Dynamical model of a single-species system in a polluted environment

Samanta, G. P.; Maiti, Alakes

doi:10.1007/bf02936164

Cited by 16 publications

(7 citation statements)

References 17 publications

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“…Thus, roughly speaking, stability and instability are consequences of stochastic perturbation of the model system under consideration. Such a conclusion is in good agreement with Prajneshu [59], Baishya and Chakrabarti [57], Samanta [32,35], Samanta and Maiti [37], Maiti and Samanta [26] and many others.…”

Section: Discussionsupporting

confidence: 86%

“…A quantum leap in the mathematical sophistication of ecological modelling occurred when May [29] introduced stochastic differential equations to investigate li-mits to niche overlap in randomly fluctuating environment. Subsequently, the sensibility of stochastic models in comparison with deterministic models is established by many researchers (see Dimentberg [33,34], Samanta [32,35], Samanta and Maiti [36,37], Bandyopadhyay and Chakrabarti [38], Bandyopadhyay and Chattopadhyay [18], Maiti and Samanta [22,39], Maiti et al [26] and references cited there in). Also, the effects of environmental fluctuations on the models with ratio-dependent functional responses have now become the focus of considerable attention in the literature (see Bandyopadhyay and Chattopadhyay [18], Maiti and Samanta [39], Mankin et al [40], and Maiti et al [26]).…”

Section: Introductionmentioning

confidence: 99%

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A Modified Holling-Tanner Model in Stochastic Environment

Maiti

Pathak²

2009

NAMC

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Recently, a modified version of the so called Holling-Tanner model is introduced in the ecological literature. A detailed account of the deterministic dynamics of this model is presented. The growth rates of the prey and predator are then perturbed by Gaussian white noises to take into account the effect of fluctuating environment. The resulting stochastic model is cultured by the technique of statistical linearization and criteria for non-equilibrium fluctuation and stability are derived. Numerical simulations are carried out. The implications of our analytical findings are addressed critically.

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Section: Discussionsupporting

confidence: 86%

Section: Introductionmentioning

confidence: 99%

A Modified Holling-Tanner Model in Stochastic Environment

Maiti

Pathak²

2009

NAMC

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“…Gard 1992;Wang 2009, 2010;Samanta and Maiti 2004). Particularly, Gard (1992) proposed a stochastic model by introducing stochastic production rate.…”

mentioning

confidence: 98%

Survival Analysis of Stochastic Competitive Models in a Polluted Environment and Stochastic Competitive Exclusion Principle

Liu

2010

Bull Math Biol

222

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Stochastic competitive models with pollution and without pollution are proposed and studied. For the first system with pollution, sufficient criteria for extinction, nonpersistence in the mean, weak persistence in the mean, strong persistence in the mean, and stochastic permanence are established. The threshold between weak persistence in the mean and extinction for each population is obtained. It is found that stochastic disturbance is favorable for the survival of one species and is unfavorable for the survival of the other species. For the second system with pollution, sufficient conditions for extinction and weak persistence are obtained. For the model without pollution, a partial stochastic competitive exclusion principle is derived.

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“…In [8], Buonomo et al studied the effect of variation of the population on the toxicant concentration in the organism and environment. Samanta and Maiti [9] studied a dynamical model of a single-species system in a polluted environment under two cases: constant exogenous input of toxicant and rapidly fluctuating random exogenous input of toxicant into the environment by means of ordinary and stochastic differential equations. There are many other works on the effects of a single toxicant on various ecosystems by using mathematical models [10][11][12][13][14][15].…”

Section: Introductionmentioning

confidence: 99%

Mathematical Analysis of a Single-Species Population Model in a Polluted Environment with Discrete Time Delays

Sharma

Samanta

2013

Journal of Mathematics

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We have discussed the dynamical behaviour of a single-species population model in a polluted environment which describes the effect of toxicants on ecological system. Boundedness, positivity, and stability analysis of the model at various equilibrium points is discussed thoroughly. We have also studied the effect of single discrete delay as well as double discrete delays on the population model. Existence conditions of the Hopf bifurcation for single time delay are investigated. The length of delay preserving the stability is also estimated. The direction and the stability criteria of the bifurcating periodic solutions are determined by using the normal form theory and the center manifold theorem. The stability of the model with double time delays is investigated by using the Nyquist criteria. By choosing one of the delays as a bifurcation parameter, the model is found to undergo a Hopf bifurcation. Some numerical simulations for justifying the theoretical results are also illustrated by using MATLAB, which shows the reliability of our model from the practical point of view.

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Dynamical model of a single-species system in a polluted environment

Cited by 16 publications

References 17 publications

A Modified Holling-Tanner Model in Stochastic Environment

A Modified Holling-Tanner Model in Stochastic Environment

Survival Analysis of Stochastic Competitive Models in a Polluted Environment and Stochastic Competitive Exclusion Principle

Mathematical Analysis of a Single-Species Population Model in a Polluted Environment with Discrete Time Delays

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