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 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 Dynamical Behaviors of a Stage-Structured Predator-Prey Model with Harvesting Effort and Impulsive Diffusion

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Lingzhi Huang ◽  
Zhichun Yang
Keyword(s):  
Dynamical System ◽  
Periodic Solution ◽  
Comparison Theorem ◽  
Discrete Dynamical System ◽  
Sufficient Conditions ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Dynamical Behaviors ◽  
Prey Model ◽  
Impulsive Diffusion

We consider a delayed predator-prey model with harvesting effort and impulsive diffusion between two patches. By the impulsive comparison theorem and the discrete dynamical system determined by the stroboscopic map, we obtain some sufficient conditions on the existence and global attractiveness of predator-eradicated periodic solution for the system. Furthermore, the permanence of the system is derived. The obtained results will modify and improve the ones in some existing publications and give the estimate for the ultimately low and upper boundedness of the systems.

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

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Bing Liu ◽  
Ling Xu ◽  
Baolin Kang
Keyword(s):  
Pest Control ◽  
Natural Enemies ◽  
Natural Enemy ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Control Model ◽  
Polluted Environment ◽  
Impulsive Delay ◽  
Impulsive Delay Differential Equation ◽  
Delay Differential

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.

scholarly journals Analysis of a Predator-Prey Model with Switching and Stage-Structure for Predator

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
T. Suebcharoen
Keyword(s):  
Differential Equation ◽  
Hopf Bifurcation ◽  
Delay Differential Equation ◽  
Stage Structure ◽  
Positive Equilibrium ◽  
Bifurcation Parameter ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Delay Differential

This paper studies the behavior of a predator-prey model with switching and stage-structure for predator. Bounded positive solution, equilibria, and stabilities are determined for the system of delay differential equation. By choosing the delay as a bifurcation parameter, it is shown that the positive equilibrium can be destabilized through a Hopf bifurcation. Some numerical simulations are also given to illustrate our 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 Permanence of Periodic Predator-Prey System with Functional Responses and Stage Structure for Prey

2008 ◽  
Vol 2008 ◽  
pp. 1-15 ◽  
Author(s):  
Can-Yun Huang ◽  
Min Zhao ◽  
Hai-Feng Huo
Keyword(s):  
Functional Response ◽  
Prey Species ◽  
Necessary Conditions ◽  
Stage Structure ◽  
Functional Responses ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Sufficient And Necessary Conditions ◽  
Holling Ii Functional Response

A stage-structured three-species predator-prey model with Beddington-DeAngelis and Holling II functional response is introduced. Based on the comparison theorem, sufficient and necessary conditions which guarantee the predator and the prey species to be permanent are obtained. An example is also presented to illustrate our main results.

scholarly journals The Dynamic Complexity of a Holling Type-IV Predator-Prey System with Stage Structure and Double Delays

2011 ◽  
Vol 2011 ◽  
pp. 1-19 ◽  
Author(s):  
Yakui Xue ◽  
Xiafeng Duan
Keyword(s):  
Time Delay ◽  
Equilibrium Point ◽  
Dynamical Behavior ◽  
Stage Structure ◽  
Point Of View ◽  
Maturation Time ◽  
Dynamic Complexity ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Type Iv

We invest a predator-prey model of Holling type-IV functional response with stage structure and double delays due to maturation time for both prey and predator. The dynamical behavior of the system is investigated from the point of view of stability switches aspects. We assume that the immature and mature individuals of each species are divided by a fixed age, and the mature predator only attacks the mature prey. Based on some comparison arguments, sharp threshold conditions which are both necessary and sufficient for the global stability of the equilibrium point of predator extinction are obtained. The most important outcome of this paper is that the variation of predator stage structure can affect the existence of the interior equilibrium point and drive the predator into extinction by changing the maturation (through-stage) time delay. Our linear stability work and numerical results show that if the resource is dynamic, as in nature, there is a window in maturation time delay parameters that generate sustainable oscillatory dynamics.

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

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Alexander Domoshnitsky ◽  
Irina Volinsky
Keyword(s):  
Differential Equation ◽  
Existence And Uniqueness ◽  
Green's Functions ◽  
Green’S Functions ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Uniqueness Of Solutions ◽  
Impulsive Delay ◽  
Impulsive Delay Differential Equation ◽  
Delay Differential

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.

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