Mathematical Biosciences and Engineering, 2008, 5(4): 859-875. doi: 10.3934/mbe.2008.5.859.

Primary: 92D25, 92D40; Secondary: 92B05

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Variation in risk in single-species discrete-time models

1. Department of Ecology, Evolution and Marine Biology, University of California at Santa Barbara, CA 93106-9610

Simple, discrete-time, population models typically exhibit complex dynamics, like cyclic oscillations and chaos, when the net reproductive rate, $R$, is large. These traditional models generally do not incorporate variability in juvenile "risk,'' defined to be a measure of a juvenile's vulnerability to density-dependent mortality. For a broad class of discrete-time models we show that variability in risk across juveniles tends to stabilize the equilibrium. We consider both density-independent and density-dependent risk, and for each, we identify appropriate shapes of the distribution of risk that will stabilize the equilibrium for all values of $R$. In both cases, it is the shape of the distribution of risk and not the amount of variation in risk that is crucial for stability.
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Keywords Ricker model; population dynamics; variation in risk; single-species models; discrete-time models; sensitivity of the equilibrium to reproduction rate

Citation: Abhyudai Singh, Roger M. Nisbet. Variation in risk in single-species discrete-time models. Mathematical Biosciences and Engineering, 2008, 5(4): 859-875. doi: 10.3934/mbe.2008.5.859


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