Modeling butterfly metapopulations: Does the logistic equation accurately model
metapopulation dynamics?
Ben Slager
Introduction
In 1798 Thomas Malthus constructed the theoretical framework for the logistic equation
in “Essay on the Principle of Population” (Berryman 1992). In his essay Malthus argued that
populations grow exponentially while resources that feed population growth stay constant or
grow arithmetically (Berryman 1992). Malthus proposed that these dynamics lead to a halt in
population growth when resource demand becomes greater than supply (Berryman 1992). In
1838 Verhulst translated Malthus’ arguments on population growth into the mathematical model
dN/dt = aN( 1 – N/K), which is known as the logistic equation (Berryman 1992). In the logistic
equation N is population density, a is the intrinsic rate of increase and K is the equilibrium
density or carrying capacity (Berryman 1992). A sigmoid curve is characteristic of the logistic
equation where population density rises exponentially to an asymptote called the carrying
capacity.
The initial uses of the logistic model investigated single species dynamics. In 1926
Volterra (and Lotka in 1932) incorporated a competition coefficient into the logistic equation and
modeled species interactions through interspecific competition (Begon et al. 2006). Lotka and
Volterra also modeled predator-prey interactions. To model predator-prey dynamics they
deviated from the logistic form and took a mass action approach (Berryman 1992). The mass
action approach assumes that population responses are proportional to biomass densities. The
predator-prey mass action equations are dN/dt = aN – bNP for the prey population and dN/dt =
cNP – dP for the predator population (Berryman 1992). In these equations a and d are per-capita
rates of change in the absence of the other species, b and c are rates of change due to interaction
and N and P are prey and predator biomass densities, respectively (Berryman 1992). In this mass
action model the prey population increases without limitation. The unrealistic assumption of
unchecked prey population growth is overcome by addition of a logistic term to the equation,
which alters the original prey equation to dN/dt = aN( 1 – N/K) – bNP. The logistic equation has
been the core or point of development for several models in ecology. One of the most recent
applications of the logistic equation to ecological modeling has been in the field of
metapopulation research.
A metapopulation is a population of subpopulations linked together through migration
(Begon et al. 2006). Implicit in this definition is the state of potential habitat in the landscape,
which is usually fragmented because the overall population is made up of subpopulations linked
together through migration which implies that the landscape is separated into several individual
habitat patches. Subpopulations of the larger total population occupy individual habitat patches
and interactions of subpopulations take place at a specific dispersal or migration rate. Below a
certain dispersal threshold subpopulations in a metapopulation framework would be considered
distinct populations. Above a specific dispersal threshold subpopulations in a metapopulation
framework would be considered one large population.
Habitat fragmentation occurs naturally and increasingly from anthropogenic sources.
Habitat fragmentation is of great concern to conservationists because it can lead to an increased
risk of species extinction due to smaller population sizes with increased vulnerability to
environmental stochasticity (Etienne & Heesterbeek 2000). In contrast to these negative effects
of fragmentation there are potential positive effects such as a greater probability of colonizing an
Slager Page 1 empty patch (Etienne and Heesterbeek 2000). These opposing lines of evidence surrounding
habitat fragmentation lead to the SLOSS debate. SLOSS stands for single large or several small
and is in reference to the amount and size of potential habitat reserves for a species that has been
targeted for conservation purposes. In order to determine an appropriate size and number of
habitat patches for a species it is necessary to understand the dynamics of the metapopulation.
By the 1940’s research had already led to the understanding that populations with
fragmented or patchy distributions may endure evolutionary implications, for example,
speciation. However, many believe that the landmark text by Andrewartha and Birch in 1954 is
the true core for the development of metapopulation dynamics (Begon et al. 2006). Andrewartha
and Birch were very aware of and advocated for population dynamics being seen as a continuous
series of colonization-extinction events (Hanski and Gilpin 1991). Additionally, Andrewartha
and Birch recognized that colonization of habitable patches in the landscape was a function of
the ability of individuals to disperse to them and that this interaction left some habitable patches
vacant (Begon et al. 2006). The concept of patchy distributions and the implications on
population dynamics that are important for conservation purposes seem to have made small and
few steps between 1954 and the late 1960’s.
