Existence of limit cycles for predator–prey systems with a class of functional responses

Hesaaraki, Mahmoud; Moghadas, Seyed M.

doi:10.1016/s0304-3800(00)00442-7

Cited by 35 publications

(22 citation statements)

References 9 publications

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“…In this paper, we continue our previous work in Hesaaraki et al [6], on the existence of limit cycles for a predator-prey system with a given functional response. In this study, we will give some conditions for the nonexistence of limit cycles for a class of functional responses.…”

Section: Introductionsupporting

confidence: 59%

“…Following Kooij et al [10], Sugie [15] presented a sufficient and necessary condition under which the system (1.1) with Ivlev's functional response has a unique limit cycle. This problem is also considered in [6] for a more general case and the necessary and sufficient condition for the existence of limit cycles under the assumption ' 000 ðxÞ > 0 is given. Hesaaraki et al [4] presented a sufficient and necessary condition for the nonexistence of limit cycles of (1.1) when the functional response is x n =ða þ x n Þ, a > 0, n > 1.…”

Section: Introductionmentioning

confidence: 99%

“…This problem is also considered in [6] for a more general case and the necessary and sufficient condition for the existence of limit cycles under the assumption ' 000 ðxÞ > 0 is given. Hesaaraki et al [4] presented a sufficient and necessary condition for the nonexistence of limit cycles of (1.1) when the functional response is x n =ða þ x n Þ, a > 0, n > 1. It is easy to see that this functional response does not satisfy the assumption ðA3Þ.…”

Section: Introductionmentioning

confidence: 99%

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Some Conditions for the Nonexistence of Limit Cycles in a Predator-Prey System

Moghadas

2002

Applicable Analysis

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In this paper, we study the problem of the existence of limit cycles for a predator-prey system with a functional response. It is assumed that the functional response is positive, increasing, concave down, and its third derivative has a unique root. A necessary condition for the nonexistence of limit cycles is presented. Some conditions are given under which the necessary condition is also the sufficient condition for the nonexistence of limit cycles.

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Section: Introductionsupporting

confidence: 59%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Some Conditions for the Nonexistence of Limit Cycles in a Predator-Prey System

Moghadas

2002

Applicable Analysis

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“…Sufficient and necessary conditions for the uniqueness of limit cycles of system (1) have been presented in Sugie et al (1997), when the functional response is h(x) = x n x n +a , for a > 0 and n > 1. The general Gause type predation model has been widely studied, establishing the stability and bifurcations (Sugie et al 1996(Sugie et al , 1997, proving the global stability of the positive equilibrium point (Huang and Zhu 2005;Sugie and Katayama 1999), and demonstrating the uniqueness (Huang and Zhu 2005;Kuang and Freedman 1988;Moghadas and Corbett 2008;Wang and Sun 2007;Xiao and Zhang 2003) or nonexistence of limit cycles (Hesaaraki and Moghadas 1999;Sugie et al 1996). However, it is not an easy task to study the quantity of limit cycles that can be generated throughout the bifurcation of a center-focus (Chicone 2006).…”

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confidence: 99%

Multiple Limit Cycles in a Gause Type Predator–Prey Model with Holling Type III Functional Response and Allee Effect on Prey

González‐Olivares

Rojas‐Palma

2010

Bull Math Biol

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This work aims to examine the global behavior of a Gause type predator-prey model considering two aspects: (i) the functional response is Holling type III and, (ii) the prey growth is affected by the Allee effect. We prove the origin of the system is an attractor equilibrium point for all parameter values. It has also been shown that it is the ω-limit of a wide set of trajectories of the system, due to the existence of a separatrix curve determined by the stable manifold of the equilibrium point (m,0), which is associated to the Allee effect on prey. When a weak Allee effect on the prey is assumed, an important result is obtained, involving the existence of two limit cycles surrounding a unique positive equilibrium point: the innermost cycle is unstable and the outermost stable. This property, not yet reported in models considering a sigmoid functional response, is an important aspect for ecologists to acknowledge as regards the kind of tristability shown here: (1) the origin; (2) an interior equilibrium; and (3) a limit cycle of large amplitude. These models have undoubtedly been rather sensitive to disturbances and require careful management in applied conservation and renewable resource contexts.

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“…Theoretical studies show that models of population dynamics often exhibit high amplitude oscillations in the sizes of populations and communities. Continuous predator-prey models, for example, may exhibit neutrally stable equilibria (Lotka 1925) and limit cycles (Baalen et al 2001;Hesaaraki and Moghadas 2001;GonsalesOlivares et al 2006). Chaotic oscillations may be found in continuous three-trophic-level models of systems with omnivory (Tanabe and Namba 2005;Namba et al 2008), in discrete maps of consumer-resource systems (Hassel 1978), and in single density-dependent population models with discrete (non-overlapping) generations (May 1974(May , 1976May and Oster 1976;Nayfe and Balachandran 1995;Hanski 1999).…”

Section: Introductionmentioning

confidence: 99%

Effects of local interaction range and mobility on the spatio‐temporal dynamics of competing animals in uniform habitats

Miller

Coll

2011

Population Ecology

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Reduction of oscillations in population size is of fundamental importance to both theoretical and applied ecology. Spatial variability in population rates among different habitat regions is known to be an important mechanism that inhibits oscillations in population size. In the current study we used an individual-based model to simulate a single population of animals whose individual members are sensitive to competition only within their vicinity (i.e., within their competition neighborhood, CN). Our model extends previous studies by exploring how local interactions reduce population oscillations in competitive systems of animals, rather than in systems of plants. Our simulations explored the effects of animal mobility and interaction range separately on population dynamics. In our model, a decrease in CN dimensions tended to reduce population oscillations at all tested animal movement speeds. Yet, movement speed affected animal distribution patterns; an increase in movement speed led to more random distributions. We also found that mean population size was affected more by CN dimensions at lower mobility levels than when it was high.

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Existence of limit cycles for predator–prey systems with a class of functional responses

Cited by 35 publications

References 9 publications

Some Conditions for the Nonexistence of Limit Cycles in a Predator-Prey System

Some Conditions for the Nonexistence of Limit Cycles in a Predator-Prey System

Multiple Limit Cycles in a Gause Type Predator–Prey Model with Holling Type III Functional Response and Allee Effect on Prey

Effects of local interaction range and mobility on the spatio‐temporal dynamics of competing animals in uniform habitats

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