scholarly journals An Impulsively Controlled Three-Species Prey-Predator Model with Stage Structure and Birth Pulse for Predator

2015 ◽  
Vol 2015 ◽  
pp. 1-13 ◽  
Author(s):  
Yanyan Hu ◽  
Mei Yan ◽  
Zhongyi Xiang
Keyword(s):  
Small Amplitude ◽  
Floquet Theory ◽  
Stage Structure ◽  
Critical Value ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Birth Pulse ◽  
Impulsive Equation ◽  
Predator Model ◽  
Predator System

We investigate the dynamic behaviors of a two-prey one-predator system with stage structure and birth pulse for predator. By using the Floquet theory of linear periodic impulsive equation and small amplitude perturbation method, we show that there exists a globally asymptotically stable two-prey eradication periodic solution when the impulsive period is less than some critical value. Further, we study the permanence of the investigated model. Our results provide valuable strategy for biological economics management. Numerical analysis is also inserted to illustrate the results.

scholarly journals Dynamics of a Beddington-DeAngelis Type Predator-Prey Model with Impulsive Effect

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Sun Shulin ◽  
Guo Cuihua
Keyword(s):  
Small Amplitude ◽  
Floquet Theory ◽  
Impulsive Control ◽  
Control Strategies ◽  
Impulsive Effect ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Impulsive Equation ◽  
Predator System ◽  
Comparison Technique

In view of the logical consistence, the model of a two-prey one-predator system with Beddington-DeAngelis functional response and impulsive control strategies is formulated and studied systematically. By using the Floquet theory of impulsive equation, small amplitude perturbation method, and comparison technique, we obtain the conditions which guarantee the global asymptotic stability of the two-prey eradication periodic solution. We also proved that the system is permanent under some conditions. Numerical simulations find that the system appears the phenomenon of competition exclusion.

scholarly journals Global Dynamics Behaviors of Viral Infection Model for Pest Management

2009 ◽  
Vol 2009 ◽  
pp. 1-16 ◽  
Author(s):  
Chunjin Wei ◽  
Lansun Chen
Keyword(s):  
Small Amplitude ◽  
Critical Value ◽  
Infection Model ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Virus Particles ◽  
All Solutions ◽  
Pest Eradication ◽  
Viral Infection Model

According to biological strategy for pest control, a mathematical model with periodic releasing virus particles for insect viruses attacking pests is considered. By using Floquet's theorem, small-amplitude perturbation skills and comparison theorem, we prove that all solutions of the system are uniformly ultimately bounded and there exists a globally asymptotically stable pest-eradication periodic solution when the amount of virus particles released is larger than some critical value. When the amount of virus particles released is less than some critical value, the system is shown to be permanent, which implies that the trivial pest-eradication solution loses its stability. Further, the mathematical results are also confirmed by means of numerical simulation.

scholarly journals Impulsive Control on Seasonally Perturbed General Holling Type Two-Prey One-Predator Model

2016 ◽  
Vol 2016 ◽  
pp. 1-20
Author(s):  
Chandrima Banerjee ◽  
Pritha Das
Keyword(s):  
Floquet Theory ◽  
Impulsive Control ◽  
Impulsive System ◽  
Critical Value ◽  
Asymptotically Stable ◽  
Functional Responses ◽  
Dynamical Behaviors ◽  
Predator Model ◽  
Conditions Of Stability ◽  
Fixed Times

We investigate the dynamical behaviors of two-prey one-predator model with general Holling type functional responses. The effect of seasonal perturbation on the model has been discussed analytically as well as numerically. The periodic fluctuation is considered in prey growth rate and the predator mortality rate of the model. The impulsive effects involving biological and chemical control strategy, periodic releasing of natural enemies, and spraying pesticide at different fixed times are introduced in the model with seasonal perturbation. We derive the conditions of stability for impulsive system using Floquet theory, small amplitude perturbation skills. A local asymptotically stable prey (pest) eradicated periodic solution is obtained when the impulsive period is less than some critical value. Numerical simulations of the model with and without seasonal disturbances exhibit different dynamics. Also we simulate numerically the model involving seasonal perturbations without impulse and with impulse. Finally, concluding remarks are given.

