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Consequences of weak Allee effect on prey in the May–Holling–Tanner predator–prey model
Author(s) -
GonzálezOlivares Edward,
GallegoBerrío Lina M.,
GonzálezYañez Betsabé,
RojasPalma Alejandro
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.3404
Subject(s) - allee effect , mathematics , limit cycle , homoclinic orbit , phase plane , attractor , equilibrium point , hopf bifurcation , limit (mathematics) , logistic function , mathematical analysis , statistical physics , differential equation , bifurcation , nonlinear system , statistics , physics , population , demography , quantum mechanics , sociology
In this work, a modified Holling–Tanner predator–prey model is analyzed, considering important aspects describing the interaction such as the following: the predator growth function is of a logistic type, and a weak Allee effect acting in the prey growth function and the functional response is of hyperbolic type. By making a change of variables and a time rescaling, we obtain a polynomial differential equations system topologically equivalent to the original one, in which the nonhyperbolic equilibrium point (0,0) is an attractor for all parameter values. An important consequence of this property is the existence of a separatrix curve dividing the behavior of trajectories in the phase plane, and the system exhibits the bistability phenomenon, because the trajectories can have different ω ‐limit sets, as an example, the origin (0,0) or a stable limit cycle surrounding an unstable positive equilibrium point. We show that, under certain parameter conditions, a positive equilibrium may undergo saddle‐node, Hopf, and Bogdanov–Takens bifurcations; the existence of a homoclinic curve on the phase plane is also proved, which breaks in an unstable limit cycle. Some simulations to reinforce our results are also shown. Copyright © 2015 John Wiley & Sons, Ltd.