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Next: Model comparison Up: A Deterministic Size-Structured Metapopulation Previous: Steady states and stability

Specific growth dynamics

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:

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

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

eqnarray237

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

displaymath2070

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

displaymath2071

where

eqnarray252

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

  equation256

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

For growth models tex2html_wrap_inline2088 and tex2html_wrap_inline2090 one can find parameters for which the trivial steady state is either stable or unstable. For model tex2html_wrap_inline2092 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. tex2html_wrap_inline2094 ) or local population growth (e.g. tex2html_wrap_inline2096 ). 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.

   figure268
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 tex2html_wrap_inline1852 . If not varied the parameters had the values r=0.12, k=0.1, a=1, l=0.01, tex2html_wrap_inline1862 , q=0.5, u=0, (f:u=0.5).

   figure282
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 tex2html_wrap_inline1870 . If not varied the parameters had the values r=1.2, k=0.1, a=1, l=0.01, tex2html_wrap_inline1862 , q=0.5, u=0.

   figure295
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 tex2html_wrap_inline1886 as a function of the parameters tex2html_wrap_inline1888 . If not varied the parameters had the values r=0.12, k=0.1, a=1, l=0.01, tex2html_wrap_inline1862 , tex2html_wrap_inline1900 .

   figure308
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 tex2html_wrap_inline1886 as a function of the parameters tex2html_wrap_inline1888 . If not varied the parameters had the values r=0.12, k=0.1, a=1, l=0.01, tex2html_wrap_inline1862 , tex2html_wrap_inline1900 .

   figure321
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 tex2html_wrap_inline1918 , as a function of the parameters tex2html_wrap_inline1920 , tex2html_wrap_inline1922 . If not varied the parameters had the values r=0.12, k=0.1, a=1, l=0.01, tex2html_wrap_inline1862 , tex2html_wrap_inline1900 , tex2html_wrap_inline1936 , and tex2html_wrap_inline1938 . In varying tex2html_wrap_inline1922 we put tex2html_wrap_inline1942 at such a value that tex2html_wrap_inline1944 .

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 tex2html_wrap_inline2192 (fig. 1a) shows that behavior similar to that for a combination of logistic growth, tex2html_wrap_inline2194 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''.


next up previous
Next: Model comparison Up: A Deterministic Size-Structured Metapopulation Previous: Steady states and stability

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