Effects of habitat destruction and resource supplementation in a predator-prey metapopulation model

We developed a mean field, metapopulation model to study the consequences of habitat destruction on a predator-prey interaction. The model complements and extends earlier work published by Bascompte and Solé (1998, J. theor. Biol.195, 383-393) in that it also permits use of alternative prey (i.e., resource supplementation) by predators. The current model is stable whenever coexistence occurs, whereas the earlier model is not stable over the entire domain of coexistence. More importantly, the current model permits an assessment of the effect of a generalist predator on the trophic interaction. Habitat destruction negatively affects the equilibrium fraction of patches occupied by predators, but the effect is most pronounced for specialists. The effect of habitat destruction on prey coexisting with predators is dependent on the ratio of extinction risk due to predation and prey colonization rate. When this ratio is less than unity, equilibrial prey occupancy of patches declines as habitat destruction increases. When the ratio exceeds one, equilibrial prey occupancy increases even as habitat destruction increases; i.e., prey "escape" from predation is facilitated by habitat loss. Resource supplementation reduces the threshold colonization rate of predators necessary for their regional persistence, and the benefit derived from resource supplementation increases in a nonlinear fashion as habitat destruction increases. We also compared the analytical results to those from a stochastic, spatially explicit simulation model. The simulation model was a discrete time analog of our analytical model, with one exception. Colonization was restricted locally in the simulation, whereas colonization was a global process in the analytical model. After correcting for differences between nominal and effective colonization rates, most of the main conclusions of the two types of models were similar. Some important differences did emerge, however, and we discuss these in relation to the need to develop fully spatially explicit analytical models. Finally, we comment on the implications of our results for community structure and for the conservation of prey species interacting with generalist predators.     

Size and scaling of predator-prey dynamics

We propose a scaled version of the Rosenzweig–MacArthur model using both Type I and Type II functional responses that incorporates the size dependence of interaction rates. Our aim is to link the energetic needs of organisms with the dynamics of interacting populations, for which survival is a result of a game-theoretic struggle for existence. We solve the scaled model of predator–prey dynamics and predict population level characteristics such as the scaling of coexistence size ranges and the optimal predator–prey size ratio. For a broad class of such models, the optimal predator–prey size ratio given available prey of a fixed size is constant. We also demonstrate how scaling predictions of prey density differ under resource limitation vs. predator drawdown. Finally, we show how evolution of predator size can destabilize population dynamics, compare scaling of predator–prey cycles to previous work, as well as discuss possible extensions of the model to multispecies communities.     

Evolutionary dynamics of prey exploitation in a metapopulation of predators

In well-mixed populations of predators and prey, natural selection favors predators with high rates of prey consumption and population growth. When spatial structure prevents the populations from being well mixed, such predators may have a selective disadvantage because they do not make full use of the prey's growth capacity and hence produce fewer propagules. The best strategy then depends on the degree to which predators can monopolize the exploitation of local prey populations, which in turn depends on the spatial structure, the number of migrants, and, in particular, the stochastic nature of the colonization process. To analyze the evolutionary dynamics of predators in a spatially structured predator-prey system, we performed simulations with a metapopulation model that has explicit local dynamics of nonpersistent populations, keeps track of the number of emigrants entering the migration pool, assumes individuals within local populations as well as within the migration pool to be well mixed, and takes stochastic colonization into account. We investigated which of the predator's exploitation strategies are evolutionarily stable and whether these strategies minimize the overall density of prey, as is the case in Lotka-Volterra-type models of competitive exclusion. This was analyzed by pairwise invasibility plots based on short-term simulations and tested by long-term simulation experiments of competition between resident and mutant predator-types that differed in one of the following parameters: the prey-to-predator conversion efficiency, the per capita prey consumption rate, or the per capita emigration rate from local populations. In addition, we asked which of these three strategies are most likely to evolve. Our simulations showed that under selection for conversion efficiency the predator-prey system always goes globally extinct yet persists under selection for consumption or emigration rates and that the evolutionarily stable (ES) exploitation strategies do not maximize local population growth rates. The most successful exploitation strategy minimizes the overall density of prey but does not make it settle exactly at the minimum. The system did not settle at the point where the mean time to co-invasion (i.e., immigration of a second predator in a local prey population) equals the mean local interaction time (an idea borne out from studies on host exploitation strategies in host-pathogen systems) but rather where the mean time to co-invasion was larger. The ES exploitation strategies represent more prudent strategies than the ones that minimize prey density. Finally, we show that-compared to consumption-emigration is a more likely target for selection to achieve prudent exploitation and that prudent exploitation strategies can evolve only provided the prey-to-predator conversion efficiency is subject to constraints.     

