Open AccessBifurcations and multistability on the May-Holling-Tanner predation model considering alternative food for the predators
Author(s) -
Eduardo GonzálezOlivares,
Claudio Arancibia–Ibarra,
Alejandro RojasPalma,
Betsabé GonzálezYañez
Publication year - 2019
Publication title -
mathematical biosciences and engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.451
H-Index - 45
eISSN - 1551-0018
pISSN - 1547-1063
DOI - 10.3934/mbe.2019213
Subject(s) - homoclinic orbit , limit cycle , multistability , mathematics , equilibrium point , phase plane , saddle point , saddle , hopf bifurcation , predation , stability theory , limit (mathematics) , mathematical analysis , bifurcation , statistical physics , mathematical optimization , physics , differential equation , geometry , ecology , quantum mechanics , nonlinear system , biology
In this paper a modified May-Holling-Tanner predator-prey model is analyzed, considering an alternative food for predators, when the quantity of prey i scarce. Our obtained results not only extend but also complement existing ones for this model, achieved in previous articles. The model presents rich dynamics for different sets of the parameter values; it is possible to prove the existence of: (i) a separatrix curve on the phase plane dividing the behavior of the trajectories, which can have different ω-limit; this implies that solutions nearest to that separatrix are highly sensitive to initial conditions, (ii) a homoclinic curve generated by the stable and unstable manifolds of a saddle point in the interior of the first quadrant, whose break generates a non-infinitesimal limit cycle, (iii) different kinds of bifurcations, such as: saddle-node, Hopf, Bogdanov-Takens, homoclinic and multiple Hopf bifurcations. (iv) up to two limit cycles surrounding a positive equilibrium point, which is locally asymptotically stable. Thus, the phenomenon of tri-stability can exist, since simultaneously can coexist a stable limit cycle, joint with two locally asymptotically stable equilibrium points, one of them over the y-axis and the other positive singularity. Numerical simulations supporting the main mathematical outcomes are shown and some of their ecological meanings are discussed.