Effects of dispersal in a tri-trophic metapopulation model

The aim of this paper is to understand how dispersal in a patchy environment influences the stability properties of tri-trophic metapopulations. Differential equation models for tri-trophic metapopulations are formulated and analysed. The patchy nature of the metapopulations is incorporated through dispersal phases. Two variants are studied: one with a dispersal phase for the top and one with a dispersal phase for the middle level. A complete characterisation of stable and unstable equilibria is given and the possibility of invasion in these food chains is studied. A dispersal phase for the middle level can destabilize the bottom level-middle level interaction, because of the delay that dispersal causes in the reaction to the resource. When the middle level is not efficiently controlled by the top level, the unstable bottom level-middle level pair can destabilize the entire food chain. Dispersal for the top level can destabilize in the same way. A characterisation of the long term behaviour of the models is given. Bistability with a stable three species equilibrium and a stable limit cycle is one of the possibilities.     

Influence of consumer mutual interference on the stabilization of a three-level trophic system

In this paper, we present a three-level (food–prey–predator) trophic food chain which includes consumer mutual interference (MIF). In contrast with other analyses, we consider the effect of both prey and predator MIF on the dynamics of a three-level trophic system. MIF is generally considered to exert a stabilizing effect on population dynamics based on the predator–prey model. However, results from analytical and numerical simulations utilizing a simple three-species food chain model suggest that while the addition of prey MIF to the model provides a stabilizing influence, as the chaotic dynamics collapse to a stable steady state, adding only predator MIF to the model can only stabilize the system at intermediate MIF values. The three-species trophic food chain is also stabilized when combination of both prey and predator MIF is added to the model. Our work serves to provide insight into the effects of MIF in the real world.     

Density-dependent dispersal complicates spatial synchrony in tri-trophic food chains

Spatial synchrony can increase extinction risk and undermines metapopulation persistence. Both dispersal and biotic interactions can strongly affect spatial synchrony. Here, we explore the spatial synchrony of a tri-trophic food chain in two patches connected by density-dependent dispersal, namely the strategies of prey evasion (PE) and predator pursuit (PP). The dynamics of the food chain are depicted by both the Hastings–Powell model and the chemostat model, with synchrony measured by the Pearson correlation coefficient. We use the density-independent dispersal in the system as a baseline for comparison. Results show that the density-independent dispersal of a species in the system can promote its dynamic synchrony. Dispersal of intermediate species in the tri-trophic food chain is the strongest synchronizer. In contrast, the density-dependent PP and PE of intermediate species can desynchronize the system. Highly synchronized dynamics emerged when the basal species has a strong PE strategy or when the top species has a moderate PP strategy. Our results reveal the complex relationship between density-dependent dispersal and spatial synchrony in tri-trophic systems.     

On the use of the logistic equation in models of food chains

In food chain models the lowest trophic level is often assumed to grow logistically. Anomalous behaviour of the solution of the logistic equation and problems with the introduction of mortality have recently been reported. As predation on the lowest trophic level is a kind of mortality, one expects problems with these food chain models. In this paper we compare two formulations for the lowest trophic level: the logistic growth formulation and the mass balance formulation with resources modelled explicitly. We examine the effects of both models on the dynamic behaviour of a tri-trophic microbial food chain in a chemostat. For this purpose bifurcation diagrams, which give the existence and stability of the equilibria of the nonlinear dynamic system, are used. It turns out that the dynamic behaviours differ in a rather large region of the control parameter space spanned by the dilution rate and the concentration of the resources in the reservoir. We urge that mass balance equations should be used in modelling food chains in chemostats as well as in ecosystems.     

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.     

Existence and bifurcation of stable equilibrium in two-prey,one-predator communities

In this paper, stability of two-prey, one-predator communities is investigated by Lyapunov's direct method and Hopf's bifurcation theory. Three patterns of three-species coexistence are possible. A globally stable non-negative equilibrium exists for the system even if two competing prey species without a predator cannot coexist. The stable equilibrium bifurcates to a periodic motion with a small amplitude when the predation rate increases. It is also shown that a chaotic motion emerges from the periodic motion when one of two prey has greater competitive abilities than the other. This predator-mediated coexistence can be realized by the intimate relationship between preferences of a predator and competitive abilities of two prey.     