Effects of density-dependent migrations on stability of a two-patch predator-prey model

We consider a predator-prey model in a two-patch environment and assume that migration between patches is faster than prey growth, predator mortality and predator-prey interactions. Prey (resp. predator) migration rates are considered to be predator (resp. prey) density-dependent. Prey leave a patch at a migration rate proportional to the local predator density. Predators leave a patch at a migration rate inversely proportional to local prey population density. Taking advantage of the two different time scales, we use aggregation methods to obtain a reduced (aggregated) model governing the total prey and predator densities. First, we show that for a large class of density-dependent migration rules for predators and prey there exists a unique and stable equilibrium for migration. Second, a numerical bifurcation analysis is presented. We show that bifurcation diagrams obtained from the complete and aggregated models are consistent with each other for reasonable values of the ratio between the two time scales, fast for migration and slow for local demography. Our results show that, under some particular conditions, the density dependence of migrations can generate a limit cycle. Also a co-dim two Bautin bifurcation point is observed in some range of migration parameters and this implies that bistability of an equilibrium and limit cycle is possible.     

Effects of Density Dependent Migrations on the Dynamics of a Predator Prey Model

We study the effects of density dependent migrations on the stability of a predator-prey model in a patchy environment which is composed with two sites connected by migration. The two patches are different. On the first patch, preys can find resource but can be captured by predators. The second patch is a refuge for the prey and thus predators do not have access to this patch. We assume a repulsive effect of predator on prey on the resource patch. Therefore, when the predator density is large on that patch, preys are more likely to leave it to return to the refuge. We consider two models. In the first model, preys leave the refuge to go to the resource patch at constant migration rates. In the second model, preys are assumed to be in competition for the resource and leave the refuge to the resource patch according to the prey density. We assume two different time scales, a fast time scale for migration and a slow time scale for population growth, mortality and predation. We take advantage of the two time scales to apply aggregation of variables methods and to obtain a reduced model governing the total prey and predator densities. In the case of the first model, we show that the repulsive effect of predator on prey has a stabilizing effect on the predator-prey community. In the case of the second model, we show that there exists a window for the prey proportion on the resource patch to ensure stability.     

Experimental beetle metapopulations respond positively to dynamic landscapes and reduced connectivity

Interactive effects of multiple environmental factors on metapopulation dynamics have received scant attention. We designed a laboratory study to test hypotheses regarding interactive effects of factors affecting the metapopulation dynamics of red flour beetle, Tribolium castaneum. Within a four-patch landscape we modified resource level (constant and diminishing), patch connectivity (high and low) and patch configuration (static and dynamic) to conduct a 2(3) factorial experiment, consisting of 8 metapopulations, each with 3 replicates. For comparison, two control populations consisting of isolated and static subpopulations were provided with resources at constant or diminishing levels. Longitudinal data from 22 tri-weekly counts of beetle abundance were analyzed using bayesian Poisson generalized linear mixed models to estimate additive and interactive effects of factors affecting abundance. Constant resource levels, low connectivity and dynamic patches yielded greater levels of adult beetle abundance. For a given resource level, frequency of colonization exceeded extinction in landscapes with dynamic patches when connectivity was low, thereby promoting greater patch occupancy. Negative density dependence of pupae on adults occurred and was stronger in landscapes with low connectivity and constant resources; these metapopulations also demonstrated greatest stability. Metapopulations in control landscapes went extinct quickly, denoting lower persistence than comparable landscapes with low connectivity. When landscape carrying capacity was constant, habitat destruction coupled with low connectivity created asynchronous local dynamics and refugia within which cannibalism of pupae was reduced. Increasing connectivity may be counter-productive and habitat destruction/recreation may be beneficial to species in some contexts.     

