A class of Leslie-Gower type predator model with a non-monotonic rational functional response and alternative food for the predators
DOI:
https://doi.org/10.17268/sel.mat.2019.02.07Keywords:
Predator-prey model, functional response, bifurcation, limit cycle, separatrix curve, stabilityAbstract
The interactions between two species are basic in the study of complex food chains, particularly the relation among the predators and their prey.
The analysis of simple models, described by continuous-time systems, in which some ecological phenomena are incorporated giving lights about this interesting interrelationship.
In this work, a Leslie-Gower type predator-prey model is analyzed, considering two aspects: the prey defends from the predation, forming group defense and the predators have an alternative food. So, a rational Holling type IV functional response and a modification of the predators carrying capacity are assumed, to describe each phenomenon.
We determine conditions on the parameter space for the existence of the equilibria and their nature.
Using the Lyapunov quantities method, we also establish conditions on the parameter values for which there exist a unique positive equilibrium point, which is stable and surrounded by two limit cycle, the innermost unstable and the outermost sable.
We conclude that the parameter indicating the existence of alternative food for predator has a great importance on the dynamic of model, because appear new equilibrum points and separatrix curves in the phase plane.
Some simulations are given to reinforce our findings the ecological interpretations of resultas are given.References
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