Persistence in predator-prey systems with ratio-dependent predator influence

This paper reports on studies of the deterministic and stochastic behaviours of a predator-prey system with prey-dependent response function. The first part of the paper deals with the deterministic analysis of uniform boundedness, permanence, stability and bifurcation. In the second part the reproductive and mortality factors of the prey and predator species are perturbed by coloured noise to simulate a random environment. Using spectral density analysis it has been shown that this system exhibits large fluctuations for high amplitude random forces. When k, the carrying capacity of prey, tends to a particular value k Ã , this system exhibits abnormally large fluctuations with increasing time with a periodic background noise. A comparative study of the deterministic and stochastic criteria of stability is also established.

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“…is said to be uniformly persistent (or permanent) [12] if there exists a positive constant 2 such that for any solution (x 1 (t), x 2 (t), . .…”

Section: Predator-prey System: Model Equations and Stationary Statesmentioning

confidence: 99%

Deterministic and stochastic analysis of a prey-dependent predator–prey system

Maiti¹,

Samanta

2005

International Journal of Mathematical Education in Science and

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“…Here, the functional response x y mþ x y is the ratio-dependent. For the mathematical aspect, after Arditi and Ginzburg's work in [1] (1) has been studied by several authors and much richer dynamics have been obtained (see, e.g., [2][3][4][5][6][7]). A traditional competition system [8] takes the form:…”

Section: Introductionmentioning

confidence: 98%

Existence of positive periodic solution for ratio-dependent N-species difference system

Ding

2009

Applied Mathematical Modelling

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a b s t r a c tIn this paper, we propose a food web model consisting of n À 1 competing preys and one predator. Firstly, following the same idea and method in [M. Fan, K. Wang, Periodic solutions of discrete time nonautonomous ratio-dependent predator-prey system, Math. Comput. Model. 35 (2002) 951-961], we derive the discrete time analogue of the model. Then, the easily verifiable sufficient criteria are established for the existence of positive periodic solutions of this analogue, the approach is based on the coincidence degree and the related continuation theorem as well as some prior estimates. Meanwhile, some numerical simulations are carried out to support the theoretical analysis of the research.

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“…For detailed justifications of (1.2) and its merits versus (1.1), see [3,33]. As for the mathematical aspect of (1.2), since [3], (1.2) has been studied by many authors and seen great progress (e.g., autonomous case [21,24,29,31,32,38]; nonautonomous continuous case [20,37]; nonautonomous discrete time case [19]). Many authors have observed that the ratio-dependent models can exhibit much richer, more complicated and more reasonable or acceptable dynamics, but it has somewhat singular behavior at low densities which has been the source of controversy and been criticized on other grounds.…”

Section: Introductionmentioning

confidence: 99%

Dynamics of a nonautonomous predator–prey system with the Beddington–DeAngelis functional response

Fan

Kuang

2004

Journal of Mathematical Analysis and Applications

178

101

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In this paper, we systematically study the dynamics of a nonautonomous predator-prey system with the Beddington-DeAngelis functional response. The explorations involve the permanence, extinction, global asymptotic stability (general nonautonomous case); the existence, uniqueness and stability of a positive (almost) periodic solution and a boundary (almost) periodic solution for the periodic (almost periodic) case. The paper ends with some interesting numerical simulations that complement our analytical findings.  2004 Elsevier Inc. All rights reserved.

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Persistence in predator-prey systems with ratio-dependent predator influence

Cited by 75 publications

References 10 publications

Deterministic and stochastic analysis of a prey-dependent predator–prey system

Deterministic and stochastic analysis of a prey-dependent predator–prey system

Existence of positive periodic solution for ratio-dependent N-species difference system

Dynamics of a nonautonomous predator–prey system with the Beddington–DeAngelis functional response

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