A diffusive predator-prey model with a protection zone

In this paper we study the effects of a protection zone Ω₀ for the prey on a diffusive predator-prey model with Holling type II response and no-flux boundary condition. We show the existence of a critical patch size described by the principal eigenvalue of the Laplacian operator over Ω₀ with homogeneous Dirichlet boundary conditions. If the protection zone is over the critical patch size, i.e., if is less than the prey growth rate, then the dynamics of the model is fundamentally changed from the usual predator-prey dynamics; in such a case, the prey population persists regardless of the growth rate of its predator, and if the predator is strong, then the two populations stabilize at a unique coexistence state. If the protection zone is below the critical patch size, then the dynamics of the model is qualitatively similar to the case without protection zone, but the chances of survival of the prey species increase with the size of the protection zone, as generally expected. Our mathematical approach is based on bifurcation theory, topological degree theory, the comparison principles for elliptic and parabolic equations, and various elliptic estimates.     

Positive solutions for a diffusive Bazykin model in spatially heterogeneous environment

In this paper, we investigate a diffusive Bazykin model in a spatially heterogeneous environment. We obtain some results on nonexistence and existence of positive solutions of the model. Moreover, the asymptotic behavior of positive solutions with respect to certain parameters is also studied.     

Strategy and stationary pattern in a three-species predator-prey model

In this paper, we study a strongly coupled system of partial differential equations which models the dynamics of a two-predator-one-prey ecosystem in which the prey exercises a defense switching mechanism and the predators collaboratively take advantage of the prey's strategy. We demonstrate the emergence of stationary patterns for this system, and show that it is due to the cross diffusion that arises naturally in the model. As far as the authors are aware, this is the first example of stationary patterns in a predator-prey system arising solely from the effect of cross diffusion.     

Realization of prescribed patterns in the competition model

We demonstrate that for any prescribed set of finitely many disjoint closed subdomains D₁,…,D_m of a given spatial domain Ω in R^N, if d₁,d₂,a₁,a₂,c,d,e are positive continuous functions on Ω and b(x) is identically zero on D?D₁∪?∪D_m and positive in the rest of Ω, then for suitable choices of the parameters λ, μ and all small ε>0, the competition model

     

A diffusive competition model with a protection zone

This paper is concerned with a two species diffusive competition model with a protection zone for the weak competitor. Our mathematical results imply that when the protection zone is above a certain critical patch size determined by the birth rate of the weak competitor, the weak species almost always survives, but it cannot survive when the protection zone is below the critical size and its competitor is strong enough. While this is the main feature of the model, the actual dynamical behavior of the reaction-diffusion system is more complicated. The key to reveal the main feature of the system lies in a detailed analysis of the attracting regions of its steady-state solutions. Our mathematical analysis shows that, compared with the predator-prey model discussed in [Yihong Du, Junping Shi, A diffusive predator-prey model with a protect zone, J. Differential Equations 226 (2006) 63-91], the protection zone has some essentially different effects on the fine dynamics of the competition model.     

The diffusive logistic model with a free boundary in heterogeneous environment

In this paper, we study the population dynamics of an invasive species in heterogeneous environment which is modeled by a diffusive logistic equation with free boundary condition. To understand the effect of the dispersal rate D and the parameter μ (the ratio of the expansion speed of the free boundary and the population gradient at the expanding front) on the dynamics of this model, we divide the heterogeneous environment into two cases: strong heterogeneous environment and weak heterogeneous environment. By choosing D and μ as variable parameters, we derive sufficient conditions for species spreading (resp. vanishing) in the strong heterogeneous environment; while in the weak heterogeneous environment, we obtain sharp criteria for the spreading and vanishing. Moreover, when spreading happens, we give an estimate for the asymptotic spreading speed of the free boundary. These theoretical results may have important implications for prediction and prevention of biological invasions.     

