Abstract
In this paper, we investigate the rich dynamics of a diffusive Holling type-Ⅱ predator-prey model with density-dependent death rate for the predator under homogeneous Neumann boundary condition. The value of this study lies in two-aspects. Mathematically, we show the stability of the constant positive steady state solution, the existence and nonexistence, the local and global structure of nonconstant positive steady state solutions. And biologically, we find that Turing instability is induced by the density-dependent death rate, and both the general stationary pattern and Turing pattern can be observed as a result of diffusion.
Highlights
We study the positive steady states of the following reaction-diffusion predator-prey model with prey-dependent Holling type-II functional response and density-dependent death rate for the predator:
The other goal of this paper is to investigate the existence, nonexistence and structure of nonconstant positive steady-state solutions to problem (1), we will concentrate on the following steady state system
For simplicity, we will focus on the stability of the unique positive steady state E∗ for ordinary differential equations (ODE) model (6) and partial differential equations (PDE) model (5), respectively
Summary
We study the positive steady states of the following reaction-diffusion predator-prey model with prey-dependent Holling type-II functional response and density-dependent death rate for the predator:. The Gause type predator-prey models with predator’s density-dependant functional response exhibit very rich dynamic behavior [17, 13]. The ODE model does not take into account either the fact that population is usually not homogeneously distributed, or the fact that predators and preys naturally develop strategies for survival. Both of these considerations involve diffusion processes which can be quite intricate as different concentration levels of predators and preys cause different population movements [16, 37, 38, 11, 36, 31].
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