Coexistence Steady State Research Articles

Global Dynamics of the Lotka‐Volterra Competition‐Diffusion System: Diffusion and Spatial Heterogeneity I

AbstractIn the first part of this series of three papers, we investigate the combined effects of diffusion, spatial variation, and competition ability on the global dynamics of a classical Lotka‐Volterra competition‐diffusion system. We establish the main results that determine the global asymptotic stability of semitrivial as well as coexistence steady states. Hence a complete understanding of the change in dynamics is obtained immediately. Our results indicate/confirm that, when spatial heterogeneity is included in the model, “diffusion‐driven exclusion” could take place when the diffusion rates and competition coefficients of both species are chosen appropriately.© 2016 Wiley Periodicals, Inc.

A remark on the global dynamics of competitive systems on ordered Banach spaces

A well-known result in [Hsu-Smith-Waltman, Trans. Amer. Math. Soc. (1996)] states that in a competitive semiflow defined on X + = X 1 + × X 2 + X^+ = X_1^+ \times X_2^+ , the product of two cones in respective Banach spaces, if ( u ∗ , 0 ) (u^*,0) and ( 0 , v ∗ ) (0,v^*) are the global attractors in X 1 + × { 0 } X_1^+ \times \{0\} and { 0 } × X 2 + \{0\}\times X_2^+ respectively, then one of the following three outcomes is possible for the two competitors: either there is at least one coexistence steady state, or one of ( u ∗ , 0 ) , ( 0 , v ∗ ) (u^*,0), (0,v^*) attracts all trajectories initiating in the order interval I = [ 0 , u ∗ ] × [ 0 , v ∗ ] I = [0,u^*] \times [0,v^*] . However, it was demonstrated by an example that in some cases neither ( u ∗ , 0 ) (u^*,0) nor ( 0 , v ∗ ) (0,v^*) is globally asymptotically stable if we broaden our scope to all of X + X^+ . In this paper, we give two sufficient conditions that guarantee, in the absence of coexistence steady states, the global asymptotic stability of one of ( u ∗ , 0 ) (u^*,0) or ( 0 , v ∗ ) (0,v^*) among all trajectories in X + X^+ . Namely, one of ( u ∗ , 0 ) (u^*,0) or ( 0 , v ∗ ) (0,v^*) is (i) linearly unstable, or (ii) linearly neutrally stable but zero is a simple eigenvalue. Our results complement the counterexample mentioned in the above paper as well as applications that frequently arise in practice.

Crucial role of strategy updating for coexistence of strategies in interaction networks.

Network models are useful tools for studying the dynamics of social interactions in a structured population. After a round of interactions with the players in their local neighborhood, players update their strategy based on the comparison of their own payoff with the payoff of one of their neighbors. Here we show that the assumptions made on strategy updating are of crucial importance for the strategy dynamics. In the first step, we demonstrate that seemingly small deviations from the standard assumptions on updating have major implications for the evolutionary outcome of two cooperation games: cooperation can more easily persist in a Prisoner's Dilemma game, while it can go more easily extinct in a Snowdrift game. To explain these outcomes, we develop a general model for the updating of states in a network that allows us to derive conditions for the steady-state coexistence of states (or strategies). The analysis reveals that coexistence crucially depends on the number of agents consulted for updating. We conclude that updating rules are as important for evolution on a network as network structure and the nature of the interaction.

Positive Coexistence of Steady States for a Diffusive Ratio-Dependent Predator-Prey Model with an Infected Prey

We examine a diffusive ratio-dependent predator-prey system with disease in the prey under homogeneous Dirichlet boundary conditions with a hostile environment at its boundary. We investigate the positive coexistence of three interacting species (susceptible prey, infected prey, and predator) and provide nonexistence conditions of positive solutions to the system. In addition, the global stability of the trivial and semitrivial solutions to the system is studied. Furthermore, the biological interpretation based on the result is also presented. The methods are employed from a comparison argument for the elliptic problem as well as the fixed-point theory as applied to a positive cone on a Banach space.

Limiting Structure of Shrinking Solutions to the Stationary Shigesada--Kawasaki--Teramoto Model with Large Cross-Diffusion

This paper is concerned with the limiting behavior of coexistence steady states of the Lotka--Volterra competition model as a cross-diffusion term tends to infinity. Under the Neumann boundary condition, Lou and Ni [J. Differential Equations, 154 (1999), pp. 157--190] derived a couple of limiting systems, which characterize the limiting behavior of coexistence steady states. One of two limiting systems characterizing the segregation of the competing species has been studied by Lou, Ni, and Yotsutani [Discrete Contin. Dyn. Syst., 10 (2004), pp. 435--458], and their work revealed the detailed bifurcation structure for the one-dimensional (1D) case. This paper focuses on the other limiting system characterizing the shrinkage of the species which is not endowed with the cross-diffusion effect. The bifurcation structure of positive solutions to the limiting system is stated. In particular, for the 1D case, we obtain a global connected set of solutions that bifurcates from a point on the line of constant solutions and blows up where the birth rate of the species is equal to the least positive eigenvalue of $-\Delta$ subject to the homogeneous Neumann boundary condition.

