapidly with
538
increased heterogeneity than variation in the small-r patch decreases. In this case, the two resource
25
539
populations remain positively correlated with higher heterogeneity, until heterogeneity becomes
540
extreme.
541
A comparison of homogeneous and heterogeneous patch systems
542
Abrams and Ruokolainen (2011) analyzed the model considered here for the case of two identical
543
("homogeneous") patches. They showed that adaptive movement frequently produced anti-phase
544
fluctuations (and thus reduced variation in) systems that would have been had perfect synchrony
545
under random movement. Alternative in-phase (i.e., perfectly synchronized) and anti-phase
546
attractors occurred over a wide range of movement function parameters for homogeneous patches.
547
The present work shows that neither of these two results extends in a simple way to systems with
548
even a very small amount of heterogeneity. Goldwyn and Hastings (2009) showed that a similar
549
dichotomy in dynamics between homogeneity and slight heterogeneity occurs in similar two-patch
550
consumer-resource models with random movement. The present results also show that the question
551
of the relative stability of systems with adaptive and random movement has qualitatively different
552
answers for homogeneous and heterogeneous systems in the context of two-patch consumer-
553
resource models; homogeneous systems appear to always be equally or less variable under adaptive
554
movement (Abrams and Ruokolainen 2011), whereas that is not the case for heterogeneous systems
555
(Fig. 6). Sufficiently high fitness sensitivity of movement always produces perfect synchrony and
556
high variation in homogeneous systems, but can minimize variation in highly heterogeneous
557
systems (e.g., Fig. 2C).
558
Additional ecological implications and future directions
559
Our results add to the growing list of cases where adaptive movement has been shown not to
560
produce spatial homogeneity in fitness (Schwinning and Rosenzweig 1990; Abrams 2000, 2007;
561
Abrams et al. 2007; Křivan et al. 2008; Amarasekare 2010). In other studies of metacommunities,
26
562
the details of spatial processes (different dispersal rates/distances and environmental heterogeneity)
563
have been shown to affect community dynamics and diversity at local and global levels (e.g.,
564
Mouquet and Loreau 2003; Maser et al. 2007; Amarasekare 2008, 2010), temporal stability of food
565
webs (e.g., Koelle and Vandermeer 2005; Rooney et al. 2006; Abrams 2007; Maser et al. 2007), as
566
well as community persistence under environmental perturbations (Roy et al. 2005; Matthews and
567
Gonzalez 2007). Explicit modeling of adaptive movement should be applied in future studies
568
addressing these community-level questions, as it seems likely to significantly alter the previous
569
findings.
570
The 2-patch models used here have a variety of potential applications. They should help
571
understand how decline or loss of predators is likely to affect aquatic systems, which often consist
572
of two coupled habitats (e.g., Vander Zanden and Vadeboncoeur 2002). If the resources are distinct
573
species, the nature of adaptive consumer movement has a large effect on apparent competition
574
between those resources (Abrams 1999; Abrams and Matsuda 2004). Two-patch models can also
575
describe the dynamics of exploited systems with protected areas; Abrams et al. (2011) show that
576
adaptive movement in the absence of information about harvesting risk in such systems can produce
577
abrupt extinction with slowly increasing harvests.
578
These models need to be extended to systems with more patches and more species. Limited
579
analysis of three and four patch models suggests that many of the same phenomena occur
580
(unpublished; see also appendix of Abrams and Ruokolainen (2011)). In the context of random
581
movement, Blasius et al. (1999) suggest that synchronization may aid persistence in large systems
582
by allowing travelling wave structures; it would be interesting to examine this in the context of
583
adaptive movement. Rowell (2009), using models of single-species growth in continuous space, has
584
shown that movement towards areas of higher fitness can have major effects on species ranges.
