Next, notice that as t becomes large, the first term on the right-hand side of (A2) becomes small. In the limit of very large t(A6)
And thus the expected value of qt given that cohort t experiences environmental state 1 is(A7)
Where from the assumptions of the model
Prob(θt-j-1 = 1∣θt = 1) = (½ + ½(1 - 2a)j+1) (1/2) (A8)
Combining these expressions and simplifying yields the expression for the expected value of qt given that cohort t experiences environmental state t, E(qt∣θt = 1), that is identical to Equation (11) except that the parameter a is substituted for m.
Taking the partial of expected individuals fitness with respect to S holding q constant yields(A9)
Next notice changing S is just a change in the scale of measurement of x. Thus
P1(S + δS,d,M) = P1(S,dS/(S + δS), MS/(S + δS) (A10)
Combining these expressions and using the fact that (12) must be an equality if d is at its optimum yields the following expression(A12)
which using the definitions of P1 and P2 becomes(A13)
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Publication information: Book title: Social Learning:Psychological and Biological Perspectives. Contributors: Thomas R. Zentall - Editor, Bennett G. Galef Jr. - Editor. Publisher: Lawrence Erlbaum Associates. Place of publication: Hillsdale, NJ. Publication year: 1988. Page number: 47.