The earliest metapopulation models, Island Biogeography (1967) and Levins (1969)
models, as well as the more recent and much more publicized Incidence Function (1994) model
are rooted in the logistic equation (Hanski and Gilpin 1991). The logistic equation has been
successfully used to model single species dynamics, interspecific competition and increased
realism in predator-prey models. However, is it possible for the logistic equation to model a
metapopulation? To what degree can a logistic based model predict real world metapopulation
dynamics? How does a logistic based model compare to other methods of metapopulation
modeling? To answer these questions I will first investigate the development and statistical
strength of logistic based models. Then I will look at the performance of a logistic based model
compared to other types of metapopulation modeling.
LOGISTIC BASED MODELS
Mainland-Island
Before the term metapopulation arrived in the literature, provided by Levins in 1970
(Hanski and Gilpin 1991), MacArthur and Wilson developed their equilibrium theory of island
biogeography (MacArthur and Wilson 1963 & Hanski and Gilpin 1991). Although MacArthur
and Wilson’s theory was built with the intentions of understanding gradients of species richness,
it is applicable to understanding metapopulation structures because it is based on the opposing
forces of colonization and extinction (Hanski and Gilpin 1991, Begon et al. 2006). MacArthur
and Wilson’s theory of island biogeography was developed in the context of real oceanic islands
but also applies to terrestrial metapopulations that experience great differences in patch size
(Harrison 1991).
MacArthur and Wilson incorporated migration and extinction into the logistic equation in
order to track the change in the fraction of islands that are occupied over time (Hanski and Gilpin
1991). The equation they used to model metapopulation dynamics was dp/dt = m(1 – p) – ep,
which is analogous to the logistic equation (Hanski and Gilpin 1991). In this model m is the rate
of migration, p is the proportion of patches occupied and e is the rate of extinction (Hanski and
Gilpin 1991). The equilibrium value, p, is p = m/(m+e). For islands experiencing any turnover
(extinctions) the equilibrium value will be positive (Hanski and Gilpin 1991).
Slager Page 2 MacArthur and Wilson stressed that local populations would fall somewhere between the
continuum of r and K selected species. Good colonizers that show quick population growth and
high rates of colonization and extinction are r selected species (Begon et al. 2006). r-species will
tend to spend large amounts of time recovering from past crashes or invading new territories
(Begon et al. 2006). Conversely, K-species are poor colonizers with long term population
persistence. These dynamics show low rates of colonization and extinction. Consequently, Kspecies spend most of their time near carrying capacity (Begon et al. 2006).
The spatial structure of Macarthur and Wilson’s island biogeography is such that there is
a single mainland and multiple islands at varying distances surrounding or adjacent to the
mainland (Figure 1 and Figure 2a). This type of mainland-island structure is similar to that of
Figure 1. Closed circles represent
habitat patches, filled = occupied,
unfilled = vacant and dashed lines
are population boundaries. Arrows
indicate colonization (Harrison
1991).
Figure 2. A continuum of metapopulation
structures (a) mainland-island, (b) intermediate
and (c) Levins (Hanski & Thomas 1994).
the Boorman and Levitt structure (Harrison et al. 1988). The number of individuals or patch
occupancy is determined by the balance between immigration and extinction, which will vary
with island area and isolation from the mainland (Hanski and Thomas 1994). From figure 3a we
can see that the rate of immigration decreases as the distance from the mainland to the island
(potential colonization site) increases. This relationship is due to the increased probability of
finding an island close in proximity to a source of colonizers (Hanski and Gilpin 1991). The
more isolated an island is the less likely a colonization event (immigration) is to occur on that
island. Similarly, as the size of the island decreases there is a smaller target for potential
colonizers and the immigration rate drops (Figure 3a). A smaller island that has similar habitat
quality as a larger island will typically have a lower carrying capacity and contain fewer
individuals in the local population. Because there are fewer individuals in the local populations
of smaller islands they are more susceptible to random extinction events (Begon et al 2006). This
dynamic is illustrated in figure 3b where there is an increased extinction rate in the smaller
islands compared to the larger ones.