Effects of prey refuges on a predator-prey model with a class of functional responses: The role of refuges

In this paper, the effects of refuges used by prey on a predator-prey interaction with a class of functional responses are studied by using the analytical approach. The refuges are considered as two types: a constant proportion of prey and a fixed number of prey using refuges. We will evaluate the effects with regard to the local stability of the interior equilibrium point, the values of the equilibrium density and the long-term dynamics of the interacting populations. The results show that the effects of refuges used by prey increase the equilibrium density of prey population while decrease that of predators. It is also proved that the effects of refuges can stabilize the interior equilibrium point of the considered model, and destabilize it under a very restricted set of conditions which is disagreement with previous results in this field.     

Single-species metapopulation dynamics: concepts, models and observations

This paper outlines a conceptual and theoretical framework for single-species metapopulation dynamics based on the Levins model and its variants. The significance of the following factors to metapopulation dynamics are explored: evolutionary changes in colonization ability; habitat patch size and isolation; compensatory effects between colonization and extinction rates; the effect of immigration on local dynamics (the rescue effect); and heterogeneity among habitat patches. The rescue effect may lead to alternative stable equilibria in metapopulation dynamics. Heterogeneity among habitat patches may give rise to a bimodal equilibrium distribution of the fraction of patches occupied in an assemblage of species (the core-satellite distribution). A new model of incidence functions is described, which allows one to estimate species' colonization and extinction rates on islands colonized from mainland. Four distinct kinds of stochasticity affecting metapopulation dynamics are discussed with examples. The concluding section describes four possible scenarios of metapopulation extinction.     

Effect of predator density dependent dispersal of prey on stability of a predator-prey system

This work presents a predator-prey Lotka-Volterra model in a two patch environment. The model is a set of four ordinary differential equations that govern the prey and predator population densities on each patch. Predators disperse with constant migration rates, while prey dispersal is predator density-dependent. When the predator density is large, the dispersal of prey is more likely to occur. We assume that prey and predator dispersal is faster than the local predator-prey interaction on each patch. Thus, we take advantage of two time scales in order to reduce the complete model to a system of two equations governing the total prey and predator densities. The stability analysis of the aggregated model shows that a unique strictly positive equilibrium exists. This equilibrium may be stable or unstable. A Hopf bifurcation may occur, leading the equilibrium to be a centre. If the two patches are similar, the predator density dependent dispersal of prey has a stabilizing effect on the predator-prey system.     

A shift from exploitation to interference competition with increasing density affects population and community dynamics

Intraspecific competition influences population and community dynamics and occurs via two mechanisms. Exploitative competition is an indirect effect that occurs through use of a shared resource and depends on resource availability. Interference competition occurs by obstructing access to a resource and may not depend on resource availability. Our study tested whether the strength of interference competition changes with protozoa population density. We grew experimental microcosms of protozoa and bacteria under different combinations of protozoan density and basal resource availability. We then solved a dynamic predator–prey model for parameters of the functional response using population growth rates measured in our experiment. As population density increased, competition shifted from exploitation to interference, and competition was less dependent on resource levels. Surprisingly, the effect of resources was weakest when competition was the most intense. We found that at low population densities, competition was largely exploitative and resource availability had a large effect on population growth rates, but the effect of resources was much weaker at high densities. This shift in competitive mechanism could have implications for interspecific competition, trophic interactions, community diversity, and natural selection. We also tested whether this shift in the mechanism of competition with protozoa density affected the structure of the bacterial prey community. We found that both resources and protozoa density affected the structure of the bacterial prey community, suggesting that competitive mechanism may also affect trophic interactions.     