Impact of vigilance on the density variations in a food chain model

Anti-predator behavior in the form of vigilance greatly influences the dynamics of a predator-prey system. In the present work, we investigate the impact of prey vigilance in a three-species food chain model where both prey and middle predator use vigilance as a survival strategy in the presence of their respective predators. We present basic mathematical results such as local and global stability, bifurcation behavior of the system. We explore the variation of the densities of the populations in different bi-parameter spaces and observe that vigilance plays a crucial role in the survival and extinction of the populations.     

Intraguild Interactions Between the Predatory Mites Neoseiulus californicus and Phytoseiulus persimilis

Species at the same trophic level may interact through competition for food, but can also interact through intraguild predation. Intraguild predation is widespread at the second and third trophic level and the effects may cascade down to the plant level. The effects of intraguild predation can be modified by antipredator behaviour in the intraguild prey. We studied intraguild predation and antipredator behaviour in two species of predatory mite, Neoseiulus californicus and Phytoseiulus persimilis, which are both used for control of the two-spotted spider mite in greenhouse and outdoor crops. Using a Y-tube olfactometer, we assessed in particular whether each of the two predators avoids odours emanating from prey patches occupied by the heterospecific predator. Furthermore, we measured the occurrence and rate of intraguild predation of different developmental stages of P. persimilis and N. californicus on bean leaves in absence or in presence of the shared prey. Neither of the two predator species avoided prey patches with the heterospecific competitor, both when inexperienced with the other predator and when experienced with prey patches occupied by the heterospecific predator. Intraguild experiments showed that N. californicus is a potential intraguild predator of P. persimilis. However, P. persimilis did not suffer much from intraguild predation as long as the shared prey was present. This is probably because N. californicus prefers to feed on two-spotted spider mites rather than on its intraguild prey.     

Evolution of dispersal in a multi‐trophic community context

Much is known about the evolution of dispersal when species interact with their resources or natural enemies, but very little is known about dispersal evolution when species interact with both resources and natural enemies. Here I investigate how the dispersal of an intermediate consumer evolves in response to its interactions with a basal resource and top predator. I find that dispersal evolution is possible even when the consumer species is not directly affected by environmental variability, but rather experiences the consequences that such variability has on its resource and predator. Spatial variation in the consumer's fitness is driven by spatial heterogeneity in resource productivity, which determines whether a predator can colonize a resource‐consumer community. Temporal variation in the consumer's fitness is driven by random disturbances that cause periodic local extinctions of the predator, followed by recolonizations that lead to transient fluctuations in consumer abundance. When spatial variation in resource productivity is low and the predator can colonize all patches in the landscape, there is no spatial variation in consumer fitness but temporal variation in fitness favors the evolution of a dispersal monomorphism. When spatial variation in resource productivity is high and the predator cannot colonize many patches in the landscape, spatial variation in fitness selects against dispersal. In this case, temporal variation can promote the evolution of a dispersal polymorphism with sedentary and mobile phenotypes, but only for certain types of tri‐trophic interactions. This finding underscores the importance of indirect interactions in shaping the evolution of dispersal. While the ecological community can provide a strong selective environment for the evolution of dispersal, the nature of interactions between trophic levels can also impose constraints on evolution.     

Age structure in predator-prey systems. I. A general model and a specific example

To study the effects of age structure in predator-prey systems, a general, analytically tractable model is formulated and solved. We demonstrate the usefulness of the model in a study of a specific system of two mites. We show that to maintain stable equilibrium between the herbaceous (pest) mite and the predacious mite, the nonintuitive strategy of reducing the growth rate of the predator may be necessary. The modelling technique allows a determination of the magnitude of the effect of age structure on stability.     

