Structured Metapopulation Model Research Articles

On biodiversity in river networks: A trade‐off metapopulation model and comparative analysis

A discrete, structured metapopulation model is coupled with the strictly hierarchical competition‐colonization trade‐off model, in which competitively superior species have lower fecundity rates and thus lower colonizing ability, to study the resulting biodiversity patterns in river networks. These patterns are then compared with those resulting from the neutral dynamics, in which every species has the same fecundity rate and is competitively equivalent at a per capita level. Significant differences exist between riparian biodiversity patterns and those predicted by theories developed for two‐dimensional landscapes. We find that dispersal directionality and network structure promote species that produce a large number of propagules at a species level; such species are considered competitively superior in the neutral model and inferior in the trade‐off model. As a result, the two key characteristics of riparian systems, dispersal directionality and network structure, lead to lower and higher overall γ diversity in the former and the latter models, respectively. The network structure, through the containment effect due to limited cross‐basin dispersal, always leads to higher between‐community, β diversity. The spatial distribution of local, α diversity becomes heterogeneous and thus important under directional dispersal and network structure. A higher degree of dividedness results in higher γ diversity for communities obeying both neutral and trade‐off models, but the increase is more dramatic in the latter.

A neutral metapopulation model of biodiversity in river networks

In this paper, we develop a stochastic, discrete, structured metapopulation model to explore the dynamics and patterns of biodiversity of riparian vegetation. In the model, individual plants spread along a branched network via directional dispersal and undergo neutral ecological drift. Simulation results suggest that in comparison to 2-D landscapes with non-directional dispersal, river networks with directional dispersal have lower local ( α ) and overall ( γ ) diversities, but higher between-community ( β ) diversity, implying that riparian species are distributed in a more localized pattern and more vulnerable to local extinction. The relative abundance patterns also change, such that higher percentages of species are in low-abundance, or rare, classes, accompanied by concave rank-abundance curves. In contrast to existing theories, the results suggest that in river networks, increased directional dispersal reduces α diversity. These altered patterns and trends result from the combined effects of directionality of dispersal and river network structure, whose relative importance is in need of continuing study. In addition, riparian communities obeying neutral dynamics seem to exhibit abrupt changes where large tributaries confluence; this pattern may provide a signature to identify types of interspecific dynamics in river networks.

Size distribution and spatial autocorrelation of subpopulations in a size structured metapopulation model

We construct simple size structured metapopulation models for single species to analyze how the dynamics of the metapopulation is affected by modes of dispersal. We assume infinite number of identical patches, each of where subpopulation grows in logistic manner. It also becomes extinct randomly at a constant rate. Some portion of subpopulations disperse among patches. We examine three types of dispersal among patches—(1) single population (SP): the dispersal rate is infinite, then we can consider the whole metapopulation as a single population, (2) global dispersal (GD): dispersal occurs at a finite rate and globally irrespective of distance and (3) local dispersal (LD): dispersal occurs only between neighboring patches. First, we compare the mean subpopulation size in equilibria among the three models. The mean subpopulation size of GD model and LD model are both smaller than SP model, but the difference among them decreases as the dispersal rate increases. The difference between GD model and LD model is totally small but not negligible when both of the dispersal rate and the intrinsic growth rate are small. We also compare the variance of the subpopulation size in equilibria of GD model and LD model. We find that the variance of GD model is larger than LD model when the growth rate and the dispersal rate are so small that the mean subpopulation sizes are small, while the former is smaller than the latter when the growth rate or dispersal rate is large. Next, we examine the subpopulation size distribution in equilibria for GD model and LD model. When the growth rate of each subpopulation is small and the dispersal rate is large, the distributions of the two model show remarkable difference with each other, because of strong spatial autocorrelation in the subpopulation size in LD model.

Evolution of Dispersal in a Structured Metapopulation Model in Discrete Time

In this article, a structured metapopulation model in discrete time with catastrophes and density-dependent local growth is introduced. The fitness of a rare mutant in an environment set by the resident is defined, and an efficient method to calculate fitness is presented. With this fitness measure evolutionary analysis of this model becomes feasible. This article concentrates on the evolution of dispersal. The effect of catastrophes, dispersal cost, and local dynamics on the evolution of dispersal is investigated. It is proved that without catastrophes, if all population-dynamical attractors are fixed points, there will be selection for no dispersal. A new mechanism for evolutionary branching is also found: Even though local population sizes approach fixed points, catastrophes can cause enough temporal variability, so that evolutionary branching becomes possible.

