Persistence and Turing instability in a cross-diffusive predator\u2013prey system with generalist predator

Baojun Miao

doi:10.1186/s13662-018-1676-x

Abstract

In this paper, we propose and investigate persistence and Turing instability of a cross-diffusion predator–prey system with generalist predator. First, by virtue of the comparison principle, we obtain sufficient conditions of persistence for a corresponding reaction–diffusion system without self- and cross-diffusion. Second, by using the linear stability analysis, we prove that under some conditions the unique positive equilibrium solution is locally asymptotically stable for the corresponding ODE system and the corresponding reaction–diffusion system without cross-diffusion and self-diffusion. Hence it does not belong to the classical Turing instability. Third, under some appropriate sufficient conditions, we obtain that the uniform positive equilibrium solution is linearly unstable for the cross-reaction–diffusion and partial self-diffusion system. The results indicate that cross-diffusion and partial self-diffusion play an important role in the study of Turing instability about reaction–diffusion systems with generalist predator. Fourth, we elaborate on the relations between the theoretical results and the cross-diffusion predator–prey system by relying on some examples. In the end, we conclude our findings and give a brief discussion.

Highlights

It is well known that ecosystems are characterized by the interaction of species with a wide range of spatial and temporal scales natural environment
We prove that a uniform equilibrium solution is linearly asymptotically stable under the same conditions for the predator–prey system with generalist predator without selfand cross-diffusion in Sect
By linear stability analysis we obtain that a unique positive equilibrium is locally asymptotically stable for ODE and PDE systems without self-and cross-diffusion under certain conditions

Summary

Introduction

It is well known that ecosystems are characterized by the interaction of species with a wide range of spatial and temporal scales natural environment. The aim of this paper is to construct a mathematical model to describe ecosystems with mutualist interaction and generalist predator, and further investigate Turing instability of the predator–prey ecosystem by virtue of mathematical analysis and numerical examples. The homogeneous Neumann boundary condition indicates that the model is self-contained and there is zero population flux across the boundary This system with cross-diffusion represents a model which involves interacting and migration in the same habitat among generalist predator u2, prey u1, and u3, while u3 and u1 are of symbiotic mutualist. Is directed toward the increasing population density of u3(or u1), that is, the two mutualist chase each other After adding these items, the cross-diffusion predator–prey system (1) means that in addition to the dispersive force, the diffusion depends on species pressure from other species.

Persistence for the PDES without self-diffusion and cross-diffusion

Conclusions and discussions