A general model of size-dependent population dynamics with nonlinear growth rate (Q1883087)

The author studies the dynamics of a continuously size-structured population, described by the following initial boundary value problem with nonlocal terms: \[ \begin{aligned} u_t + (V(x,P(t))u)_x &= G(u(\cdot,t))(x), \quad x\in [0,l), \;(0\leq )a \leq t \leq T,\\ V(0,P(t))u(0,t) &= C(t)+F(u(\cdot,t)), \quad a \leq t \leq T,\\ u(x,a) &= u_a(x), \quad x\in [0,l),\\ P(t) &= \int_0^l w(x) u(x,t) \,dx, \end{aligned} \] where \(u(x,t)\) denotes the density of individuals sized \(x\) at time \(t\) and \(P(t)\) is a (weighed) total population density. The functions \(F\), \(G\), \(w\) and \(C\) are assumed to satisfy certain (not very restrictive) conditions. The Gurtin-McNamy model is an example of such problem. The results generalize those of \textit{A. Calsina} and \textit{J. Saldana} [J. Math. Biol. 33, No. 4, 335--364 (1995; Zbl 0828.92025)]. The author proves the existence of a local solution using Schauder's fixed point theorem and, if the functions \(F\) and \(G\) satisfy some additional conditions, also uniqueness and continuous dependence on initial data. Moreover if \(G\) has a certain form, the solutions of the problem with initial value \(u_0\in L^1\) can be expressed analytically using a family of evolution operators.