Global dynamics of a discrete two-species Lottery-Ricker competition model

In this article, we study the global dynamics of a discrete two-dimensional competition model. We give sufficient conditions on the persistence of one species and the existence of local asymptotically stable interior period-2 orbit for this system. Moreover, we show that for a certain parameter range, there exists a compact interior attractor that attracts all interior points except Lebesgue measure zero set. This result gives a weaker form of coexistence which is referred to as relative permanence. This new concept of coexistence combined with numerical simulations strongly suggests that the basin of attraction of the locally asymptotically stable interior period-2 orbit is an infinite union of connected components. This idea may apply to many other ecological models. Finally, we discuss the generic dynamical structure that gives relative permanence.     

Global stability, local stability and permanence in model food webs

The dynamical theory of food webs has been based typically on local stability analysis. The relevance of local stability to food web properties has been questioned because local stability holds only in the immediate vicinity of the equilibrium and provides no information about the size of the basin of attraction. Local stability does not guarantee persistence of food webs in stochastic environments. Moreover, local stability excludes more complex dynamics such as periodic and chaotic behaviors, which may allow persistence. Global stability and permanence could be better criteria of community persistence. Our simulation analysis suggests that these three stability measures are qualitatively consistent in that all three predict decreasing stability with increasing complexity. Some new predictions on how stability depends on food web configurations are generated here: a consumer-victim link has a smaller effect on the probabilities of stability, as measured by all three stability criteria, than a pair of recipient-controlled and donor-controlled links; a recipient-controlled link has a larger effect on the probabilities of local stability and permanence than a donor-controlled link, while they have the same effect on the probability of global stability; food webs with equal proportions of donor-controlled and recipient-controlled links are less stable than those with different proportions.     

Qualitative permanence of Lotka–Volterra equations

In this paper, we consider permanence of Lotka-Volterra equations. We investigate the sign structure of the interaction matrix that guarantees the permanence of a Lotka-Volterra equation whenever it has a positive equilibrium point. An interaction matrix with this property is said to be qualitatively permanent. Our results provide both necessary and sufficient conditions for qualitative permanence.     

Alternative food, switching predators, and the persistence of predator-prey systems

Sigmoid functional responses may arise from a variety of mechanisms, one of which is switching to alternative food sources. It has long been known that sigmoid (Holling's Type III) functional responses may stabilize an otherwise unstable equilibrium of prey and predators in Lotka-Volterra models. This poses the question of under what conditions such switching-mediated stability is likely to occur. A more complete understanding of the effect of predator switching would therefore require the analysis of one-predator/two-prey models, but these are difficult to analyze. We studied a model based on the simplifying assumption that the alternative food source has a fixed density. A well-known result from optimal foraging theory is that when prey density drops below a threshold density, optimally foraging predators will switch to alternative food, either by including the alternative food in their diet (in a fine-grained environment) or by moving to the alternative food source (in a coarse-grained environment). Analyzing the population dynamical consequences of such stepwise switches, we found that equilibria will not be stable at all. For suboptimal predators, a more gradual change will occur, resulting in stable equilibria for a limited range of alternative food types. This range is notably narrow in a fine-grained environment. Yet, even if switching to alternative food does not stabilize the equilibrium, it may prevent unbounded oscillations and thus promote persistence. These dynamics can well be understood from the occurrence of an abrupt (or at least steep) change in the prey isocline. Whereas local stability is favored only by specific types of alternative food, persistence of prey and predators is promoted by a much wider range of food types.     

