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A General Theory of Evolution
By selection by density dependent competitive interactions

Fisher's fundamental theorem of natural selection

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Selection by density dependent competitive interactions provides a solution to a long-standing problem relating to Fisher's (1930) fundamental theorem of natural selection. Defining fitness as the population dynamic growth rate (r), also known as the Malthusian parameter, Fisher introduced the fundamental theorem as

The rate of increase in fitness of any organism at any time is equal to its genetic variance in fitness at that time.
From this formulation there grew the misinterpretation that the fundamental theorem deals with an overall increase in the average Malthusian parameter of a population (e.g., Wright, 1930, 1955; Li, 1955; Kempthorne, 1957; Crow and Kimura, 1956; Kimura, 1958, 1965; Kojima and Kelleher, 1960; Ewens, 1969). Consequently, the theorem was soon seen as a special case that failed for most natural populations (see Kimura, 1958; Li, 1967; Karlin and Feldman, 1970; Turner, 1970; Elandt-Johnson, 1971; Jacquard, 1974; reviewed Edwards, 1994).

But Price (1972a) made it clear that the fundamental theorem makes statements only about a partial increase in r caused by natural selection. Hence the population evolves toward an equilibrium where the partial increase is balanced against a partial decline caused by the populations impair of the environment. The result is an average Malthusian parameter that takes a rather stable value around zero while the population equilibrium, or carrying capacity, increases steadily, as suggested also by k-selection.

After Price the fundamental theorem regained some of its glory as a correct mathematical statement (e.g., Frank and Slatkin, 1992; Edwards, 1994; Burt, 1995). But selection by density dependent competitive interactions shows that the distinction between a partial and a total change in r does not save Fisher's idea of a partial increase in r (Witting, 2000a). Instead of a partial increase we may expect a partial decline when the level of interactive competition is sufficiently high. It is only when the organism lives in a density independent environment that the fundamental theorem seems to hold as a general principle. At this limit the theorem defines a law of hyper-exponential increase in the population abundance (Witting, 2000b); a law that includes the Malthusian law of exponential increase (Malthus, 1798) as the special case with no evolutionary potential.

References

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