Dynamics of a Leslie-Gower predator-prey system with cross-diffusion

A Leslie–Gower predator–prey system with cross-diffusion subject to Neumann boundary conditions is considered. The global existence and boundedness of solutions are shown. Some sufficient conditions ensuring the existence of nonconstant solutions are obtained by means of the Leray–Schauder degree theory. The local and global stability of the positive constant steady-state solution are investigated via eigenvalue analysis and Lyapunov procedure. Based on center manifold reduction and normal form theory, Hopf bifurcation direction and the stability of bifurcating timeperiodic solutions are investigated and a normal form of Bogdanov–Takens bifurcation is determined as well

Dynamics of a Leslie-Gower predator-prey system with cross-diffusion

A Leslie–Gower predator–prey system with cross-diffusion subject to Neumann boundary conditions is considered. The global existence and boundedness of solutions are shown. Some sufficient conditions ensuring the existence of nonconstant solutions are obtained by means of the Leray–Schauder degree theory. The local and global stability of the positive constant steady-state solution are investigated via eigenvalue analysis and Lyapunov procedure. Based on center manifold reduction and normal form theory, Hopf bifurcation direction and the stability of bifurcating time-periodic solutions are investigated and a normal form of Bogdanov-Takens bifurcation is determined as well

Coexistence of heterogenous predator-prey systems with density-dependent dispersal

This paper is concerned with existence, non-existence and uniqueness of positive (coexistence) steady states to a predator-prey system with density-dependent dispersal. To overcome the analytical obstacle caused by the cross-diffusion structure embedded in the density-dependent dispersal, we use a variable transformation to convert the problem into an elliptic system without cross-diffusion structure. The transformed system and pre-transformed system are equivalent in terms of the existence or non-existence of positive solutions. Then we employ the index theory alongside the method of the principle eigenvalue to give a nearly complete classification for the existence and non-existence of positive solutions. Furthermore we show the uniqueness of positive solutions and characterize the asymptotic profile of solutions for small or large diffusion rates of species. Our results pinpoint the positive role of density-dependent dispersal on the population dynamics for the first time by showing that the density-dependent dispersal is a beneficial strategy promoting the coexistence of species in the predator-prey system by increasing the chance of predator's survival.Comment: 28 pages, 2 figure

Enhancing Population Persistence by a Protection Zone in a Reaction-Diffusion Model with Strong Allee Effect

Protecting endangered species has been an important issue in ecology. We derive a reaction-diffusion model for a population in a one-dimensional bounded habitat, where the population is subjected to a strong Allee effect in its natural domain but obeys a logistic growth in a protection zone. We establish the conditions for population persistence and extinction via the principal eigenvalue of an associated eigenvalue problem and investigate the dependence of this principal eigenvalue on the location (i.e., the starting point and the length) of the protection zone. The results are used to design the optimal protection zone under different boundary conditions, that is, to suggest the starting point and length of the protection zone with respect to different population growth rate in the protection zone, in order for the population to persist in a long term

The Dynamics of a Diffusive Nutrient-Algae Model Based upon the Sanyang Wetland

The stability and spatiotemporal dynamics of a diffusive nutrient-algae model are investigated mathematically and numerically. Mathematical theoretical studies have considered the positivity and boundedness of the solution and the existence, local stability, and global stability of equilibria. Turing instability has also been studied. Furthermore, a series of numerical simulations was performed and a complex Turing pattern found. These results indicate that the nutrient input rate has an important influence on the density and spatial distribution of algae populations. This may help us to obtain a better understanding of the interactions of nutrient and algae and to investigate plankton dynamics in aquatic ecosystems