5 10 15 20
Fig. 4.4. Attractors in the Ricker model with k =1 for different values of the parameter r.
f (x2)x2 = x1 and xi = x2. This two-periodic orbit is stable for e2 < r <r2, where r2 « 12.509. At r2 another period-doubling bifurcation occurs, and a four-periodic orbit appears. When the parameter r is increased further, a period-doubling route to chaos is observed (See Figure 4.4). The parameter k does not affect dynamics qualitatively, and is thus only a scaling factor.
4.2.2. Finite number of patches with the Ricker model
Let us next study a metapopulation model with n patches, and with local dynamics as described above. This model has been extensively studied by Ref. 15. Concerning migration, it is assumed that a migrating individual survives migration with probability F and immigrates immediately into a patch. The population density in patch i in the next time step will thus be
Xi,t+1 = (1 - rn)fi(xiit)xiit H--'^mfj(x^t)x^t. (4.4)
Such a metapopulation does not necessarily have only one feasible at-tractor. Take as an example a situation, where the parameters r are chosen such that in an isolated patch there would be a two-cyclic orbit. At least for small values of the migration parameter m, the metapopulation can be either in an in-phase (Figure 4.5a) or an out-of-phase (Figure 4.5b) cycle.
If local population sizes are large in one time step and small in the next a) In-phase cycle a) In-phase cycle
b) Out-of-phase cycle b) Out-of-phase cycle
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