An Impulsive Three-Species Model with Square Root Functional Response and Mutual Interference of Predator

Zhen Wang,Yuanfu Shao,Xianjia Fang,Xiangmin Ma

doi:10.1155/2016/3897234

Zhen Wang, Yuanfu Shao + Show 2 more

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https://doi.org/10.1155/2016/3897234

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Abstract

An impulsive two-prey and one-predator model with square root functional responses, mutual interference, and integrated pest management is constructed. By using techniques of impulsive perturbations, comparison theorem, and Floquet theory, the existence and global asymptotic stability of prey-eradication periodic solution are investigated. We use some methods and sufficient conditions to prove the permanence of the system which involve multiple Lyapunov functions and differential comparison theorem. Numerical simulations are given to portray the complex behaviors of this system. Finally, we analyze the biological meanings of these results and give some suggestions for feasible control strategies.

Highlights

Introduction and Model FormulationIn real world, the study on models of three or more species is very popular, such as food-chain and food webs systems, which have extremely rich dynamics [1, 2]
Liu et al [5] gave the following Holling type II functional response which describes the relations of one prey and one predator: x1󸀠
The main purpose of this paper is to investigate the dynamical behaviors of an impulsive one-predator twoprey model with mutual interference, square root functional response, and integrated control methods

Summary

Introduction and Model Formulation

The study on models of three or more species is very popular, such as food-chain and food webs systems, which have extremely rich dynamics [1, 2]. Ajraldi et al [8] pointed out that, by using the terms of the square root of the prey population, the response functions of prey that exhibited herd behavior are more properly modeled. In this respect, Braza [9] gave the following predator-prey model: α√x1 (t)x2 (t) , 1 + w√x1 (t). We give the following preypredator system with square root functional response and mutual interference of the predator: α√x1 (t)x2 1 + w√x1. The main purpose of this paper is to investigate the dynamical behaviors of an impulsive one-predator twoprey model with mutual interference, square root functional response, and integrated control methods.

Preliminaries

Boundedness

Stability of Prey-Eradication Periodic Solution

Single Chemical Control

Examples and Simulations

Conclusion