Abstract
In recent years, the impulsive population systems have been studied by many researchers. However, seasonal effects on prey are rarely discussed. Thus, in this paper, the dynamics of the Holling-type IV two-competitive-prey one-predator system with impulsive perturbations and seasonal effects are analyzed using the Floquet theory and comparison techniques. It is assumed that the impulsive perturbations act in a periodic fashion, the proportional impulses (the chemical controls) for all species and the constant impulse (the biological control) for the predator at different fixed time but, the same period. In addition, the intrinsic growth rates of prey population are regarded as a periodically varying function of time due to seasonal variations. Sufficient conditions for the local and global stabilities of the two-prey-free periodic solution are established. It is proven that the system is permanent under some conditions. Moreover, sufficient conditions, under which one of the two preys is extinct and the remaining two species are permanent, are also found. Finally, numerical examples and conclusion are given.
Highlights
It is assumed that the impulsive perturbations act in a periodic fashion, the proportional impulses the chemical controls for all species and the constant impulse the biological control for the predator at different fixed time but, the same period
It is of great interest to study dynamical properties for impulsive perturbations in population dynamics
Impulsive prey-predator population systems have been discussed by a number of researchers 1–8 and, what is more, there are many literatures on simple multispecies systems consisting of a three-species food chain with impulsive perturbations 7, 9–18
Summary
It is of great interest to study dynamical properties for impulsive perturbations in population dynamics. Song and Li 13 studied dynamical behavior of a Holling type II two-prey one-predator system with impulsive effect concerning biological control and chemical control strategies at fixed time. It has been studied that dynamical systems with simple dynamical behavior may display complex dynamical behavior when they have periodic perturbations 20–22. For this reason, in this paper, we consider the intrinsic growth rates A of prey population as a periodically varying function of time due to seasonal variations. We develop the Holling-type IV two-competitive-prey one-predator system with seasonality by introducing a proportional periodic impulsive poisoning spraying pesticide for all species and a constant periodic releasing, or immigrating, for the predator at different fixed time as follows: x1 t a1.
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