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The effect of adaptive change in the prey on the dynamics of an
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exploited predator population
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Peter A. Abrams, Department of Zoology, University of Toronto, 25 Harbord Street,
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Toronto, Ontario M5S 3G5, Canada (phone 416-978-1014; fax 416-978-8532; email
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abrams@zoo.utoronto.ca)
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Hiroyuki Matsuda, Faculty of Environment & Information Sciences, Yokohama National
University , 79-7, Tokiwadai, Hodogaya-ku, Yokohama, Kanagawa 240-8501, Japan
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Abstract
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Simple mathematical models are analyzed to determine the relationship between
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harvesting effort and stock size for a predator in a system in which the prey adapt to the
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risk of predation. Two types of model are studied: in the first, the prey has a tradeoff
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between increasing its own net reproductive rate and increasing its vulnerability to the
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predator. In the second class of models, there are two prey species that differ in their
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vulnerability to the predator. Each prey species has fixed, non-evolving characteristics,
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but changes in the average characteristics within the prey trophic level can occur via
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shifts in the relative abundance of the two species. In both classes of models, the
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equilibrium predator population can increase with increasing harvest of the predator. In
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the case of the 2-prey model, the predator's equilibrium population always increases with
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an increased harvest rate if the two prey coexist and share a single resource. The
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predator's equilibrium population often decreases from its maximum size to zero over a
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very small range of harvest rates, once those rates become high enough. Because
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increased stock size is often used to justify increased harvest rates, this type of
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relationship poses a risk that harvest rate will increase to the point where the stock
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quickly collapses. The results are relevant to understanding and predicting the changes in
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population size of species experiencing declining environmental conditions.
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Keywords: food web, predator-prey system, prey adaptation, sustainable yield
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Fisheries models have largely employed a single-species perspective.
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Multispecies models (reviewed by Hollowed et al. 2000; Latour et al. 2003) have
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generally assumed that the characteristics of individual species do not change through
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time. Although one popular simulation package (Ecosim; Walters et al. 2000)
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incorporates both a multi-species approach and an assumption of transitions from
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vulnerable to invulnerable states, we currently lack a good understanding of the
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circumstances when a 2- or more-species system incorporating adaptive evolution or
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behaviour will exhibit dynamics that are fundamentally different from those of single
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species systems. Understanding the properties of models with small numbers of species
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is important in determining what factors should be included in larger scale simulation
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models, and in understanding the dynamics of those larger models.
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An earlier article (Matsuda and Abrams 2004) described how population cycles
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and/or adaptive change in the predator could qualitatively change the relationship
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between harvesting effort applied to the predator and predator population size. One of
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the main results of that analysis was that, unlike single-species models, stock size could
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increase as harvest increased in both stable and unstable predator-prey systems. The
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present article will extend the previous one to examine the effect of adaptation in the prey
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population on the harvesting effort – stock size relationship for the predator. Studies of
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several fish species have shown that prey often exhibit costly shifts in either foraging
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behaviour (Turner and Mittelbach 1990; Fraser and Gilliam 1992; Eklöv and Persson
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1995), or genetically determined life history traits (Reznick and Bryga 1990) in response
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to greater abundance of piscivorous predators. The foraging shifts are examples of life
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history plasticity, since they involve increased survival at the expense of decreased
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growth or reproduction. There is mounting evidence that adaptive changes, both
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behavioural and evolutionary, affect the dynamics of exploited fish populations (Heino
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and Godø 2002). Here we take a general theoretical approach to determine the potential
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population dynamical consequences of prey adaptation in systems in which the predator
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is exploited.
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We begin by noting some of the properties of single-species models in which the
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harvested stock has a relatively homogeneous population whose members have fixed
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characteristics, and whose limiting dynamic in the absence of environmental variability is
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a stable equilibrium. Such a population exhibits a decrease in population size in response
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to an increased per capita harvest rate. The initial and ultimate responses to increased
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harvest are both negative in sign. In such systems, feedback control of harvest effort
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represents a safe method for maintaining a healthy stock size, provided stock size can be
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measured, fishing effort can be controlled, and the target stock size is not too small.
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These results are straightforward conclusions of the models presented in standard
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fisheries texts (e.g., Hilborn and Walters 1992; Quinn and DeRiso 1999). The remainder
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of this article will show how adaptive change in the vulnerability of a prey species to a
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predator can alter these apparently intuitive conclusions.
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The relationship between predator population size and predator
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mortality in a predator-prey system without prey adaptation
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The analysis here assumes a simple predator-prey model in which the populations
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are not structured. The prey has density dependent population growth, with population
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size R, and per capita growth rate f(R). An average predator individual consumes prey at
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a rate given by CRg(CR), where C is a per capita attack rate, and g(CR) is a decreasing
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function (the 'satiation function') describing the proportional reduction in the predator's
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consumption rate as the encounter rate with prey increases; this may occur because of
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handling time or other factors. We assume the functional response, CRg(CR), increases
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with R. The consumed prey is converted into new predators according to a birth rate