Although local populations (islands) of the mainland-island metapopulation structure are
subject to extinction events the metapopulation as a whole is not (Harrison et al. 1988, Hanski
and Gilpin 1991, Hanski 1994). This immunity from metapopulation collapse is due to the
permanent mainland source of potential colonizers (Harrison et al. 1988, Hanski and Gilpin
1991, Hanski 1994). From time to time the mainland-island metapopulation structure is confused
Slager Page 3 with source-sink dynamics (Hanski and Gilpin 1991). In the mainland-island structure the
differences between mainland and island are seen as stochastic while deterministic habitat
differences are a distinct characteristic of source-sink population structures (Hanski and Gilpin
1991).
Figure 4. Distribution of serpentine grassland habitat for butterfly populations. Arrows
indicate the seven populations measured in 1987 (Harrison et al. 1988).
The
first full scale butterfly metapopulation study was carried out by Harrison et al. (Harrison et al.
1988) on the bay checkerspot butterfly (Euphydryas editha) in California (Hanski and Gaggiotti
2004). The bay checkerspot butterfly uses Plantago erecta as the predominate larval host plant
and Orthocarpus as an alternative host (Harrison et al. 1988). The nectar resources for the bay
checkerspot include Lasthenia chrysostoma, Lomatium macrocarpum, Layia playglossa and
Linanthus androsaceus (Harrison et al. 1988). Host plants and nectar resources of the bay
checkerspot can be found on grasslands based on serpentine soil (Harrison et al 1988).
Harrison et al. identified a potential study area of 15 by 30 km on serpentine outcrops in
southern Santa Clara County, California within the vicinity of Morgan Hill (Figure 4). This area
was framed by natural topography and a metropolitan area. Within this study area there was one
very large patch (ca. 2000 ha), named Morgan Hill, and 59 additional smaller patches of
serpentine grassland ranging in area from 0.1 to 250 ha (Harrison et al. 1988). The smaller
patches surrounded Morgan Hill on its entire western side (Figure 4).
A survey in 1986 by Harrison et al. found that only eight of the small patches within 1.44.4 km from Morgan Hill were occupied by the bay checkerspot butterfly (Harrison et al. 1988).
Alternatively, no adults were present on 15 apparently suitable habitat patches that were 4.9-20.8
km from Morgan Hill (Harrison et al. 1988). This could be explained by distance-dependent
colonization indicative of the mainland-island metapopulation structure (Figures 5 & 6)
(Harrison et al. 1988). The bay checkerspot population was found to be controlled by an absolute
threshold of distance and habitat quality (Figure 7).
Harrison et al. used the data they collected from the bay checkerspot as input into their
logistic based mainland-island model to predict times to extinction for local patch populations
within the metapopulation. The authors predicted that this particular metapopulation will “wink
in and out” around an equilibrium population of 8-11 patches. These predicted patches had an
equilibrium probability greater than 50% and included all 8 of the observed occupied patches
from the survey in 1986 (Table 1).
Although statistical tests indicating the strength of model predictions are missing from
this study it is nonetheless impressive how similar the model predictions were in comparison
Slager Page 4 with observed patch occupancy. I feel that this is a good indication that the logistic equation is
capable representing metapopulation dynamics. However, having statistics to back up these
rough comparisons would strengthen the argument that a logistic based model can be used to
investigate metapopulation dynamics.
Figure 6. Predicted probabilities of
colonization over time for patches located at
differing distances from the main source
(MH) (Harrison et al. 1988).
Slager Figure 5. Predicted yearly immigration
rates as a function of distance. D´ is the
dispersal constant (Harrison et al. 1988).
Figure 7. Thresholds for distance from MH
and habitat patch quality (Harrison et al.
1988).
Page 5 Table 1. Predicted probabilities of colonization, extinction and equilibrium
occupancy for potential habitat patches (Harrison et al. 1988).
From Levins to Incidence Function
Richard Levins coined the term metapopulation in 1970 while constructing a simple
model that would investigate the dynamics of an assemblage of local populations connected to
each other by occasional migration (Levins 1969, Levins & Culver 1971, Hanski and Gilpin
1991, Hanski 1994, Braak et al. 1998 and Begon et al. 2006). Levins’s original motivation and
application of his metapopulation model was to a pest control situation (Levins 1969 and Hanski
and Gilpin 1991). Levins’s metapopulation model didn’t experience the same early success as
the island biogeographic theory (Figure 8). Hanski and Gaggiotti speculate that this may be
because MacArthur and Wilson were already widely respected scientists and publishing in top
tier journals (Hanski and Gaggiotti 2004). Maybe more importantly, MacArthur and Wilson’s
model was attached to the species-area relationship, which allowed for ecologists to use it in
their research (Hanski and Gaggiotti 2004). Levins’s metapopulation model did not provide a
similar opportunity for empirical research and is probably in part responsible for it being largely
shelved for 20 years (Hanski and Gaggiotti 2004).