27
585
This suggests that outcomes in spatially explicit metapopulations with interacting species are likely
586
to be quite complex. (See Rowell 2010 for an example with competing species.). Even in two-
587
patch models, the type of movement can alter the direction of response of trophic level abundances
588
to predator mortality and/or resource enrichment (Abrams 2007, 2008; Abrams and Ruokolainen
589
2011). Better understanding of adaptive movement in multi-species, multi-patch systems should
590
enhance predictions of the responses to perturbation in any spatially subdivided system.
591
592
593
Acknowledgments
This work was supported by a Strategic Project Grant and several Discovery Grants from the
594
Natural Sciences and Engineering Research Council of Canada. We thank P. Amarasekare and A.
595
J. Golubski and several anonymous reviewers for comments on earlier drafts.
28
596
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723
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724
Wolfram Research. 2011. Mathematica. 8.0.1. Champaign-Urbana, IL.
34
Table 1
Mechanisms Producing Low Variation
Mechanism
Necessary Conditions
Facilitating factors
Delayed movement
Low-moderate heterogeneity
small baseline movement, m
Narrow range of relatively low  values Heterogeneity in C
One patch is stable in
isolation
Stable patch has a consumer population Heterogeneity in C
comparable to or larger than the unstable
patch
Moderately low predator
Stable patch is not on the verge of
efficiency (e.g., high mortality,
instability
low conversion efficiency)
The fluctuations in the unstable patch are
sufficiently small in amplitude
35
Online Appendices; Supplemental results
725
726
A. Methods
The dynamical system described by Eqs. (1) was numerically integrated with Mathematica
727
v. 7.0.1 or 8.0.1 (Wolfram Research 2009, 2011) using the NDSolve routine with AccuracyGoal set
728
to Infinity. For most parameter combinations the system was integrated for at least 8000 time units
729
and the population statistics were calculated using the last half of that period. Most simulations had
730
converged to their ultimate dynamics well before 4000 time units. In all cases, the CVs and
731
correlations were calculated for the last 2000 time units to look for directional trends. Any case of a
732
large difference resulted in recalculating the statistics for much longer simulations. Some cases
733
with long transients involved chaotic dynamics and occurred for relatively low heterogeneities. The
734
existence of alternative attractors was explored using runs with randomly chosen initial densities for
735
extreme parameter values and by continuation of attractors (using final densities from one run to
736
initialize the next simulations having the next larger or smaller value of the parameter being varied).
737
All figures showing dynamics as a function of C1 or  were run from low to high values and from
738
high to low values of the parameter on the x-axis to check for alternative attractors. The potential
739
for larger amplitude cycles was then checked for all parameters that exhibited short-period anti-
740
phase cycles by additional simulations starting with densities of both resources at 10-5 times their
741
carrying capacity. Because alternative attractors were relatively rare (see below), the figures
742
presented here are only based on simulations where attractors were continued from low to high
743
parameter values.
744
The set of simulations using reference parameters (as in fig. 2) from  = 90 to  = 500 for a
745
set of ten different heterogeneities only revealed alternative attractors for C1 values of 1.1, 1.2, 1.4,
746
and 1.5. The widest range occurred for C1 = 1.5, where alternatives occurred for 128 <  < 155.
36
747
The only one of this set of simulations where more than two alternatives was observed was for C1 =
748
1.2, where three alternatives occurred for 124 <  < 128, and two or more existed for 118 <  <
749
128 Figure A-1 illustrates the dynamics of the two alternatives for one  value in the system with
750
C1 = 1.5, and shows how the CVs and correlations differ between attractors across the full range of
751
 exhibiting alternatives for this level of heterogeneity.
