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Existence and asymptotic behavior of solutions for a mathematical ecology model with herd behavior. (English) Zbl 1446.35234
Summary: 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.
MSC:
35Q92 PDEs in connection with biology, chemistry and other natural sciences
35K57 Reaction-diffusion equations
35B40 Asymptotic behavior of solutions to PDEs
34K18 Bifurcation theory of functional-differential equations
92D25 Population dynamics (general)
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