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Under directional selection, the mean phenotype is predicted to eventually evolve at rate k, but lag behind the mean environment �� by k/s. If the lag between the mean phenotype and the environment becomes too large, population growth rate becomes negative, at which point the population rapidly goes extinct. The expected population growth rate at time t is given as: (eqn 3) where http://www.selleck.cn/products/PD-0332991.html the maximal population growth rate can be given as, , where B offspring per capita when the mean phenotype is equal to its environment, that is, , and where V��,t, is equal to the sum: . In the simplest case of density dependence, for a population size of N breeding adults limited by a carrying capacity K, of the N expr offspring produced per generation only a maximum of K will survive; when N expr? http://www.selleckchem.com/products/a-1155463.html reproducing organisms). To evaluate the relative contribution of adaptation to reducing extinction risk, mean time to extinction of an adapting population can be compared with that of a population where the mean phenotype is otherwise held constant, that is, . An alternative measure of extinction risk that can be derived analytically is the theoretical maximum rate of environmental change at which a population could continue http://www.selleckchem.com/products/cb-5083.html to just replace itself (i.e. rmax?=?0) under directional selection. This measure of extinction risk is referred to as the critical rate of environmental change, kc (Lynch & Lande 1993). Beyond this rate, population growth rate becomes negative leading to rapid extinction. B��rger & Lynch (1995) derive the critical rate of change as: (eqn 4) where V��?=?V��,�� and . Obviously, no trait can be expected to evolve indefinitely. But on the basis that long-term selection experiments on small populations have shown responses of ten or more phenotypic standard deviations, a low critical rate of change could sustain an adaptive response to selection for hundreds, perhaps even thousands of generations before pleiotropic constraints are encountered (Lynch & Lande 1993; B��rger & Lynch 1995). Note that while these solutions can provide good estimates of mean time to extinction, they fail to account for the skewed distributions of extinction times expected to occur at rates of environmental change close to the critical rate. Caution should be taken therefore when interpreting the risk of extinction. Huey & Kingsolver (1993) showed that if fitness is subject to a specialist-generalist trade-off then thermal specialists should have a lower risk of extinction, all else being equal.