## More Than One Birth State

When individuals can be born in different birth states (think, for example, of different patches), the previous results generalize with little change. Let Xj (a) be the average rate (or ratio) at which an individual born in state j gives birth to offspring in state i, and A (a) = (X j (a)). Then r has to be determined from

or as the rightmost solution of determinant (I - A (r)) = 0 [20a'']

e raAjj(a)da

For ease of reference, denote A(0) as A, and denote the dominant right eigenvector of A as U and the dominant left eigenvector as VT. Both eigenvectors are supposed to be normalized such that 1 TU = 1 and VTU 1. U can then be interpreted as a generation-wise stable birth state distribution, and the components of V as generation-wise reproductive values.

The average lifetime offspring number can be found by averaging the total number of offspring begotten by individuals born in the various birth states:

Relation [17] applies without change, while [18b] and [16] a2 in [19a] have to be modified. In the case of mutants that are not too different from the resident:

with the index r indicating the resident values, and

where v, and Uh denote the stochastic number of offspring in state i produced by a resident individual that itself was born in state h, respectively, the corresponding components of Vr and Ur .

Cyclic environments can be treated like constant ones through the ploy of making the phase of the cycle a component of the birth state.