In this section we will derive the characteristic equation from which the
stability of the equilibria (Eqs. 22-24) of the
metapopulation model (Eqs. 19-21) can be determined.
To do so we first linearize system (Eqs. 19-21)
around *b* and . Thereafter we insert an exponential trial
solution, which directly gives the wanted equation.

For the linearization define the deviations from the steady states as

Substitution in (Eqs. 19-21) and Taylor expansion up to first order results in

and

If we now insert the exponential trial solution

define , and , and
use the fact that are only dependent
on *D* through *x*, we get the characteristic equations

and,

in which

are quantities still left to be computed. If we define

and follow the derivations laid out in appendix C, then (Eqs. 47,49,50) are computed from the differential equations

in which

The initial conditions for these ode's are

Equation (52) can be solved resulting in

Now Eq. (48) can be written as

If is constant and if there are only total disasters, then and the linearized equations simplify to an uncoupled system of characteristic equations: one equation for the patch dynamics

which has dominant eigenvalue , and therefore the steady-state age distribution is neutrally stable, and another equation for the population living on this neutrally stable distribution

Wed Feb 26 07:30:07 EST 1997