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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 [10].

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 tex2html_wrap_inline2368 as


When we only allow total disasters, i.e. tex2html_wrap_inline2370 , the nasty integral containing tex2html_wrap_inline2372 disappears in both Eqs. (36 and 37), and tex2html_wrap_inline2064 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 tex2html_wrap_inline2376 is sufficiently close to zero. A secant method is then used to find the root of equation (36).

If the disasters are partial, both tex2html_wrap_inline2064 and the steady-state density distribution tex2html_wrap_inline2380 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 tex2html_wrap_inline2380 we computed tex2html_wrap_inline2380 from the ode


in which tex2html_wrap_inline2386 is the maximum attainable local population size obtained by solving tex2html_wrap_inline2388 . In the iterative scheme we started with an initial guess of tex2html_wrap_inline2064 , then we computed a starting distribution tex2html_wrap_inline2380 by solving Eq. (42) with tex2html_wrap_inline2394 . Next we computed from tex2html_wrap_inline2380 a lookup table for


and a new value of tex2html_wrap_inline2064 by solving the ode's


In the subsequent steps we computed a new distribution and tex2html_wrap_inline2064 by using Eq. (42) with the lookup table from the previous iteration. The iteration process was stopped when the subsequent tex2html_wrap_inline2064 's were sufficiently close. We were not able to find unstable steady states in this way.

To compute the dominant eigenvalue tex2html_wrap_inline2404 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 tex2html_wrap_inline2404 -dependent ode's to solve with tex2html_wrap_inline2064 , given a lookup table for tex2html_wrap_inline2380 . The secant method is used again to find the dominant real root of Eqs. (45-46).

next up previous
Next: Characteristic equations for the Up: A Deterministic Size-Structured Metapopulation Previous: References

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