   Next: Specific growth dynamics Up: A Deterministic Size-Structured Metapopulation Previous: A Stieltjes renewal equation

For a determination of the fate of a metapopulation we have to analyze the stability of the size (or age) distribution of the patches, the disperser density and the fraction of unsuitable patches in steady state. In the following, steady states will be denoted by a bar. Before concentrating on some specific realizations of v(x) and we derive some general results.

The steady state disperser density is obtained by putting Eq. (3) equal to zero. We get From Eq. (20) we obtain the steady state fraction of empty patches At last, the steady state age distribution is given by in which is the solution of Eq. (2) starting from with density , and experiencing the steady state disperser density on the interval [0,t].

Due to conservation of patches the steady state age distribution must satisfy and so Equations (1,22, 23, and 25), should provide the means for calculating the actual steady state values. For computational convenience however, we determined the steady states numerically using Eqs. (1,22, and 25), in combination with the steady state size distribution which was obtained after integration of Eq.(10) (see appendix A for the numerical scheme).

The asymptotic behavior around the steady state is dependent on the value of the dominant eigenvalue of the resolvent of Eqs. (19-21). In appendix B, Eqs. (45-46), the characteristic equation is given from which the dominant eigenvalue is computed.

Equation (24) provides us the steady age distribution: the distribution of patch lifetimes between two consecutive disasters. In a field situation one would be more interested in the mean lifetime of a patch before it is struck by a disaster wiping out the whole population, which is given by the equation    Next: Specific growth dynamics Up: A Deterministic Size-Structured Metapopulation Previous: A Stieltjes renewal equation

John Val
Wed Feb 26 07:30:07 EST 1997