Lotka-Volterra equations: Decomposition,stability, and structure

Summary The major objective of this paper is to propose a new decomposition-aggregation framework for stability analysis of Lotka-Volterra equations employing the concept of vector Liapunov functions. Both the disjoint and the overlapping decompositions are introduced to increase flexibility in constructing Liapunov functions for the overall system. Our second objective is to consider the Lotka-Volterra equations under structural perturbations, and derive conditions under which a positive equilibrium is connectively stable. Both objectives of this paper are directed towards a better understanding of the intricate interplay between stability and complexity in the context of robustness of model ecosystems represented by Lotka-Volterra equations. Only stability of equilibria in models with constant parameters is considered here. Nonequilibrium analysis of models with nonlinear time-varying parameters is the subject of a companion paper.Research supported by U.S. Department of Energy under the Contract EC-77-S-03-1493.On leave from Kobe University, Kobe, Japan.     

Diffusion-mediated persistence in two-species competition Lotka-Volterra model

We consider a system composed of two Lotka-Volterra patches connected by diffusion. Each patch has two competitors. Conditions for persistence of the system are given. It is proved that the system can be made persistent under appropriate diffusion coefficients ensuring the instability of boundary equilibria, even if each species is not persistent within each patch. The choice of the coefficients depends closely on the patch dynamics without diffusion.     

Imitation dynamics of vaccine decision-making behaviours based on the game theory

Based on game theory, we propose an age-structured model to investigate the imitation dynamics of vaccine uptake. We first obtain the existence and local stability of equilibria. We show that Hopf bifurcation can occur. We also establish the global stability of the boundary equilibria and persistence of the disease. The theoretical results are supported by numerical simulations.     

Increasing community size and connectance can increase stability in competitive communities

The relationship between community complexity and stability has been the subject of an enduring debate in ecology over the last 50 years. Results from early model communities showed that increased complexity is associated with decreased local stability. I demonstrate that increasing both the number of species in a community and the connectance between these species results in an increased probability of local stability in discrete-time competitive communities, when some species would show unstable dynamics in the absence of competition. This is shown analytically for a simple case and across a wider range of community sizes using simulations, where individual species have dynamics that can range from stable point equilibria to periodic or more complex. Increasing the number of competitive links in the community reduces per-capita growth rates through an increase in competitive feedback, stabilising oscillating dynamics. This result was robust to the introduction of a trade-off between competitive ability and intrinsic growth rate and changes in species interaction strengths. This throws new light on the discrepancy between the theoretical view that increased complexity reduces stability and the empirical view that more complex systems are more likely to be stable, giving one explanation for the relative lack of complex dynamics found in natural systems. I examine how these results relate to diversity-biomass stability relationships and show that an analytical solution derived in the region of stable equilibrium dynamics captures many features of the change in biomass fluctuations with community size in communities including species with oscillating dynamics.     

Global dynamics of a discrete two-species Lottery-Ricker competition model

In this article, we study the global dynamics of a discrete two-dimensional competition model. We give sufficient conditions on the persistence of one species and the existence of local asymptotically stable interior period-2 orbit for this system. Moreover, we show that for a certain parameter range, there exists a compact interior attractor that attracts all interior points except Lebesgue measure zero set. This result gives a weaker form of coexistence which is referred to as relative permanence. This new concept of coexistence combined with numerical simulations strongly suggests that the basin of attraction of the locally asymptotically stable interior period-2 orbit is an infinite union of connected components. This idea may apply to many other ecological models. Finally, we discuss the generic dynamical structure that gives relative permanence.     

Competitive exclusion and coexistence of pathogens in a homosexually-transmitted disease model

A sexually-transmitted disease model for two strains of pathogen in a one-sex, heterogeneously-mixing population has been studied completely by Jiang and Chai in (J Math Biol 56:373-390, 2008). In this paper, we give a analysis for a SIS STD with two competing strains, where populations are divided into three differential groups based on their susceptibility to two distinct pathogenic strains. We investigate the existence and stability of the boundary equilibria that characterizes competitive exclusion of the two competing strains; we also investigate the existence and stability of the positive coexistence equilibrium, which characterizes the possibility of coexistence of the two strains. We obtain sufficient and necessary conditions for the existence and global stability about these equilibria under some assumptions. We verify that there is a strong connection between the stability of the boundary equilibria and the existence of the coexistence equilibrium, that is, there exists a unique coexistence equilibrium if and only if the boundary equilibria both exist and have the same stability, the coexistence equilibrium is globally stable or unstable if and only if the two boundary equilibria are both unstable or both stable.     

