Bifurcations of cycles in nonlinear semelparous Leslie matrix models.

This paper develops a method for studying bifurcations that occur in a neighborhood of the extinction equilibrium in nonlinear semelparous Leslie matrix models. The method uses a Lotka-Volterra equation with cyclic symmetry to detect the existence and to evaluate the stability of bifurcating equilibria and cycles. An application of the method provides sharp stability conditions for both a single-class cycle and a positive equilibrium bifurcating from the extinction equilibrium. The stability condition for a bifurcating single-class cycle confirms that the periodicity observed in periodical insects occurs if competition is more severe between than within age-classes. The developed method is also used to investigate two examples of nonlinear semelparous Leslie matrix models incorporating predator satiation. The investigation shows that a single-class cycle, which is associated with the periodicity in periodical insects, is a unique stable cycle in a neighborhood of the extinction equilibrium if the density effects in survival probabilities are identical among age-classes.