Credit market imperfection, financial market globalization, and catastrophic transition

We analyze a two-country overlapping generations model with integrated financial markets. We assume heterogeneous countries with respect to the population size, to the technology and to the level of credit market imperfection. We show that a subcritical Neimark–Sacker bifurcation may occur and that, before its destabilization, a stable steady state coexists with two invariant closed curves – one attracting and one repelling. In this way we reinforce existing results on the implications of the credit market imperfection that not only causes amplification and persistence of macroeconomic shocks, but also leads to significant changes in the long run behavior of the economy (i.e., catastrophic transition).

Dynamical Analysis of a Plateau Pika with Cross-Diffusion under Contraception Control

A plateau pika model with spatial cross-diffusion is investigated. By analyzing the corresponding characteristic equations, the local stability of an coexistence steady state is discussed whend21is small enough. However, whend21is large enough, the model shows Turing bifurcation ifB2 -4AC > 0. Furthermore, it is proved that if,R > R0, βK > dand cross-diffusion rates are zero, the positive coexistence steady state is globally asymptotically stable. A nonconstant positive solution bifurcates from the coexistent steady state by the Leray-Schauder degree theory. Numerical simulations are carried out to illustrate the main results.

Global dynamics of zooplankton and harmful algae in flowing habitats

This paper is devoted to the study of two advection–dispersion–reaction models arising from the dynamics of harmful algae and zooplankton in flowing-water habitats where a main channel is coupled to a hydraulic storage zone, representing an ensemble of fringing coves on the shoreline. For the system modeling the dynamics of algae and their toxin that contains little limiting nutrient, we establish a threshold type result on the global attractivity in terms of the basic reproduction ratio for algae. For the model with zooplankton that eat the algae and are inhibited by the toxin produced by algae, we show that there exists a coexistence steady state and the zooplankton is uniformly persistent provided that two basic reproduction ratios for algae and zooplankton are greater than unity.

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.

The effects of diffusion and spatial variation in Lotka–Volterra competition–diffusion system II: The general case

It is well known that the interactions between diffusion and spatial heterogeneity could create very interesting phenomena. In this series of two papers, using the classical Lotka–Volterra competition system, we will illustrate the combined effects of dispersal and spatial variations on the outcome of the competition.In Part I, with the total resources being fixed at exactly the same level, we first show that a heterogeneous distribution of resources is usually superior to its homogeneous counterpart in the presence of diffusion. Then we study the more general case when both species have heterogeneous carrying capacities, but still with the same total resources. Limiting behaviors of co-existence steady states as the dispersal rates tend to 0 or ∞ are also obtained.In Part II, we continue our investigation but under much broader situations – including different strengths and distributions of the resources, and with different competition abilities.

Seasonal forcing and multi-year cycles in interacting populations: lessons from a predator–prey model

Many natural systems are subject to seasonal environmental change. As a consequence many species exhibit seasonal changes in their life history parameters--such as a peak in the birth rate in spring. It is important to understand how this seasonal forcing affects the population dynamics. The main way in which seasonal models have been studied is through a two dimensional bifurcation approach. We augment this bifurcation approach with extensive simulation in order to understand the potential solution behaviours for a predator-prey system with a seasonally forced prey growth rate. We consider separately how forcing influences the system when the unforced dynamics have either monotonic decay to the coexistence steady state, or oscillatory decay, or stable limit cycles. The range of behaviour the system can exhibit includes multi-year cycles of different periodicities, parameter ranges with coexisting multi-year cycles of the same or different period as well as quasi-periodicity and chaos. We show that the level of oscillation in the unforced system has a large effect on the range of behaviour when the system is seasonally forced. We discuss how the methods could be extended to understand the dynamics of a wide range of ecological and epidemiological systems that are subject to seasonal changes.

The effects of diffusion and spatial variation in Lotka–Volterra competition–diffusion system I: Heterogeneity vs. homogeneity

It is well known that the interactions between diffusion and spatial heterogeneity could create very interesting phenomena. In this series of two papers, using the classical Lotka–Volterra competition system, we will illustrate the combined effects of dispersal and spatial variations on the outcome of the competition.In Part I, with the total resources being fixed at exactly the same level, we first show that a heterogeneous distribution of resources is usually superior to its homogeneous counterpart in the presence of diffusion. Then we study the more general case when both species have heterogeneous carrying capacities, but still with the same total resources. Limiting behaviors of co-existence steady states as the dispersal rates tend to 0 or ∞ are also obtained.In Part II, we continue our investigation but under much broader situations – including different strengths and distributions of the resources, and with different competition abilities.

Traveling Wave Solutions for a Class of Predator–Prey Systems

We use a shooting method to show the existence of traveling wave fronts and to obtain an explicit expression of minimum wave speed for a class of diffusive predator–prey systems. The existence of traveling wave fronts indicates the existence of a transition zone from a boundary equilibrium to a co-existence steady state and the minimum wave speed measures the asymptotic speed of population spread in some sense. Our approach is a significant improvement of techniques introduced by Dunbar. The advantage of our method is that it does not need the notion of Wazewski’s set and LaSalle’s invariance principle used in Dunbar’s approach. In our approach, we convert the equations for traveling wave solutions to a system of first order equations by a “non-traditional transformation”. With this converted new system, we are able to construct a Liapunov function, which gives an immediate implication of the boundedness and convergence of the relevant class of heteroclinic orbits. Our method provides a more efficient way to study the existence of traveling wave solutions for general predator–prey systems.