The role of size-specific predation in the evolution and diversification of prey life histories

Some of the best empirical examples of life-history evolution involve responses to predation. Nevertheless, most life-history theory dealing with responses to predation has not been formulated within an explicit dynamic food-web context. In particular, most previous theory does not explicitly consider the coupled population dynamics of the focal species and its predators and resources. Here we present a model of life-history evolution that explores the evolutionary consequences of size-specific predation on small individuals when there is a trade-off between growth and reproduction. The model explicitly describes the population dynamics of a predator, the prey of interest, and its resource. The selective forces that cause life-history evolution in the prey species emerge from the ecological interactions embodied by this model and can involve important elements of frequency dependence. Our results demonstrate that the strength of the coupling between predator and prey in the community determines many aspects of life-history evolution. If the coupling is weak (as is implicitly assumed in many previous models), differences in resource productivity have no effect on the nature of life-history evolution. A single life-history strategy is favored that minimizes the equilibrium resource density (if possible). If the coupling is strong, then higher resource productivities select for faster growth into the predation size refuge. Moreover, under strong coupling it is also possible for natural selection to favor an evolutionary diversification of life histories, possibly resulting in two coexisting species with divergent life-history strategies.     

Foraging for intermittently refuged prey: theory and field observations of a parasitoid

Many insect herbivores feed in concealed locations but become accessible intermittently, creating windows of greater vulnerability to attack, and generating a proportion of the prey population that is readily accessible to foraging natural enemies. We incorporated accessible prey into an extant optimal foraging model, and found that this addition allowed opportunistic exploitation of prey that have already emerged from refugia (the leaving strategy) as a viable strategy, in addition to waiting at refugia for prey to emerge (the waiting strategy). We parameterized the model empirically for the parasitoid Macrocentrus grandii and its host, Ostrinia nubilalis, under field conditions. The model predicted that M. grandii should adopt a leaving strategy when host patch density is high (travel time between patches is short), but a waiting strategy when host patch density is low (travel time between patches is long). Field observations of M. grandii patch tenure were consistent with model predictions, indicating that M. grandii exhibited flexible behaviour based on experience within a foraging bout, and that these behavioural shifts improved foraging efficiency. Behaviour of M. grandii was responsive to heterogeneity in host emergence rates, and appeared to be driven by the relatively small proportion of the host population that became accessible at a fast rate. Therefore understanding forager responses to intermittently refuged prey may require characterization of the behaviour of a subset of the prey population, rather than the average prey individual. The model can potentially be used as a framework for comparative studies across forager taxa, to understand when foragers on intermittently accessible prey should adopt fixed waiting or leaving strategies vs. a flexible strategy that is responsive to the current environment.     

Impact of dispersal on population growth: the role of inter-patch distance

Dispersal can play a key role in the dynamics of patchy populations through patch colonization, and generally this leads to distance-dependent colonization. Less recognised are the roles of dispersal and inter-patch distance on the growth of a population after colonization. We use a laboratory mite model system in which both juveniles and adults can disperse to explore the impact of dispersal, and particularly inter-patch distance, on population dynamics. We examine the dynamics of patches after colonization by manipulating the presence of a dispersal corridor to a source patch at two inter-patch distances. Consistent with many field studies, the results show colonization was slower in more distant patches. Following colonization, the effect of the dispersal corridor on dynamics was dependent on inter-patch distance. In patches near the source, the number of adults tended to increase at a faster rate, and juveniles at a slower rate when connected with a dispersal corridor. In contrast, adult numbers grew slower and juveniles tended to grow faster when connected with a corridor in more distant patches. In the long-term, equilibrium adult numbers were lower in patches connected to the source patch at both distances. These results are likely to be driven by the effects of inter-patch distance on dispersal mortality, and the effects of dispersal on patch abundance and within-patch competition. These results confirm that distance is important for patch colonization and also show that distance can affect population density after colonization. The effects of dispersal and distance on local dynamics could be important in the dynamics of patchy populations in increasingly fragmented landscapes.     

