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# Social Learning: Psychological and Biological Perspectives

By: Thomas R. Zentall; Bennett G. Galef Jr. | Book details

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Page 47

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(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.

APPENDIX 2

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)

and therefore

(A11)

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)

-47-

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