Abstract

The goal of this chapter is to present a general theory for the asymptotic dynamics of nonlinear matrix equations as they apply to the dynamics of structured populations. The point of view taken is that of bifurcation theory, i.e. changes in dynamics are studied as a function of a single model parameter. This book chapter shows how a biologically significant parameter (the inherent net reproductive number of the population) can always be defined, introduced into the model equations, and successfully used as a bifurcation parameter in a completely general setting. Several rewards can be gained from this point of view. First, the approach establishes general results concerning the existence and stability of equilibrium distributions, which are then available for any particular application. Second, it makes available powerful analytical techniques for obtaining results about the asymptotic dynamics of what might otherwise be intractable model equations. Finally, the approach serves to organize one's study of any particular model in terms of a general, biologically meaningful parameter. Moreover, even when one or more of the general results fail to apply to a specific model (because of the failure of some required hypothesis or other), the approach often gives insights into what exceptional phenomena occur and what analytical steps should next be taken.