Slager Page 6 Figure 8. Number of citations in the BIOSIS database to the keywords indicated and
divided by the total number of citations in a particular year (Hanski & Gagiotti 2004).
Levins used dp/dt = mp(1 – p) – ep as the core of his metapopulation model, which has
been demonstrated to be structurally analogous to the logistic equation (Hanski and Gilpin 1991).
For this equation m is the migration rate, p is the proportion of habitat patches that are occupied
and e is the extinction rate (Hanski and Gilpin 1991). In the Levins model it is assumed that
habitat patches are all of equal size and that the local populations are either extinct or at carrying
capacity (Hanski and Gilpin 1991) (Figure 2c). This leaves local dynamics to be largely ignored
other than the presence or absence of local populations. In the Levins model, the spatial
arrangement of patches is not taken into consideration and so the rate of migration from any one
patch to another is the same (Levins and Culver 1971, Hanski and Gilpin 1991). Also, the
probability of extinction is constant and the same for all local populations (Hanski and Gilpin
1991). This characteristic of extinction prevents the presence of a local population obtaining
“mainland status” and exemption from extinction (Begon et al 2006).
In the early 1990’s Ilkka Hanski relaxed many of the constraining assumptions of the
Levins metapopulation model to form an intermediate model that he called the incident function
model. In his model Hanski developed the use of the logistic equation from the Levins model to
incorporate an increasing number of environmental inputs affecting metapopulation dynamics
(Hanski and Thomas 1994). Hanski’s incidence function model is considered to be a discrete
time stochastic patch occupancy model (SPOM); meaning that it takes the random demographic
events into consideration while determining the presence or absence of a species in habitat
patches (Hanski and Gaggiotti 2004). The incidence function model is perhaps the most well
known of the SPOM’s and possibly of metapopulation models in general (van Nouhuys 2009).
In the incidence function model extinction is modeled by the equation; Ei = e / Aix
(Hanski 1994a). The rate of extinction is population size dependent so, Ei is a function of Ai
(Hanski 1994a). This model allows for some variability in the dependence of extinction risk on
patch size (Hanski 1994a). Variability is measured by X, if X > 1 there is a range of patch sizes
where extinction becomes increasingly unlikely and if X < 1 there is no critical patch size and all
sizes of populations and patches face a great risk of extinction (Hanski 1994a). In the extinction
equation above habitat quality is assumed to be equal throughout all habitat patches, which gives
them all the same equilibrium density. However, habitat quality can be modeled as linearly
related to population density. To incorporate this latter characteristic area (Ai) would be replaced
by quality (Qi) (Hanski 1994a).
Slager Page 7 Colonization probability (Ci) is a function of the number of immigrants arriving at patch i
per year (Mi) and is modeled by the equation Ci = Mi2/ (Mi2+ y2) (Hanski 1994a). Mi is largely
ignored because the change in the fraction of patches occupied is quite small at equilibrium
(Hanski 1994a). Colonization increases in an S shape with increasing number of immigrants and
y determines the speed to equilibrium (Hanski 1994a). To determine the number of immigrants
(Mi) there are a variety of components taken into consideration for the variables Si and β for the
equation Mi = βSi. Si is influenced by the distance between patches moved and the survival rate
of migrants over the distance moved (Hanski 1994a). β is determined by the density of
individuals in patches, the rate of emigration and the fraction of emigrants moving toward a
specific patch (Hanski 1994a). With these influences incorporated the equation for colonization
becomes Ci = 1 / (1 + [y’ / Si]2). Here y’ describes the colonization ability of the organism where
a low value of y’ negates the affects of isolation (Hanski 1994a). So, the probability of a patch
being occupied is a function of patch size and it’s spatial location in terms of other occupied
patches (Hanski 1994a). The essential data to construct the incidence function model are patch
area, spatial locations of patches and the occupancy (presence/absence) of the present set of
patches. With this set of data the remaining parameter values can be estimated and
metapopulation dynamics can be predicted for most systems.
