Variation in risk in single-species discrete-time models

  • Received: 01 January 2008 Accepted: 29 June 2018 Published: 01 October 2008
  • MSC : Primary: 92D25, 92D40; Secondary: 92B05

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

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  • 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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    © 2008 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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