Existence and asymptotic behavior of solutions for a mathematical ecology model with herd behavior

Abstract

In this work, we consider a prey-predator model with herd behavior under Neumann boundary conditions. For the system without diffusion, we establish a sufficient condition to guarantee the local asymptotic stability of all nontrivial equilibria and prove the existence of limit cycle of our proposed model. For the system with diffusion, we consider the long time behavior of the model including global attractor and local stability, and the Hopf and steady-state bifurcation analysis from the unique homogeneous positive steady state are carried out in detail. Furthermore, some numerical simulations to illustrate the theoretical analysis are performed to expand our theoretical results. KEYWORDSasymptotic behavior of solutions, bifurcation theory, prey-predator model, reaction-diffusion MSC CLASSIFICATION 35K57; 35B40; 34K18Math Meth Appl Sci. 2020;43:5629-5644.wileyonlinelibrary.com/journal/mma

Steady-state bifurcation and Turing instability in diffusion prey-predator models with Allee effect in predator population

“…On the other hand, the spatial diffusion maybe more appropriate for animals with an unstructured lifecycle and the movement must be considered together with the population multiplication [10]. The importance of spatial models has been recognized by biologists for a long time and the study on the dynamics of the spatial models has been one of the dominant themes in the field of biological mathematics [14,[25][26][27][28][29]. In this paper, incorporating the spatial diffusion and Allee effect (1.3) into (1.1), we shall consider the nonlinear reaction-diffusion prey-predator population dynamics, governed by the following system…”

Nonlinear dynamics in a discrete-time predator-prey system

“…In recent years, great attention has been paid to the predator-prey system, which has important application in biological model. Many scholars have carried out the study of different predator-prey models (see [1][2][3][4][5][6]). Among the variety of models, the predator-prey models with different Holling type functional response are refined so as to better reflect the specific characteristics of the different populations or economical needs.…”

Nonlinear dynamics in a discrete-time predator-prey system

“…In recent years, great attention has been paid to the predator-prey system, which has important application in biological model. Many scholars have carried out the study of different predator-prey models (see [1][2][3][4][5][6]). Among the variety of models, the predator-prey models with different Holling type functional response are refined so as to better reflect the specific characteristics of the different populations or economical needs.…”