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A simple Gause‐type predator–prey model considering social predation
Author(s) -
GonzálezOlivares Eduardo,
ValenzuelaFigueroa Sebastián,
RojasPalma Alejandro
Publication year - 2018
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.5372
Subject(s) - predation , equilibrium point , mathematics , population , ecology , predator , theoretical ecology , competition (biology) , mathematical economics , statistical physics , differential equation , mathematical analysis , biology , physics , demography , sociology
The consumer–resource relationships are among the most fundamental of all ecological relationships and have been the focus of ecology since its beginnings. Usually are described by nonlinear differential equation systems, putting the emphasis in the effect of antipredator behavior (APB) by the prey; nevertheless, a minor quantity of articles has considered the social behavior of predators. In this work, two predator–prey models derived from the Volterra model are analyzed, in which the equation of predators is modified considering cooperation or collaboration among predators. It is well known that competition among predators produces a stabilizing effect on system describing the model, since there exists a wide set in the parameter space where the system has a unique equilibrium point in the phase plane, which is globally asymptotically stable. Meanwhile, the cooperation can originate more complex and unusual dynamics. As we will show, it is possible to prove that for certain subset of parameter values the predator population sizes tend to infinite when the prey population goes to extinct. This apparently contradicts the idea of a realistic model, when it is implicitly assumed that the predators are specialist, ie, the prey is its unique source of food. However, this could be a desirable effect when the prey constitutes a plague. To reinforce the analytical result, numerical simulations are presented.