Abstract
Temporal synchronization in population dynamics is a common ecological phenomenon that can have important consequences for conservation and management strategies. Though many environmental factors may cause cycling, it is also possible for cycles to occur due to intrinsic properties of the population, such as competition between individuals at different developmental stages. For discrete-time matrix models, competition between stages may result in synchronous cycles in which stages are temporally separated. In this paper we define a class of matrix models which exhibit a simultaneous bifurcation of positive equilibria and synchronous 2-cycles when the extinction equilibrium destabilizes. We also show that a dynamic dichotomy exists between the two steady states such that, if the steady states are bifurcating forward, then they have opposite stability conditions. Which of the two steady states is stable depends on measures of the within-class and between-class competition. These results extend the fundamental bifurcation theorem to a class of matrix models whose inherent projection matrix has index of imprimitivity two.
Published Version
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