752
B. Impact of heterogeneity for relatively low lower fitness sensitivities
753
Most of the analysis in the text assumed a high fitness sensitivity;  = 500. The four panels
754
of Figure B-1 are equivalent to those of the text Fig. 3, illustrating the impact of C-heterogeneity on
755
coefficients of variation of population sizes (CVs) and between-patch correlations. Figure B-1
756
assumes  = 180 (fig. 3 assumes  = 500). The main qualitative difference between the figures here
757
and in the text is that increased heterogeneity does not greatly reduce the CVs for low to moderate
758
heterogeneities when d = 0.04, in spite of anti-phase consumer movements. This reflects the
759
reduced stabilizing effect of slower movement in the context of anti-phase dynamics; this is also
760
shown in Fig. 2C in the text.
761
C. More detailed description of the dynamics for several alternative systems
762
C.1. Other parameter heterogeneities in the reference system
763
Figures C-1 and C-2 are similar in form to Figures 3 and 5 in the text, and show CVs and
764
correlations for the global populations of both consumers and resources for the cases of linked
765
heterogeneity in both r and C (i.e.; C1 = r1; Fig. C-1) and heterogeneity in d (Fig. C-2). In Figure
766
C-1, low consumer mortality (d = 0.02; Fig. C-1A) produces results that are similar to asymmetry in
767
r alone, so there is no major decrease in variability with higher heterogeneity. Here again, the 'low'
768
parameter value patch is stable in isolation when r = C < 0.737, but the higher population size and
769
large variance in the large r patch overwhelm any stabilization caused by the low r patch. When d =
37
770
0.04 (Fig. C-1B), stabilization in isolation occurs when r = C < 0.974. The greater stability of the
771
low patch combined with less pronounced cycles in the high patch allow anti-phase cycles with low
772
variability in total densities to occur at intermediate heterogeneities; high heterogeneity again
773
produces much greater population variability than in nearly homogeneous conditions.
774
The range of heterogeneities in consumer death rate, d, illustrated in Figure C-2 is lower
775
than in comparable figures for other parameters because a mortality rate very close to zero is
776
biologically impossible, and would produce unrealistically large amplitude cycles. When the mean
777
death rate is relatively low (d = 0.02; Fig. C2-A), greater heterogeneity decreases variation and
778
decreases the consumer's density correlation. Resource correlation decreases slowly with
779
heterogeneity, but is large and positive at all heterogeneities. In this case, both patches exhibit large
780
consumer-resource cycles in isolation, limiting the potential for reduced variation via adaptive
781
consumer movement. When d1 + d2 = 0.08 (Fig. C2-B), increasing heterogeneity in d implies that
782
the high-d patch becomes stable in isolation at a relatively low heterogeneity (larger d > 0.0417),
783
and, at greater heterogeneity (larger d > 0.0625), it is a sink. This produces a more typical U-
784
shaped relationship between heterogeneity and variability, similar to Fig. 3B in the text.
785
Figure C-3 shows similar results for heterogeneity in the resource density dependence, k,
786
which has a mean value of one for the reference parameter set. Panel B assumes that r and k have
787
identical differences between patches. Both panels assume a relatively high death rate (d = 0.04),
788
but they still do not exhibit any cases of clear anti-phase dynamics. In panel A this is because the
789
more stable, high-k patch has a smaller consumer population size; even when it is stable in isolation,
790
the large-amplitude cycles in the low-k patch result in highly variable populations. In panel B there
791
are no cases with a stable patch that would be stable in isolation, and the cycles are only changed
792
slightly by the larger r and k.
38
793
C.2. Larger m
794
A larger baseline movement rate (m) increases movement proportionally for all fitness
795
differences. A larger m means that there is less of a time lag in the response of the consumer's
796
distribution when the relative fitnesses in the two patches reverse. In most cases, a larger m results
797
in greater variation (higher CVs) when there are approximately synchronized predator-prey cycles
798
in the two patches, and reduced variation when there are simple anti-phase cycles due to the one-
799
stable patch mechanism. In fig. 6, a larger m did not greatly change the pattern of variability as a
800
function of between-patch heterogeneity, although relatively stable anti-phase dynamics were often
801
changed to a stable point equilibrium. A large m reduces the potential for anti-phase cycles at
802
moderately low , such as those that occur for a range of  < 200 in figs. 2A, B. These systems
803
with m = 0.05 always have positive correlations of resource densities, with CVs that increase slowly
804
as  increases.