Dispersal rates affect species composition in metacommunities of Sarracenia purpurea inquilines

Dispersal among local communities can have a variety of effects on species composition and diversity at local and regional scales. Local conditions (e.g., resource and predator densities) can have independent effects, as well as interact with dispersal, to alter these patterns. Based on metacommunity models, we predicted that local diversity would show a unimodal relationship with dispersal frequency. We manipulated dispersal frequencies, resource levels, and the presence of predators (mosquito larvae) among communities found in the water-filled leaves of the pitcher plant Sarracenia purpurea. Diversity and abundance of species of the middle trophic level, protozoa and rotifers, were measured. Increased dispersal frequencies significantly increased regional species richness and protozoan abundance while decreasing the variance among local communities. Dispersal frequency interacted with predation at the local community scale to produce patterns of diversity consistent with the model. When predators were absent, we found a unimodal relationship between dispersal frequency and diversity, and when predators were present, there was a flat relationship. Intermediate dispersal frequencies maintained some species in the inquiline communities by offsetting extinction rates. Local community composition and the degree of connectivity between communities are both important for understanding species diversity patterns at local and regional scales.     

An acarine predator-prey metapopulation system inhabiting greenhouse cucumbers

The two-spotted spider mite ( Tetranychus urticae ) is a serious pest on greenhouse cucumbers, but can be controlled by the phytoseiid predator Phytoseiulus persimilis. The two mite species exhibit considerable fluctuations in overall population densities but within acceptable limits. The system appears to be persistent at a regional (greenhouse) scale in spite of frequent local extinctions (e.g. at individual plants). Experimental evidence indicates that the mites form a metapopulation system characterized by 'shifting mosaic' dynamics. A stochastic simulation model is used to analyse the role played by dispersal in the dynamics and persistence of the system. It shows that demographic stochasticity generates sufficient endogenous 'noise' to counteract the synchronizing effect of density-dependent dispersal, provided dispersal rates are not too high and the system is not too small. Low dispersal rates, on the other hand, increase the risk of local outbreaks of spider mites that may cause destruction of plants.     

On the role of body size in a tri-trophic metapopulation model

 A particular tri-trophic (resource, prey, predator) metapopulation model with dispersal of preys and predators is considered in this paper. The analysis is carried out numerically, by finding the bifurcations of the equilibria and of the limit cycles with respect to prey and predator body sizes. Two routes to chaos are identified. One is characterized by an intriguing cascade of flip and tangent bifurcations of limit cycles, while the other corresponds to the crisis of a strange attractor. The results are summarized by partitioning the space of body sizes in eight subregions, each one of which is associated to a different asymptotic behavior of the system. Emphasis is put on the possibility of having different modes of coexistence (stationary, cyclic, and chaotic) and/or extinction of the predator population. Received 1 August 1995; received in revised form 8 January     

Host-plant species modifies the diet of an omnivore feeding on three trophic levels

The diet choice of omnivores feeding on two adjacent trophic levels (either plants and herbivores or herbivores and predators) has been studied extensively. However, omnivores usually feed on more than two trophic levels, and this diet choice and its consequences for population dynamics have hardly been studied. We report how host-plant quality affects the diet choice of western flower thrips feeding on three trophic levels: plants (cucumber or sweet pepper), eggs of spider mites and eggs of a predatory mite that attacks spider mites. Spider mites feed on the same host plants as thrips and produce a web that hampers predator mobility. To assess the indirect effects of spider mites on predation by thrips, the thrips were offered spider-mite eggs and predatory-mite eggs on cucumber or sweet pepper leaf discs that were either clean, damaged by spider mites but without spider-mite web, or damaged and webbed. We show that, overall, thrips consumed more eggs on sweet pepper, a plant of low quality, than on cucumber, a high quality host plant. On damaged and webbed leaf discs (mimicking the natural situation), thrips killed more predator eggs than spider-mite eggs on sweet pepper, but they killed equal numbers of eggs of each species on cucumber. This is because web hampered predation on spider-mite eggs by thrips on sweet pepper, but not on cucumber, whereas it did not affect predation on predatory-mite eggs. We used the data obtained to parameterize a model of the local dynamics of this system. The model predicts that total predation by the omnivore has little effects on population dynamics, whereas differential attack of predator eggs and spider-mite eggs by the omnivore has large effects on the dynamics of both mite species on the two host plants.     

