On the Dynamic Dichotomy between Positive Equilibria and Synchronous 2-cycles in Matrix Population Models

Abstract

For matrix population models with nonnegative, irreducible and primitive inherent projection matrices, the stability of the branch of positive equilibria that bifurcates from the extinction equilibrium as the dominant eigenvalue of the inherent projection matrix increases through one is determined by the direction of bifurcation. However, if the inherent projection matrix is imprimitive this bifurcation becomes more complicated. This is the result of the simultaneous departure of multiple eigenvalues from the unit complex circle. Matrix models with imprimitive projection matrices commonly appear in models of semelparous species, which are characterized by one reproductive event that is often followed by death. Due to the imprimitivity of the projection matrix, semelparous Leslie models exhibit two contrasting dynamics, either equilibria in which all age classes are present or synchronized cycles in which age classes are separated temporally. The two-stage semelparous Leslie model has index of imprimitivity two, meaning that two eigenvalues simultaneously leave the unit circle when the dominant eigenvalue increases past one. This model exhibits a dynamic dichotomy in which the two steady states have opposite stability properties. We show that this dynamic dichotomy is a general feature of synchrony models which are characterized by the simultaneous creation of a branch of positive equilibria and a branch of synchronous 2-cycles when the extinction equilibrium destabilizes (Chapter 3). A synchrony model must, necessarily, have index of imprimitivity two but is not limited to models of semelparous species. We provide a specific example of a synchrony model for an iteroparous species which is motivated by observations of a cannibalistic gull population (Chapter 2). We also extend the study of the synchrony model to a Darwinian model which couples population dynamics with the dynamics of a suite of evolving phenotypic traits (Chapter 4). For the evolutionary synchrony model, we show that the dynamic dichotomy occurs provided that fitness, as measured by the spectral radius, is maximized. In addition, we examine the dynamic dichotomy for semelparous species in a continuous-time setting (Chapter 5).