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

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.

Stability of a Three-Species Cooperative System with Time Delays and Stochastic Perturbations

“…If (H 1 ) holds, then the zero solution of system (12) is stable in probability; that is, system (4) around the equilibrium state of (1) is stable in probability. Proof.…”

“…Proof. For system (12), define V 1 , V 2 , and V 3 as before and assume that there exists a number δ > 0 such that sup t≥τ |x i (s)| < δ, i � 1, 2, 3. By the Itô formula, we calculate LV 1 , LV 2 , and LV 3 along system (12), respectively, then x 2 3 (s)ds,…”

“…It is essential to take the influence of time delays into account in mathematical modelling [5][6][7][8][9]. Furthermore, in view of the complexity of natural world, single-species or two-species ecological models often cannot describe some natural phenomena accurately and many vital behaviours can only be exhibited by systems with three or more species [10][11][12][13]. Considering the effect of time delays to threespecies cooperative system, we propose the following deterministic model:…”

Spatiotemporal Patterns Induced by Hopf Bifurcations in a Homogeneous Diffusive Predator-Prey System

“…In this paper, we are mainly concerned with the spatiotemporal oscillatory patterns of a particular reactiondiffusion predator-prey system arising from ecology. In the model of our concern, the prey is assumed to exhibit herd behavior, so that the predator interacts with the prey along the outer corridor of the herd of prey, and the interaction terms are supposed to use the square root of the prey population rather than simply the prey population, where the use of the square root properly represents the assumption that the interactions occur along the boundary of the population (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]). The reaction-diffusion system takes in the following form:…”