Judy A. Stamps, V. V. Krishnan, and Mary L. Reid. 2005. Search costs and habitat selection by dispersers. Ecology 86:510–518.

Appendix C. Effect of deferred search costs on t*. A pdf file of this appendix is also available.

As noted in Appendix B, the optimal switching time, t*, is a solution to the equation,  ,where w(t_s) was shown in Appendix B to be

.

  

If we are interested in finding the conditions under which t* is independent of the deferred search cost per unit time, ,  we need

which yields

.

Noting that (h-g) = e_B and that , we can simplify the above equation to

.

Case 1:  = 0

If mortality rate is = 0, then , and the above equation reduces further to

.

Since e_B 0, this leads to the condition  _.

On the other hand, under conditions when the optimal switching time is independent of the deferred search cost, the optimal switching time is also the solution to the following equation (Ward  1987):

.

Since , when = 0 the above equation reduces to the condition

.

We can now combine the above condition with the earlier condition, , to obtain

.

Substituting h = (e_A+e_B+), g = (e_A+), and = 0 in the equation above, we arrive at the condition   . If this condition is true, then  has no effect on t*.  On the other hand, when  ,  we have , thus resulting in an increase of w’(t_s) as  increases. 

When w(t_s) is maximum, i.e., at  t_s = t*,  we have  w’(t_s) = 0 and . If  there is an increase in w’(t_s) due to an increase in  , it implies that t* will also increase as  increases. It can be shown, using similar reasoning, that when  , the effect of  is reversed and that t* decreases as  increases.

Case 2: 0

If we assume that the mortality rate, , is sufficiently small in comparison to e_A and e_B     

so that  ,  and ,  then the equation   leads to the following  relationship:

.

The conditions under which an increase in produces an increase in t* will, of course, depend upon the specific values of e_A, e_B, T, F_A and F_B. However, over the range of  parameter values which are relevant to habitat selection by dispersers,  this equation shows that when > 0, a larger value of e_AT is required to create a situation in which an increase in leads to an increase in t* , and, conversely, that a smaller value of e_AT  is required to produce a situation in which an increase in  results in a decrease in t*.

LITERATURE CITED

Ward, S. A. 1987. Optimal habitat selection in time-limited dispersers. American Naturalist 129:568–579.