On a limiting system of the triangular SKT competition model with large cross-diffusion and random-diffusion

This paper mainly focus on the limiting system which arises from the study of SKT competition model when both the cross-diffusion rate and the random-diffusion rate tend to infinity. For multi-dimensional domains, the existence of positive steady states bifurcating from a double eigenvalue can be proved by applying the bifurcation argument with some special transformation when the birth rate of one species is near some critical value. Further by virtue of the spectral perturbation argument based on the Lyapunov–Schmidt decomposition method, we prove the spectral instability of such nontrivial positive steady states for the limiting system.     

Steady state in a cross-diffusion predator–prey model with the Beddington–DeAngelis functional response

The purpose of this paper is to study the existence of steady state in a linear cross-diffusion predator–prey model with Beddington–DeAngelis functional response. The proofs mainly rely on Fixed point index theory and analytical techniques.     

Global solutions to a class of multi-species reaction-diffusion systems with cross-diffusions arising in population dynamics

In this paper, an n$n$-species strongly coupled cooperating diffusive system is considered in a bounded smooth domain, subject to homogeneous Neumann boundary conditions. Employing the method of energy estimates, we obtain some conditions on the diffusion matrix and inter-specific cooperatives to ensure the global existence and uniform boundedness of a nonnegative solution. The globally asymptotical stability of the constant positive steady state is also discussed. As a consequence, all the results hold true for multi-species Lotka–Volterra type competition model and prey–predator model.     

Stability and cross-diffusion-driven instability in a diffusive predator–prey system with hunting cooperation functional response

This paper presents a qualitative study of a diffusive predator–prey system with the hunting cooperation functional response. For the system without diffusion, the existence, stability and Hopf bifurcation of the positive equilibrium are explicitly determined. It is shown that the hunting cooperation affects not only the existence of the positive equilibrium but also the stability. For the diffusive system, the stability and cross-diffusion driven Turing instability are investigated according to the relationship of the self-diffusion and the cross-diffusion coefficients. Stability and cross-diffusion instability regions are theoretically determined in the plane of the cross-diffusion coefficients. The technique of multiple time scale is employed to deduce the amplitude equation of Turing bifurcation and then pattern dynamics driven by the cross-diffusion is also investigated by the corresponding amplitude equation.     

Non-constant positive steady states of a prey-predator system with cross-diffusions

In this paper, we study a strongly coupled elliptic system arising from a Lotka-Volterra prey-predator system, where cross-diffusions are included in such a way that the prey runs away from the predator and the predator moves away from a large group of preys. We establish the existence and non-existence of its non-constant positive solutions. Our results show that if m₁b<a<2m₁b/(1−m₁m₂) when 0<m₁m₂<1 or a>m₁b when m₁m₂?1, , d₂>0, d₃?0 and , then there exists (d₁,d₂,d₃,d₄) such that the stationary problem admits non-constant positive solutions. Otherwise, the stationary problem has no non-constant positive solution. In particular, the results indicate that its non-constant positive solutions are mainly created by the cross-diffusion d₄.     

Strategies for the existence of spatial patterns in predator–prey communities generated by cross-diffusion

Fear of predators is an important drive for predator–prey interactions, which increases survival probability but cost the overall population size of the prey. In this paper, we have extended our previous work spatiotemporal dynamics of predator–prey interactions with fear effect by introducing the cross-diffusion. The conditions for cross-diffusion-driven instability are obtained using the linear stability analysis. The standard multiple scale analysis is used to derive the amplitude equations for the excited modes near Turing bifurcation threshold by taking the cross-diffusion coefficient as a bifurcation parameter. From the stability analysis of amplitude equations, the conditions for the emergence of various ecologically realistic Turing patterns such as spot, stripe, and mixture of spots and stripes are identified. Analytical results are verified with the help of numerical simulations. Turing bifurcation diagrams are plotted taking diffusion coefficients as control parameters. The effect of the cross-diffusion coefficients on the homogeneous steady state and pattern structures of the self-diffusive model is illustrated using the simulation techniques. It is also observed that the level of fear has stabilizing effect on the cross-diffusion induced instability and spot patterns change to stripe, then a mixture of spots and stripes and finally to the labyrinthine type of patterns with an increase in the level of fear.     

Deterministic particle method approximation of a contact inhibition cross-diffusion problem

We use a deterministic particle method to produce numerical approximations to the solutions of an evolution cross-diffusion problem for two populations.According to the values of the diffusion parameters related to the intra- and inter-population repulsion intensities, the system may be classified in terms of an associated matrix. When the matrix is definite positive, the problem is well posed and the finite element approximation produces convergent approximations to the exact solution.A particularly important case arises when the matrix is only positive semi-definite and the initial data are segregated: the contact inhibition problem. In this case, the solutions may be discontinuous and hence the (conforming) finite element approximation may exhibit instabilities in the neighborhood of the discontinuity.In this article we deduce the particle method approximation to the general cross-diffusion problem and apply it to the contact inhibition problem. We then provide some numerical experiments comparing the results produced by the finite element and the particle method discretizations.     

Modeling spatial adaptation of populations by a time non-local convection cross-diffusion evolution problem

In [19], Sighesada et al. presented a system of partial differential equations for modeling spatial segregation of interacting species. Apart from competitive Lotka-Volterra (reaction) and population pressure (cross-diffusion) terms, a convective term modeling the populations attraction to more favorable environmental regions is included. In this article, we introduce a modification of their convective term to take account for the notion of spatial adaptation of populations. After describing the model we briefly discuss its well-possedness and propose a numerical discretization in terms of a mass-preserving time semi-implicit finite differences scheme. Finally, we provide the results of two biologically inspired numerical experiments showing qualitative differences between the original model of [19] and the model proposed in this article.     

Global behavior of solutions in a Lotka–Volterra predator–prey model with prey-stage structure

In this paper, the global behavior of solutions is investigated for a Lotka–Volterra predator–prey system with prey-stage structure. First, we can see that the stability properties of nonnegative equilibria for the weakly coupled reaction–diffusion system are similar to that for the corresponding ODE system, that is, linear self-diffusions do not drive instability. Second, using Sobolev embedding theorems and bootstrap arguments, the existence and uniqueness of nonnegative global classical solution for the strongly coupled cross-diffusion system are proved when the space dimension is less than 10. Finally, the existence and uniform boundedness of global solutions and the stability of the positive equilibrium point for the cross-diffusion system are studied when the space dimension is one. It is found that the cross-diffusion system is dissipative if the diffusion matrix is positive definite. Furthermore, cross diffusions cannot induce pattern formation if the linear diffusion rates are sufficiently large.     

种群动力学中椭圆系统在零解处的分支正解

考虑齐次Dirichlet边界条件下具有交错扩散压力的广义Lotka-Volterra两种群竞争反应扩散稳态系统. 首先借助Lyapunov-Schmidt约化方法考虑了系统在零解处小分支正解的存在性, 然后借助标准的线性化方法研究了这些分支正解的稳定性.