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function, b, which is an increasing function of food intake rate (i.e., the functional
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response). The per capita death rate of predators is given by d. If predator population
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size is denoted N, these assumptions result in the following population dynamical model:
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dR/dt = R[f(R) – CNg(CR)]
(1a)
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dN/dt = N[b(CRg(CR)) – d]
(1b)
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The equilibrium predator abundance, N*, is determined by eq (1a), and is simply
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N* = f(R)/[Cg(CR)]
(2)
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Increased mortality (d) of the predator is expected to increase the equilibrium resource
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population based on the equilibrium condition for eq (1b). A greater R decreases the
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resource per capita growth rate, and increases predator satiation, which means a smaller
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g. Thus, the equilibrium predator population may increase with its own mortality if g
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decreases more rapidly than f as the result of its mortality. This is possible when the
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predator has a saturating (i.e., Holling type 2 or 3) functional response. However, as is
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well-known (Rosenzweig and MacArthur 1963), the fact that N* increases with d implies
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that the vertical predator isocline intersects the prey isocline where the latter has a
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positive slope. This in turn means that the equilibrium is an unstable focus, and that
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cycles will occur. Numerical results show that, although the mean predator population
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size differs from N*, the mean also increases with the predator's own mortality (Abrams
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et al 1997; Abrams 2002).
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Thus, the conclusion from simple predator-prey models with fixed prey
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characteristics is that the predator population will only increase with its own mortality
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rate when there are cycles. This conclusion must be modified if the predator's per capita
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growth rate is directly and negatively affected by the predator population size; e.g., if d is
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replaced by d0 + I(N), where d0 is a constant and I is an increasing function of predator
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population size. In this case it is possible for the equilibrium predator population, N*, to
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increase with the predator’s death rate, d0 in a stable system, although this typically
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requires rather careful balancing of parameter values, and only occurs over a narrow
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range of d0. An increase in fishing mortality seems most likely to affect density
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independent mortality, d0, rather than changing the density dependent death rate, I(N).
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The effect of adaptive change of prey vulnerability on the relationship
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between predator population size and predator mortality
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There are several ways in which adaptive change in prey characteristics can alter
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the manner in which the predator's population size changes with its own mortality. Prey
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vulnerability helps determine the attack rate, C, and changes in C are expected to have
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correlated effects on prey growth rate (Lima 1998). The per capita growth function of
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prey is assumed to increase with C and to decrease with R. Therefore, f(R) in eqs (1, 2)
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must be replaced by f(R, C). If f increases due to an increase in C following greater
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predator mortality, equation (2) shows that this could cause an increase in N* with
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increased d. In addition, an increase in both C and R following greater predator mortality
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can decrease the function, g, which also increases N* (see eq (2)). The increased
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predator satiation (smaller g) also makes it adaptive for the prey to reduce costly
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defences, again implying an increase in both C and the per capita growth, f. The
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remainder of this section develops this qualitative argument by augmenting the
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population dynamics model (eqs. (1)) with an equation for adaptive change in the value
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of C, assuming that lower C implies lower f. We assume that this trait is subject to
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stabilizing selection in the absence of the predator, so that 2f/C2 < 0.
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We assume that the prey's per capita growth rate f is an increasing function of
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vulnerability, C at values of C near the evolutionary equilibrium, implying a cost to
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defense. A model that describes adaptive change in an approximate manner for many
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circumstances (Abrams et al. 1993; Abrams 2001) assumes that the rate of change of a
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trait (e.g., C) in the population is proportional to the rate of change of a particular
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individual's fitness with its own trait value, evaluated at the population mean. This
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corresponds to the general finding that behaviour changes more rapidly when the reward
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from a unit change is greatest. Similarly, genetically determined traits evolve most
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rapidly when the slope of the relationship between the trait and fitness is most steep. For
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prey adaptation in the context of Equations (1a,b), the vulnerability trait changes
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according to:
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dC/dt = v[f/C – Ng(CR)]
(3)
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where v is an adaptive rate constant that scales the rate of change in the trait relative to
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the rate of change of population densities, and the expression in brackets is the derivative
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of an individual’s fitness with respect to its trait, C. The derivative of individual fitness
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with respect to the trait determines the rate of change in the mean value of the trait in
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models of mutation-limited evolution (Dieckmann and Law 1996), quantitative genetic
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models (Abrams et al. 1993; Abrams and Matsuda 1997), and behavioural models
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(Taylor and Day 1997; Abrams 2001). In all of these models, equation (3) is an
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approximation based on the assumption that the amount of variation in the trait C is
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sufficiently small. If the change is genetic, v is the additive genetic variance of the trait.
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If genetic or behavioural variation is depleted when C approaches extreme values, v will
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be a function of C (Abrams 2001).
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The system consisting of eqs (1a,b) and eq (3) can be analyzed without adopting
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specific functional forms for f and g. It is possible to determine when the equilibrium
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will be locally stable, and to determine how the equilibrium values of the variables
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change with the predator mortality, d. The details of this analysis are provided in the
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appendix. The condition required for N* to increase with d is that the following
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derivative be positive:
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 2 2 f
 f
 2 f f
 2 f 

