Appendix B. Optimal switching time with deferred search costs. A pdf version of this appendix is also available.
Assuming a sequential, finite time search model, and ignoring deferred costs, we can use the model developed by Ward (1987) to show that the expected fitness for an individual that switches from being selective to being nonselective at time ts during the search period is given by
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where P0(t) = 
for
0 
t < ts
and P0(t) = 
for
ts t T.
Also, g, h are constants
given by 
and 
.
(For definitions of terms, see Appendix A).
Substituting the appropriate expressions for P0(t) and evaluating the integrals leads to
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However, the above fitness does
not include the deferred cost of search. In our derivation below, we include
the deferred costs under the assumption that 
is
the deferred cost of search per unit time.
If the animal is in selective mode, the probability of the animal finding habitat A in the time interval (t, t+dt) is eA P0(t)dt, and the expected deferred cost of search is
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Using similar reasoning, the expected deferred search cost when the disperser is in nonselective mode can be derived as
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Integrating the above two expressions and summing them up yields the total expected deferred cost of search as
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The expected fitness, including deferred search cost of a disperser arriving at a habitat, as a function of switching time, ts, is then
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The optimal switching time, t*,
may be found by solving 
and
verifying that the resulting fitness at ts= t* is a
maximum and not a minimum.
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This further simplifies to
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Since 
,
we have
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The above equation has no simple analytic solution for ts. We use a numerical method to search for a solution in the interval (0,T). If such a solution exists, it is a possible value of t*. However, this solution may represent the switching time for either a maximum or a minimum fitness. Therefore, we need to simulate w(ts) in the neighborhood of the solution t* in order to numerically determine whether or not t* represents the switching time for maximum fitness.