Cross-Diffusion Limit for a Reaction-Diffusion System with Fast Reversible Reaction

We consider a reaction-diffusion system which models a fast reversible reaction of type C ₁ + C ₂?C ₃ between mobile reactants inside an isolated vessel. Assuming mass action kinetics, we study the limit when the reaction speed tends to infinity in case of unequal diffusion coefficients and prove convergence of a subsequence of solutions to a weak solution of an appropriate limiting pde-system, where the limiting problem turns out to be of cross-diffusion type. The proof combines the L ²-approach to reaction-diffusion systems having at most quadratic reaction terms with a thorough exploitation of the entropy functional for mass action systems. The limiting cross-diffusion system has unique local strong solutions for sufficiently regular initial data, while uniqueness of weak solutions is in general open but is shown to be valid under restrictions on the diffusivities.     

Global existence of classical solutions for a class of diffusive ecological models with two free boundaries and cross-diffusion

In this paper we consider a class of diffusive ecological models with two free boundaries and with cross-diffusion and self-diffusion in one space dimension. The systems under consideration are strongly coupled, and the position of each free boundary is determined by the Stefan condition. We first show local existence of the solutions for the ecological models under some assumptions, and then prove the global existence of the solutions under extra assumptions. Our approach to the problem is by suitable changes, fixed point theorems and various estimates. Applications of these results are given to a two-species diffusive predator–prey model and a two-species diffusive competition model.     

A free boundary problem describing S–K–T competition ecological model with cross-diffusion

In this paper we investigate a free boundary problem describing S–K–T competition ecological model with two competing species and with cross-diffusion and self-diffusion in one space dimension, where one species is made up of two groups separated by a free boundary, and the other has a single group. The system under consideration is strongly coupled and the coefficients of the equations are allowed to be discontinuous. We first show the global existence and uniqueness of the solutions for the corresponding diffraction problem by approximation method, Galerkin method and Schauder fixed point theorem, and then prove the local existence of the solutions for the free boundary problem by Schauder fixed point theorem.     

Turing instability and coexistence in an extended Klausmeier model with nonlocal grazing

In this paper, we study the coexistence of an extended Klausmeier model with cross-diffusion and nonlocal sustained grazing. First, we analyze a saddle–node bifurcation of spatially homogeneous system. Second, we focus on the reaction–diffusion system with nonlocal sustained grazing. Our main result is that nonlocal terms promote linear stability, and the system may produce pattern under the influences of self-diffusion and cross-diffusion. Moreover, both the grazing parameter and rainfall rate can induce transitions among bare soil state, vegetation pattern state and homogeneous vegetation state. Finally, we address the nonlocal reaction–diffusion system as a bifurcation problem, and analyze the existence and stability of bifurcation solutions. Furthermore, numerical simulations have been illustrated to verify our theoretical findings.     

Pattern formation in a predator–prey model with spatial effect

In this paper, spatial dynamics in the Beddington–DeAngelis predator–prey model with self-diffusion and cross-diffusion is investigated. We analyze the linear stability and obtain the condition of Turing instability of this model. Moreover, we deduce the amplitude equations and determine the stability of different patterns. Numerical simulations show that this system exhibits complex dynamical behaviors. In the Turing space, we find three types of typical patterns. One is the coexistence of hexagon patterns and stripe patterns. The other two are hexagon patterns of different types. The obtained results well enrich the finding in predator–prey models with Beddington–DeAngelis functional response.     

Uniform boundedness and convergence of solutions to the systems with a single nonzero cross-diffusion

Uniform boundedness and convergence of global solutions are proved for quasilinear parabolic systems with a single nonzero cross-diffusion in population dynamics. Gagliardo-Nirenberg type inequalities are used in the estimates of solutions in order to establish W¹₂-bounds uniform in time. By using the uniform bound, convergence of solutions are established for systems with large diffusion coefficients in the weak competition case.     

Global solutions for a general strongly coupled prey–predator model

This work investigates global solutions for a general strongly coupled prey–predator model that involves (self-)diffusion and cross-diffusion, where the cross-diffusion is of the form v/(1+u^ℓ) with ℓ≥1. Very few mathematical results are known for such models, especially in higher spatial dimensions.