FCLT and MDP for Stochastic Lotka–Volterra Model

In this paper, we continue an asymptotic analysis of a stochastic version of the Lotka–Volterra model for predator–prey interactions. While the fluid approximation and large deviations were shown in Klebaner and Liptser (Ann. Appl. Probab. 11, 1263–1291, 2001) here we establish the diffusion approximation and moderate deviations.     

Bifurcation Analysis of a Predator–Prey System with Generalised Holling Type III Functional Response

We consider a generalised Gause predator–prey system with a generalised Holling response function of type III: . We study the cases where b is positive or negative. We make a complete study of the bifurcation of the singular points including: the Hopf bifurcation of codimensions 1 and 2, the Bogdanov–Takens bifurcation of codimensions 2 and 3. Numerical simulations are given to calculate the homoclinic orbit of the system. Based on the results obtained, a bifurcation diagram is conjectured and a biological interpretation is given.      

Well‐posedness,positivity, and time asymptotics properties for a reaction–diffusion model of plankton communities,involving a rational nonlinearity with singularity

In this work, we consider a reaction–diffusion system, modeling the interaction between nutrients, phytoplankton, and zooplankton. Using a semigroup approach in , we prove global existence, uniqueness, and positivity of the solutions. The nonlinearity is handled by providing estimates in , allowing to deal with most of the functional responses that describe predator/prey interactions (Holling I, II, III, Ivlev) in ecology. The paper finally exhibits some time asymptotic properties of the solutions.     

Global stability of a periodic Holling‐Tanner predator‐prey model

This paper is concerned with the global dynamics of a Holling‐Tanner predator‐prey model with periodic coefficients. We establish sufficient conditions for the existence of a positive solution and its global asymptotic stability. The stability conditions are first given in average form and afterward as pointwise estimates. In the autonomous case, the previous criteria lead to a known result.     

Graph‐theoretic method on the periodicity of multipatch dispersal predator‐prey system with Holling type‐II functional response

We study the periodicity of multipatch dispersal predator‐prey system with Holling type‐II functional response in this paper. By providing a new method, we overcome the difficulty to get the priori bounds estimation of unknown solutions of operator equation Lu=λNu. Graph theory with coincidence degree theory is used, and a sufficient criterion for the periodicity of the system is obtained. The criterion presented in this paper is closely related with topological structure of dispersal network and can be verified easily. Finally, a numerical example is also provided to verify the effectiveness of theoretical results.     

Optimal harvesting policy for a stochastic predator–prey model

Optimal harvesting of a stochastic predator–prey model is considered in this paper. Sufficient and necessary criteria for the existence of optimal harvesting strategy are obtained. At the same time, the optimal harvest effort and the maximum of sustainable yield are given.     

Periodic traveling wave train and point-to-periodic traveling wave for a diffusive predator–prey system with Ivlev-type functional response

A diffusive predator–prey system with Ivlev-type scheme is investigated in this article. The existences of a small amplitude periodic traveling wave train _Γp${\Gamma }_p$ and the traveling wave solution connecting the boundary equilibrium _Eu(1,0)$E_u(1,0)$ to the periodic traveling wave _Γp${\Gamma }_p$ are obtained. The existence of this point-to-periodic solution reveals that the predator invasion leads to the periodic population densities in the coexistence domain, and thus plays a mild role in the evolution of predator–prey communities. The techniques used here are the Hopf bifurcation theorem, the improved shooting method combining with the geometric singular perturbation method.     

Analysis of a class of Kolmogorov systems

A two-dimensional Kolmogorov system depending on two independent parameters and having a degenerate condition is studied in this work. We obtain local analytical properties of the system when the parameters vary in a sufficiently small neighborhood of the origin. The behavior of the system is described by bifurcation diagrams. Applications of Kolmogorov systems can be found particularly in modeling population dynamics in biology and ecology.     

How do dispersal rates affect the transition from periodic to irregular spatio-temporal oscillations in invasive predator–prey systems?

When one considers the spatial aspects of a cyclic predator–prey interaction, ecological events such as invasions can generate periodic travelling waves (PTWs)—sometimes known as wavetrains. In certain instances PTWs may destabilise into spatio-temporal irregularity due to convective type instabilities, which permit a fixed width band of PTWs to develop behind the propagating invasion front. In this paper, we detail how one can locate this transition when one has unequal predator and prey dispersal rates. We do this by using absolute stability theory combined with a recent derivation of the amplitude of PTWs behind invasion. This work is applicable to a wide range of reaction–diffusion type predator–prey models, but in this paper we apply it to a specific set of equations (the Leslie–May model). We show that the width of PTW band increases/decreases when the ratio of prey and predator dispersal rates is large/small.     

A Random Continuous Model for Two Interacting Populations

A system of two interacting populations is considered where each population density follow a stochastic partial differential equation. The stability behaviour of both populations is studied. Accepted 29 August 2001. Online publication 21 December 2001.