A spatially structured metapopulation model with patch dynamics

Metapopulation models that incorporate both spatial and temporal structure are studied in this paper. The existence and stability of equilibria are provided, and an extinction threshold condition is derived which depends on patch dynamics (patch destruction and creation) and metapopulation dynamics (patch colonization and extinction). These results refine threshold conditions given by previous metapopulation models. By comparing landscapes with different spatial heterogeneities with respect to weighted long-term patch occupancies, we conclude that the pattern of a landscape is of overwhelming importance in determining metapopulation persistence and patch occupancy. We show that the same conclusion holds when a rescue effect is considered. We also derive a stochastic differential equations (SDE) model of the Itô type based on our deterministic model. Our simulations reveal good agreement between the deterministic model and the SDE model.

Which traits promote persistence of feral GM crops? Part 2: implications of metapopulation structure

Transgenes may spread from crops into the environment via the establishment of feral populations, often initiated by seed spill from transport lorries or farm machinery. Locally, such populations are often subject to large environmental variability and usually do not persist longer than a few years. Because secondary feral populations may arise from seed dispersal to adjacent sites, the dynamics of such populations should be studied in a metapopulation context. We study a structured metapopulation model with local dispersal, mimicking a string of roadside subpopulations of a feral crop. Population growth is assumed to be subject to local disturbances, introducing spatially random environmental stochasticity. Our aim is to understand the role of dispersal and environmental variability in the dynamics of such ephemeral populations. We determine the effect of dispersal on the extinction boundary and on the distribution of persistence times, and investigate the influence of spatially correlated disturbances as opposed to spatially random disturbances. We find that, given spatially random disturbances, dispersal slows down the decline of the metapopulation and results in the occurrence of long‐lasting local populations which remain more or less static in space. We identify which life history traits, if changed by genetic modification, have the largest impact on the population growth rate and persistence times. For oilseed rape, these are seed bank survival and dormancy. Combining our findings with literature data on transgene‐induced life history changes, we predict that persistence is promoted by transgenes for oil‐modifications (high stearate or high laurate) and, possibly, for insect resistence (Bt). Transgenic tolerance to glufosinate herbicide is predicted to reduce persistence.

Extinction risk of a meta-population: aggregation approach

Aggregation of variables of a complex mathematical model with realistic structure gives a simplified model which is more suitable than the original one when the amount of data for parameter estimation is limited. Here we explore use of a formula derived for a single unstructured population (canonical model) in predicting the extinction time for a population living in multiple habitats. In particular we focus multiple populations each following logistic growth with demographic and environmental stochasticities, and examine how the mean extinction time depends on the migration and environmental correlation. When migration rate and/or environmental correlation are very large or very small, we may express the mean extinction time exactly using the formula with properly modified parameters. When parameters are of intermediate magnitude, we generate a Monte Carlo time series of the population size for the realistic structured model, estimate the “effective parameters” by fitting the time series to the canonical model, and then calculate the mean extinction time using the formula for a single population. The mean extinction time predicted by the formula was close to those obtained from direct computer simulation of structured models. We conclude that the formula for an unstructured single-population model has good approximation capability and can be applicable in estimating the extinction risk of the structured meta-population model for a limited data set.

Convergence of a structured metapopulation model to Levins?s model

We consider a structured metapopulation model describing the dynamics of a single species, whose members are located in separate patches that are linked through migration according to a mean field rule. Our main aim is to find conditions under which its equilibrium distribution is reasonably approximated by that of the unstructured model of Levins (1969). We do this by showing that the (positive) equilibrium distribution converges, as the carrying capacity of each population goes to infinity together with appropriate scalings on the other parameters, to a bimodal distribution, consisting of a point mass at 0, together with a positive part which is closely approximated by a shifted Poisson centred near the carrying capacity. Under this limiting regime, we also give simpler approximate formulae for the equilibrium distribution. We conclude by showing how to compute persistence regions in parameter space for the exact model, and then illustrate all our results with numerical examples. Our proofs are based on Stein's method.

Metapopulation viability analysis of the bog fritillary butterfly using RAMAS/GIS

Many species living in man‐shaped landscapes are restricted to small natural habitat patches and form metapopulations; predicting their future is a central issue in applied ecology. We examined the viability of the bog fritillary butterfly Proclossiana eunomia Esper, a specialist glacial relict species, in a highly fragmented landscape (<1% of suitable habitat in 10 km2), by way of population viability analysis. We used comprehensive data from a long‐term study in which a patchy population was monitored during ten consecutive years to parameterise a spatially structured metapopulation model using commercially available platform RAMAS/GIS 3.0. Population growth rate was density‐dependent and modulated by various climatic variables acting on different developmental stages of the butterfly. Density dependence was probably related to larval parasitism by a specific parasitoid. Population size was negatively affected by an increase in the mean temperature. Dispersal was modelled as the observed proportion of movements between patches, taking into account the probability of emigration out of a given patch. Our model provided results close to the picture of the system drawn from the field data and was considered as useful in making predictions about the metapopulation. Demographic parameters proved to have a far higher impact on metapopulation persistence than dispersal or correlation of local dynamics. Scenarios simulating both global warming and management of habitat patches by rustic herbivore grazing indicated a decrease in the viability of the metapopulation. Our results prompted the regional nature conservation agency to modify the planned management regime. We urge conservation biologists to use structured population models including local population dynamics for viability analysis targeted to such threatened metapopulations in highly fragmented landscapes.

