Persistence in food webs—I Lotka-Volterra food chains

“…Persistence refers to the tendency of populations to remain within acceptable limits of size despite disturbances (Reddingius 1971, Holling 1973, Innis 1974, Botkin and Sobel 1974, Chesson 1978, Harrison 1979, Gard and Hallam 1979. Persistence refers to the tendency of populations to remain within acceptable limits of size despite disturbances (Reddingius 1971, Holling 1973, Innis 1974, Botkin and Sobel 1974, Chesson 1978, Harrison 1979, Gard and Hallam 1979.…”

Section: System Stabilitymentioning

confidence: 99%

Equilibrium and Nonequilibrium Concepts in Ecological Models

Jiang¹,

Waterhouse

1987

Ecological Monographs

518

254

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Mathematical models and empirical studies have revealed two potentially disruptive influences on ecosystems; (1) instabilities caused by nonlinear feedbacks and time-lags in the interactions ofbiological species, and (2) stochastic forcings by a fluctuating environment. Because both of these phenomena can severely affect system survival, ecologists are confronted with the question of why complex ecosystems do, in fact, exist. Our study analyzes the basic themes of this research and identifies five general hypotheses that, in recent years, theoretical ecologists have built into models to increase their stability against disruptive feedback and stochasticity. To counter feedback instabilities, theoreticians have considered (1) functional interactions between species that act as stabilizers, (2) disturbance patterns that interrupt adverse feedback effects, and (3) the stabilizing effect of integrating small-spatial-scale systems into large landscapes. To decrease the influence of stochasticity, modelers have hypothesized (4) compensatory mechanisms operating at low population densities, and (5) the moderating effect of spatial extent and heterogeneity. We show that modeling based on these ideas can be organized in a systematic way. We also show that the stable equilibrium state should not be viewed as a fundamental property of ecological systems, but as a property that can emerge asymptotically from extrapolation to sufficiently large spatial scales.

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“…However, mathematically it means that strictly positive solutions do not have omega limit set on the boundary of the non-negative cone [11].Accordingly, if the dynamic system does not persist, then the solution have omega limit set on the boundary of the nonnegative cone, and hence the dynamic system faces extinction. Now, before examine the persistence of stage structure model given by system (2) by using the method of average Lyapunov function as given in [12], we need to study the global dynamics in the boundary planes xz , xw , and in the 3 .…”

Section: The Persistence Of System (2)mentioning

confidence: 99%