Global asymptotic stability of a ratio-dependent predator–prey system with diffusion

Fan, Yong-Hong; Li, Wan–Tong

doi:10.1016/j.cam.2005.04.007

Cited by 47 publications

(18 citation statements)

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“…Berryman 1992) because it leaves out a possibility of predators' interference. A few authors considered a model where the per capita predation rate is ratio dependent (Arditi and Ginzburg 1989;Kuang and Beretta 1998;Bartumeus et al 2001;Xiao and Ruan 2001;Fan and Li 2006). Although trophic interactions in real ecosystems are likely to be more complicated (e.g.…”

Section: Introductionmentioning

confidence: 99%

Self-organised spatial patterns and chaos in a ratio-dependent predator–prey system

2010

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Mechanisms and scenarios of pattern formation in predator-prey systems have been a focus of many studies recently as they are thought to mimic the processes of ecological patterning in real-world ecosystems. Considerable work has been done with regards to both Turing and non-Turing patterns where the latter often appears to be chaotic. In particular, spatiotemporal chaos remains a controversial issue as it can have important implications for population dynamics. Most of the results, however, were obtained in terms of 'traditional' predator-prey models where the per capita predation rate depends on the prey density only. A relatively new family of ratio-dependent predator-prey models remains less studied and still poorly understood, especially when space is taken into account explicitly, in spite of their apparent ecological relevance. In this paper, we consider spatiotemporal pattern formation in a ratio-dependent predator-prey system. We show that the system can develop patterns both inside and outside of the Turing parameter domain. Contrary to widespread opinion, we show that the interaction between two different type of instability, such as the TuringHopf bifurcation, does not necessarily lead to the onset of chaos; on the contrary, the emerging patterns remain stationary and almost regular. Spatiotemporal chaos can only be observed for parameters well inside the Turing-Hopf domain. We then investigate the relative importance of these two instability types on the onset of chaos and show that, in a ratio-dependent predatorprey system, the Hopf bifurcation is indeed essential for the onset of chaos whilst the Turing instability is not.

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Section: Introductionmentioning

confidence: 99%

Self-organised spatial patterns and chaos in a ratio-dependent predator–prey system

2010

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“…The two-species SKT competing system and its overall behavior continue to be of great interest in the literature to both mathematical analysis and real-life modelling, but recently more and more attention has been focused on three or multi-species systems and the SKT model in any space dimension due to their more complicated patterns. Meanwhile, the SKT models with other types of reaction terms are also proposed and investigated [4][5][6][7][8][9][10][11]. The obtained results mainly concentrate on the stability analysis of positive constant steady state and the existence of nonconstant positive steady states (stationary patterns) [6][7][8][9][10][11][12][13][14].…”

Section: Introductionmentioning

confidence: 97%

Global solutions to a class of multi-species reaction-diffusion systems with cross-diffusions arising in population dynamics

Wen

2009

Journal of Computational and Applied Mathematics

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a b s t r a c tIn this paper, an n-species strongly coupled cooperating diffusive system is considered in a bounded smooth domain, subject to homogeneous Neumann boundary conditions. Employing the method of energy estimates, we obtain some conditions on the diffusion matrix and inter-specific cooperatives to ensure the global existence and uniform boundedness of a nonnegative solution. The globally asymptotical stability of the constant positive steady state is also discussed. As a consequence, all the results hold true for multispecies Lotka-Volterra type competition model and prey-predator model.

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“…One typical and classical method is to describe the spatial migration of individuals by Laplace operator (see [4,9]) and such an operator can give rise to some important spatiotemporal pattern, for example Turing instability (or the diffusion-driven instability, see [30]). Particularly, during the past decades, diffusive predator-prey systems have been extensively studied, and one can refer to [10,15,19,20,24,25,26,27,28,32] and references cited therein.…”

Section: Introductionmentioning

confidence: 99%

Qualitative Analysis of a Diffusive Food Web Consisting of a Prey and Two Predators

Shi¹

2013

Bulletin of the Korean Mathematical Society

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Abstract. This paper is concerned with the positive steady states of a diffusive Holling type II predator-prey system, in which two predators and one prey are involved. Under homogeneous Neumann boundary conditions, the local and global asymptotic stability of the spatially homogeneous positive steady state are discussed. Moreover, the large diffusion of predator is considered by proving the nonexistence of non-constant positive steady states, which gives some descriptions of the effect of diffusion on the pattern formation.

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Global asymptotic stability of a ratio-dependent predator–prey system with diffusion

Cited by 47 publications

References 18 publications

Self-organised spatial patterns and chaos in a ratio-dependent predator–prey system

Self-organised spatial patterns and chaos in a ratio-dependent predator–prey system

Global solutions to a class of multi-species reaction-diffusion systems with cross-diffusions arising in population dynamics

Qualitative Analysis of a Diffusive Food Web Consisting of a Prey and Two Predators

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