805
C.3. Other parameter sets and/or movement functions based on Eqs. (1)
806
A more detailed parameter exploration of the homogeneous patch version of Eqs. (1) is
807
presented in Abrams and Ruokolainen (2011). That analysis suggested that there were some
808
differences in the variability/synchrony vs. heterogeneity patterns produced by larger handling times
809
and different relative speeds of consumer and resource dynamics. These parameter differences have
810
less impact in heterogeneous systems. For example, slowing resource dynamics by multiplying
811
both the rs and ks by 1/4 did not significantly change Figs. 3 and 5. Tripling the handling time and
812
reducing mortality rates to compensate produced very high amplitude cycles. This meant that
813
variation in systems with anti-phase cycles was also greater than in the reference example.
814
However such cycles still greatly reduced variation and occurred over roughly similar ranges of
815
heterogeneity when movement rates were high. Systems with large handling times that were close
39
816
to the stability threshold (Hopf) point within a patch often had alternative attractors at low
817
heterogeneities; one a simple nearly anti-phase cycle, and the other a complex long-period cycle
818
with positively correlated resource populations.
819
Because of its exponential form, the movement model considered here allows extremely
820
rapid redistribution of consumer individual when the fitness difference between patches is large and
821
 is also large. In many cases, it may be more realistic to assume that the per capita movement rate
822
approaches a maximum rate as the difference between patches becomes larger. This may be done
823
by using the following function, introduced in the appendix of Abrams and Ruokolainen (2011):
824
mExp[  (W j  Wi )]
1   mExp[  (W j  Wi )]
825
where  is a small (<< 1) positive constant. The movement rate approaches a maximum of 1/ as
826
mExp[(Wj - Wi)] becomes large; here it is referred to as a “constrained rate” movement function.
827
The second term in the denominator is small except when the fitness difference is large, but large
828
differences do not occur once the system has reached its limiting dynamics. Thus, figures based on
829
0 <  < 0.1 produce minimal changes in the results for the reference parameters values shown in
830
Figs. 3 and 5. For example,  = 0.05 reduces the maximum ratio of movement rate to per capita
831
growth rate in a homogeneous system with reference parameters,  = 500, and d = 0.02 from 1010 to
832
approximately 470. However, in both models the largest realized movement on the attractor is
833
smaller than either of these values. This shows that the results are not artefacts of the hypothetical
834
large movement rates that could occur with large fitness differences under Eq. (1b)
835
Our model could be modified in many other ways. Some of those that we have briefly
836
explored are: adding a third trophic level, assuming either type-3 or predator-dependent consumer
837
functional responses, or using a movement functional where the consumer never goes to a lower
40
838
fitness patch (see Abrams and Ruokolainen 2011). Three-level systems will be considered in more
839
detail in a future article. However, in systems where fluctuations are driven by the interaction of the
840
top two levels (i.e., the bottom two levels had a stable equilibrium in the absence of the mobile top
841
predator) the response to heterogeneity is similar to the 2-level models explored here. Both type-3
842
responses and predator-dependent responses increase stability generally, so they make it more likely
843
that adaptive movement decreases variation by producing simple anti-phase cycles or stable points.
844
These alternative functional responses also increase the range of parameters and heterogeneities
845
producing stable equilibrium points for both patches in isolation. Adaptive movement in such cases
846
does not change that stability. The no-error movement function generally produced results similar
847
to those reported here.
