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J Biol Dyn. 2012;6 Suppl 2:80-102. Epub 2012 Aug 31.

Stable bifurcations in semelparous Leslie models.

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Department of Mathematics and Interdisciplinary Program in Applied Mathematics, 617 N Santa Rita, University of Arizona, Tucson, AZ 85721, USA.


In this paper, we consider nonlinear Leslie models for the dynamics of semelparous age-structured populations. We establish stability and instability criteria for positive equilibria that bifurcate from the extinction equilibrium at R (0)=1. When the bifurcation is to the right (forward or super-critical), the criteria consist of inequalities involving the (low-density) between-class and within-class competition intensities. Roughly speaking, stability (respectively, instability) occurs if between-class competition is weaker (respectively, stronger) than within-class competition. When the bifurcation is to the left (backward or sub-critical), the bifurcating equilibria are unstable. We also give criteria that determine whether the boundary of the positive cone is an attractor or a repeller. These general criteria contribute to the study of dynamic dichotomies, known to occur in lower dimensional semelparous Leslie models, between equilibration and age-cohort-synchronized oscillations.

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