Open AccessCompetition among Predators and Allee Effect on Prey, Their Influence on a Gause-Type Predation Model
Author(s) -
Eduardo GonzálezOlivares,
Javier Cabrera-Villegas,
Fernando Córdova–Lepe,
Alejandro RojasPalma
Publication year - 2019
Publication title -
mathematical problems in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.262
H-Index - 62
eISSN - 1026-7077
pISSN - 1024-123X
DOI - 10.1155/2019/3967408
Subject(s) - allee effect , predation , competition (biology) , equilibrium point , population , stability (learning theory) , competition model , mathematics , limit cycle , ecology , limit (mathematics) , statistical physics , mathematical economics , biology , economics , computer science , physics , microeconomics , differential equation , mathematical analysis , demography , profit (economics) , machine learning , sociology
Interference or competition among predators (CAP) has often been ruled out in depredation models, although there are varied mathematical forms to describe and incorporate it into this interaction. In this work, we present the most known of these descriptions and one of them will be used in a modified Volterra model. Moreover, of this ecological phenomenon, a simple and strong Allee effect affecting the prey population will be considered in the relationship. An important feature of the new model is to have until two positive equilibrium points, to the difference with the Volterra model (without Allee effect); hence different and interesting dynamic situations appear in the system. Conditions for the existence and local stability of equilibria are determined. The boundedness of solutions, the existence of a limit cycle and a separatrix curve are also proven. Besides, the main properties of the model are examined from an ecological point of view. To make a comparative discussion of our results, an Appendix is added with the main properties of models, in which neither the Allee effect nor the competition among predators is considered. Some simulations are shown to endorse our results.