Stability regions in predator-prey systems with constant-rate prey harvesting

Brauer, Fred; Soudack, A. C.

doi:10.1007/bf00280586

Cited by 146 publications

(76 citation statements)

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“…There is a positive equilibrium (x * , y * ) = (20,15). By Corollary 33, there is a critical value τ 0 = 0.8256, the equilibrium (x * , y * ) is stable when τ < 0.8256.…”

Section: Gause Models With Prey Harvesting and Delayed Prey Specific mentioning

confidence: 90%

“…As Nakaoka et al [69] showed that a LotkaVolterra predator-prey model with delays in the specific growth terms for both species can exhibit chaotic behavior, I expect that other predator-prey models with multiple delays could have similar complex dynamics. (ii) Delayed predator-prey models with both predator and prey harvesting (see Brauer and Soudack [14,15], Myerscough et al [68], and Hogarth et al [53] for some ODE models). I suspect that more degenerate bifurcations, such as fold-Hopf, Hopf-Hopf, and degenerate Bogdanov-Takens bifurcations (Kuznetsov [57]), may occur.…”

Section: Discussionmentioning

confidence: 99%

“…Predator-prey systems with constant-rate harvesting have been studied extensively and very complex dynamical behaviors, such as the existence of multiple equilibria, homoclinic loop, Hopf bifurcation and Bogdanov-Takens bifurcation, have been observed, see, for example, Brauer and Soudack [14,15,16], Beddington and Cooke [5], Dai and Tang [27], Hogarth et al [53], Myerscough et al [68], Xiao and Ruan [91]. Brauer [11] was the first to consider the combined effects of time delay and constant harvesting on predator-prey models.…”

Section: Gause-type Predator-prey Systems With Discrete Delay and Harmentioning

confidence: 99%

“…When τ = 0, the ODE model was studied by Brauer and Soudack [14,15]; when H = 0, the delayed predator-prey model has been analyzed by May [61], Hassard et al [48], Ruan [74], etc. The stability of system (5.7) was investigated by Brauer [17].…”

Section: Gause Models With Prey Harvesting and Delayed Prey Specific mentioning

confidence: 99%

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On Nonlinear Dynamics of Predator-Prey Models with Discrete Delay

Ruan

2009

Math. Model. Nat. Phenom.

156

100

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Abstract. In this survey, we briefly review some of our recent studies on predator-prey models with discrete delay. We first study the distribution of zeros of a second degree transcendental polynomial. Then we apply the general results on the distribution of zeros of the second degree transcendental polynomial to various predator-prey models with discrete delay, including Kolmogorov-type predator-prey models, generalized Gause-type predator-prey models with harvesting, etc. Bogdanov-Takens bifurcations in delayed predator-prey models with nonmonotone functional response and in delayed predator-prey model with predator harvesting are also introduced.

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“…There is a positive equilibrium (x * , y * ) = (20,15). By Corollary 33, there is a critical value τ 0 = 0.8256, the equilibrium (x * , y * ) is stable when τ < 0.8256.…”

Section: Gause Models With Prey Harvesting and Delayed Prey Specific mentioning

confidence: 90%

Section: Discussionmentioning

confidence: 99%

Section: Gause-type Predator-prey Systems With Discrete Delay and Harmentioning

confidence: 99%

Section: Gause Models With Prey Harvesting and Delayed Prey Specific mentioning

confidence: 99%

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On Nonlinear Dynamics of Predator-Prey Models with Discrete Delay

Ruan

2009

Math. Model. Nat. Phenom.

156

100

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“…There are several papers that deal with predator-prey system subject to a harvest (Brauer & Soudack 1978, 1979a, 1979b, 1981, 1982.…”

