Carol S. Thornber and Steven D. Gaines. 2004. Population demographics in species with biphasic life cycles. Ecology 85:1661–1674.

Appendix B. Stability of the logistic models.

We explored the stability of equilibria for a range of per capita parameters of the logistic models. Simulations were run for various per capita rates, and their resulting equilibria were identified as stable or unstable. To analytically determine their stability, Jacobian matrices were constructed from the equilibria, and their resulting eigenvalues were compared with the population's observed behavior (the mathematics are much too lengthy and complex to be included here). Eigenvalues with an absolute magnitude less than one indicate a population with a stable equilibrium; if the magnitude is greater than this, the equilibrium is unstable. As expected, populations generally became less stable the larger their intrinsic rate of increase. However, there were some local fluctuations in stability for particular per capita rates; in some cases as the per capita mortality rate increased, the system became less stable (Fig. B1, below). This complexity of patterns of stability indicates that it is not easy to predict stability in haploid-diploid species with discrete growth.

FIG. B1. In this example, we measured stability by determining the dominant eigenvalue of the Jacobian matrix for Eqns. 11, 12 (see paper) for a range of values for . Values used were . As  increases, the stability first decreases and then increases.