Biomass and biodiversity in species-rich tritrophic communities

We consider a tritrophic system with one basal and one top species and a large number of primary consumers, and derive upper and lower bounds for the total biomass of the middle trophic level. These estimates do not depend on dynamical regime, holding for fixed point, periodic, or chaotic dynamics. We have two kinds of estimates, depending on whether the predator abundance is zero. All these results are uniform in a self-limitation parameter, which regulates prey diversity in the system. For strong self-limitation, diversity is large; for weak self-limitation, it is small. Diversity depends on the variance of species’ parameter values. The larger this variance, the lower the diversity, and vice versa. Moreover, variation in the parameters of the Holling type II functional response changes the bifurcation character, with the equilibrium state with nonzero predator abundance losing stability. If that variation is small then the bifurcation can lead to oscillations (the Hopf bifurcation). Under certain conditions, there exists a supercritical Hopf bifurcation. We then find a connection between diversity and Hopf bifurcations. We also show that the system exhibits top-down regulation and a hump-shaped diversity-productivity curve.We then extend the model by allowing species to experience self-regulation. For this extended model, explicit estimates of prey diversity are obtained. We study the dynamics of this system and find the following. First, diversity and system dynamics crucially depend on variation in species parameters. We show that under certain conditions, the system undergoes a supercritical Hopf bifurcation. We also establish a connection between diversity and Hopf bifurcations. For strong self-limitation, diversity is large and complex dynamics are absent. For weak self-limitation, diversity is small and the equilibrium with non-zero predator abundance is unstable.     

Three-species ecosystems in a solvable model

A detailed discussion of the three-species ecosystems is presented in an exactly solvable model with interactions of the Gompertz form. Three different possibilities, namely, a one-prey-two-predator system, a two-prey-one-predator system and a three-step prey-predator food chain are considered. These systems are studied not only when they include their basic prey-predator interactions, but also when various self-interactions as well as competition between like species, in different possible combinations, are included. It is then inferred, by obtaining and examining the exact solutions, as to when these systems possess stable equilibrium and when not, or when they are purely oscillatory, etc. We also study, within our model, the two-species versus three-species situation. It is seen that there are situations when the three-species system possesses stable equilibrium even under circumstances under which the corresponding two-species system is unstable. We also come across cases when the addition of the third species destroys the possibility of stable equilibrium which the initial two-species system possessed. Some other results also follow. Of particular interest is the one where the initial two-species system is purely oscillatory but the enlarged system, which is a three-step prey-predator chain, has the first and the last populations of the chain rising indefinitely and the middle population remains oscillatory. A comparison of our results with results of other authors, wherever possible, has also been made.     

Trophic supplements to intraguild predation

Intraguild predation (IGP) is a dominant community module in terrestrial food webs that occurs when multiple consumers feed both on each other and on a shared prey. This specific form of omnivory is common in terrestrial communities and is of particular interest for conservation biology and biological control given its potential to disrupt management of threatened or pest species. Extensive theory exists to describe the dynamics of three-species IGP, but these models have largely overlooked the potential for other, exterior interactions, to alter the dynamics within the IGP module. We investigated how three forms of feeding outside of the IGP module by intraguild predators (i.e. trophic supplementation) affect the dynamics of the predators (both IG predator and IG prey) and their shared resource. Specifically, we examined how the provision of a constant donor-controlled resource, the availability of an alternative prey species, and predator plant-feeding affect the dynamics of IGP models. All three forms of trophic supplements modified the basic expectations of IGP theory in two important ways, and their effects were similar. First, coexistence was possible without the IG prey being a superior competitor for the original shared resource if the IG prey could effectively exploit one of the types of trophic supplements. However, supplements to the IG predator restricted the potential for coexistence. Second, supplements to the IG prey ameliorated the disruptive effects of the IG predator on the suppression of the shared resource, promoting effective control of the resource in the presence of both predators. Consideration of these three forms of trophic supplementation, all well documented in natural communities, adds substantial realism and predictive power to intraguild predation theory.     

