Bifurcations of a Ratio-Dependent Predator-Prey System with Constant Rate Harvesting

Xiao, Dongmei; Jennings, Leslie

doi:10.1137/s0036139903428719

Cited by 200 publications

(114 citation statements)

References 19 publications

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“…The result shows that stationary distribution does not rely on the existence and the stability of the positive equilibrium in the deterministic system. There is a periodic phenomenon in a non-autonomous periodic system: when r 1 …”

Section: Numerical Simulations and Conclusionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…The optimal management of renewable resources, which has a direct relationship to sustainable development, has been studied extensively by many authors (see [1,2] and the references cited therein). Xiao [1] pointed out that the aim is to determine how much we can harvest without dangerously altering the harvested population. According to Clark in [2], the management of renewable resources has been based on the maximum sustainable yield, with the property that any larger harvest rate will lead to the depletion of the population.…”

Section: Introductionmentioning

confidence: 99%

“…In addition, Gupta et al [9] presented three types of harvesting functions (constant harvesting, proportional harvesting, nonlinear harvesting), and showed that nonlinear harvesting is more realistic than constant harvesting and proportional harvesting. Motivated by [1][2][3][4][5][6][7][8][9][10][11], in this paper, we propose the following ratio-dependent predator-prey system with Holling III type functional response and nonlinear harvesting terms, which therefore is described as: ), (1.1) where x(t) and y(t) represent the population densities at time t, respectively. All parameters involved in the model are positive.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Dynamical analysis of a ratio-dependent predator–prey model with Holling III type functional response and nonlinear harvesting in a random environment

Lan

Wei

et al. 2018

Adv Differ Equ

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The objective of this paper is to study the dynamics of the stochastic ratio-dependent predator-prey model with Holling III type functional response and nonlinear harvesting. For the autonomous system, sufficient conditions for globally positive solution and stochastic permanence are established. Then, applying comparison theorem for stochastic differential equation, sufficient conditions for extinction and persistence in the mean are obtained. In addition, we prove that there exists a unique stationary distribution and it has ergodicity under certain parametric restrictions. For the periodic system, we obtain conditions for the existence of a nontrivial positive periodic solution. Finally, numerical simulations are carried out to substantiate the analytical results.

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Section: Numerical Simulations and Conclusionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Dynamical analysis of a ratio-dependent predator–prey model with Holling III type functional response and nonlinear harvesting in a random environment

Lan

Wei

et al. 2018

Adv Differ Equ

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“…On the other hand, in recent years, many researchers have studied dynamical behaviors of predator-prey systems with nonzero harvesting rates of either species or both species simultaneously(see [3,7,12,15,21,22,23]). In general, human needs are not immutable for a long time.…”

Section: Introductionmentioning

confidence: 99%

Dynamics of a Discrete Ratio-Dependent Predator-Prey System Incorporating Harvesting

Baek¹,

Ha²,

Hyun³

et al. 2015

East Asian mathematical journal

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ABSTRACT. In this paper, we consider a discrete ratio-dependent predator-prey system with harvesting effect. In order to investigate dynamical behaviors of this system, first we find out all fixed points of the system and then classify their stabilities by using their Jacobian matrices and local stability method. Next, we display some numerical examples to substantiate theoretical results and finally, we show numerically, by means of a bifurcation diagram, that various dynamical behaviors including cycles, periodic doubling bifurcation and chaotic bands can be existed.

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Multiplicity of positive almost periodic solutions in a delayed Hassell-Varley-type predator-prey model with harvesting on prey

Tian-Wei-Tian

2013

Math. Meth. Appl. Sci.

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In this paper, we consider a delayed Hassell–Varley‐type predator–prey model with harvesting on prey. By means of Mawhin's continuation theorem of coincidence degree theory, some new sufficient conditions are obtained for the existence of at least two positive almost periodic solutions for the aforementioned model. To the best of the author's knowledge, so far, the result of this paper is completely new. An example is employed to illustrate the result of this paper. Copyright © 2013 John Wiley & Sons, Ltd.

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Bifurcations of a Ratio-Dependent Predator-Prey System with Constant Rate Harvesting

Cited by 200 publications

References 19 publications

Dynamical analysis of a ratio-dependent predator–prey model with Holling III type functional response and nonlinear harvesting in a random environment

Dynamical analysis of a ratio-dependent predator–prey model with Holling III type functional response and nonlinear harvesting in a random environment

Dynamics of a Discrete Ratio-Dependent Predator-Prey System Incorporating Harvesting

Multiplicity of positive almost periodic solutions in a delayed Hassell-Varley-type predator-prey model with harvesting on prey

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