Permanence and the dynamics of biological systems

Hutson, V.; Schmitt, Klaus

doi:10.1016/0025-5564(92)90078-b

Cited by 312 publications

(211 citation statements)

References 51 publications

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“…Moreover, system (1.2), exhibiting Armstrongand McGehee-type coexistence is only weakly persistent, but not persistent. Hence, it is not permanent, otherwise it would exhibit componentwise positive equilibria (see [27]). …”

Section: Introductionmentioning

confidence: 99%

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Coexistence in competition models with density-dependent mortality

Ruan

Ardito

Ricciardi

et al. 2007

Comptes Rendus. Biologies

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We consider a two-competitor/one-prey model in which both competitors exhibit a general functional response and one of the competitors exhibits a density-dependent mortality rate. It is shown that the two competitors can coexist upon the single prey.As an example, we consider a two-competitor/one-prey model with a Holling II functional response. Our results demonstrate that density-dependent mortality in one of the competitors can prevent competitive exclusion. Moreover, by constructing a Liapunov function, the system has a globally stable positive equilibrium.

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Section: Introductionmentioning

confidence: 99%

“…For further discussion about various definitions of persistence and permanence and their connections, we refer to [14,27,43,48].…”

Section: Introductionmentioning

confidence: 99%

Coexistence in competition models with density-dependent mortality

Ruan

Ardito

Ricciardi

et al. 2007

Comptes Rendus. Biologies

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“…10 12 ; B 2 ) and λ max ( f u 10 12 ; B 2 ) are independent of u 10 For g ∈ Z 0 denote For g 12 ∈ Y 1 0 and s ∈ R we put (g 12 …”

Section: Resultsmentioning

confidence: 99%

“…Uniform persistence for autonomous and time periodic nonlinear systems of parabolic partial differential equations of second order has been well studied (see [1][2][3][4][5][6]12,16,28], etc.). It has also been recently intensively studied for various nonautonomous nonlinear systems of parabolic equations (see [8], [15,22,24,26,27], etc.).…”

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confidence: 99%

Persistence in Forward Nonautonomous Competitive Systems of Parabolic Equations

Mierczyński

Shen

2010

J Dyn Diff Equat

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“…In practice, much of the research into the question of long term coexistence of species has regarded this as an equilibrium problem. From a theoretical perspective it has been the tractability of local (also known as asymptotic or neighbourhood) stability analysis that has ensured the pervasiveness of the equilibrium view point: 'even if other definitions of stability are more attractive, if they are not tractable then the ecologist cannot adopt them with profit' (Hutson and Schmitt, 1992). The deficiencies of local stability analysis are numerous and well known (Anderson et al, 1992;Berlow et al, 2004;Haydon, 1994;Jansen and Sigmund, 1998) and there is little reason to believe that the natural world is in equilibrium.…”

Section: Introductionmentioning

confidence: 99%

On the generality of stability–complexity relationships in Lotka–Volterra ecosystems

Townsend

Haydon

Matthews

2010

Journal of Theoretical Biology

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Understanding how complexity persists in nature is a long-standing goal of ecologists. In theoretical ecology, local stability is a widely used measure of ecosystem persistence and has made a major contribution to the ecosystem stability-complexity debate over the last few decades. However, permanence is coming to be regarded as a more satisfactory definition of ecosystem persistence and has relatively recently become available as a tool for assessing the global stability of Lotka -Volterra communities. Here we document positive relationships between permanence and Lotka -Volterra food web complexity and report a positive correlation between the probability of local stability and permanence. We investigate further the frequency of discrepancy (attributed to fragile systems that are locally stable but not permanent or locally unstable systems that are permanent and have cyclic or chaotic dynamics), associate non-permanence with the local stability or instability of equilibria on the boundary of the state-space, and investigate how these vary with aspects of ecosystem complexity. We find that locally stable interior equilibria tend to have all locally unstable boundary equilibria. Since a locally stable boundary is inconsistent with permanent dynamics, this can explain the observed positive correlation between local interior stability and permanence. Our key finding is that, at least in Lotka -Volterra model ecosystems, local stability may be a better measure of persistence than previously thought.

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Permanence and the dynamics of biological systems

Cited by 312 publications

References 51 publications

Coexistence in competition models with density-dependent mortality

Coexistence in competition models with density-dependent mortality

Persistence in Forward Nonautonomous Competitive Systems of Parabolic Equations

On the generality of stability–complexity relationships in Lotka–Volterra ecosystems

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