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PROG which means that the same sequence of random numbers is used for each combination: thus I replicate exactly the same environmental variation for each simulation. grid to generate the necessary variation and pass this to MAIN. PROG. However, if I do this exactly as before I will get an error message, because I need to pass only one combination at a time. 9 Contour and perspective plots for model 5. mig. Step 20: Adding inheritance: functions pnorm, dnorm and numerical integration The observation from the above analysis of the potential importance of migration in a heterogeneous environment raises the question of what is the optimal migration rate.

This paradoxical behavior can be illustrated with a simple example: suppose that l can take two values, 0 or 3, with equal frequency. 5, and hence the expected population size increases without bound as t increases. 00098, a very small probability indeed! The geometric mean is always smaller s2 than the arithmetic and the two are related by the approximation Eðln lÞ % ln l À2ll2 (Lewontin and Cohen 1969), where E(lnl) is the geometric mean,  l is the arithmetic 2 mean, and sl is the variance.

If this is the case then perhaps the arithmetic mean growth rate is also not an appropriate measure of fitness in a stochastic environment. Haldane and Jayakar (1963) and Cohen (1966) showed that the appropriate measure is the geometric rate of increase. The reason for this resides in the difference between the geometric and arithmetic means (Lewontin and Cohen 1969). In our model population size at time t is given by 40 MODELING EVOLUTION Ntþ1 ¼ N0 l1 l2 l3 . lt ¼ N0 t Y li ð1:32Þ i¼1 l. The expected We assumed that li is a random, uncorrelated variable with mean  population size at time t is then given by the product of the initial population size, N0, times the expectation of the product l1l2l3 .

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