To test this model Hanski collected the necessary information (described above), fit it to
the incidence function model and then compared the model output to observed values (Hanski
1994a, Hanski and Thomas 1994). Hanski (1994a) gathered data for three butterfly
metapopulations (M. cinxia, H. comma and S. orion) and compared observed values of critical
patch areas (A0) and turnover events (colonization and extinction) to model predictions (Table 2).
From observation of table 2 one will quickly notice that the predicted values of Ao of M.cinxia
and H.comma are approximately half that of the observed values. However, the author believes
that due to population dynamics these are reasonable estimates of critical patch area (Hanski
1994a). The author holds a similar belief for the predicted and observed turnover rates in both M.
cinxia and H. comma (Table 2) (Hanski 1994a). The agreement between the model and
Table 2. Results of butterfly metapopulation modeling (Hanski 1994a).
observations for S. orion seem to be much better (Table 2) (Hanski 1994a). However the issue
arises again that there are no supporting statistics to confirm the conclusions that were made. So,
although I feel that it is reasonable to conclude that the model predictions and observations are
Slager Page 8 similar it is difficult to make definitive statements about the accuracy of the model without
statistical testing.
Table 3. The nine parameters used for model simulation
(Hanski & Thomas 1994).
Hanski and Thomas added additional detail to the incidence function model by adding six
environmental parameters, which in total incorporated nine parameter values for three butterfly
metapopulations (M. cinxia, H. comma and P.argus) (Table 3) (Hanski and Thomas 1994). After
simulation of the enhanced incidence function model the authors found that observations and
predictions matched up well in terms of magnitude, particularly the fraction of patches occupied
(Table 4). This round of investigation into the incidence function model included statistical
evidence (t-values in table 4). From observation of the t-values in table 4 it can be seen that the
majority of parameters included in the model were found to have a significant impact of the
predictability of metapopulation dynamics. The statistical strength of this nine parameter model
and the close relation between the observed and predicted fraction of occupied patches leads me
to believe that the logistic equation is a capable base for metapopulation modeling and a
potential platform to inform conservationists and ecologists alike.
As an explanation for the discrepancy in predicted and observed values of occupied
patches for H. comma (Table 4) the authors suggested that the metapopulation potentially was
not in equilibrium and thus throwing off the predicted values (Hanski and Thomas 1994). The
assumption that the metapopulation is at an equilibrium state may draw the most criticism to the
incidence function model (Baguette 2004). It is not that other models are free from this critique,
but rather the popularity of the incidence function model brings more attention to it (Baguette
2004 & Hanski 2004).
Slager Page 9 Based on the success of statistical testing and the close correspondence between predicted
and observed patch occupancy I conclude that the incidence function model serves as a reliable
tool for metapopulation modeling. A benefit of this model is that it can be made as simple or
complex as the data allow (Hanski and Gilpin 1991). This characteristic of the incidence function
model makes it more accessible to conservationists that may not have multiple years of data
available or the amount of time required to derive equations. Additionally, it is relatively
straightforward to manipulate some of the assumptions underlying the structure of the model to
better fit a target species, making this model applicable to most any organism (Hanski et al.
2000).
Table 4. Comparison of observed and predicted effects of isolation and patch area on
patch occupancy and local density. (t-values are included in the brackets) (Hanski and
Thomas 1994).
MODEL COMAPARISONS
Structured Metapopulation Model
A structured metapopulation model can incorporate a variety of parameters but central to
this type of model is local population size. In Schtickzelle and Baguette (2004) the authors
created a structured model to investigate the metapopulation dynamics of the bog fritillary
butterfly. At the core of their structured model were population growth rates and dispersal ability
of the bog fritillary butterfly. The authors included two additional descriptive components to
their model, time and space. A time structure component was included with the addition of
seasonal weather information for each year. Spatial structure was added into the model by
incorporating information about patch location. The authors used data available from a 10 year
field study of the bog fritillary butterfly as input for commercial software (RAMAS/GIS), which
set the parameters for the metapopulation model. The resulting metapopulation model was run
for 1000 simulations in order to illustrate the long term metapopulation dynamics of the bog
fritillary.
