Infinite Dilution

The previous deliberations dealt with concept engineering. To arrive at a general fitness definition also an assumption is needed: any influences of the focal population on the environment can be neglected. This will be referred to as the infinite dilution assumption. It is needed to justify speaking of the environment as a second independent variable coordinate with the traits.

Although infinite dilution of population effects is sufficient for delimiting a framework within which to define invasion fitness, to make the resulting concept into a useful tool a more refined variant is needed: The influence single individuals exert on the environment is negligible, although for large populations the effect of the added individual contributions may be substantial. This individual-centered concept of infinite dilution has the advantage that it encompasses both large resident and small invader populations. Under the assumption that individual effects are infinitely diluted, the effect of any small focal (sub)population, be it a mutant population or any other subpopulation having our attention, will also be infinitely diluted. The individual-centered infinite dilution assumption will be operative throughout the next three sections, to be replaced by a slightly extended version thereafter.

Of course, a growing focal population will not forever stay infinitely diluted. Yet, for populations starting from only a few founders, as is, inter alia, the case for mutants, the infinite dilution approximation often is sufficiently good for a sufficiently long time that many biological conclusions can be based on it. The fact that in most modeling contexts small invader populations as well as populations on the brink of extinction are close to infinitely diluted, allows bringing many fluctuating physiologically and spatially structured population models almost fully in line with simple viability selection counterparts.

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