Abstract
The bifurcation that occurs from the extinction equilibrium in a basic discrete time, nonlinear juvenile-adult model for semelparous populations, as the inherent net reproductive number R0 increases through 1, exhibits a dynamic dichotomy with two alternatives: an equilibrium with overlapping generations and a synchronous 2-cycle with non-overlapping generations. Which of the two alternatives is stable depends on the intensity of competition between juveniles and adults and on the direction of bifurcation. We study this dynamic dichotomy in an evolutionary setting by assuming adult fertility and juvenile survival are functions of a phenotypic trait u subject to Darwinian evolution. Extinction equilibria for the Darwinian model exist only at traits u• that are critical points of R0(u). We establish the simultaneous bifurcation of positive equilibria and synchronous 2-cycles as the value of R0(u•) increases through 1 and describe how the stability of these dynamics depend on the direction of bifurcation, the intensity of between-class competition, and the extremal properties of R0(u) at u•. These results can be equivalently stated in terms of the inherent population growth rate r(u).
Highlights
A model for the dynamics of a population structured by juvenile and adult classes is described by the equationsJ = f φ (J, A) A (1a)A = sσ (J, A) J (1b) where J and A denote juvenile and adult densities, respectively, and J and A denote these densities after one unit of time
Following the methodology of evolutionary game theory (EGT), we model the dynamics of the mean phenotypic u by assuming that its change in time is proportional to the change in fitness as a function of u, which is taken to be the population growth rate ln r where r = r (J, A, u) is the spectral radius of the projection matrix f (u) φ (J, A, u)
Since the variance v does not appear in the equilibrium equations, it follows that the bifurcating branch of positive equilibria in Theorem 4.1 do not depend on v
Summary
A model for the dynamics of a population structured by juvenile (immature) and adult (mature) classes is described by the equations. Our results provide criteria for the evolutionary stability of either equilibria with overlapping generations or oscillations with non-overlapping generations These criteria will show how the trait dependence of adult fertility and juvenile survival rates, in addition to density effects on these rates, determine which of these two alternatives occurs in an evolutionary setting, at least as implied by the EGT-JA model (3). By Theorem 2.2 both branches of positive equilibria and synchronous 2-cycles bifurcate to the right and one branch is stable and the other is unstable In this dynamic dichotomy, the positive equilibria are stable if |cw (u)| > |cb (u)| , i.e. the magnitude of within-class competition intensity is larger than that of between-class competition.
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