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Existence and asymptotic behavior of solutions for a mathematical ecology model with herd behavior
Author(s) -
Yang Wenbin
Publication year - 2020
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.6301
Subject(s) - mathematics , attractor , neumann boundary condition , hopf bifurcation , limit cycle , allee effect , stability (learning theory) , herd behavior , steady state (chemistry) , bifurcation , work (physics) , exponential stability , boundary value problem , boundary (topology) , limit (mathematics) , mathematical analysis , nonlinear system , computer science , population , chemistry , herding , sociology , forestry , engineering , quantum mechanics , machine learning , mechanical engineering , physics , demography , geography
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.