Premium
Limit cycles in a Gause‐type predator–prey model with sigmoid functional response and weak Allee effect on prey
Author(s) -
GonzálezOlivares Eduardo,
RojasPalma Alejandro
Publication year - 2012
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.2509
Subject(s) - allee effect , mathematics , limit cycle , limit (mathematics) , functional response , predation , type (biology) , equilibrium point , sigmoid function , bistability , mathematical economics , saddle point , saddle , statistical physics , predator , control theory (sociology) , mathematical analysis , mathematical optimization , ecology , economics , computer science , physics , population , control (management) , differential equation , biology , geometry , management , quantum mechanics , machine learning , artificial neural network , demography , sociology
The goal of this work is to examine the global behavior of a Gause‐type predator–prey model in which two aspects have been taken into account: (i) the functional response is Holling type III; and (ii) the prey growth is affected by a weak Allee effect. Here, it is proved that the origin of the system is a saddle point and the existence of two limit cycles surround a stable positive equilibrium point: the innermost unstable and the outermost stable, just like with the strong Allee effect. Then, for determined parameter constraints, the trajectories can have different ω − limit sets. The coexistence of a stable limit cycle and a stable positive equilibrium point is an important fact for ecologists to be aware of the kind of bistability shown here. So, these models are undoubtedly rather sensitive to disturbances and require careful management in applied contexts of conservation and fisheries. Copyright © 2012 John Wiley & Sons, Ltd.