Deterministic approximation of a stochastic metapopulation model

We analyse the limit behaviour of a stochastic structured metapopulation model as the number of its patches goes to infinity. The sequence of probability measures associated with the random process, whose components are the proportions of patches with different number of individuals, is tight. The limit of every convergent subsequence satisfies an infinite system of ordinary differential equations. The existence and the uniqueness of the solution are shown by semigroup methods, so that the whole random process converges weakly to the solution of the system.

Deterministic approximation of a stochastic metapopulation model

We analyse the limit behaviour of a stochastic structured metapopulation model as the number of its patches goes to infinity. The sequence of probability measures associated with the random process, whose components are the proportions of patches with different number of individuals, is tight. The limit of every convergent subsequence satisfies an infinite system of ordinary differential equations. The existence and the uniqueness of the solution are shown by semigroup methods, so that the whole random process converges weakly to the solution of the system.

The Effects of Small Dispersal Rates on Extinction Times in Structured Metapopulation Models

Habitat destruction is a critical factor that affects persistence in several taxa, including Pacific salmon. Salmon are noted for their ability to home to their natal streams for reproduction. Since straying (i.e., spawners reproducing in nonnatal streams) is typically low in salmon, its effects have not been appreciated. In this article, we develop both a general analytical model and a simple simulation model describing structured metapopulations to study how weak connections between subpopulations affect the ability of a species to tolerate habitat destruction and/or declines in habitat quality. Our goals are to develop general principles and to relate these principles to salmon population dynamics. The analytical model describes the dynamics of two density-dependent subpopulations, connected by dispersal, whose growth rates fluctuate in response to environmental and demographic stochasticity. We find that, for moderate levels of environmental variability, small dispersal rates can significantly increase mean extinction times. This effect declines with increasing habitat quality, increasing temporal correlation, and increasing spatial correlation, but it is still significant for realistic parameter values. The simulation model shows there is a threshold rate of dispersal that minimizes extinction probabilities. These results cannot be seen in classical metapopulation models and provide new insights into the rescue effect.

Vertically Transmitted Symbionts in Structured Host Metapopulations

We develop a structured metapopulation model for vertically transmitted symbionts in natural host populations. We focus primarily on two questions: • Are mutualism and high transmission probability prerequisites for the survival of symbionts in structured host metapopulations? • What are the ecological conditions under which coexistence of infected and uninfected hosts is possible? We start with studying in depth the case of qualitatively identical patches and derive conditions for invasion and coexistence of uninfected and infected hosts. Our model predicts that, in a qualitatively uniform environment, coexistence is possible only if the symbionts increase the fitness of their host, so the mutualism is indeed needed for coexistence. We also prove that evolution selects for 100% infection frequency in the metapopulation. Then we generalize the model for different patch qualities and get conditions for invasion in a virgin environment.

Evolutionary suicide and evolution of dispersal in structured metapopulations.

We study the evolution of dispersal in a structured metapopulation model. The metapopulation consists of a large (infinite) number of local populations living in patches of habitable environment. Dispersal between patches is modelled by a disperser pool and individuals in transit between patches are exposed to a risk of mortality. Occasionally, local catastrophes eradicate a local population: all individuals in the affected patch die, yet the patch remains habitable. We prove that, in the absence of catastrophes, the strategy not to migrate is evolutionarily stable. Under a given set of environmental conditions, a metapopulation may be viable and yet selection may favor dispersal rates that drive the metapopulation to extinction. This phenomenon is known as evolutionary suicide. We show that in our model evolutionary suicide can occur for catastrophe rates that increase with decreasing local population size. Evolutionary suicide can also happen for constant catastrophe rates, if local growth within patches shows an Allee effect. We study the evolutionary bifurcation towards evolutionary suicide and show that a discontinuous transition to extinction is a necessary condition for evolutionary suicide to occur. In other words, if population size smoothly approaches zero at a boundary of viability in parameter space, this boundary is evolutionarily repelling and no suicide can occur.

