# American Institute of Mathematical Sciences

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

## 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, United States, United States

Received  January 2008 Revised  June 2008 Published  October 2008

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