. Advances in herpetology and evolutionary biology : essays in honor of Ernest E. Williams. Williams, Ernest E. (Ernest Edward); Herpetology; Evolution. 684 Advances in Herpetologij and Evulutionary Biology population is varying cyclically or cha- otically, how does immigration modify the distribution of densities shown by the population over time? As immigration increases, the slope of the growth curve at an equilibrium where X' < 0 changes according to dk' ^ /d2GN /dN*) dl ^dN2 dl ^' Given a growth curve that is concave downward. d£G dN2 0. Therefore the effect of immigration upon populat
. Advances in herpetology and evolutionary biology : essays in honor of Ernest E. Williams. Williams, Ernest E. (Ernest Edward); Herpetology; Evolution. 684 Advances in Herpetologij and Evulutionary Biology population is varying cyclically or cha- otically, how does immigration modify the distribution of densities shown by the population over time? As immigration increases, the slope of the growth curve at an equilibrium where X' < 0 changes according to dk' ^ /d2GN /dN*) dl ^dN2 dl ^' Given a growth curve that is concave downward. d£G dN2 0. Therefore the effect of immigration upon population stability is determined by the concavity of the growth curve G(N). Standard discrete-time models provide ready examples of both effects. One model for which immigration is destabi- lizing is the discrete logistic equation N(t+1) = N(t) (1 + r ^ N(t)) + I (3) where r is the intrinsic growth rate. The stability properties of this model without immigration are well-understood (May and Oster, 1976; Roughgarden, 1979). In the Appendix, I outline the stability char- acter of the discrete logistic with immi- gration. Figure 2 depicts the stability domains of this model. (The dashed and dotted lines are explained in the Appen- dix.) The overall impression from this figure is that immigration reduces or eliminates stability in the peripheral population. By contrast, in other models immigra- tion may stabilize an intrinsically un- stable population. Figure 3 shows the re- sults of a local stability analysis of the following model: N(t+1) = N(t) exp[r(l - N(t)/K)] + I (4) With no immigration and r < 3, the dy- namics of this model resemble those of (3) May (1976). As immigration increases, the domain of unstable behavior dimin- ishes. The contrast between Figures 2 and 3 is striking. In a more detailed analysis it can be shown that the oppos- ing consequences of immigration for the two models are due to the downward concavity of (3) and the upward concavity of (4) at high values o
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