848
C.4. Movement decisions based only on current patch quality
849
Many organisms have limited ability to detect conditions in a distinct patch. If leaving
850
decisions are based only on the quality of the current patch, the analogue of Eqs. (1) represents per
851
capita movement out of patch i by mExp[–Wi]. When there is no food in a patch, this rate
852
(mExp[[d]) should be much greater than the death rate d. To obtain ratios of movement rate to
853
population change comparable to those in Figs. 3 and 4 for a similar  value, the baseline movement
854
parameter m must be larger than in the main reference example. A range of different types of
855
heterogeneity were simulated using m = 0.005 and  = 300 or 500. The main difference produced
856
by this rule is that consumers are quickly homogenized across patches when both patches have low
857
fitness (e.g., both have low resource densities). Nevertheless, most of the graphs for comparable
858
parameter values are similar to those in Figs. 3, 5 in the text and figures C1-3 in this appendix. This
859
local rule produced low-variability systems with anti-phase dynamics for some parameter sets
860
where the comparison model considered above did not; this was often an alternative attractor to one
41
861
with larger amplitude cycles. For example, when d = 0.04, systems with r-heterogeneity could
862
produce low-variability anti-phase dynamics for values of the larger r-patch from approximately
863
1.01 to 1.31, although there was an alternative attractor with larger, positively correlated cycles for
864
all of this range. For systems with C heterogeneity, the local movement rule produced dynamics
865
similar to those in Figs. 2 and 3, but an alternative low-variance anti-phase attractor was also
866
present at low heterogeneities when d = 0.04. There were often quantitative differences between
867
this movement rule and the patch-comparison rule. However, reduced variation via adaptive
868
movement usually occurred in the same two basic scenarios involving either low or high s with
869
one stable patch.
870
C.5 Movement based on a switching function
871
Much of the previous literature on adaptive patch choice of consumers has assumed that the
872
abundances of consumers in the two patches adjusts instantaneously based on current densities
873
using a switching function (Post et al. 2000; McCann et al. 1998, 2005; Rooney et al. 2006, 2008).
874
Here, we carried out some comparisons with a model that used a variant of the switching function
875
used in Post et al. (2000). Their switching function implies that the relative number of consumers
876
instantaneously matches the relative rewards in each patch. We used the relative attack rates (Ci) in
877
each patch as Post et al.'s (2000) 'preference' parameter. This means that the proportions of
878
consumers in patches 1 and 2 at each point in time is given by C1R1/( C1R1 + C2R2) and C2R2/( C1R1
879
+ C2R2) respectively. This allows consumer dynamics to be represented by a single variable of total
880
population size. Otherwise, the population dynamics are identical to Eqs. (1), with logistic resource
881
growth and an independent type-II response in each patch.
882
883
A reanalysis of cases with heterogeneity in C, r, or d was carried out for the reference
parameter set. Switching models often predict decreased variation for intermediate heterogeneity,
42
884
as in the analyses presented above. However, the switching model predicted no effect of
885
heterogeneity in d; all systems have the same dynamics as the homogeneous patch system in Fig. C-
886
2. Modifying the switching model to make distribution a function of fitness rather than resource
887
density would improve its performance. Additional differences result from the switching model's
888
assumption that extremely small fitness differences can have a large effect on consumer distribution
889
when fitness in both patches is low (potentially producing a large ratio). Finally, because switching
890
is instantaneous, there were no cases corresponding to the anti-phase systems with low , such as
891
those in Fig. 1B.
892
43
893
Legends for Appendix Figures
894
Figure A-1. Alternative attractors for the case of C1 = 1.5 and C2 = 0.5 from Figure 2B in the text.
895
Panel A show the CVs and correlations coefficients of consumers and resources across the range of
896
 where alternative attractors are observed. Panels B and C illustrate the temporal dynamics of the
897
two attractors for  = 140.
898
Figure B-1. The impact of heterogeneity in attack rate C on dynamics, given a lower fitness
899
sensitivity than in Fig. 3 in the text. Solid lines denote resources and dashed lines denote
900
consumers. All parameters are in the text Fig. 3 except that  = 180 rather than  = 500. Results
901
for the same two death rates illustrated in Fig. 3 are shown.
