A delayed epidemic model with stage-structure and pulses for pest management strategy
ABSTRACT From a biological pest management standpoint, epidemic diseases models have become important tools in control of pest populations. This paper deals with an impulsive delay epidemic disease model with stage-structure and a general form of the incidence rate concerning pest control strategy, in which the pest population is subdivided into three subgroups: pest eggs, susceptible pests, infectious pests that do not attack crops. Using the discrete dynamical system determined by the stroboscopic map, we obtain the exact periodic susceptible pest-eradication solution of the system and observe that the susceptible pest-eradication periodic solution is globally attractive, provided that the amount of infective pests released periodically is larger than some critical value. When the amount of infective pests released is less than another critical value, the system is shown to be permanent, which implies that the trivial susceptible pest-eradication solution loses its attractivity. Our results indicate that besides the release amount of infective pests, the incidence rate, time delay and impulsive period can have great effects on the dynamics of our system.
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ABSTRACT: Insect pests are common but undesirable elements in ecosystems and represent thorny problems for most developing countries. To prevent pest outbreaks, growers often resort to insect–pathogenic viruses rather than to pesticides which affect human health and the environment. The purpose of this paper is to investigate a new age-structured pest management model which describes the interaction between susceptible insect pests, infected insect pests, pathogenic viruses and defence immunity mechanisms. A feature of this model is that it accounts for the dependence of the amount of pathogenic viruses released and of the efficiency of the defence mechanisms upon the so-called age of infection. First, the asymptotic behavior of the system is established via a monotonicity argument which makes use of several integral inequalities, being shown that the infection ultimately dies out, while under certain circumstances the susceptible pests also become extinct. By means of the Michailov criterion, one then analyzes the linearized stability of the trivial equilibrium and of the semi-trivial infected pest-free equilibrium. In this regard, it is observed that the defence mechanisms and maximal length of the infective period play important roles in the dynamics of the system. Several pest controls strategies are further investigated by means of numerical simulations, which show that when the dose of pathogenic viruses released initially is larger than a certain amount the profile of the response of defence mechanisms can be modified by changing this dose. Finally, the paper is concluded with a discussion on the biological significance of the mathematical results and framework.Mathematical and Computer Modelling 07/2010; 52:37-54. · 1.42 Impact Factor
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ABSTRACT: The authors deal with the homogeneous Dirichlet problem -(|u ' (t)| p-2 u ' (t)) ' =f(t,u(t),u ' (t)),Δu ' (t i )=I i (u(t i )),i=1,2,⋯,ℓ,u(0)=u(T)=0, where p≥2, 0<t 1 <⋯<t ℓ <T, Δu ' (t i )=(|u ' | p-2 u ' )(t i + )-(|u ' | p-2 u ' )(t i - ). The existence of a nontrivial solution is obtained using variational and iterative methods.Nonlinear Analysis Real World Applications 10/2010; 11(5). · 2.20 Impact Factor
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ABSTRACT: The authors consider the boundary value problem y '' (t)=f(t,y(t)),a.e.t∈[0,1],t≠t i ,Δy| t=t i =I i (y(t i -)),i=1,⋯,mΔy| t=t i =I ¯ i (y(t i -)),i=1,⋯,my(0)-k 1 y ' (0)=∫ 0 1 h 1 (s,y(s))ds,y(1)+k 2 y ' (1)=∫ 0 1 h 2 (s,y(s))ds, where 0<t 1 <⋯<t m <1, f,h 1 ,h 2 :[0,1]×ℝ→ℝ, I i , I ¯ i :ℝ→ℝ, k 1 , k 2 ≥0. Using Banach’s fixed-point theorem, an existence and uniqueness result for the BVP is obtained. Using nonlinear alternative of Leray-Schauder type, an obtained existence result is obtained.Dynamic Systems and Applications 01/2010; 19(3). · 0.40 Impact Factor