Premium
Dynamical behavior for a class of predator–prey system with general functional response and discontinuous harvesting policy
Author(s) -
Cai Zuowei,
Huang Lihong,
Zhang Lingling,
Hu Xiaolian
Publication year - 2015
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.3379
Subject(s) - uniqueness , mathematics , lipschitz continuity , dynamical systems theory , convergence (economics) , exponential stability , equilibrium point , lyapunov function , stability theory , dynamical system (definition) , stability (learning theory) , function (biology) , class (philosophy) , differential equation , control theory (sociology) , mathematical analysis , computer science , nonlinear system , evolutionary biology , artificial intelligence , economics , biology , physics , control (management) , quantum mechanics , machine learning , economic growth
In this paper, we introduce a class of predator–prey system with general functional response, whose harvesting policy is modeled by a discontinuous function. Based on the differential inclusions theory, topological degree theory in set‐valued analysis and generalized Lyapunov approach, we analyze the existence, uniqueness and global asymptotic stability of positive periodic solution. In particular, a series of useful criteria on existence, uniqueness and global asymptotic stability of the positive equilibrium point are established for the autonomous system corresponding to the non‐autonomous biological and mathematical model with a discontinuous right‐hand side. Moreover, some new sufficient conditions are provided to guarantee the global convergence in measure of harvesting solution and convergence in finite time of any positive solution for the autonomous discontinuous biological system. The obtained results, which improve and generalize previous works on dynamical behavior in the literature, are of interest for understanding and designing biological system with not only continuous or even Lipschitz continuous but also discontinuous harvesting function. Finally, we give three examples with numerical simulations to show the applicability and effectiveness of our main results. Copyright © 2015 John Wiley & Sons, Ltd.