An Impulsive Predator-Prey System With Beddington–deangelis Functional Response and Time Delay

Zhang, Hong; Georgescu, Paul; Chen, Lansun

doi:10.1142/s1793524508000072

Cited by 56 publications

(33 citation statements)

References 21 publications

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“…Similarly to Zhang et al [18], it can be shown that I (t) is globally asymptotically stable by using stroboscopic map. As a consequence, system (4.1) always has a pest-free periodic solution ( I (t), 0).…”

Section: Impulsive Release Of the Entomopathogenic Nematodesmentioning

confidence: 70%

Dynamic complexity of microbial pesticide model

Wang

Chen

2009

Nonlinear Dyn

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In this paper, a mathematical model for the entomopathogenic nematode attacking the pest is investigated. This novel theoretical framework could result in an objective criterion on how to release the entomopathogenic nematode in order to control the pest population under the economic threshold (ET) which indicates the maximally admissible pest densities. Firstly, continuous release of the entomopathogenic nematode is taken. By using qualitative analysis method, the sufficient condition of the global stability of the positive equilibria and the existence and uniqueness of limit cycle of the system are obtained. Secondly, impulsive release of the entomopathogenic nematode is also considered. Using the Floquet's theorem and small-amplitude perturbation, we obtain that the pest-free periodic solution is locally stable if some conditions are satisfied. In a certain limiting case, it is shown that a nontrivial periodic solution emerges via a supercritical bifurcation. Finally, our findings are confirmed by means of numerical simulations.

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Section: Impulsive Release Of the Entomopathogenic Nematodesmentioning

confidence: 70%

Dynamic complexity of microbial pesticide model

Wang

Chen

2009

Nonlinear Dyn

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“…Especially, based on the economic theory proposed by [17], an algebraic equation is established to investigate the effect of harvest effort on some population in the ecosystem. Compared with the models proposed in the references [46,32,57,13], the advantages of the models proposed in [59,60,61,33] are that these models not only investigate the interaction mechanism of the prey-predator with stage structure for predator but also offer a simpler way to study the effect of harvest effort on the ecosystem from an economic perspective. As mentioned above, [60,61,59] have investigated some biological-economic systems by using the differential-algebraic model.…”

Section: Model Formulationmentioning

confidence: 99%

Bifurcations and Control in a Singular Biological Economic Model

Zhang

Liu

Zhang

2012

Complexity, Analysis and Control of Singular Biological Systems

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In the natural world, there are many species whose individuals have a life history that takes them through two stages, juvenile stage and adult stage . Individuals in each stage are identical in biological characteristics, and some vital rates (rates of survival, development, and reproduction) of individuals in a population almost always depend on stage structure. Furthermore, there is a strong interaction relationship between the mature population and the immature population , which is to some extent relevant to the persistence and extinction of the related population. Consequently, it is constructive to investigate the dynamics of such ecosystem without ignorance of stage structure for population.Before 1990, the stage structure has attracted some attention from several authors [6,20,21,23,24,31,48,53,29]. The well-known biological system of single species with stage structure is established by O. Aiello and H. Freedman [1], which represents a milestone in this research field. In the following decades, much research efforts are put into the investigation of biological dynamical systems with stage structure for population. Some related research results can be found in [1,2,3,10,14,26,8,39,34,35,36,37,38,40,44,41,42,43,45,47,51,52,54,25,16,9] and the references therein.Recently, the dynamics of a class of stage-structured prey-predator models with discrete time delay have been studied by several authors [4,55,18,58,5] and the references therein.Arino et al. [4] suggested that the time delay to adulthood should be state dependent and careful formulation of such state dependent time delays can lead to models that produce periodic solutions. Xu et al. [55] studied the persistence and stability of a delayed prey-predator model with stage structure for predator. Gourley et al. [18] formulated a general and robust prey predator model with stage-structure with constant maturation time delay and performed a systematic mathematical and computational Q. Zhang et al.: Complex

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“…Furthermore, the functional response also has an important impact on the behavior of biological dynamical systems [25][26][27][28][29]. We note the fact that, in biology, very high nutrient concentration may inhibit the growth of microorganisms actually, and the microorganisms will die eventually as the nutrient concentration increasing unlimitedly.…”

Section: Introductionmentioning

confidence: 96%

Dynamic behaviors of a turbidostat model with Tissiet functional response and discrete delay

Yao

Xiang

et al. 2018

Adv Differ Equ

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In this paper, dynamic behaviors of a turbidostat model with Tissiet functional response, linear variable yield and time delay are investigated. The existence and boundedness of solutions, the local asymptotic stability of its equilibria and the phenomenon of Hopf bifurcation for this system are considered. Using the Liapunov-LaSalle invariance principle, we show that the washout equilibrium is global asymptotic stability for any time delay. Furthermore, based on some knowledge of limit set, we show the necessary and sufficient conditions of permanent of the turbidostat model. Finally, numerical simulations are offered to support our results.

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An Impulsive Predator-Prey System With Beddington–deangelis Functional Response and Time Delay

Cited by 56 publications

References 21 publications

Dynamic complexity of microbial pesticide model

Dynamic complexity of microbial pesticide model

Bifurcations and Control in a Singular Biological Economic Model

Dynamic behaviors of a turbidostat model with Tissiet functional response and discrete delay

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