Ng
'
C

R

C



2
C 2
C R  
N * C R
 R


d
 2 f
 f
 2 f 
gb '  g  CRg ' C 2

R

C


2
C R  
 R
 C
(4)
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The appendix discusses the signs of the terms in this expression. If the equilibrium of the
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system is locally stable, both the quantity in square brackets in the denominator, and the
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entire denominator must be negative (see appendix). In addition we assume that f/R <
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0, because the prey’s per capita growth rate declines with its own population size, and
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2f/C2 < 0, as mentioned above. Furthermore, g + CRg' > 0, because the functional
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response increases with CR (i.e., it increases with the availability of prey). These
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considerations mean that, if there is no satiation (g' = 0), the numerator of eq (4) must be
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positive, eq (4) must be negative, and the predator must therefore decrease in population
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size with increases in its own death rate. However, with satiation (g' < 0) the second of
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the two terms in the numerator of eq (4), having the form Ng'[…], is usually negative,
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making it possible for the predator to increase as its own mortality increases. One factor
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that contributes to an increase in predator density in the increase in prey growth, due to a
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larger vulnerability, C. The appendix shows that C*/d has the sign of –[f/R –
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C2f/(CR)]. Thus, unless there is a large negative interaction of the effects of C and R
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on prey growth, C* will increase with d. The greater per-individual productivity of prey
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provides the basis for the predator population to increase.
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Some of the necessary conditions for stability of eqs (1a, 1b, 3) have a simple
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form. Unfortunately, the full necessary and sufficient local stability conditions for the
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system consisting of eqs (1a, 1b, 3) is sufficiently complicated to be of little use in
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understanding stability, unless specific forms are adopted for the functions in the model.
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Here we simply examine stability numerically using specific forms for the three
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unspecified functions in eqs. (1a, 1b, 3). We assume:
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f(R,C) = rm + r1C – r2C2 – (R/K)
(5a)
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g(CR) = 1/(1 + hCR)
(5b)
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b(CRg(CR)) = b0CRg(CR)
(5c)
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Here, the prey’s per capita growth rate, f, changes in a unimodal fashion with the trait, C,
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and has a minimum value of rm. Equation (5b) implies that the predator has a Holling
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(1959) disk equation (type 2) functional response, with handling time h. Equation (5c)
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means that the birth rate function is linear; births are directly proportional to food intake
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with a proportionality constant, b0. In the absence of the predator, the prey fitness is
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given by f, and there is quadratic stabilizing selection on the prey’s trait, C. The
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functions given by eqs (5) also permit an analytical solution for the equilibrium R, N, and
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C, although the formulas for the equilibrium values are quite lengthy. The equilibrium
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predator population is plotted as a function of predator death rate for a particular set of
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parameters in figure 1; it is clear that the equilibrium predator population increases with
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mortality over the vast majority of the range of potential mortalities. The equilibrium C
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increases at an accelerating rate over the same range of mortalities in this example. The
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figure legend discusses the stability of these equilibrium values. This model is similar to
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a model whose dynamics were studied in Abrams and Matsuda (1997), and there are
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often alternative attractors (see legend of fig. 1). Because the trait dynamics given by eq
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(3) alone can result in biologically unrealistic (either very large or negative) values of the
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trait, the numerical results assume that there is selection for less extreme values if C
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approaches zero or a maximum value (see legend of fig. 1).
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It is important to note that the prediction of an increased predator population is
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based on the change in equilibrium population size. The initial response of the predator
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to an increase in its mortality is always a decrease in population size. This is followed by
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an increase in both the prey's abundance and its trait value, which eventually allow the
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predator to increase above its initial population size. Figure 2 shows an example of the
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transient responses of both densities and traits to a doubling of the predator's per capita
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mortality rate for the system described in figure 1.
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The effect of adaptive change in the prey community composition on the
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relationship between predator population size and predator mortality
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A predator-prey system that contains more than a single prey species can exhibit
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adaptive change at the level of the prey assemblage, even when no species within that
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assemblage undergoes evolutionary change. Abrams and Matsuda (1997) showed that a
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system with two competing prey that differed in their growth rates and predator
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vulnerabilities could exhibit dynamics very similar to those of a single-prey model in
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which the single prey species had an adaptive tradeoff between growth rates and predator
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vulnerability. This section will examine whether and when such systems can exhibit the
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type of increase of predator abundance with predator harvest demonstrated in the
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previous section.
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The first model considered here consists of two prey species with a single shared
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resource as well as a single shared predator. This 'diamond food web' has been analyzed
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in different ways by various authors (e.g. Armstrong 1979; Holt et al. 1994; Leibold
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1996). Neither these studies, nor more recent ones (Grover and Holt 1998; Abrams
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1999), have pointed out the implications of the food web structure for the response of the
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predator to increased mortality. If the predator has linear functional responses to both
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prey, and the prey do not have any direct effects of population size on their own growth
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rate, then the predator's equilibrium population is independent of its own mortality; this
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was noted by Leibold (1996) for a similar model of the diamond web, and is a
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consequence of the fact that the more vulnerable of the two prey types increases at the
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expense of the less vulnerable prey when mortality is imposed on the predator. When the
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predator's functional response is nonlinear, its population always increases with an
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increase in its own mortality as long as the two prey are able to coexist.
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Although we have assumed that there is no adaptive change in either C1 or C2, the
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relative frequencies of the prey change when mortality is imposed on the predator. The
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resulting increase in the more vulnerable prey increases predator satiation, reducing the
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effective attack rates on both prey. It is these reduced attack rates that allow the predator
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to increase in abundance. These results can be demonstrated by an analysis of a general
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model with three trophic levels. Having three levels entails a shift in notation; we now
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denoted the predator population size by P, the two prey populations by Ni, and the
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resource population by R. This yields a model having the following form:
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dR/dt = R[f(R) – C1N1 – C2N2]
(6a)
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dN1/dt = N1[b1(C1R) – d1 – s1PG(s1N1+es2N2)]
(6b)
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dN2/dt = N2[b2(C2R) – d2 – s2PG(s1N1+es2N2)]
(6c)
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dP/dt = P[B{(s1N1 + es2N2)G(s1N1+es2N2)} – D]
(6d)
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The symbols that are common to this system and eqs (1) have the same meaning. The
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predator’s satiation function is denoted G. The parameter e measures the energetic
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content (or nutritional value) of prey 2 relative to prey 1, and it is assumed that energetic
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content (or nutritional value) of prey determines satiation. N1 and N2 represent the
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populations of the two prey species. The per capita death rates of the two prey and the
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predator are di and D respectively, while the conversion functions of resource to prey and
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prey to predator are bi and B respectively. Note that the parameters of both prey are
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constant; adaptive change can only occur at the prey trophic level by shifts in the
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abundances of the two species. The two prey species are assumed to have linear
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functional responses to the resource to simplify the algebra and to reduce the possibility
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of complex dynamics. The same techniques described in the appendix for eqs (1a, 1b, 3)
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can be applied to this system, and the resulting expression for the change in P* with D is,
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P *
 PG '