Section: Introductionmentioning

confidence: 99%

Control of Nonlinear Dynamic Models of Predator-Prey Type

Bhaya

Meza

2012

Oecol Aust

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In the ecological context, a large class of population dynamics models can be written as dynamical systems of one or two variables, i.e., each variable represents a population density of a species. When one or more species is removed from the system (harvested), it is necessary to introduce a control (harvest policy) in order to avoid the extinction of species, due to harvesting. A threshold policy (TP) and a threshold policy with hysteresis (TPH) reviewed and discussed in this paper can be used to avoid the collapse of population densities, governed by predator-prey models. A threshold policy changes the dynamics of a predator-prey dynamical system in such a way that a stable positive equilibrium point is achieved. In other words, coexistence of both species occurs. A threshold policy with hysteresis changes the dynamics so that a limit cycle (bounded oscillation) is achieved, i.e., coexistence of species with a bounded oscillation in population densities occurs. This paper studies the continuous and discrete logistic model for one species and the Lotka-Volterra and RosenzweigMacArthur models for two species. The TP and TPH are seen to be versatile and useful in renewable resources management, being simple to design and implement, with some advantages in a situation of overexploitation, as well as in the presence of different types of uncertainties. The design of the policies is carried out by appropriate choice of virtual equilibria in a simple and intuitive manner, and the mathematics used is simple. Keywords: Nonlinear predator-prey models; threshold policy; threshold policy with hysteresis. RESUMO CONTROLE DE MODELOS DINÂMICOS NÃO LINEARES DO TIPO PREDADOR-PRESA.No contexto ecológico, uma ampla classe de modelos de dinâmica populacional pode ser escrita como sistemas dinâmicos não lineares em uma ou duas variáveis, i.e., cada variável representa uma densidade populacional de uma espécie. Quando uma ou mais espécies são removidas do sistema (colheita), é necessário introduzir um controle (política de colheita) para evitar a extinção das espécies devido à colheita. Uma política de limiar (TP do inglês) e uma política de limiar com histerese (TPH do inglês) revisadas e discutidas neste artigo podem ser utilizadas para evitar o colapso das densidades populacionais governadas pelos modelos do tipo predadorpresa. Uma política de limiar muda a dinâmica de um sistema predador-presa de maneira que um ponto de equilíbrio positivo estável é alcançado. Em outras palavras, permite a coexistência das espécies. Uma política de limiar com histerese muda a dinâmica de maneira que um ciclo limite é alcançado (oscilação limitada), isto é, ocorre a coexistência das espécies com oscilação limitada nas densidades populacionais. Este artigo estuda o modelo logístico contínuo e discreto de uma espécie e os modelos Lotka-Volterra e Rosenzweig-MacArthur de duas espécies. A TP e a TPH parecem ser versáteis e úteis na gestão de recursos renováveis , sendo simples para projetar e implementar, com algumas vantagens em uma situaç...

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Bogdanov‐Takens bifurcations of codimensions 2 and 3 in a Leslie‐Gower predator‐prey model with Michaelis‐Menten–type prey harvesting

Kong

2017

Math Methods in App Sciences

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Michaelis-Menten-type prey harvesting were studied. In the paper "Diff. Equ. Dyn. Syst. 20(2012), 339-366," Gupta et al proved that the Leslie-Gower predator-prey model with Michaelis-Menten-type prey harvesting has rich dynamics. Some equilibria of codimension 1 and their bifurcations were discussed. In this paper, we find that the model has an equilibrium of codimensions 2 and 3. We also prove analytically that the model undergoes Bogdanov-Takens bifurcations (cusp cases) of codimensions 2 and 3. Hence, the model can have 2 limit cycles, coexistence of a stable homoclinic loop and an unstable limit cycle, supercritical and subcritical Hopf bifurcations, and homoclinic bifurcation of codimension 1 as the values of parameters vary. Moreover, several numerical simulations are conducted to illustrate the validity of our results. KEYWORDS Bogdanov-Takens bifurcation of codimension 2 or 3, Michaelis-Menten-type harvesting, phase portrait, predator-prey model Math Meth Appl Sci. 2017;40:6715-6731.wileyonlinelibrary.com/journal/mma

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Stability regions in predator-prey systems with constant-rate prey harvesting

Cited by 146 publications

References 2 publications

On Nonlinear Dynamics of Predator-Prey Models with Discrete Delay

On Nonlinear Dynamics of Predator-Prey Models with Discrete Delay

Control of Nonlinear Dynamic Models of Predator-Prey Type

Bogdanov‐Takens bifurcations of codimensions 2 and 3 in a Leslie‐Gower predator‐prey model with Michaelis‐Menten–type prey harvesting

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