Sup- Together With Each Other We Can Help Make AR-A014418 A Lot Better !
A female will search for a mate provided AR-A014418 chemical structure that c is less than the mean male quality �� because she otherwise has no incentive to engage in the search process. In particular, her expected net gain if she engages in search is negative whenever c?>?��. The cumulative distribution of the male indicator character X, namely F, is more realistically expected to fluctuate from generation to generation and in the next section of the paper we establish sufficient conditions for the evolutionary stability of a genetically determined t* to perturbations of F. 3.?Evolutionary stability of a genetically fixed threshold The optimal threshold t* is a function of the distribution F. If t* is continuous with respect to F, then under some conditions, which we will establish, it is stable to generational perturbations of F. In particular, we shall establish conditions on perturbations of F across generations that allow an unlearned, genetically determined t* to be optimal over an evolutionary time scale in comparison to any other genetically determined threshold. Imagine a sequence of distributions Fn?=?F1, F2, F3, �� of X on the interval [0, ��) for which u has finite mean ��n?>?c, where Fn is an evolutionary sequence of distributions, experienced over a sequence of n generations of searchers, that converges, as will be specified, on F. For each Fn there is an optimal threshold criterion tn*. If these threshold criteria converge Cofactor on the optimal threshold t* under F, then t* is optimal over the evolutionary sequence Fn. Thus, our objective is to establish conditions on Fn which imply that tn* converges on t*. Theorem?1 If Fn converges to F uniformly on the interval [0, ��) of X and u is bounded on [0, ��), then tn* converges to t*. Proof It cannot be true that tn* is unbounded. Imagine that the subsequence tni* of tn* tends toward infinity. Selleckchem Mdm2 inhibitor Then c=��tni*��u��x1?Fnixdx=��tni*��u��x1?Fx+Fx?Fnixdx=��tni*��u��x1?Fxdx+��tni*��u��xFx?Fnixdx�ܡ�tni*��u��x1?Fxdx+��tni*��u��xFx?Fnixdx�ܡ�tni*��u��x1?Fxdx+supux?supFx?Fnix. (4) The first term on the right-hand side approaches zero as i increases to infinity because u has a finite mean with respect to F. The second term converges to zero because Fni converges uniformly to F. This contradicts the assumption that c?>?0. The sequence tn* must consequently be bounded. The Bolzano�CWeierstrass Theorem then assures us that tn* converges to some number [24]. Suppose that tn* converges to the limit S, where S?��?t*. Then for i sufficiently large 0=c?c=��tni*��u��x1?Fnixdx?��t*��u��x1?Fxdx=��tni*t*u��x1?Fnixdx+��t*��u��x1?Fnixdx?��t*��u��x1?Fxdx=��tni*t*u��x1?Fnixdx+��t*��u��xFx?Fnixdx. (5) The second integral converges to zero as i increases to infinity. The Dominated Convergence Theorem applies to the first integral, with the integrable dominating function u(t) [25]. The conclusion is that 0=��St*u��x1?Fxdx. (6) Because u strictly increases with X and F(t*?)?