Persistence increases with diversity and connectance in trophic metacommunities

Background

We are interested in understanding if metacommunity dynamics contribute to the persistence of complex spatial food webs subject to colonization-extinction dynamics. We study persistence as a measure of stability of communities within discrete patches, and ask how do species diversity, connectance, and topology influence it in spatially structured food webs.

Methodology/Principal Findings

We answer this question first by identifying two general mechanisms linking topology of simple food web modules and persistence at the regional scale. We then assess the robustness of these mechanisms to more complex food webs with simulations based on randomly created and empirical webs found in the literature. We find that linkage proximity to primary producers and food web diversity generate a positive relationship between complexity and persistence in spatial food webs. The comparison between empirical and randomly created food webs reveal that the most important element for food web persistence under spatial colonization-extinction dynamics is the degree distribution: the number of prey species per consumer is more important than their identity.

Conclusions/Significance

With a simple set of rules governing patch colonization and extinction, we have predicted that diversity and connectance promote persistence at the regional scale. The strength of our approach is that it reconciles the effect of complexity on stability at the local and the regional scale. Even if complex food webs are locally prone to extinction, we have shown their complexity could also promote their persistence through regional dynamics. The framework we presented here offers a novel and simple approach to understand the complexity of spatial food webs.     

Stability analysis of pathogen-immune interaction dynamics

The paper considers models of dynamics of infectious disease in vivo from the standpoint of the mathematical analysis of stability. The models describe the interaction of the target cells, the pathogens, and the humoral immune response. The paper mainly focuses on the interior equilibrium, whose components are all positive. If the model ignores the absorption of the pathogens due to infection, the interior equilibrium is always asymptotically stable. On the other hand, if the model does consider it, the interior equilibrium can be unstable and a simple Hopf bifurcation can occur. A sufficient condition that the interior equilibrium is asymptotically stable is obtained. The condition explains that the interior equilibrium is asymptotically stable when experimental parameter values are used for the model. Moreover, the paper considers the models in which uninfected cells are involved in the immune response to pathogens, and are removed by the immune complexes. The effect of the involvement strongly affects the stability of the interior equilibria. The results are shown with the aid of symbolic calculation software.     

Persistence and patchiness of predator-prey systems induced by discrete event population exchange mechanisms

We consider a spatially distributed predator-prey system in which population exchange between cells can occur at only certain points in the predation cycle. Studying the characteristics of this discrete event model, by both analytic and simulation methods, we show that indefinite persistence of populations is possible over a wide range of parameter settings, even though the cells all tend to rapid extinction in isolation. The persistence is shown to take two forms: one in which a constant distribution of cells both in phase and space is maintained, and one in which this homogeneity of distribution is replaced by dynamically stable patterns of patches and waves. The former is shown to be neatly describable by the classical Lotka-Volterra equation. In contrast, persistence of locally unstable Lotka-Volterra predator-prey systems in both forms is shown to be impossible when the discrete population exchange mechanism is replaced by one of the continuous linear diffusion type.     

Multispecies food webs with diffusion

Using Liapunov's direct method, in this paper, it has been shown that the general Lotka-Volterra food web is stable without and with diffusion under each case of homogeneous reservoir and flux boundary conditions. However, for a three species food web with Holling's functional response the above general result regarding stability is not necessarily true. In such a case, conditions and regions for non-linear stability, without and with diffusion, have been derived. It is shown that such an otherwise unstable system may become stable with diffusion at least in a subregion of the positive octant of the state space.     

