Age structure in predator-prey systems: Intraspecific carnivore interaction,passive diffusion,and the paradox of enrichment

An existing arthropod predator-prey model incorporating age structure in the carnivore through the use of the von Foerster equation is extended to include the effects of intraspecific carnivore interaction and passive diffusion or migration. A linear stability analysis of the community equilibrium point of that differential-integral equation system is performed and the resulting secular equation analyzed by the method of D-partitions. These stability results are then compared to those obtained by employing an analogous differential equation model without age structure, in particular as they relate to the so-called paradox of enrichment. In the absence of passive diffusion, it is shown that, unlike for a differential equation model, the paradox of enrichment can occur even with a carnivore which exhibits intraspecific competition. This destabilizing effect of age structure is seen to occur most dramatically when interspecific interactions are large, while the effect of passive diffusion is to offset that tendency and restabilize the system. These predictions are in accordance with relevant experimental evidence involving mites.     

Dispersal delays, predator-prey stability, and the paradox of enrichment

It takes time for individuals to move from place to place. This travel time can be incorporated into metapopulation models via a delay in the interpatch migration term. Such a term has been shown to stabilize the positive equilibrium of the classical Lotka-Volterra predator-prey system with one species (either the predator or the prey) dispersing. We study a more realistic, Rosenzweig-MacArthur, model that includes a carrying capacity for the prey, and saturating functional response for the predator. We show that dispersal delays can stabilize the predator-prey equilibrium point despite the presence of a Type II functional response that is known to be destabilizing. We also show that dispersal delays reduce the amplitude of oscillations when the equilibrium is unstable, and therefore may help resolve the paradox of enrichment.     

Functional response,numerical response,and stability in arthropod predator-prey ecosystems involving age structure

Summary A general model of arthropod predator-prey systems incorporating age structure in the predator is employed to study the role of functional and numerical responses on stability and the paradox of enrichment. The destabilizing effect of age structure leads to both qualitatively and quantitatively new results for an environment which has an infinite prey carrying capacity, including a lower bound to prey density for a stable equilibrium, a feature not present in models without age structure. When applied to an environment with finite prey carrying capacity, the effect of age structure is to reinforce the arguments implicit to the paradox of enrichment originally developed for traditional models lacking age structure.     

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.     

Age structure and stability in models of prey-predator systems

The effect on stability of making growth and interaction parameters stage-dependent was investigated by comparing models of the prey-predator type of interaction. An extensive simulation experiment was designed to study this effect in stochastic as well as deterministic models. The results showed that age structure could enhance stability, but that the form of age structure and parameter values determine the end result.     

Global stability of Gause-type predator-prey systems

In this paper, we present some global stability results obtained from comparison analysis, Bendixson-Dulac criterion or limit cycle stability analysis for the general Gause-type predator-type systems.Research supported in part by a FGIA grant from the Arizona State University Research Fund AMS (MOS) subject classifications. Primary 34C05; secondary 34C25, 92A15.     

Absolute stability and dynamical stabilisation in predator-prey systems

Many ecological systems exhibit multi-year cycles. In such systems, invasions have a complicated spatiotemporal structure. In particular, it is common for unstable steady states to exist as long-term transients behind the invasion front, a phenomenon known as dynamical stabilisation. We combine absolute stability theory and computation to predict how the width of the stabilised region depends on parameter values. We develop our calculations in the context of a model for a cyclic predator-prey system, in which the invasion front and spatiotemporal oscillations of predators and prey are separated by a region in which the coexistence steady state is dynamically stabilised.     

