### Mathematical Biosciences and Engineering

2008, Issue 4: 859-875. doi: 10.3934/mbe.2008.5.859

# 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

### Related Papers:

• 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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沈阳化工大学材料科学与工程学院 沈阳 110142

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