Abstract
In this paper, Turing patterns and steady state bifurcation of a diffusive Beddington–DeAngelis-type predator–prey model with density-dependent death rate for the predator are considered. We first investigate the stability and Turing instability of the unique positive equilibrium point for the model. Then the existence/nonexistence, the local/global structure of nonconstant positive steady state solutions, and the direction of the local bifurcation are established. Our results demonstrate that a Turing instability is induced by the density-dependent death rate under appropriate conditions, and both the general stationary pattern and Turing pattern can be observed as a result of diffusion. Moreover, some specific examples are presented to illustrate our analytical results.
Highlights
Understanding the dynamical relationship between predator and prey is a central research subject in ecology, and one significant component of the predator–prey relationship is the predator’s rate of feeding upon prey, i.e., the so-called functional response
Prey dependence means that the functional response is only a function of the prey’s density, while predator dependence means that the functional response is a function of both the prey’s and the predator’s densities
(2019) 2019:102 argued that in many situations, especially when predators have to search for food, the functional response in a predator–prey model should be predator-dependent
Summary
Understanding the dynamical relationship between predator and prey is a central research subject in ecology, and one significant component of the predator–prey relationship is the predator’s rate of feeding upon prey, i.e., the so-called functional response. 4, the local and global structure of nonconstant positive steady state are established, and the direction of the local bifurcation is given. Such that model has no nonconstant positive steady state provided that d1 ≥ d1∗, d2 ≥ d2∗.
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