Towards resolving the paradox of enrichment: the impact of zooplankton vertical migrations on plankton systems stability

Eutrophication, often resulting from human activity, is a serious threat to aquatic communities. Theoretical analysis of this phenomenon, based on conceptual mathematical models, leads to controversial predictions known as Rosenzweig's paradox of enrichment. At the same time, field observations demonstrate that real plankton communities exhibit various mechanisms of self-regulation which can buffer negative effects of enrichment. In this paper, we study potential effects of zooplankton vertical migration on stability of plankton systems functioning. We consider an intrinsically unstable plankton model, which is characterized by an unlimited phytoplankton multiplication and population oscillations of increasing amplitude, and investigate whether vertical migrations of zooplankton can stabilize such a system at low plankton densities. By means of developing two different models accounting for different ecological situations, e.g. deep waters and shallow waters, we show that vertical migrations of zooplankton can result in stabilization of eutrophic plankton systems. Thus, we show that this mechanism, rarely taken into account in models of plankton dynamics, may be important for resolving the paradox of enrichment in plankton communities.     

Spatial heterogeneity and the stability of predator-prey systems: Predator-mediated coexistence

A model for two competing prey and one predator in an environment consisting of many discrete, identical patches is developed. Conditions are derived determining which species will be present as a function of the dispersal abilities of the three species. One of the major predictions of the model is that intermediate levels of predation give the greatest chance of coexistence. This, and other results, are related to studies of natural communities in the intertidal and elsewhere.     

Local spatial structure and predator-prey dynamics: counterintuitive effects of prey enrichment

The Lotka-Volterra predator-prey model with prey density dependence shows the final prey density to be independent of its vital rates. This result assumes the community to be well mixed so that encounters between predators and prey occur as a product of the landscape densities, yet empirical evidence suggests that over small spatial scales this may not be the normal pattern. Starting from an individual-based model with neighborhood interactions and movements, a deterministic approximation is derived, and the effect of local spatial structure on equilibrium densities is investigated. Incorporating local movements and local interactions has important consequences for the community dynamics. Now the final prey density is very much dependent on its birth, death, and movement rates and in ways that seem counterintuitive. Increasing prey fecundity or mobility and decreasing the coefficient of competition can all lead to decreases in the final density of prey if the predator is also relatively immobile. However, analysis of the deterministic approximation makes the mechanism for these results clear; each of these changes subtly alters the emergent spatial structure, leading to an increase in the predator-prey spatial covariance at short distances and hence to a higher predation pressure on the prey.     

The stability of ecosystems: A brief overview of the paradox of enrichment

In theory, enrichment of resource in a predator-prey model leads to destabilization of the system,thereby collapsing the trophic interaction,a phenomenon referred to as "the paradox of enrichment". After it was first pro posed by Rosenzweig (1971), a number of subsequent studies were carried out on this dilemma over many decades. In this article, we review these theoretical and experimental works and give a brief overview of the proposed solutions to the paradox. The mechanisms that have been discussed are modifications of simple predator -prey models in the presence of prey that is inedible, invulnerable, unpalatable and toxic. Another class of mechanisms includes an incorporation of a ratio-dependent functional form,inducible defence of prey and density-dependent mortality of the predator. Moreover, we find a third set of explanations based on complex population dynamics including chaos in space and time. We conclude that,although any one of the various mechanisms proposed so far might potentially prevent destabilization of the predator-prey dynamics following enrichment, in nature different mechanisms may combine to cause stability, even when a system is enriched. The exact mechanisms,which may differ among systems,need to be disentangled through extensive field studies and laboratory experiments coupled with realistic theoretical models.     

The stability of predator-prey systems subject to the Allee effects

In recent years, many theoreticians and experimentalists have concentrated on the processes that affect the stability of predator-prey systems. But few papers have addressed the Allee effect with focus on the their stability. In this paper, we select two classical models describing predator-prey systems and introduce the Allee effects into the dynamics of both the predator and prey populations in these models, respectively. By combining mathematical analysis with numerical simulation, we have shown that the Allee effect may be a destabilizing force in predator-prey systems: the equilibrium point of the system could be changed from stable to unstable or otherwise, the system, even when it is stable, will take much longer time to reach the stable state. We also conclude that the equilibrium of the prey population will be enlarged due to the Allee effect of the predator, but the Allee effects of the prey may decrease the equilibrium value of the predator, or that of both the predator and prey. It should also be pointed out that the impact of the Allee effects of predator and prey due to different mechanisms on different predator-prey systems could also vary.     

