  This equation is also subject to some further conditions Pence and Ramsey argue are biologically reasonable, for example that selection is density dependent, that there is an extinction threshold of some population size below which extinction obtains, and most important that the population dynamics are non-chaotic. Pence and Ramsey show how their definition avoids all of the problems discussed above, as well as some other counterexamples to previous proposals, owing to its incorporating and combining all possible daughter populations over indefinitely many reproductive generations. They also show that a general version of the same equation has been independently derived by mathematical biologists, and they argue that so defined, fitness has a substantial role to play in biology, especially the field of “adaptive dynamics”. They show how standard operation fitness measures biologists employ can be derived from the definition, including the simple geometric and arithmetic means, along with their higher moments (variance, skew, etc.) when multi-generational effects are present. Thus, for example, Pence and Ramsey’s proposal has the result that an organism with a probability of reproducing one offspring per generation equal to 1 will be fitter than an organism with a .5 probability of 0 offspring and a .5 probability of 2 offspring, since the second organism’s possible lines of descent will all have a probability of extinction that approaches 1 as time approaches infinity.