D B ' G  G  G '(s1 N1  es2 N 2 ) 
(7)
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This expression must be positive because G' is negative and B' positive by definition, and
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the quantity in parentheses in the denominator is the derivative of the predator's
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functional response with respect to food intake, which must also be positive at a stable
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equilibrium. It should be noted that a sufficiently large magnitude increase or decrease in
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D is likely to eliminate one of the prey species, and further changes in D will only have
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the intuitive effects on the average value of P (i.e., higher D means smaller P) if the
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system is stable. Note also that without predator satiation (G = 1; G' = 0), the predator's
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equilibrium abundance does not change with an increase in its own per capita mortality.
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Thus, the harvesting rate of predators would not change the equilibrium abundance as
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long as both prey are present.
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The local stability conditions of an equilibrium of eqs (6) having all four species,
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are quite complicated. However, the equilibrium must be stable when there is no
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satiation (G = 1), since the Jacobian matrix then satisfies qualitative stability conditions
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(Jefferies 1974). Simulations have shown that a wide range of parameters will produce a
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stable system, as long as the predator's death rate is not too low and the satiation function
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is not too strongly concave.
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The quantitative change in predator population size with its own mortality can be
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illustrated for a particular version of the diamond web model. Here, we introduce some
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simplifications of eqs (6) that allow the equilibrium predator population size to be
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determined analytically. The numerical response functions, bi and B in eqs (6) are all
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linear, resource growth is logistic (f(R) = r(1 – (R/K))), and the satiation function has the
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form of Holling's (1959) disk equation; G(x) = 1/(1 + hx). In addition, we assume that
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the resource has sufficiently rapid dynamics relative to the prey that it can be assumed to
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always have its steady-state abundance of (K/r)(r – C1N1 – C2N2). This allows the
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dynamics of the two prey to be reformulated as Lotka-Volterra competition equations,
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with the product of the two competition coefficients being unity (see Abrams 1999). We
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also assume that s2 = 0; i.e., prey 2 is immune to predation. Thus, the model becomes:
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dN1/dt = r1N1[1 – (N1 + 12N2) / K1] – s1N1P/(1+hs1N1)
(8a)
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dN2/dt = r2N2[1 – (N2 + 21N1) / K2]
(8b)
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dP/dt = P[(B0s1N1) / (1+hs1N1) – D]
(8c)
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where B0 is a constant, ij is the competition coefficient of prey j on prey i, and the
308
parameters ri and Ki now refer to the prey rather than the resource. Equations (8) have a
309
relatively simple analytical expression for the equilibrium predator population:
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311
P* 
K1s12  B0  Dh 
if D 
312
313
B0 r1  B0 s1  K1  12 K 2   D 1  s1hK1  12  21  s1hK 2   
P* 
2
K 2 s1 B0
 21  K 2 s1h