902
Figure C-1. The impact of correlated heterogeneity in both r and C on system dynamics for two
903
different mortality rates. Solid lines denote resources and dashed lines denote consumers. The
904
movement sensitivity () is set to 350;  = 500 caused problems with numerical integration. The x-
905
axis gives the r and C value of the faster growing patch (r1 + r2 = 2; C1 + C2 = 2, with r1 = C1).
906
Figure C-2. The coefficients of variation (CVs) of density and correlations between patches in
907
system with heterogeneity in the consumer death rate. Solid lines denote resources and dashed lines
908
denote consumers. The fitness sensitivity () is again set to 350. In Panel A, the x-axis gives the
909
death rate of the higher death rate patch for a system in which d1 + d2 = 0.04, giving a mean d of
910
0.02. In panel B, d1 + d2 = 0.08.
911
Figure C-3. The coefficients of variation (CVs) of density and correlations between patches in
912
system with heterogeneity in the consumer death rate. Solid lines denote resources and dashed lines
913
denote consumers. The movement parameters are:  = 500 and m = 0.0005. The death rate is d =
914
0.04. In Panel A, only k differs between patches, while in panel B, r and k are equal to each other
44
915
within a patch, but differ between patches. The r/k ratio within a patch (the equilibrium resource
916
density in the absence of consumers) remains constant with changes in r.
917
45
918
Fig A-1
919
A.
920
921
922
923
924
925
926
927
928
929
CVs (R solid; N dashed)
Correlations (R solid; N dashed)


B. Attractor with higher CV at  = 140 (solid = patch 1; dashed = patch 2)
Time
Time
C. Attractor with lower CV at  = 140 (solid = patch 1; dashed = patch 2)
Time
Time
46
Fig. B-1
Dashed line = consumer; Solid line = resource
CVs
Correlations
A. d = 0.02
B. d = 0. 04
Larger C value
Larger C value
47
Fig. C-1
Dashed line = consumer; Solid line = resource
CVs
A. d = 0.02
Correlations
B. d = 0.04
Larger r and C values
Larger r and C values
48
Fig. C-2
Dashed line = consumer; Solid line = resource
CVs
A. Mean d = 0.02
Correlations
B. Mean d = 0.04
Larger d value
Larger d value
49
Fig. C-3
Dashed line = consumer; Solid line = resource
CV’s
Correlations
A. k-asymmetry
B. k- and r- asymmetry
Larger k value
930
Larger k value
50
931
932
933
Figure Legends
934
(Eqs. (1)). All of the examples assume asymmetry in the attack rate, C. All involve the
935
representative parameter set for those parameters that are equivalent in both patches (r = 1; k = 1;
936
h = 3; b = 0.25; and d = 0.02). Panels A, B, and C are for C2 = 0.5, which implies C1 = 1.5.
937
Case D shows dynamics that do not occur for these C values; here C2 = 0.9 and C1 = 1.1. In all
938
cases, the solid line represents patch 1 and the dashed line represents patch 2. The long-dashed
939
line in the left hand side of panel B is the mean consumer population (1/2 the total).
940
Figure 2. Dynamics as a function of  for several examples of the reference system whose
941
dynamics are illustrated in Fig. 1 (d = 0.02). The solid line represents the total (global) resource
942
population and the dashed line is the consumer population. The left hand side shows the
943
coefficient of variation of total consumer density (resource CV follows a very similar trajectory,
944
so was omitted). The right hand side shows the between-patch correlation coefficient of
945
population densities for both species. The three panels reflect different amounts of heterogeneity
946
in the consumer's attack rate. Note the different scaling of the y-axis in the three panels.