Habitat Deterioration, Habitat Destruction, and Metapopulation Persistence in a Heterogenous Landscape

Levins's unstructured metapopulation model predicts that the equilibrium fraction of empty habitat patches is a constant function of the fractionhof suitable patches in the landscape and that this constant equals the threshold value for metapopulation persistence. Levins's model thus suggests that the minimum amount of suitable habitat necessary for metapopulation persistence can be estimated from the fraction of empty patches at steady state. In this paper we construct several more realistic structured metapopulation models that include variation in patch quality and the rescue effect. These models predict both positive and negative correlations between the fractions of suitable patches and empty patches. The type of correlation depends in an intricate manner on the strength of the rescue effect and on the quality distribution of the patches to be destroyed. Empty patches can be considered as the resource limiting metapopulation growth. Our results demonstrate that the correlation between the fractions of suitable patches and empty patches is positive if and only if the average value of the resource decreases as the number of patches increases.     

Host–parasitoid persistence over variable spatio‐temporally susceptible habitats: bottom–up effects of ephemeral resources

We experimentally and theoretically investigated the persistence of hosts and parasitoids interacting in a metapopulation structure consisting of ephemeral local patches (MELPs). We used a host–parasitoid system consisting of necrophagous Diptera species and their pupal parasitoids. The basal resources used by the host species were assumed to be ephemeral, supporting only one generation of individuals before completely disappearing from the environment. We experimentally measured the host–parasitoid persistence and the effects of local demographic processes in two scenarios: 1) constant occurrence of basal resources at a single site (no dispersion or colonization of other sites) and 2) variable occurrence of basal resources between two sites (colonization of a new patch requiring species dispersal). The experimental setup and findings were then formalized into a mathematical model describing the interaction dynamics in a MELP structure. We evaluated the contribution of several factors to the host–parasitoid coexistence, such as resource allocation probability (probability of resource appearance in a site), variation in resource size and number of sites available to receive resources in the MELP. We found that demographic fluctuations and environmental stochasticity affected the density of migrants, patch habitat connectivity, persistence and spatial distribution of interacting species.     

Not attackable or not crackable—How pre‐ and post‐attack defenses with different competition costs affect prey coexistence and population dynamics

It is well‐known that prey species often face trade‐offs between defense against predation and competitiveness, enabling predator‐mediated coexistence. However, we lack an understanding of how the large variety of different defense traits with different competition costs affects coexistence and population dynamics. Our study focusses on two general defense mechanisms, that is, pre‐attack (e.g., camouflage) and post‐attack defenses (e.g., weaponry) that act at different phases of the predator—prey interaction. We consider a food web model with one predator, two prey types and one resource. One prey type is undefended, while the other one is pre‐ or post‐attack defended paying costs either by a higher half‐saturation constant for resource uptake or a lower maximum growth rate. We show that post‐attack defenses promote prey coexistence and stabilize the population dynamics more strongly than pre‐attack defenses by interfering with the predator's functional response: Because the predator spends time handling “noncrackable” prey, the undefended prey is indirectly facilitated. A high half‐saturation constant as defense costs promotes coexistence more and stabilizes the dynamics less than a low maximum growth rate. The former imposes high costs at low resource concentrations but allows for temporally high growth rates at predator‐induced resource peaks preventing the extinction of the defended prey. We evaluate the effects of the different defense mechanisms and costs on coexistence under different enrichment levels in order to vary the importance of bottom‐up and top‐down control of the prey community.     