K 2 s1 B0
B0 r
D
1 
 if D 
 21  K 2 s1h
s1  B0  Dh   s1 K1  B0  Dh  
(9a)
(9b)
314
315
where the second solution corresponds to the case where prey 2 is excluded. Figure 3
316
plots eq (9a,b) as a function of the predator’s per capita death rate, D; the different lines
317
are characterized by different relative carrying capacities of the two prey. As can be seen
318
from the figure, the greatest proportional increase in predator population over the widest
319
range of the potential mortality rates occurs when the invulnerable 'prey' species has only
320
a slightly smaller carrying capacity (i.e., a slight competitive disadvantage) relative to the
321
prey species that is vulnerable to the competitor. However, the predator is relatively rare
322
under these circumstances. The abrupt change in slope of the relationships shown in fig.
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3 occurs when the predator mortality is such that the less vulnerable prey is excluded by
324
the more vulnerable prey; at still higher mortalities, the equilibrium predator population
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decreases.
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The above results suggest that a slightly more complex foodweb might be capable
of producing similar phenomena. In particular, it is of interest to determine how strong
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328
the competition between the two prey must be in order to produce the result we have
329
stressed above; the predator's population size increases as its own per capita mortality
330
increases. This can be investigated by analyzing a model similar to eqs (6), but with a
331
second resource added, and some differences between the two prey in their relative
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utilization rates of those resources. This involves simply adding a second resource with
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logistic growth, characterized by attack rates by the two prey species that differ in their
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ratio from the two attack rates on resource 1. The simplest situation involves attack rates
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of prey 1 on the two resources (C11 and C12) that are mirror images of those of the prey 2
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(C21 and C22); i.e., C11 = C22 and C12 = C21. This set of consumption rates results in
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competition coefficients that are equal to each other, provided both resources remain in
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the system. Such a system can be approximated by equations (8), with competition
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coefficients 12 = 21 =  < 1. The equilibrium predator population is plotted against its
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mortality rate for different values of  in figure 4. As  is decreased from unity, the
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predator mortality rate required to exclude the less vulnerable prey 2 increases. A
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decrease in  also decreases the range of mortality rates over which predator population
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size increases with increasing mortality. Thus, predator population size is often relatively
344
insensitive to its own mortality over a wide range of mortality rates, as shown in the
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bottom panel of fig. 4. When the competition coefficients are small enough ( < K2 =
346
0.5, given the other parameters in fig. 4), the equilibrium predator population decreases
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monotonically with its own mortality rate. Of course, if one or both prey were capable of
348
adaptive adjustment of a trait related to both intrinsic growth rate and predator
349
vulnerability, the predator could decrease with harvest even when there was no
350
competition between the prey; this case would be similar to the first model analyzed here.
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Discussion
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We will refer to the increase in population size that is caused by higher mortality
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as the ‘hydra effect’, in honour of the mythological creature that grew two new heads for
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every one that was removed. The implications for fisheries policy of the hydra effect are
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quite profound. Such increases are likely to be interpreted as a license to increase
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harvesting rates further. However, the increase in the predator in these situations also
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entails a decrease in the remaining capacity of the prey to increase in population size
359
and/or vulnerability. There is also a decrease in the capacity of the predator to assimilate
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more prey because its functional response approaches its saturation level. Thus, at some
