Existence of solutions to models of age-dependent populations with finite life span (Q1081560)

The McKendrick-von Foerster population balance equation \[ D\rho =- \lambda \rho,\text{ where } D\rho =\lim_{h\to 0}[\rho (a-h,t-h)-\rho (a,t)]/h, \] and \(\rho\) is the population density, a is age, \(\lambda\) is the death modulus and t is time is considered. Supposing the death modulus dependent on the population density \(\rho\) this equation is nonlinear in \(\rho\). Contrary to previous analyses, situations when individuals do not live beyond some age L are analyzed. The maximum life span L is allowed to depend on time t and on total population density P(t), \[ P(t)=\int^{L}_{0}\rho (a,t)\quad da. \] The first case simulates some insect populations, where life span is controlled by (time-dependent) temperature while the second simulates populations whose maximum life spans are limited by the availability of resources. The Banach fixed point theorem is used to establish the existence and uniqueness of solutions of the McKendrick-von Foerster equation under the above mentioned assumptions.