Turing patterns of a strongly coupled predator-prey system with diffusion effects

The main purpose of this work is to investigate the effects of cross-diffusion in a strongly coupled predator-prey system. By a linear stability analysis we find the conditions which allow a homogeneous steady state (stable for the kinetics) to become unstable through a Turing mechanism. In particular, it is shown that Turing instability of the reaction-diffusion system can disappear due to the presence of the cross-diffusion, which implies that the cross-diffusion induced stability can be regarded as the cross-stability of the corresponding reaction-diffusion system. Furthermore, we consider the existence and non-existence results concerning non-constant positive steady states (patterns) of the system. We demonstrate that cross-diffusion can create non-constant positive steady-state solutions. These results exhibit interesting and very different roles of the cross-diffusion in the formation and the disappearance of the Turing instability.     

Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a prey refuge

We study a predator-prey model with Holling type II functional response incorporating a prey refuge under homogeneous Neumann boundary condition. We show the existence and non-existence of non-constant positive steady-state solutions depending on the constant m∈(0,1], which provides a condition for protecting (1−m)u of prey u from predation. Moreover, we investigate the asymptotic behavior of spacially inhomogeneous solutions and the local existence of periodic solutions.     

A strongly coupled predator–prey system with non-monotonic functional response

The predator–prey system with non-monotonic functional response is an interesting field of theoretical study. In this paper we consider a strongly coupled partial differential equation model with a non-monotonic functional response—a Holling type-IV function in a bounded domain with no flux boundary condition. We prove a number of existence and non-existence results concerning non-constant steady states (patterns) of the underlying system. In particular, we demonstrate that cross-diffusion can create patterns when the corresponding model without cross-diffusion fails.     

Analysis of diffusive two-competing-prey and one-predator systems with Beddington-Deangelis functional response

In this paper, a diffusive two-competing-prey and one-predator system with Beddington-DeAngelis functional response is considered. The sufficient and necessary conditions for the existence of coexistence states are provided using the fixed point index theory developed. In addition, the stability and uniqueness of coexistence states are investigated. Finally, this paper discusses the sufficient conditions for extinction and permanence of the time-dependent system.     

Non-constant positive steady states of the Sel'kov model

This paper deals with the reaction-diffusion system known as the Sel'kov model with the homogeneous Neumann boundary condition. This model has been applied to various problems in chemistry and biology. We first give a priori estimates (positive upper and lower bounds) of positive steady states, and then study the non-existence, bifurcation and global existence of non-constant positive steady states as the parameters λ and θ are varied.     

Asymptotic profiles of the positive steady state for an SIS epidemic reaction-diffusion model. Part I

To capture the impact of spatial heterogeneity of environment and movement of individuals on the persistence and extinction of a disease, Allen et al. in [L.J.S. Allen, B.M. Bolker, Y. Lou, A.L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Discrete Contin. Dyn. Syst. Ser. A 21 (1) (2008) 1-20] proposed a spatial SIS (susceptible-infected-susceptible) reaction-diffusion model, and studied the existence, uniqueness and particularly the asymptotic behavior of the endemic equilibrium as the diffusion rate of the susceptible individuals goes to zero in the case where a so-called low-risk subhabitat is created. In this work, we shall provide further understanding of the impacts of large and small diffusion rates of the susceptible and infected population on the persistence and extinction of the disease, which leads us to determine the asymptotic behaviors of the endemic equilibrium when the diffusion rate of either the susceptible or infected population approaches to infinity or zero in the remaining cases. Consequently, our results reveal that, in order to eliminate the infected population at least in low-risk area, it is necessary that one will have to create a low-risk subhabitat and reduce at least one of the diffusion rates to zero. In this case, our results also show that different strategies of controlling the diffusion rates of individuals may lead to very different spatial distributions of the population; moreover, once the spatial environment is modified to include a low-risk subhabitat, the optimal strategy of eradicating the epidemic disease is to restrict the diffusion rate of the susceptible individuals rather than that of the infected ones.     

Effect of cross-diffusion on the stationary problem of a prey-predator model with a protection zone

This paper is concerned with the stationary problem of a prey-predator cross-diffusion system with a protection zone for the prey. We discuss the existence and non-existence of coexistence states of the two species by using the bifurcation theory. As a result, it is shown that the cross-diffusion for the prey has beneficial effects on the survival of the prey when the intrinsic growth rate of the predator is positive. We also study the asymptotic behavior of positive stationary solutions as the cross-diffusion coefficient of the prey tends to infinity.     