947
Figure 3. The between-patch correlation and coefficient of variation as a function of
948
heterogeneity in C when m = 0.0005 and  = 500. Dashed lines show results for consumer and
949
solid lines show results for resources. The x-axis gives the higher C value. The lowest
950
coefficients of variation reflect the simple anti-phase cycles shown in Fig. 1B. In both systems,
951
cycle amplitudes begin to increase at the highest level of heterogeneity. Even quite small values
952
of the C2 result in movement to patch 2 when R1 is very rare, and this greatly reduces the
953
amplitude and period of cycles in patch 1.
954
Figure 4. Factors influencing the occurrence of anti-phase dynamics. Panel A compares two sets
955
of limits on dynamics. The dashed line shows both the upper and lower heterogeneities in attack
Figure 1. Illustration of the most common types of dynamics observed for the primary model
51
956
rate, C, which produce anti-phase dynamics in a system with parameter values other than
957
mortality (d) as in Fig. 3. The solid lines are mortality and heterogeneity combinations where
958
the small-C patch becomes a sink (above the upper line) and where the small-C patch becomes
959
unstable (below the lower line). Panel B shows the population size of the more stable (small
960
parameter value) patch for a system with reference parameters and d = 0.02, for heterogeneity in
961
C (solid line) and heterogeneity in r (dashed line).
962
Figure 5. The impact of heterogeneity in resource intrinsic growth rate r on variability and
963
between-patch correlations for consumers (dashed lines) and resources (solid lines). This
964
assumes the reference parameter set and applies the same two levels of mortality shown in Fig.
965
3. The movement sensitivity () is set to 350 because 500 caused difficulties with numerical
966
integration at high heterogeneities. The x-axis gives r in the faster growing patch (r1 + r2 = 2).
967
Figure 6. A comparison of population variability produced by random and adaptive movement.
968
Part A shows the CVs of total consumer and resource populations as a function of between-patch
969
heterogeneity in attack rate, C, for four sets of assumptions about movement when consumers
970
are relatively efficient (d = 0.02), while part B does the same for d = 0.04. Adaptive movement
971
assumes m = 0.0005 with  = 500 or m = 0.05 with  = 250. Random movement assumes the
972
same pair of m values with  = 0. Other parameters are those given in the legend of Fig. 3.
52
Dashed line = patch 2; Solid line = patch 1
Fig. 1
Consumer populations
A.  = 100; C1 = 1.5; C2 = 0.5
Resource populations
B.  = 180; C1 = 1.5; C2 = 0.5
C.  = 500; C1 = 1.5; C2 = 0.5
D.  = 500; C1 = 1.1 C2 = 0.9
Time
Time
53
Fig. 2
Dashed line = consumer; Solid line = resource
Coefficient of variation of N
A. C1 = 1.2; C2 = 0.8
Between-patch correlations
B. C1 = 1.5; C2 = 0.5
C. C1 = 1.8; C2 = 0.2
Fitness sensitivity, 
Fitness sensitivity, 
54
Fig. 3
Dashed line = consumer; Solid line = resource
CVs
Correlations
A. d = 0.02
B. d = 0.04
Larger C value
Larger C value
55
Fig. 4
Dashed line = boundaries of anti-phase dynamics
Upper solid line = patch 2 becomes a sink
Lower solid line = patch 2 becomes stable
Higher attack rate C1
A.
Consumer per capita mortality, d
Ratio of equilibrium N2/N1
B.
Dashed line = ratio for r-heterogeneity
Solid line = ratio for C-heterogeneity
Larger parameter value (C1 or r1)
56
Fig. 5
Dashed line = consumer; Solid line = resource
CVs
Correlations
A. d = 0.02
B. d = 0.04
Larger r value
Larger r value
57
Fig 6
solid = adaptive, m = 0.0005
short dashed = adaptive, m = 0.05
medium dashed = random, m = 0.05
long dashed = random, m = 0.0005
A. d = 0.02
CV of N
CV of R
Larger attack rate, C1
B. d = 0.04
CV of N
CV of R
Larger attack rate, C1