Discrete consumers,small scale resource heterogeneity,and population stability

We present a consumer-resource model in which individual consumers subsist on a continuum of resource distributed over a very large number of small “bite-sized” patches, each patch being sufficiently small that all its resource is eaten whenever a consumer visits. This form of consumer–resource interaction forces a heterogeneous distribution of resource among the patches, and may dampen out the large amplitude, consumer-resource cycles that are predicted by traditional models of well-mixed, spatially homogeneous systems. The resource equilibrium does not increase with enrichment, a prediction that distinguishes this model from models that invoke direct or indirect consumer density dependence as a stabilizing mechanism.     

Patterns of dispersal and dynamics among habitat patches varying in quality

Both source-sink theory and extensions of optimal foraging theory ("balanced dispersal" theory) address dispersal and population dynamics in landscapes where habitat patches vary in quality. However, studying dispersal mechanisms empirically has proven difficult, and dispersal is rarely tied back to long-term spatial dynamics. We used a manipulable laboratory system consisting of bacteria and protozoa to investigate the ability of source-sink and optimal foraging theories to explain both dispersal and emergent spatial dynamics. Consistent with source-sink models and contrary to balanced dispersal models, there was a consistent net flux of protist individuals from high to low resource patches. However, unlike the simplest source-sink models, intermediate rates of dispersal led to highest abundances in low resource patches. Side experiments found strong density dependence in local population dynamics and differences in average protist body size in high and low resource patches. Parameterization and analysis of a two-patch model showed that high migration from high to low resource patches could have depressed population density in low resource patches, creating pseudosinks. The movement of individuals and biomass from sources to sinks (a form of ecosystem subsidy) resulted in the convergence of body size and population densities in sources and sinks. Our results indicate a need to carefully consider movement patterns and interaction with local dynamics in potential source-sink systems.     

Extension of ideal free resource use to breeding populations and metapopulations

The concept of an ideal and free use of limiting resources is commonly invoked in behavioural ecology as a null model for predicting the distribution of foraging consumers across heterogeneous habitat. In its original conception, however, its predictions were applied to the longer timescales of habitat selection by breeding birds. Here I present a general model of ideal free resource use, which encompasses classical deterministic models for the dynamics in continuous time of feeding aggregations, breeding populations and metapopulations. I illustrate its key predictions using the consumer functional response given by Holling's disc equation. The predictions are all consistent with classical population dynamics, but at least two of them are not usually recognised as pertaining across all scales. At the fine scale of feeding aggregations, the steady state of an equal intake for all ideal free consumers may be intrinsically unstable, if patches are efficiently exploited by individuals with a non-negligible handling time of resources. At coarser scales, classical models of population and metapopulation dynamics assume exploitation of a homogeneous environment, yet they can yield testable predictions for heterogeneous environments too under the assumption of ideal free resource use.     

From complex spatial dynamics to simple Markov chain models: do predators and prey leave footprints?

In this paper we present a concept for using presence–absence data to recover information on the population dynamics of predator–prey systems. We use a highly complex and spatially explicit simulation model of a predator–prey mite system to generate simple presence–absence data: the number of patches with both prey and predators, with prey only, with predators only, and with neither species, along with the number of patches that change from one state to another in each time step. The average number of patches in the four states, as well as the average transition probabilities from one state to another, are then depicted in a state transition diagram, constituting the "footprints" of the underlying population dynamics. We investigate to what extent changes in the population processes modeled in the complex simulation (i.e. the predator's functional response and the dispersal rates of both species) are reflected by different footprints
The transition probabilities can be used to forecast the expected fate of a system given its current state. However, the transition probabilities in the modeled system depend on the number of patches in each state. We develop a model for the dependence of transition probabilities on state variables, and combine this information in a Markov chain transition matrix model. Finally, we use this extended model to predict the long-term dynamics of the system and to reveal its asymptotic steady state properties.