Positive solutions to logistic type equations with harvesting

We use comparison principles, variational arguments and a truncation method to obtain positive solutions to logistic type equations with harvesting both in RN and in a bounded domain Ω⊂RN, with N?3, when the carrying capacity of the environment is not constant. By relaxing the growth assumption on the coefficients of the differential equation we derive a new equation which is easily solved. The solution of this new equation is then used to produce a positive solution of our original problem.     

Diffusive Logistic Equations with Degenerate Boundary Conditions

The purpose of this paper is to provide a careful and accessible exposition of static bifurcation theory for a class of degenerate boundary value problems for diffusive logistic equations with indefinite weights that model population dynamics in environments with spatial heterogeneity. We discuss the changes that occur in the structure of the positive solutions as a parameter varies near the first eigenvalue of the linearized problem, and prove that the most favorable situations will occur if there is a relatively large favorable region (with good resources and without crowding effects) located some distance away from the boundary of the environment.Dedicated to Professor Mitsuru Ikawa on the occasion of his 60th birthday     

Stability of steady-state solutions to a prey-predator system with cross-diffusion

This paper is concerned with a cross-diffusion system arising in a prey-predator population model. The main purpose is to discuss the stability analysis for coexistence steady-state solutions obtained by Kuto and Yamada (J. Differential Equations, to appear). We will give some criteria on the stability of these coexistence steady states. Furthermore, we show that the Hopf bifurcation phenomenon occurs on the steady-state solution branch under some conditions.     

Effect of a protection zone in the diffusive Leslie predator-prey model

In this paper, we consider the diffusive Leslie predator-prey model with large intrinsic predator growth rate, and investigate the change of behavior of the model when a simple protection zone Ω₀ for the prey is introduced. As in earlier work [Y. Du, J. Shi, A diffusive predator-prey model with a protection zone, J. Differential Equations 229 (2006) 63-91; Y. Du, X. Liang, A diffusive competition model with a protection zone, J. Differential Equations 244 (2008) 61-86] we show the existence of a critical patch size of the protection zone, determined by the first Dirichlet eigenvalue of the Laplacian over Ω₀ and the intrinsic growth rate of the prey, so that there is fundamental change of the dynamical behavior of the model only when Ω₀ is above the critical patch size. However, our research here reveals significant difference of the model's behavior from the predator-prey model studied in [Y. Du, J. Shi, A diffusive predator-prey model with a protection zone, J. Differential Equations 229 (2006) 63-91] with the same kind of protection zone. We show that the asymptotic profile of the population distribution of the Leslie model is governed by a standard boundary blow-up problem, and classical or degenerate logistic equations.     

Pullback permanence in a non-autonomous competitive Lotka-Volterra model

The goal of this work is to study in some detail the asymptotic behaviour of a non-autonomous Lotka-Volterra model, both in the conventional sense (as t→∞) and in the “pullback” sense (starting a fixed initial condition further and further back in time). The non-autonomous terms in our model are chosen such that one species will eventually die out, ruling out any conventional type of permanence. In contrast, we introduce the notion of “pullback permanence” and show that this property is enjoyed by our model. This is not just a mathematical artifice, but rather shows that if we come across an ecology that has been evolving for a very long time we still expect that both species are represented (and their numbers are bounded below), even if the final fate of one of them is less happy. The main tools in the paper are the theory of attractors for non-autonomous differential equations, the sub-supersolution method and the spectral theory for linear elliptic equations.     

Note on “Bifurcation of a predator-prey model with undercrowding effect”

Inexistence and existence of limit cycles in a predator-prey model with undercrowding effect are studied. The number of limit cycles obtained by Zheng Weiwei in 2000 is corrected.     

Advection-mediated competition in general environments

We consider a reaction–diffusion–advection system of two competing species with one of the species dispersing by random diffusion as well as a biased movement upward along resource gradient, while the other species by random diffusion only. It has been shown that, under some non-degeneracy conditions on the environment function, the two species always coexist when the advection is strong. In this paper, we show that for general smooth environment function, in contrast to what is known, there can be competitive exclusion when the advection is strong, and, we give a sharp criterion for coexistence that includes all previously considered cases. Moreover, when the domain is one-dimensional, we derive in the strong advection limit a system of two equations defined on different domains. Uniqueness of steady states of this non-standard system is obtained when one of the diffusion rates is large.