We now concentrate on some specific mechanistic assumptions on local growth and disaster rate. We consider three models for local growth commonly used in single population models:

- , i.e. exponential growth with migration.
- , i.e. logistic growth with migration.
- , i.e. migration with logistic growth in which growth is dependent on the contact rate between two organisms.

Disasters can happen independent or dependent on local population size. The following rates are used.

- , i.e. there is an exponential distribution for the waiting time to the next disaster, and population size has no influence on disasters.
- , i.e. an increase of population size enhances the probability for a patch of being hit by a disaster.
- , i.e. an increase of population size decreases the probability of being hit by a disaster. This model is assumed to reflect the effect of demographic stochasticity on small populations.

Furthermore a disaster may be total with probability *q*. In case of a
partial disaster the new population size is taken from a homogeneous
distribution. Thus

Lastly, we make a distinction between (a) a situation in which patches are always suitable for habitation and (b) one in which a patch might become unsuitable for habitation after the occurrence of a total disaster.

The trivial steady state

is a solution of Eqs. (19-21) for any combination of the models above. The dominant eigenvalue related to the trivial steady-state is given by the largest root of

where

The trivial equilibrium is unstable when this root is positive, which is guaranteed when

i.e. the expected per capita death caused by a disaster must be smaller than the net expected per capita reproduction.

For growth models and one can find parameters for which the trivial steady state is either stable or unstable. For model the trivial steady state is always stable (figs. 1-5).

Whether a non-trivial equilibrium exists depends not only on the parameter values, but also on the local dynamics and disaster rate (table 1, and figs. 1-5. A first result we put forward is that there is no upper bound to population size in the metapopulation unless there is a negative influence of local population size on patch survival (e.g. ) or local population growth (e.g. ). So random disasters and demographic stochasticity in combination with exponential growth cannot keep a metapopulation within bounds. If the population stays within bounds, only three possible outcomes were found with all combinations and parameter values we tried. One in which there is but one stable non-trivial equilibrium, one in which there is an unstable and a (larger) stable non-trivial equilibrium, and in cases where the bifurcation of the trivial equilibrium is subcritical both former outcomes are present. We never observed an oscillatory behavior, however we were not able to explore a large parameter set in case of partial disasters, since the numerical method for computing steady states and stability is difficult and time consuming.

**Figure 1:** Mean population density (solid line) and disperser density
(dotted line) of the steady-state population density distribution for
logistic local growth as a function of the parameters . If
not varied the parameters had the values *r*=0.12, *k*=0.1, *a*=1, *l*=0.01,
, *q*=0.5, *u*=0, (f:*u*=0.5).

**Figure 2:** Mean population density (fat line) and disperser density
(thin dashed line) of the steady-state population density distribution for
local growth with Allee-effect, as a function of the parameters . If not varied the parameters had the values *r*=1.2, *k*=0.1, *a*=1, *l*=0.01, , *q*=0.5, *u*=0.

**Figure 3:** Mean population density (solid line) and disperser density
(dashed line) of the steady-state population density distribution for
exponential local growth and density dependent disasters as a function of the parameters . If not varied the
parameters had the values *r*=0.12, *k*=0.1, *a*=1, *l*=0.01, , .

**Figure 4:** Mean population density (solid line) and disperser density
(dashed line) of the steady-state population density distribution for
logistic local growth and density dependent disasters
as a function of the parameters . If not varied the
parameters had the values *r*=0.12, *k*=0.1, *a*=1, *l*=0.01, , .

**Figure 5:** Mean population density (fat line) and disperser density
(thin dashed line) of the steady-state population density distribution for
local logistic growth density dependent rate of disasters , as a function of the parameters , . If not varied the parameters had the values *r*=0.12, *k*=0.1, *a*=1, *l*=0.01, , , , and . In
varying we put at such a value that .

For a given local growth in case of total disasters at constant rate the
bifurcation pattern is similar to bifurcation patterns for that local growth
when isolated from the metapopulation context. So nothing new here.
More interesting is that in a metapopulation context various combinations of
local growth and extinction and partial disasters do not behave directly
according to their local growth dynamics but show dynamics similar
to other local dynamics. Exponential growth in combination with a disaster rate
increasing with population size is similar to logistic growth. If in
addition this disaster rate is decreasing in population size at small
population size the dynamics resembles that of an Allee effect.
A bifurcation diagram in case of partial disasters for the combination of
logistic growth and constant disaster rate (fig. 1a) shows that
behavior similar to that for a combination of logistic growth,
decreasing in *x*, and total disasters (fig. 1a). The survival of a part of
the population after a disaster results in larger population sizes, and
causes a region where there are two non trivial steady states.
So in general we observe that processes resembling a rescue effect, i.e.
a disaster rate decreasing with population size and partial disasters,
add an Allee effect to the metapopulation dynamics, and a disaster rate
increasing with population size adds ``density dependent growth''.

Wed Feb 26 07:30:07 EST 1997