July 9, 2008 - 3:30 p.m. Graeme Wake
Centre for Mathematics in Industry, Massey University
@ Auckland,
New Zealand

A model for phenotype
change in a stochastic framework
In some species, an inducible secondary phenotype
will develop some time after the environmental change
that evokes it. Nishimura (EER 8:553) showed how
an individual organism should optimize the time
it takes to respond to such an environmental change.
Question: If the optimal response time is considered
to act over the population, what are the implications
for the expected value of the mean fitness in that
population?
Model: A stochastic predator-prey model in which
the prey have a fixed initial energy budget. Prey
individuals trade off reduction in the probability
of predation against increase in the energy required
to maintain a phenotype with improved defense. Fitness
is assessed as the product of survival probability
and the energy that remains for non-defensive purposes.
Key assumption: The response time in the population
is a normally distributed random variable because
of biological variance inherent in mounting the
response.
Conclusions: No unique values for mean response
time and variance in response time maximize fitness.
Rather, maximum fitness is described by pairs of
these values lying on a straight line in the plane
of mean time and variance. Populations with equal
mean fitness will show different frequency distributions
of response times, which may allow different populations
to explore different environments. If a significant
proportion of the population delays an inducible
response, the population may suffer no loss of fitness
from either its predation level or the energy expenditures
it requires to maintain plasticity.
This work is in conjunction with the Liggins Institute
(Professor Peter Gluckman) and AgResearch (Tony
Pleasants).

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