捕食者有病的生态-流行病模型的分析

建立并分析了捕食者具有疾病且有功能反应的生态-流行病(SI)模型,讨论了解的有界性．应用特征根法得到了平衡点局部渐近稳定的充分条件,进一步分析了平衡点的全局稳定性,得到了边界平衡点和正平衡点全局稳定的充分条件。     

Robust permanence and impermanence for stochastic replicator dynamics

Garay and Hofbauer (SIAM J. Math. Anal. 34 (2003)) proposed sufficient conditions for robust permanence and impermanence of the deterministic replicator dynamics. We reconsider these conditions in the context of the stochastic replicator dynamics, which is obtained from its deterministic analogue by introducing Brownian perturbations of payoffs. When the deterministic replicator dynamics is permanent and the noise level small, the stochastic dynamics admits a unique ergodic distribution whose mass is concentrated near the maximal interior attractor of the unperturbed system; thus, permanence is robust against small unbounded stochastic perturbations. When the deterministic dynamics is impermanent and the noise level small or large, the stochastic dynamics converges to the boundary of the state space at an exponential rate.     

Coexistence in MacArthur-style consumer-resource models

The nature of and conditions for permanent coexistence of consumers and resources are characterized in a family of models that generalize MacArthur's consumer-resource model. The generalization is of the resource dynamics, which need not be of Lotka-Volterra form but are subject only to certain restrictions loose enough to admit many resource dynamics of biological interest. For any such model, (1) if there is an interior equilibrium, then it is globally attracting, else some boundary equilibrium is globally attracting-thus permanent coexistence is coexistence at a globally attracting equilibrium; (2) there is an interior equilibrium if and only if for any species, the equilibrium approached in the absence of that species and the presence of the others is invasible by that species--thus permanent coexistence is equivalent to mutual invasibility; (3) for resources without direct interactions, the conditions for permanent coexistence of the consumers admit an instructive formulation in terms of regression statistics. The significance and limitations of the models and results are discussed.     

具有年龄结构的食蚜蝇-蚜虫捕食模型

现有的具有年龄结构的捕食-食饵模型总是假设只有成年捕食者捕食猎物,这与实际情况不符。建立了一个幼年捕食者捕食食饵的具有年龄结构的食蚜蝇-蚜虫模型,应用微分方程定性理论,讨论了系统平衡点及其稳定性:其中平衡点E1(0,0,0)为不稳定的;满足一定条件时,边界平衡点E2(K,0,0)及正平衡点E3(x*,y1*,y2*)为局部渐近稳定的;且应用一致持续生存理论得到了系统永久持续生存的条件,为有害生物综合治理提供了理论依据。     

Species dynamics alter community diversity–biomass stability relationships

The relationship between community diversity and biomass variability remains a crucial ecological topic, with positive, negative and neutral diversity–stability relationships reported from empirical studies. Theory highlights the relative importance of Species–Species or Species–Environment interactions in driving diversity–stability patterns. Much previous work is based on an assumption of identical (stable) species‐level dynamics. We studied ecosystem models incorporating stable, cyclic and more complex species‐level dynamics, with either linear or non‐linear density dependence, within a locally stable community framework. Species composition varies with increasing diversity, interacting with the correlation of species' environmental responses to drive either positive or negative diversity–stability patterns, which theory based on communities with only stable species‐level dynamics fails to predict. Including different dynamics points to new mechanisms that drive the full range of diversity–biomass stability relationships in empirical systems where a wider range of dynamical behaviours are important.     

A stage-structured predator–prey model with distributed maturation delay and harvesting

ABSTRACT

A stage-structured predator–prey system with distributed maturation delay and harvesting is investigated. General birth and death functions are used. The local stability of each feasible equilibria is discussed. By using the persistence theory, it is proven that the system is permanent if the coexistence equilibrium exists. By using Lyapunov functional and LaSalle invariant principle, it is shown that the trivial equilibrium is globally stable when the other equilibria are not feasible, and that the boundary equilibrium is globally stable if the coexistence equilibrium does not exist. Finally, sufficient conditions are derived for the global stability of the coexistence equilibrium.