Open AccessGlobal stability in a modified Leslie-Gower type predation model assuming mutual interference among generalist predators
Author(s) -
Eduardo GonzálezOlivares,
Alejandro RojasPalma
Publication year - 2020
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.2020392
Subject(s) - predation , equilibrium point , mathematics , lyapunov function , type (biology) , jacobian matrix and determinant , generalist and specialist species , function (biology) , differentiable function , competition (biology) , control theory (sociology) , mathematical analysis , ecology , computer science , physics , biology , control (management) , differential equation , nonlinear system , quantum mechanics , evolutionary biology , artificial intelligence , habitat
In the ecological literature, mutual interference (self-interference) or competition among predators (CAP) to effect the harvesting of their prey has been modeled through different mathematical formulations. In this work, the dynamical properties of a Leslie-Gower type predation model is analyzed, incorporating one of these forms, which is described by the function $g\left(y\right) =y^{\beta }$, with $0<\beta <1$. This function $g$ is not differentiable for $y=0$, and neither the Jacobian matrix of the system is not defined in the equilibrium points over the horizontal axis ($x-axis$). To determine the nature of these points, we had to use a non-standard methodology. Previously, we have shown the fundamental properties of the Leslie-Gower type model with generalist predators, to carry out an adequate comparative analysis with the model where the competition among predators (CAP) is incorporated. The main obtained outcomes in both systems are: (i) The unique positive equilibrium point, when exists, is globally asymptotically stable (g.a.s), which is proven using a suitable Lyapunov function. (ii) There not exist periodic orbits, which was proved constructing an adequate Dulac function.