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point, the predator population will start to decrease rapidly in response to further
362
increases in harvest, as in figure 1. Of course, the equilibrium population size (shown in
363
figs. 1, 3 and 4) is not attained immediately; this time lag is illustrated in figure 2. In
364
addition, if mortality of the predator increases continuously over time, the population lags
365
behind its equilibrium value, and the final decrease in the population does not have as
366
steep a slope as these graphs of equilibrium densities might suggest. This is shown in
367
Abrams (2002), which compares the change in actual predator population and
368
equilibrium for several models of cycling predator-prey systems in which neither species
369
has adaptive change. That article also shows the population size of a predator may be a
370
large fraction of its original equilibrium size at the point where a steadily increasing
371
mortality passes the level where its equilibrium population size is zero. This means that
372
there is a risk that restrictions on harvest will come too late to prevent a catastrophic
18
373
decline or extinction. This same message applies to the models with adaptive change
374
discussed here.
375
Even if a stock never exhibits the hydra effect (an increase in population size with
376
harvesting), it is likely that both adaptive changes in its characteristics and its interactions
377
with other species will affect population dynamics. Thus, both factors may often need to
378
be studied to predict responses to harvesting or to environmental changes. There have
379
been other calls for more consideration of behavioural effects in understanding fisheries
380
systems (e.g. Dill et al. 2003), but we have previously had few examples of when and
381
why adaptive behaviour might alter the predictions of single-species approaches.
382
The predator's functional response has been shown to play an important role in the
383
results obtained here. If the predator's response is linear (Holling type 1), neither of the
384
categories of model studied here (adaptive prey or replacement of competing prey)
385
predicts a hydra effect in the predator population. This is also true of at least some other
386
functional responses that lack satiation. Here we briefly describe some unpublished
387
results on the effects of changing the 2-prey model (eqs (6)) so that it has the simplest
388
predator functional responses used in the popular computer package, Ecosim (Walters et
389
al. 1997, 2000). In this case, the functional response of a predator with population P to
390
prey species i, with population Ni is assumed to be, CiNi/(Qi + wiP), where Ci, Qi, and wi
391
are constants. This is based on a model of random movement between vulnerable and
392
protected states (Abrams and Walters 1996), rather than the adaptive defense assumed by
393
eq (3) in our analysis. Because of the lack of predator satiation in this response, the
394
effect of mortality on the predator is similar to that in a comparable model with linear
395
functional responses; i.e., no change in equilibrium predator density. We could also
19
396
modify eqs (6) so that the two prey species as well as the predator, have the simplest
397
possible Ecosim functional responses. Because of the dependence of prey functional
398
responses on their own population density, competition between the prey is reduced and
399
the model becomes more similar to eqs (8). As in that model, when there is no satiation,
400
the equilibrium predator density always declines with increases in mortality. Ecosim
401
includes options for including both variable foraging and handling time. These use an
402
iterative numerical procedure to determine handling and foraging times (Walters and
403
Martell 2004). While the mathematical form of the resulting functional response differs
404
from the responses used here, it shares the property of approaching saturation at high
405
population levels. Any such response will often lead to predator populations that increase
406
with increasing harvest rates, given parameter values and food web configurations similar
407
to those explored here. There is at least some evidence for strong satiation in some fish
408
species (e.g. Eby et al. 1997), and Koen-Alonzo and Yodzis (unpub. results) have found
409
at least one instance of stock size increasing with harvesting in an ecosystem model with
410
saturating functional responses.
411
The models analyzed here have all been quite simple, and it is important to
412
determine to what extent our conclusions depend on that simplicity. One question is
413
whether competitive interactions with other predators can prevent the kind of increase in
414
abundance with increased mortality that is demonstrated here. An increase in a second
415
predator following increased mortality of the first might prevent any increase in prey
416
abundance or vulnerability from increasing the first predator. Our models have also
417
assumed that the populations are unstructured; all individuals within a species are
418
identical with respect to both their interactions with other species and their susceptibility
20
419
