## From Lifetime Offspring Number to Intrinsic Rate of Increase and Invasion Probability

Most ESS calculations consider situations where the environment can be assumed constant over community dynamical time, as these are about the only cases where analytical results can be obtained. For the remainder of this section, the same assumption will be operative.

In nonfluctuating environments, life histories with everybody born equal always allow an age representation, characterized by an average age-dependent effective birth rate (or ratio) A(a) (often seen decomposed into an age-dependent survival l(a) and conditional fecundity b(a) as A(a) = l (a)b(a)), from which the intrinsic rate of increase r can be calculated by solving with

fi i i \ J eTra A(a)da (respectively 1 — ^ e - raA(a)J

(From here on only the continuous time formulas will be displayed.) Lifetime offspring production equals

The two quantities are related through

For not too large |ln(R0)|, or when births concentrate around a single parental age, r « ln(Ro)/rb [18a]

with Tb, the average age when giving birth, pi /

For small positive ln(50) the establishment probability, p, can be approximated, under some mild conditions on the offspring number distribution, as p « 2ln(Ro)/a2

[19a with a the variance of the lifetime offspring numbers. For mutations with small effect the more easily obtained resident values can be substituted for Tb and a2, without affecting the order of the approximation.

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