scholarly journals A STAGE-STRUCTURED PREDATOR-PREY SI MODEL WITH DISEASE IN THE PREY AND IMPULSIVE EFFECTS

2013 ◽  
Vol 18 (4) ◽  
pp. 505-528 ◽  
Author(s):  
Tongqian Zhang ◽  
Xinzhu Meng ◽  
Yi Song ◽  
Tonghua Zhang
Keyword(s):  
Pest Management ◽  
Analytical Techniques ◽  
Stage Structure ◽  
Threshold Level ◽  
Pest Population ◽  
Predator Prey ◽  
Impulsive Delay Differential Equations ◽  
Si Model ◽  
Impulsive Delay ◽  
Delay Differential

This paper aims to develop a high-dimensional SI model with stage structure for both the prey (pest) and the predator, and then to investigate the dynamics of it. The model can be used for the study of Integrated Pest Management (IPM) which is a combination of constant pulse releasing of animal enemies and diseased pests at two different fixed moments. Firstly, we use analytical techniques for impulsive delay differential equations to obtain the conditions for global attractivity of the ‘pest-free’ periodic solution and permanence of the population model. It shows that the conditions strongly depend on time delay, impulsive release of animal enemies and infective pests. Secondly, we present a pest management strategy in which the pest population is kept under the economic threshold level (ETL) when the pest population is permanent. Finally, numerical analysis is presented to illustrate our main conclusion.

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

2010 ◽  
Vol 2010 ◽  
pp. 1-17 ◽  
Author(s):  
Jianjun Jiao
Keyword(s):  
Discrete Dynamical System ◽  
Stage Structure ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Discrete Time Delay ◽  
Impulsive Diffusion ◽  
Impulsive Delay ◽  
Globally Attractive ◽  
Impulsive Delay Differential Equation ◽  
Delay Differential

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.

scholarly journals Harvesting Control for a Stage-Structured Predator-Prey Model with Ivlev's Functional Response and Impulsive Stocking on Prey

2007 ◽  
Vol 2007 ◽  
pp. 1-15
Author(s):  
Kaiyuan Liu ◽  
Lansun Chen
Keyword(s):  
Functional Response ◽  
Global Attractivity ◽  
Sufficient Conditions ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
All Solutions ◽  
Impulsive Delay Differential Equations ◽  
Impulsive Delay ◽  
Delay Differential

We investigate a delayed stage-structured Ivlev's functional response predator-prey model with impulsive stocking on prey and continuous harvesting on predator. Sufficient conditions of the global attractivity of predator-extinction periodic solution and the permanence of the system are obtained. These results show that the behavior of impulsive stocking on prey plays an important role for the permanence of the system. We also prove that all solutions of the system are uniformly ultimately bounded. Our results provide reliable tactical basis for the biological resource management and enrich the theory of impulsive delay differential equations.

scholarly journals Dynamics of a Stage Structure Pest Control Model with Impulsive Effects at Different Fixed Time

2009 ◽  
Vol 2009 ◽  
pp. 1-15 ◽  
Author(s):  
Bing Liu ◽  
Ying Duan ◽  
Yinghui Gao
Keyword(s):  
Time Delay ◽  
Pest Control ◽  
Natural Enemies ◽  
Stage Structure ◽  
Threshold Level ◽  
Fixed Time ◽  
Control Model ◽  
Pest Population ◽  
Maturation Time ◽  
Maturation Time Delay

Many existing pest control models, which control pests by releasing natural enemies, neglect the effect that natural enemies may get killed. From this point of view, we formulate a pest control model with stage structure for the pest with constant maturation time delay (through-stage time delay) and periodic releasing natural enemies and natural enemies killed at different fixed time and perform a systematic mathematical and ecological study. By using the comparison theorem and analysis method, we obtain the conditions for the global attractivity of the pest-eradication periodic solution and permanence of the system. We also present a pest management strategy in which the pest population is kept under the economic threshold level (ETL) when the pest population is uniformly permanent. We show that maturation time delay, impulsive releasing, and killing natural enemies can bring great effects on the dynamics of the system. Numerical simulations confirm our theoretical results.

scholarly journals Exponential Stability of Impulsive Delay Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
G. L. Zhang ◽  
M. H. Song ◽  
M. Z. Liu
Keyword(s):  
Differential Equations ◽  
Exponential Stability ◽  
Delay Differential Equations ◽  
Sufficient Conditions ◽  
Euler Method ◽  
Exponentially Stable ◽  
Impulsive Delay Differential Equations ◽  
Numerical Process ◽  
Impulsive Delay ◽  
Delay Differential

The main objective of this paper is to further investigate the exponential stability of a class of impulsive delay differential equations. Several new criteria for the exponential stability are analytically established based on Razumikhin techniques. Some sufficient conditions, under which a class of linear impulsive delay differential equations are exponentially stable, are also given. An Euler method is applied to this kind of equations and it is shown that the exponential stability is preserved by the numerical process.

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