to harvest. There are many circumstances when structured population models have
420
qualitatively different dynamics than do unstructured population models (e.g., Iwasa et al.
421
1987). Finally, it is important to investigate how coupled adaptive changes in both
422
predator and prey are likely to affect the responses of the populations of either or both
423
species to exploitation. It will require additional work to determine whether models
424
incorporating these additional features differ systematically in the likelihood that a
425
harvested species will increase over a significant range of harvest rates.
426
Although there are many examples of fisheries that have been overexploited, we
427
do not know of any for which there is clear evidence that increased fishing pressure
428
caused an increase in the population size of top predator species, or conversely, that
429
decreased fishing pressure resulted in a decrease in stock size. We have searched R.A.M.
430
Myers’ online database (http://ram.biology.dal.ca/~myers/data.html), which provides
431
estimates of both stock size and fishing mortality for a number of species. There are
432
some examples where decreases in stock size occurred simultaneously with a decrease in
433
fishing mortality (e.g., monkfish, Lophius piscatorius, data from the Report of the ICES
434
Working Group on the Assessment of Southern Shelf Demersal Stocks). However, this
435
pattern is also consistent with a negative effect of harvesting on stock size, provided the
436
decrease in fishing mortality is not sufficiently large. For most top predators in Myers’
437
database, stock estimates are unavailable until after fishing pressure had been intense for
438
some time. Thus, in most of the data sets having estimates of fishing mortality, the
439
mortality was already relatively high when it was first measured (or at least reported).
440
As far as we have been able to determine, past records of harvest and stock size do not
441
provide clear evidence for the phenomena we have predicted here.
21
442
It is possible that increases in stock size due to harvesting has never occurred
443
because of some complexities of natural communities that are missing from the models
444
considered here. It is also possible that such increases occurred, but were not observed
445
because reliable information on stock size is often not available during the early stages of
446
a fishery on a given species. Because of environmental variability and variation in
447
fishing rates, it may be difficult to assign a cause to an observed increase in abundance.
448
It is also possible that the high initial rate and rapid increase in harvest rates for many fish
449
stocks have meant that the fishery collapsed before any of the compensatory processes
450
discussed here had time to operate. Restrepo et al. (1998) argue that the natural mortality
451
rate may be used as a reasonable proxy for the MSY fishing mortality rate. Harvest rates
452
in modern fisheries are usually much larger than the natural mortality rate, and they have
453
increased rapidly when stock sizes have remained reasonably high (Pauly et al. 2002). In
454
addition, modern fisheries have often involved harvesting of several trophic levels, or
455
sequential harvest of different predatory species (Jackson et al. 2001). Any of these
456
conditions may have prevented an increase in predator stock sizes in response to changes
457
in the population densities and/or defensive traits of their prey.
458
Recent documentation of extensive declines in predatory fish species have
459
brought about calls for greater consideration of the maintenance of biological diversity in
460
fisheries regulation (e.g. Pauly et al. 2002). Such a shift would likely require that future
461
fisheries not increase harvest rates as rapidly as has been typical in the past. This would
462
make it possible to determine whether the ‘hydra effects’ described here and in Matsuda
463
and Abrams (2004) are likely to occur in natural fish communities.
464
465
22
466
Acknowledgements
467
The work was supported by a Strategic Project grant from the Natural Sciences and
468
Engineering Research Council of Canada to P.A. and a grant from the Japan Society for
469
Promotion of Science to H. M. We thank C. J. Walters and D. L. DeAngelis for their
470
comments on an earlier draft.
471
23
472
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567
Appendix
Stability and responses of equilibrium values to predator mortality in Eqs (1a, 1b, 3)
To determine the change in the equilibrium R, N, and C with an increase in d, we
568
569
set the left hand sides of equations (1a, 1b, 3) equal to zero, and differentiate the resulting
570
equations with respect to d, noting that the equilibrium values of the variables are implicit
571
functions of d. This yields 3 simultaneous linear equations for the three quantities,
572
N*/d, C*/d, and R*/d. As an example, differentiating equation (1a) at
573
equilibrium yields the following equation:
574
0 f '
575
Solving this, together with the corresponding equations derived from the equilibrium
576
conditions for equations (1b) and (3), yields expressions showing how the equilibrium
577
values of the three variables change with the predator's death rate, d:
578
 2 2 f
 f
 2 f f
 2 f 

