Three stage semelparous Leslie models.

Nonlinear Leslie matrix models have a long history of use for modeling the dynamics of semelparous species. Semelparous models, as do nonlinear matrix models in general, undergo a transcritical equilibrium bifurcation at inherent net reproductive number R(0) = 1 where the extinction equilibrium loses stability. Semelparous models however do not fall under the purview of the general theory because this bifurcation is of higher co-dimension. This mathematical fact has biological implications that relate to a dichotomy of dynamic possibilities, namely, an equilibration with over lapping age classes as opposed to an oscillation in which age classes are periodically missing. The latter possibility makes these models of particular interest, for example, in application to the well known outbreaks of periodical insects. While the nature of the bifurcation at R(0) = 1 is known for two-dimensional semelparous Leslie models, only limited results are available for higher dimensional models. In this paper I give a thorough accounting of the bifurcation at R(0) = 1 in the three-dimensional case, under some monotonicity assumptions on the nonlinearities. In addition to the bifurcation of positive equilibria, there occurs a bifurcation of invariant loops that lie on the boundary of the positive cone. I describe the geometry of these loops, classify them into three distinct types, and show that they consist of either one or two three-cycles and heteroclinic orbits connecting (the phases of) these cycles. Furthermore, I determine stability and instability properties of these loops, in terms of model parameters, as well as those of the positive equilibria. The analysis also provides the global dynamics on the boundary of the cone. The stability and instability conditions are expressed in terms of certain measures of the strength and the symmetry/asymmetry of the inter-age class competitive interactions. Roughly speaking, strong inter-age class competitive interactions promote oscillations (not necessarily periodic) with separated life-cycle stages, while weak interactions promote stable equilibration with overlapping life-cycle stages. Methods used include the theory of planar monotone maps, average Lyapunov functions, and bifurcation theory techniques.