A Unifying Framework for Metapopulation Dynamics

Many biologically important processes, such as genetic differentiation, the spread of disease, and population stability, are affected by the (natural or enforced) subdivision of populations into networks of smaller, partly isolated, subunits. Such "metapopulations" can have extremely complex dynamics. We present a new general model that uses only two functions to capture, at the metapopulation scale, the main behavior of metapopulations. We show how complex, structured metapopulation models can be translated into our generalized framework. The metapopulation dynamics arising from some important biological processes are illustrated: the rescue effect, the Allee effect, and what we term the "antirescue effect." The antirescue effect captures instances where high migration rates are deleterious to population persistence, a phenomenon that has been largely ignored in metapopulation conservation theory. Management regimes that ignore a significant antirescue effect will be inadequate and may actually increase extinction risk. Further, consequences of territoriality and conspecific attraction on metapopulation-level dynamics are investigated. The new, simplified framework can incorporate knowledge from epidemiology, genetics, and population biology in a phenomenological way. It opens up new possibilities to identify and analyze the factors that are important for the evolution and persistence of the many spatially subdivided species.

The Effects of Small Dispersal Rates on Extinction Times in Structured Metapopulation Models

The Effects of Small Dispersal Rates on Extinction Times in Structured Metapopulation Models

Spatially Structured Metapopulation Models: Global and Local Assessment of Metapopulation Capacity

We model metapopulation dynamics in finite networks of discrete habitat patches with given areas and spatial locations. We define and analyze two simple and ecologically intuitive measures of the capacity of the habitat patch network to support a viable metapopulation. Metapopulation persistence capacity λM defines the threshold condition for long-term metapopulation persistence as λM>δ, where δ is defined by the extinction and colonization rate parameters of the focal species. Metapopulation invasion capacity λI sets the condition for successful invasion of an empty network from one small local population as λI>δ. The metapopulation capacities λM and λI are defined as the leading eigenvalue or a comparable quantity of an appropriate “landscape” matrix. Based on these definitions, we present a classification of a very general class of deterministic, continuous-time and discrete-time metapopulation models. Two specific models are analyzed in greater detail: a spatially realistic version of the continuous-time Levins model and the discrete-time incidence function model with propagule size-dependent colonization rate and a rescue effect. In both models we assume that the extinction rate increases with decreasing patch area and that the colonization rate increases with patch connectivity. In the spatially realistic Levins model, the two types of metapopulation capacities coincide, whereas the incidence function model possesses a strong Allee effect characterized by λI=0. For these two models, we show that the metapopulation capacities can be considered as simple sums of contributions from individual habitat patches, given by the elements of the leading eigenvector or comparable quantities. We may therefore assess the significance of particular habitat patches, including new patches that might be added to the network, for the metapopulation capacities of the network as a whole. We derive useful approximations for both the threshold conditions and the equilibrium states in the two models. The metapopulation capacities and the measures of the dynamic significance of particular patches can be calculated for real patch networks for applications in metapopulation ecology, landscape ecology, and conservation biology.

On fitness in structured metapopulations.

In this paper we derive a general expression measuring fitness in general structured metapopulation models. We apply the theory to a model structured by local population size and in which local dynamics is explicitly modelled. In particular, we calculate the evolutionarily stable dispersal strategy for individuals that can assess the local population density in the case where only dispersal is subject to evolutionary control but all other model in gredients are assumed fixed. We show that there exists a threshold size such that at ESS everyone should stay as long as the population size is below the threshold and everyone should disperse immediately as the population size reaches the threshold.

Necessary and Sufficient Conditions for Evolutionary Suicide

Evolutionary suicide is an evolutionary process where a viable population adapts in such a way that it can no longer persist. It has already been found that a discontinuous transition to extinction is a necessary condition for suicide. Here we present necessary and sufficient conditions, concerning the bifurcation point, for suicide to occur. Evolutionary suicide has been found in structured metapopulation models. Here we show that suicide can occur also in unstructured population models. Moreover, a structured model does not guarantee the possibility of suicide: we show that suicide cannot occur in age-structured population models of the Gurtin–MacCamy type. The point is that the mutant’s fitness must explicitly depend not only on the environmental interaction variable, but also on the resident strategy.

How should we define fitness in structured metapopulation models? Including an application to the calculation of evolutionarily stable dispersal strategies

We define a fitness concept applicable to structured metapopulations consisting of infinitely many equally coupled patches. In addition, we introduce a more easily calculated quantity Rm that relates to fitness in the same manner as R0 relates to fitness in ordinary population dynamics: the Rm of a mutant is only defined when the resident population dynamics converges to a point equilibrium and Rm is larger (smaller) than 1 if and only if mutant fitness is positive (negative). Rm corresponds to the average number of newborn dispersers resulting from the (on average less than one) local colony founded by a newborn disperser. Efficient algorithms for calculating its numerical value are provided. As an example of the usefulness of these concepts we calculate the evolutionarily stable conditional dispersal strategy for individuals that can account for the local population density in their dispersal decisions. Below a threshold density x, at which staying and leaving are equality profitable, everybody should stay and above x everybody should leave, where profitability is measured as the mean number of dispersers produced through lines of descent consisting of non-dispersers.