Graph theoretical criteria for stability and boundedness of predator-prey systems

We introduce a graphical approach in the study of the qualitative behavior ofm species predator-prey systems. We prove that tree graphs imply global stability for Volterra models and local stability for general models; furthermore, we derive sufficient conditions so that loop graphs imply stability and boundedness of the solutions.     

Predator-prey systems with group defence: The paradox of enrichment revisited

The main concern of this paper is with survival or extinction of predators in models of predator-prey systems exhibiting group defence of the prey. It is shown that if there is no mutual interference among predators, enrichment could result in their extinction. However, if there is mutual interference, the predator population survives (at least deterministically). Research partially supported by the Natural Sciences and Engineering Research Council of Canada, Grant No. NSERC A 4823. Research partially supported by a Natural Sciences and Engineering Research Council of Canada postdoctoral fellowship.     

Imperfect optimal foraging and the paradox of enrichment

We show that the paradox of enrichment can be theoretically resolved in a flexible predator–prey system in which the predator practices imperfect optimal foraging. A previous study showed that perfect optimal foraging can mitigate increases in the amplitude of population oscillations associated with enrichment, but it did not show a stabilization pattern. Our results show that imperfect optimal foraging can stabilize the system and resolve the paradox of enrichment under nonequilibrium dynamics. Furthermore, the degree of stabilization with enrichment was stronger when the imperfection of optimal foraging was larger.     

富食悖论研究进展

Michael Rosenzweig于1971年首次提出＂富食悖论＂这一概念：在简单的被捕食者-捕食者系统中,随着营养物质供应的增加,系统变得不再稳定,并产生大振幅波动,最终导致系统内的物种灭亡。然而,许多实验结果并不支持Rosenzweig的理论。综述＂富食悖论＂的定义、理论和实验研究,同时综述各种解释实际与理论之间矛盾的机制,最后探讨＂富食悖论＂的研究前景。     

A resolution of the paradox of enrichment

Theoretical studies have shown a paradoxical destabilizing response of predator-prey ecosystems to enrichment, but there is the gap between the intuitive view of nature and this theoretical prediction. We studied a minimal predator-prey system (a two predator-two prey system) in which the paradox of enrichment pattern can vanish; the destabilization with enrichment is reversed, leading to stabilization (a decrease in the amplitude of oscillation of population densities). For resolution of the paradox, two conditions must be met: (1) the same prey species must be preferred as a dietary item by both predator species, creating the potential for high exploitative competition between the predator species, and (2), while both predators are assumed to select their diet in accordance with optimal diet utilization theory, one predator must be a specialist and the other a generalist. In this system, the presence of a less profitable prey species can cause the increase in population oscillation amplitudes associated with increasing enrichment to be suppressed via the optimal diet utilization of the generalist predator. The resulting stabilization is explained by the mitigating effect of the less profitable prey showing better population growth with increasing enrichment on the destabilization underlying the specialist predator and prey relation, thus resolving the paradox of enrichment.     

Unpalatable prey resolves the paradox of enrichment

Enrichment is an increasingly serious trend in natural ecosystems. A theoretical model of a predator–prey system with a natural assumption of satiation in predation predicts that enrichment causes the populations to fluctuate to stochastic extinction. However, this ''paradox of enrichment'' does not always occur in experimental and natural communities. Here we present a theoretical model that describes a novel mechanism for resolving the paradox in the case of a predator with optimal selective feeding. Specifically, a less profitable but edible (thus `unpalatable'') prey species sharply reduces the amplitude of population oscillations and firmly prevents the minimum abundances of species from falling below certain values. The presence of such an unpalatable prey thus guarantees the robustness of the system against enrichment.     

The volterra model for three species predator-prey systems: Boundedness and stability

Summary The classical Volterra equations modelling three species predatorprey interactions are considered with the intention of developing global properties of their solutions. Thirty-four different cases are exhibited. Global asymptotic stability is established for all non-loop cases. Global boundedness is established for all but one kind of loop case, and a condition is exhibited which insures boundedness for that case also. Finally a particular case is shown to have unbounded solutions.