Adaptive Diversification by Doebeli Michael;

Author:Doebeli, Michael;
Language: eng
Format: epub
Publisher: Princeton University Press

FIGURE 7.2. The period-doubling route to chaos in the difference equation G(n + 1) = g(n) · n with g(n) given by eq. (7.4). The x-axis is the parameter c = dG/dn(n*), where n* is the nonzero solution of G(n*) = n* given by eq. (7.7). For the plot, the parameter λ was fixed at λ = 5, and for a given value of c, eq. (7.8) was used to calculate the parameter b, and then eq. (7.7) was used to calculate the value of the parameter a under the assumption that the carrying capacity n* = 2000. Using these parameter values, the dynamics was run for 1,000 transient time steps, and then the next 1,000 time steps were plotted on the y-axis. This was done for 200 equidistant c-values between −0.5 and −4.5.

Based on the local dynamics (eq. (7.2)), a two-patch metapopulation can be constructed by assuming that after reproduction has occurred in each patch, a fraction d of the offspring migrate to the other patch. In slight abuse of terminology, I call d the dispersal rate. (Strictly speaking, d is a dispersal probability, and the term “rate” is usually reserved for events occurring per unit time in continuous time models.) In addition, one can assume that dispersal has a cost, so that only a fraction s of all dispersers actually arrive at the new location. If n1(t) and n2(t) denote the population densities in the two patches at the start of generation t, and g1 and g2 denote the per capita reproductive output functions in the two patches, then the population densities in the two patches after reproduction, but before dispersal are given by g1(n1(t)) · n1(t) and g2(n2(t)) · n2(t), respectively. During the dispersal phase, a fraction (1 − d) of these local populations stays in their patch, and a fraction d disperses, of which a fraction s arrives in the other patch. This leads to the following system of coupled difference equations for the discrete-time population dynamics in the two patches:

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