Stability of nonlinear age structured models

Stability and bifurcations of nonlinear Leslie type models are discussed. Age-specific fertilities are assumed to be negatively affected by population density, measured as a linear combination of the number of individuals in each age class ( equation in PDF) The effect of population density on the reproductive output is measured by g (equation in PDF) where g is a function satisfying a few hypothesis that are also met by virtually every density dependent functional relation encountered in the ecological literature. Sufficient conditions for boundedness of solutions are presented. Conditions for existence and uniqueness of trivial and nontrivial equilibria are established. For the trivial equilibrium, necessary and sufficient conditions for stability are obtained while for the positive equilibrium a variety of sufficient conditions are discussed. Causes determining the loss of stability of the positive equilibrium are sought. The reasons why a period doubling bifurcation may occur are completely determined. However, the problem of determination of causes of a Hopf bifurcation is only partially resolved. The connection between Hopf bifurcation, oscillatory behavior and inherited delays imposed by biological constraints is explored. Finally, an approximation for the bifurcation period is obtained. The error bound is found to be dependent only on the mean and variance of distributions closely related with reproductive activity and density dependent influence through age. The methods used in obtaining the approximation for the bifurcation period are employed to approximate the boundary between stable and unstable behavior in the parameter space.