Oscillation and Periodicity of a Second Order Impulsive Delay Differential Equation with a Piecewise Constant Argument

This paper concerns with the existence of the solutions of a second order impulsive delay differential equation with a piecewise constant argument. Moreover, oscillation, nonoscillation and periodicity of the solutions are investigated.

About Positivity of Green's Functions for Nonlocal Boundary Value Problems with Impulsive Delay Equations

The impulsive delay differential equation is considered(Lx)(t)=x′(t)+∑i=1mpi(t)x(t-τi(t))=f(t), t∈[a,b], x(tj)=βjx(tj-0), j=1,…,k, a=t0<t1<t2<⋯<tk<tk+1=b, x(ζ)=0, ζ∉[a,b],with nonlocal boundary conditionlx=∫abφsx′sds+θxa=c, φ∈L∞a,b; θ, c∈R.Various results on existence and uniqueness of solutions and on positivity/negativity of the Green's functions for this equation are obtained.

Dynamics of a Stage Structured Pest Control Model in a Polluted Environment with Pulse Pollution Input

By using pollution model and impulsive delay differential equation, we formulate a pest control model with stage structure for natural enemy in a polluted environment by introducing a constant periodic pollutant input and killing pest at different fixed moments and investigate the dynamics of such a system. We assume only that the natural enemies are affected by pollution, and we choose the method to kill the pest without harming natural enemies. Sufficient conditions for global attractivity of the natural enemy-extinction periodic solution and permanence of the system are obtained. Numerical simulations are presented to confirm our theoretical results.

The Effect of Impulsive Diffusion on Dynamics of a Stage-Structured Predator-Prey System

We investigate a predator-prey model with impulsive diffusion on predator and stage structure on prey. The globally attractive condition of prey-extinction periodic solution of the system is obtained by the stroboscopic map of the discrete dynamical system. The permanent condition of the system is also obtained by the theory of impulsive delay differential equation. The results indicate that the discrete time delay has influence on the dynamical behaviors of the system. Finally, some numerical simulations are carried out to support the analytic results.

Global attractivity of positive periodic solutions for an impulsive delay periodic “food limited” population model

We will consider the following nonlinear impulsive delay differential equationN′(t)=r(t)N(t)((K(t)−N(t−mw))/(K(t)+λ(t)N(t−mw))), a.e.t>0,t≠tk,N(tk+)=(1+bk)N(tk),K=1,2,…, wheremis a positive integer,r(t),K(t),λ(t)are positive periodic functions of periodicω. In the nondelay case(m=0), we show that the above equation has a unique positive periodic solutionN*(t)which is globally asymptotically stable. In the delay case, we present sufficient conditions for the global attractivity ofN*(t). Our results imply that under the appropriate periodic impulsive perturbations, the impulsive delay equation preserves the original periodic property of the nonimpulsive delay equation. In particular, our work extends and improves some known results.

Oscillation of impulsive delay differential equations and applications to population dynamics

The main result of this paper is that the oscillation and nonoscillation properties of a nonlinear impulsive delay differential equation are equivalent respectively to the oscillation and nonoscillation of a corresponding nonlinear delay differential equation without impulse effects. An explicit necessary and sufficient condition for the oscillation of a nonlinear impulsive delay differential equation is obtained.