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# Characteristic equations for the steady states.

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