American Institute of Mathematical Sciences

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2010, 7(3): 719-728. doi: 10.3934/mbe.2010.7.719

A model for phenotype change in a stochastic framework

Graeme Wake 1, , Anthony Pleasants 2, , Alan Beedle 3, and  Peter Gluckman 3,

1. 

National Research Centre for Growth and Development & Institute of Information and Mathematical Sciences, Massey University, Private Bag 102904, Albany, Auckland, New Zealand
2. 

National Research Centre for Growth and Development & AgResearch Limited, Ruakura Research Centre, Private Bag 3123, Hamilton, New Zealand
3. 

National Research Centre for Growth and Development & Liggins Institute, University of Auckland, Private Bag 92019, Auckland, New Zealand, New Zealand

Received  April 2009 Revised  February 2010 Published  June 2010

In some species, an inducible secondary phenotype will develop some time after the environmental change that evokes it. Nishimura (2006) [4] showed how an individual organism should optimize the time it takes to respond to an environmental change ("waiting time''). If the optimal waiting time is considered to act over the population, there are implications for the expected value of the mean fitness in that population. A stochastic predator-prey model is proposed in which the prey have a fixed initial energy budget. Fitness is the product of survival probability and the energy remaining for non-defensive purposes. The model is placed in the stochastic domain by assuming that the waiting time in the population is a normally distributed random variable because of biological variance inherent in mounting the response. It is found that the value of the mean waiting time that maximises fitness depends linearly on the variance of the waiting time.
Keywords: waiting time., fitness, stochasticity, inducible defence, phenotypic plasticity.
Mathematics Subject Classification: Primary: 92D15, 60G15, 65K10; Secondary: 60G40, 60K4.
Citation: Graeme Wake, Anthony Pleasants, Alan Beedle, Peter Gluckman. A model for phenotype change in a stochastic framework. Mathematical Biosciences & Engineering, 2010, 7 (3) : 719-728. doi: 10.3934/mbe.2010.7.719
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