scholarly journals Stability property of the prey free equilibrium point

2019 ◽  
Vol 17 (1) ◽  
pp. 646-652
Author(s):  
Qin Yue
Keyword(s):  
Differential Inequality ◽  
Prey Species ◽  
Food Resource ◽  
Stage Structure ◽  
Asymptotically Stable ◽  
Suitable Assumption ◽  
Globally Asymptotically Stable ◽  
Predator Species ◽  
Inequality Theory ◽  
Predator Model

Abstract We revisit a prey-predator model with stage structure for predator, which was proposed by Tapan Kumar Kar. By using the differential inequality theory and the comparison theorem of the differential equation, we show that the prey free equilibrium is globally asymptotically stable under some suitable assumption. Our study shows that although the predator species has other food resource, if the amount of the predator species is too large, it could also do irreversible harm to the prey species, and this could finally lead to the extinction of the prey species. Our result supplement and complement some known results.

COMPLEX DYNAMICS OF ONE-PREY MULTI-PREDATOR SYSTEM WITH DEFENSIVE ABILITY OF PREY AND IMPULSIVE BIOLOGICAL CONTROL ON PREDATORS

2005 ◽  
Vol 08 (04) ◽  
pp. 483-495 ◽  
Author(s):  
YONGZHEN PEI ◽  
CHANGGUO LI ◽  
LANSUN CHEN ◽  
CHUNHUA WANG
Keyword(s):  
Biological Control ◽  
Control Strategy ◽  
Complex Dynamics ◽  
Impulsive Control ◽  
Critical Value ◽  
Asymptotically Stable ◽  
Behavior Dynamics ◽  
Impulsive Control Strategy ◽  
Predator Model ◽  
Predator System

This work investigates the dynamic behaviors of one-prey multi-predator model with defensive ability of the prey by introducing impulsive biological control strategy. By using the Floquent theorem and the small amplitude perturbation method, it is proved that there exists an asymptotically stable pest-eradication periodic solution when the impulsive period is less than some critical value, and a permanence condition is established via the method of comparison involving multiple Liapunov functions. It is shown that the multi-predator impulsive control strategy is more effective than the classical one and makes the behavior dynamics of the system more complex.

scholarly journals Méthode d'agrégation des variables appliquée à la dynamique des populations

2006 ◽  
Vol Volume 5, Special Issue TAM... ◽  
Author(s):  
Pierre Auger ◽  
Abderrahim El Abdllaoui ◽  
Rachid Mchich
Keyword(s):  
Stable Limit Cycle ◽  
Stable Limit ◽  
Dynamique Des Populations ◽  
Globally Asymptotically Stable ◽  
Density Dependent ◽  
Migration Rates ◽  
First Case ◽  
International Audience ◽  
Predator Model ◽  
Predator System

International audience We present the method of aggregation of variables in the case of ordinary differential equations. We apply the method to a prey - predator model in a multi - patchy environment. In this model, preys can go to a refuge and therefore escape to predation. The predator must return regularly to his terrier to feed his progeny. We study the effect of density-dependent migration on the global stability of the prey-predator system. We consider constant migration rates, but also density-dependent migration rates. We prove that the positif equilibrium is globally asymptotically stable in the first case, and that its stability changes in the second case. The fact that we consider density-dependent migration rates leads to the existence of a stable limit cycle via a Hopf bifurcation. Nous présentons les grandes lignes de laméthode d'agrégation des variables dans les systèmes d'équations différentielles ordinaires. Nous appliquons laméthode à un modèle proie-prédateur spatialisé. Dans ce modèle, les proies peuvent échapper à la prédation en se réfugiant sur un site. Le prédateur doit aussi retourner régulièrement dans son terrier pour nourrir sa progéniture. Nous étudions les effets de migration dépendant de la densité des populations sur la stabilité globale du système proie-prédateur. Nous considérons des taux de migration constants, puis densité-dépendants. Dans le cas de taux constants il existe un équilibre positif toujours stable alors que dans le cas de taux de migration densité-dépendants, il existe un cycle limite stable via une bifurcation de Hopf.

