12.4. Metapopulation models

   Metapopulation is a set of local populations connected by migrating individuals.

   Local populations usually inhabit isolated patches of resources, and the degree
   of isolation may vary depending on the distance among patches:

                                     [metpop.gif]

   Metapopulation models consider local populations as individuals. Dynamics of
   local populations is either not considered at all, or is considered in a very
   abbreviated way. Most of metapopulation models are based on
   colonization-extinction equilibrium.

   One of the first metapopulation models was developed by MacArthur and Wilson
   (1967). They considered immigration of organisms (e.g., birds) from a continent
   to islands in the ocean. The proportion of islands colonized by a species, p,
   changes according to the equation:

                                    [eqmetpp1.gif]

   Equilibrium proportion of colonized islands can be found by solving the equation
   dp/dt = 0:

                                    [eqmetpp2.gif]

   Assume that extinction rate declines with increasing island diameter S:

                                    [eqmetpp3.gif]

   and colonization rate declines with increasing distance D from the continent:

                                    [eqmetpp4.gif]

   Now, the proportion of colonized islands becomes a function of island size and
   its distance from the continent:

                                    [eqmetpp5.gif]

   If there is a group of species with similar biology and similar migration
   capabilities, then the proportion of colonized islands is proportional to the
   number of species that live on an island. Now the model can be tested using
   regression:

                                    [eqmetpp6.gif]

   where

                                    [eqmetpp7.gif]

   Example: There are 100 bird species on the continent and the number of species on
   islands is in the following table:

   Island No. Diameter, km (S) Distance from continent, km (D) Number of bird
   species Proportion of species (p*) ln(p*/(1-p*))
   1 10 30 40 0.40 -0.405
   2 3 50 5 0.05 -2.944
   3 20 100 20 0.20 -1.386

   Using linear regression of ln(p*/(1-p*) against two factors: S and D we get the
   following model parameters: a = 0.229; b = 0.0467; and g = -1.29. These
   parameters can be used to predict the number of species on other islands using
   information about island size and its distance from the continent.

   A similar metapopulation model describes metapopulation dynamics without a
   continent. In this case, neighboring islands become a source of colonization. The
   proportion of colonized islands changes according to differential equation:

                                    [eqmetpp8.gif]

   where c = colonization rate per 1 island, and e = extinction rate. Equilibrium
   proportion of colonized islands can be found by solving the equation dp/dt = 0:

                                    [eqmetpp9.gif]

   This model can be expanded by incorporating island size and degree of isolation
   in the same way as the previous model.

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   [4]Alexei Sharov 11/16/95

References

   1. http://alexei.nfshost.com/PopEcol/lec12/stratdsp.html
   2. http://alexei.nfshost.com/PopEcol/lec12/dispers.html
   3. http://alexei.nfshost.com/PopEcol/lec13/outbreak.html
   4. http://alexei.nfshost.com/~sharov/alexei.html