The results of the 1000 simulations were presented in comparison to the observational
field data without statistical testing (Table 5). The authors only commented on the results in
terms of the correspondence between observed and model predicted values of population size,
similar to what we have previously encountered with the island biogeographic model and the
Slager Page 10 first investigation of the incidence function model presented in this paper. Hanski (2004) has
pointed out that structured simulation models are nearly impossible to statistically validate
(Hanski 2004). Additionally, he points out that it is inappropriate to validate a model through
rough comparison of the same empirical data that were used to set the model parameters (Hanski
2004). Although there is an appropriate time and place for the use of simulation models, in
general I believe that due to it’s ability to perform and the outcome of statistical analysis of the
incidence function model, it is a much more robust method for developing models of
metapopulation dynamics than the structured model presented here.
Table 5. Model validation through comparison of observed
and predicted population size (NPL) distributions
(Schtickzelle & Baguette 2004).
Table 6. Model validation through comparison of observed and predicted data-set
parameter estimates (Schtickzelle & Baguette 2004).
Individual Based model
An individual based model was constructed by Heino and Hanski (2000) that
incorporated 3 models (biological, statistical and long term), which collaborated to predict the
metapopulation dynamics of Melitaea diamina. The biological model was based on a landscape
containing many habitat patches of different sizes and spatial qualities (Hanski et al. 2000).
Slager Page 11 Immigration and emigration were seen as patch area dependent, while migration mortality was
based on isolation of the focal patch (Hanski 2000). The statistical model was constructed around
the dependent assumptions of migration and patch size as well as mortality and patch isolation
(Hanski 2000). The previous two models set the parameters for the third model, the long term
model. This final model forced out the metapopulation dynamics (Heino & Hanski 2001).
However, this third model is the incidence function model and is the source of my doubts about
the individual based approach seen here.
The results of the individual based metapopulation model illustrated similar dynamics to
those of the observed Melitaea diamina metapopulation, as well as to the results of the incidence
function model (Table 7). As mentioned previously, Hanski (2004) pointed out in a critique of
Schtickzelle and Baguette’s (2004) structured model, it is inappropriate to validate a model’s
results by comparing them to the empirical data that they came from. Similar to this I think that it
is inappropriate to compare the results of essentially the same model bearing two separate names.
To me, this individual based model is the incidence function model but presented with an
alternative way to parameterize the data. Because the last step of the individual based model is to
run the data through the incidence function model I believe these results are further evidence that
the incidence function approach is a well proven model. However, I feel that nothing is really
proven here about the ability of the individual based model to represent metapopulation
dynamics.
Table 7. Comparison of the properties of the incidence function model
and the individual-based model for Melitaea diamina. (The observed
patch occupancy was 0.37) (Values in parentheses are standard errors)
(Heino & Hanski 2001).
Conclusions
From review of the literature I feel that although I am confident in saying the incidence
function model has been empirically tested and is statistically robust, not all logistic based
models are. The Levins model posed a problem with being empirically tested and the statistics
were missing from the predictions of the island biogeogrpahic model. Although these models
were a necessary bridge to the incidence function model, I feel they are no better of a
conservation tool than the individual based metapopulation model or the structured model.
The individual based metapopulation model made accurate predictions of metapopulation
dynamics but this was brought about by use of the incidence function model. To me, what this
evidence shows is that not only is the incidence function model a great predictor of
metapopulation dynamics but that the individual based model may be an alternative way to
parameterize it. This alternative method may enable conservationists to use data previously
collected or provide a simpler method of collecting the appropriate data. However, by calling this
model something other than the incidence function model is misleading.
Slager Page 12 The structured model experienced a similar barrier as the island biogeographic model and
the initial presentation of the incidence function model, it is difficult to rigorously test and leaves
validation up to rough comparison. Perhaps in some cases a rough comparison would be
satisfactory. However, in this particular case the model predictions were compared to the
observed data that set the model parameters to begin with. In a circular situation like this it seems
that only strong statistical testing is appropriate to validate the model.
All metapopulation models are not equally capable of modeling metapopulation
dynamics. The Levins model was unable to be empirically tested, while the island biogeographic
and structured models were unable to be statistically tested. However, from further development
of the logistic based models the incidence function model was developed and eventually made to
be statistically tested and shown to accurately represent metapopulation dynamics.
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