scholarly journals Impulsive Biological Pest Control Strategies of the Sugarcane Borer

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Marat Rafikov ◽  
Alfredo Del Sole Lordelo ◽  
Elvira Rafikova
Keyword(s):  
Pest Control ◽  
Floquet Theory ◽  
Control Strategies ◽  
Egg Parasitoid ◽  
Asymptotically Stable ◽  
Biological Pest Control ◽  
Globally Asymptotically Stable ◽  
Larval Stages ◽  
Sugarcane Borer ◽  
Impulsive Release

We propose an impulsive biological pest control of the sugarcane borer (Diatraea saccharalis) by its egg parasitoidTrichogramma galloibased on a mathematical model in which the sugarcane borer is represented by the egg and larval stages, and the parasitoid is considered in terms of the parasitized eggs. By using the Floquet theory and the small amplitude perturbation method, we show that there exists a globally asymptotically stable pest-eradication periodic solution when some conditions hold. The numerical simulations show that the impulsive release of parasitoids provides reliable strategies of the biological pest control of the sugarcane borer.

scholarly journals Periodicity and Permanence of a Discrete Impulsive Lotka-Volterra Predator-Prey Model Concerning Integrated Pest Management

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Chang Tan ◽  
Jun Cao
Keyword(s):  
Discrete System ◽  
Numerical Experiments ◽  
Critical Value ◽  
Euler Method ◽  
Impulsive Effect ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model

By piecewise Euler method, a discrete Lotka-Volterra predator-prey model with impulsive effect at fixed moment is proposed and investigated. By using Floquets theorem, we show that a globally asymptotically stable pest-eradication periodic solution exists when the impulsive period is less than some critical value. Further, we prove that the discrete system is permanence if the impulsive period is larger than some critical value. Finally, some numerical experiments are given.

scholarly journals Complex Behavior in a Fish Algae Consumption Model with Impulsive Control Strategy

2011 ◽  
Vol 2011 ◽  
pp. 1-17 ◽  
Author(s):  
Jin Yang ◽  
Min Zhao
Keyword(s):  
Control Strategy ◽  
Small Amplitude ◽  
Impulsive Control ◽  
Asymptotically Stable ◽  
Perturbation Techniques ◽  
Globally Asymptotically Stable ◽  
Impulsive Equations ◽  
Consumption Model ◽  
The Rich ◽  
Impulsive Control Strategy

This paper investigates a dynamic mathematical model of fish algae consumption with an impulsive control strategy analytically. It is proved that the system has a globally asymptotically stable algae-eradication periodic solution and is permanent by using the theory of impulsive equations and small-amplitude perturbation techniques. Numerical results for impulsive perturbations demonstrate the rich dynamic behavior of the system. Further, we have also compared biological control with chemical control. All these results may be useful in controlling eutrophication.

scholarly journals On the Study of Chemostat Model with Pulsed Input in a Polluted Environment

2007 ◽  
Vol 2007 ◽  
pp. 1-12 ◽  
Author(s):  
Zhong Zhao ◽  
Xinyu Song
Keyword(s):  
Periodic Solution ◽  
Critical Value ◽  
Asymptotically Stable ◽  
Polluted Environment ◽  
Chemostat Model ◽  
Globally Asymptotically Stable ◽  
Floquet Theorem

Chemostat model with pulsed input in a polluted environment is considered. By using the Floquet theorem, we find that the microorganism eradication periodic solution is globally asymptotically stable if the impulsive periodTis more than a critical value. At the same time, we can find that the nutrient and microorganism are permanent if the impulsive periodTis less than the critical value.

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