Coupling planktonic ecosystem and fisheries food web models for a pelagic ecosystem: Description and validation for the subarctic Pacific

We provide a modeling framework that fully couples a one-dimensional physical mixed layer model, a biogeochemical model, and an upper trophic level fisheries model. For validation purposes, the model has been parameterized for the pelagic Eastern Pacific Subarctic Gyre ecosystem. This paper presents a thorough description of the model itself, as well as an ensemble-based parameterization process that allows the model to incorporate the high level of uncertainty associated with many upper trophic level predator-prey processes. Through a series of model architecture experiments, we demonstrate that the use of a consistent functional response for all predator-prey interactions, as well as the use of density-dependent mortality rates for planktonic functional groups, are important factors in reproducing annual and seasonal observations. We present the results of a 50-year climatological simulation, which demonstrates that under contemporary physical forcing, the model is capable of reproducing long-term seasonal dynamics in primary production and biogeochemical cycling, while maintaining steady-state coexistence of upper trophic level functional groups at levels consistent with observations.

Coupling effect of grazing pressure and nutrient enrichment on system stability

A three dimensional nutrient-plant–herbivore model was proposed and conditions for boundedness, positive invariance, existence and stability of different equilibrium points, Hopf-bifurcation and global stability were obtained. We performed numerical simulations to observe the simultaneous effect of the top-down and the bottom-up mechanism on the system. It was found that nutrient enrichment destroyed the coexistence steady state of the system. This nutrient enrichment could be due to high nutrient input rate or high nutrient recycling rate. In both cases the system showed instability. Moreover, these results were independent of the grazing pressure and the predation functional form.

Features of Fast Living: On the Weak Selection for Longevity in Degenerate Birth-Death Processes

Deterministic descriptions of dynamics of competing species with identical carrying capacities but distinct birth, death, and reproduction rates predict steady state coexistence with population ratios depending on initial conditions. Demographic fluctuations described by a Markovian birth-death model break this degeneracy. A novel large carrying capacity asymptotic theory confirmed by conventional analysis and simulations reveals a weak preference for longevity in the deterministic limit with finite-time extinction of one of the competitors on a time scale proportional to the total carrying capacity.

Uniqueness and Complete Dynamics in Heterogeneous Competition-Diffusion Systems

In this paper we study the interactions between diffusion and heterogeneity of the environment in the classical diffusive Lotka--Volterra competition systems. In the weak competition case, we establish the uniqueness, hence the global asymptotic stability, of coexistence steady states under various circumstances, and thereby we obtain a complete understanding of the change in dynamics when one of the interspecific competition coefficients is small.

On a reaction-diffusion model for sterile insect release method with release on the boundary

We consider a partial differential equation model that describes the sterile insect release method (SIRM) in a bounded 1-dimensional domain (interval). Unlike everywhere-releasing in the domain as considered in previous works [17] and [14] , we propose the mechanism of releasing on the boundary only. We show existence of the fertile-free steady state and prove its stability under some conditions. By using the upper-lower solution method, we also show that under some other conditions there may exist a coexistence steady state. Biological implications of our mathematical results are that the SIRM with releasing only on the boundary can successfully eradicate the fertile insects as long as the strength of the sterile releasing is reasonably large, while the method may also fail if the releasing is not sufficient.

Analysis of a model for the effects of an external toxin on anaerobic digestion

Anaerobic digestion has been modeled as a two-stage process using coupled chemostat models with non-monotone growth functions, [9]. This study incorporates the effects of an external toxin. After reducing the model to a 3-dimensional system, global stability of boundary and interior equilibria is proved using differential inequalities and comparisons to the corresponding toxin-free model. Conditions are given under which the behavior of the toxin-free model is preserved. Introduction of the toxin results in additional patterns such as bistabilities of coexistence steady states or of a periodic orbit and an interior steady state.

Competition between species can stabilize public-goods cooperation within a species.

Competition between species is a major ecological force that can drive evolution. Here, we test the effect of this force on the evolution of cooperation within a species. We use sucrose metabolism of budding yeast, Saccharomyces cerevisiae, as a model cooperative system that is subject to social parasitism by cheater strategies. We find that when cocultured with a bacterial competitor, Escherichia coli, the frequency of cooperator phenotypes in yeast populations increases dramatically as compared with isolated yeast populations. Bacterial competition stabilizes cooperation within yeast by limiting the yeast population density and also by depleting the public goods produced by cooperating yeast cells. Both of these changes induced by bacterial competition increase the cooperator frequency because cooperator yeast cells have a small preferential access to the public goods they produce; this preferential access becomes more important when the public good is scarce. Our results indicate that a thorough understanding of species interactions is crucial for explaining the maintenance and evolution of cooperation in nature.