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# Numerical methods

In this section we will present the numerical schemes with which the steady states and the stability of these were computed. Most of the numerical analysis on the steady states were performed with Mathematica@. The results were checked by numerical integration of the pde model (Eqs. 10-16) with the escalator box car train method .

The central equation in computing the steady-states will be Using Eq. (26), Eq. (34) can be rewritten as In steady-state we can think of the population size in a patch being originated from growth in an undisturbed patch, i.e. population size can be obtained by solving the ode Using this relation we can rewrite the integral in Eq. (35) as a fictive age integral Combing Eqs. (20, 23, and 26) we can write as When we only allow total disasters, i.e. , the nasty integral containing disappears in both Eqs. (36 and 37), and is the only quantity to be determined. In this case the integrals on the rhs. of (36) are computed by simultaneously solving the following set of ode's up to the age at which is sufficiently close to zero. A secant method is then used to find the root of equation (36).

If the disasters are partial, both and the steady-state density distribution are unknown. It turned out that the following iterative scheme converged slowly, to stable steady states. Instead of using Eq. ( 26) as a substitute for we computed from the ode in which is the maximum attainable local population size obtained by solving . In the iterative scheme we started with an initial guess of , then we computed a starting distribution by solving Eq. (42) with . Next we computed from a lookup table for and a new value of by solving the ode's In the subsequent steps we computed a new distribution and by using Eq. (42) with the lookup table from the previous iteration. The iteration process was stopped when the subsequent 's were sufficiently close. We were not able to find unstable steady states in this way.

To compute the dominant eigenvalue from the characteristic equations (Eqs 45-46) we used the same procedure as in the computation of the steady states in case of total disasters. Again we change to fictive patch age, and build a system of -dependent ode's to solve with , given a lookup table for . The secant method is used again to find the dominant real root of Eqs. (45-46).   Next: Characteristic equations for the Up: A Deterministic Size-Structured Metapopulation Previous: References

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