Explicit model for searching behavior of predator

The authors present an approach for explicit modeling of spatio-temporal dynamics of predator-prey community. This approach is based on a reaction-diffusion-adjection PD (prey dependent) system. Local kinetics of population is determined by logistic reproduction function of prey, constant natural mortality of predator and Holling type 2 trophic function. Searching behavior of predator is described by the advective term in predator balance equation assuming the predator acceleration to be proportional to the prey density gradient. The model was studied with zero-flux boundary conditions. The influence of predator searching activity on the community dynamics, in particular, on the emergence of spatial heterogeneity, has been investigated by linear analysis and numerical simulations. It has been shown how searching activity may effect the persistence of species, stabilizing predator-prey interactions at very low level of pest density. It has been demonstrated that obtaining of such dynamic regimes does not require the use of complex trophic functions.     

Differential predation drives overyielding of prey species in a patchy environment

In environments characterized by regional heterogeneity among patches, competitor diversity can enhance ecosystem functions such as biomass production. Studies that have addressed the strength of diversity effects in heterogeneous environments have primarily considered a patchy distribution of resources. However, in many systems, top–down effects influence competitor productivity and composition. We use a three‐trophic level consumer–resource model to ask how differential responses to predation influence consumer diversity effects at two scales; 1) in patches with and without predator populations, and 2) at a ‘regional’ scale, consisting of one patch with‐ and one patch without a predator population. At the local scale, the strength and direction of consumer diversity effects depended on the strength of the differential response to predation. Positive or negative influences of consumer richness on equilibrium consumer biomass were the result of a selection effect of diversity. At the regional scale, we observed transgressive overyielding driven by a positive complementarity effect for parameters that define a strong differential response to predation. Given the prevalence of spatially and temporally heterogeneous top–down effects on competitor composition in many ecosystems and trophic levels, we advocate consideration of differential predation as an important step towards incorporating realistic trophic complexity into diversity–function studies.     

Immigration can destabilize tri-trophic interactions: implications for conservation of top predators

Top predators often have large home ranges and thus are especially vulnerable to habitat loss and fragmentation. Increasing connectance among habitat patches is therefore a common conservation strategy, based in part on models showing that increased migration between subpopulations can reduce vulnerability arising from population isolation. Although three-dimensional models are appropriate for exploring consequences to top predators, the effects of immigration on tri-trophic interactions have rarely been considered. To explore the effects of immigration on the equilibrium abundances of top predators, we studied the effects of immigration in the three-dimensional Rosenzweig-MacArthur model. To investigate the stability of the top predator equilibrium, we used MATCONT to perform a bifurcation analysis. For some combinations of model parameters with low rates of top predator immigration, population trajectories spiral towards a stable focus. Holding other parameters constant, as immigration rate is increased, a supercritical Hopf bifurcation results in a stable limit cycle and thus top predator populations that cycle between high and low abundances. Furthermore, bistability arises as immigration of the intermediate predator is increased. In this case, top predators may exist at relatively low abundances while prey become extinct, or for other initial conditions, the relatively higher top predator abundance controls intermediate predators allowing for non-zero prey population abundance and increased diversity. Thus, our results reveal one of two outcomes when immigration is added to the model. First, over some range of top predator immigration rates, population abundance cycles between high and low values, making extinction from the trough of such cycles more likely than otherwise. Second, for relatively higher intermediate predator migration rates, top predators may exist at low values in a truncated system with impoverished diversity, again with extinction more likely.