Ng
'
C

R

C



2
C 2
C R  
N * C R
 R


d
 2 2 f
 f
 2 f 
gb '  g  CRg ' C
 R
C

2
C R  
 R
 C
(A1)
579
 f
2 f 
C


R
CR 
C *


d
 2 2 f
 f
 2 f 
b '  g  CRg ' C
 R
C

2
CR  
 R
 C
(A2)
580
 2 f 
C

CR 
R *


d
 2 2 f
 f
 2 f 
b '  g  CRg ' C
 R
C

2
CR  
 R
 C
(A3)
R *
R * 
N *
C *
 C *
 CNg '  R
C
 Ng
  Cg
d
d 
d
d
 d
29
581
The conditions for the local stability of the equilibrium point(s) are found from the
582
Jacobian matrix of the dynamic system eqs (1a, 1b, 3) evaluated at the equilibrium point.
583
Unfortunately, the full conditions for stability (conditions that the eigenvalues of this
584
matrix have negative real parts) are very complex. However, one of the necessary
585
conditions is that the quantity in square brackets in the denominators of the above three
586
expressions must be negative. (This quantity is equal to a positive constant multiplied by
587
the determinant of the Jacobian matrix.)
30
588
Figure Legends
589
590
Figure 1. The equilibrium predator population size as a function of its own per capita
591
mortality rate for a system in which dynamics are described by equations (1a, 1b, and 3),
592
using the functional forms given by eqs (5). The model also included minimum and
593
maximum values of 0 and 5 respectively for the trait C. The trait was prevented from
594
exceeding these bounds by adding a function, /C2 - /(5 – C)2, to the right hand side of
595
eq (3), with  = 10-6. This provides a force that pushes C away from these boundaries,
596
but has a negligible effect when C is significantly different from 0 or 5. The parameter
597
values from eqs (5) are: rm = 2; r1 = 2; r2 = 0.2; K = 1; h = 0.5; b0 = 1. If adaptation is
598
slow (approximately v < 0.01), the equilibrium is stable over most the entire range of
599
mortality rates shown (approximately d > 0.15). More rapid adaptation produces limit
600
cycles over a broader range of low mortality rates. For example, if v = 0.5 the system
601
always exhibits cycles for d < 1.37, and there are alternative cyclic and noncyclic
602
attractors for approximately 1.37 < d < 1.68.
603
604
Figure 2. The transient responses of prey, predator and trait, to a doubling of the predator
605
mortality at time 0, for the system discussed in figure 1. The rate constant for change in
606
C was v = 0.05. The initial system was at equilibrium for the parameter values given in
607
fig. 1 with a predator mortality rate of d = 0.75. This mortality was doubled to d = 1.5 at
608
time 0.
609
31
610
Figure 3. The equilibrium abundance of the predator as a function of its own death rate.
611
The underlying model eqs (8), in which the resource dynamics are not explicit and the
612
prey compete via Lotka-Volterra competition. The parameters are, 12 = 21 = 1; K1 = r1
613
= r2 = 1; h = 1; s1 = 1; B0 = 1. Each line shows the relationship for the value of K2 given
614
above the line.
615
616
Figure 4. The equilibrium abundance of the predator as a function of its own death rate,
617
for a range of different strengths of competition. The model is given by equations (8) and
618
it is assumed that the two competition coefficients, ij, are both 0.5. The other parameter
619
values are the same as in figure 3.