Stability of equilibria for a nonlinear population model with age- and spatial-structure

Abstract

In this thesis we study a nonlinear partial differential equation which models the time evolution of a population with age- and spatial-structure. In an abstract setting, this model reads

\begin{eqnarray}\label{Pabstract} \partial_t u + \partial_a u +A(a)u &=& - \mu(u,a)u, \qquad t > 0, \, a \in (0,a_m), \nonumber\\ u(t,0) &=& \int_0^{a_m} b(u,a)u(t,a)\, \text{d}a, \qquad t > 0, \\ u(0,a) &=& u_0 (a), \qquad a \in (0,a_m), \nonumber \end{eqnarray} where $u: [0,T) \to \mathbb{E}_0 $ is interpreted as the density function of the population, taking values in an appropriate function space $\mathbb{E}_0$, $b=b(u,a) \geq 0$ and $\mu = \mu (u,a) \geq 0$ are the birth and mortality rates, and $A(a): E_1 \subset E_0 \to E_0$ is a closed operator on the real Banach lattice $E_0$, for each $a \in J:= [0,a_m)$. In our considerations to follow we fix $p \in [1, \infty)$ and set $ \mathbb{E}_0:= L_p(J, E_0). $

In the first part we consider the semilinear model of age- and spatial-structured population dynamics, which is obtained when the birth law is assumed to be linear. Put differently, the birth rate in problem (0.1) is supposed to be a function of the age-parameter only, i.e. b = b(a). Assuming A generates a parabolic evolution operator, it is then shown that this semilinear structure allows to formulate problem (0.1) as a semilinear Cauchy problem in the Banach space $\mathbb{E}_0$. In particular, we can study mild solutions, their asymptotic behaviour, and convergence to equilibria, and we will see that the stability analysis can be reduced to the linearised problem. In a subsequent step, the spectral theory of positive compact operators is applied to this linear problem, and as a result we will see that the stability behaviour is completely determined by a single quantity, namely the spectral radius of an associated operator. It should be noted that essential ingredients for this result are the assumptions of maximal Lp-regularity of the spatial diffusion operator A, and the positivity of the parabolic evolution operator generated by A.

In a subsequent part, we introduce a weak solution concept for problem (0.1). Assuming A generates a parabolic evolution operator, these so called integral solutions are constructed for the linearised problem in a first step. In the second step we apply a fixed-point argument in order to establish existence of integral solutions for the nonlinear problem. Furthermore we carry out a detailed analysis of the linear inhomogeneous problem, which serves as a preparation for the last part.

In the final part we study the stability behaviour of equilibria to problem (0,1). It has to be pointed out that the birth rate is now allowed to depend on the density, i.e. b = b(u,.), and consequently we lose the semilinear structure, as considered in the first part. In particular, mild solutions are not available any longer, and this is where the integral solutions of the foregoing part come into play. More precisely, we will prove that problem (0,1) is well-posed within this framework. Finally, it is shown that a principle of linearised stability is available within this setting.