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

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dc.contributor.advisor Walker, Christoph DE Zehetbauer, Josef ger 2020-01-14T09:05:20Z 2020-01-14T09:05:20Z 2019
dc.identifier.citation Zehetbauer, Josef: Stability of equilibria for a nonlinear population model with age- and spatial-structure. Hannover : Gottfried Wilhelm Leibniz Universität, Diss., 2019, 93 S. DOI: ger
dc.description.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. ger
dc.language.iso eng ger
dc.publisher Hannover : Institutionelles Repositorium der Leibniz Universität Hannover
dc.rights Es gilt deutsches Urheberrecht. Das Dokument darf zum eigenen Gebrauch kostenfrei genutzt, aber nicht im Internet bereitgestellt oder an Außenstehende weitergegeben werden. ger
dc.subject Evolution operator eng
dc.subject Well-posedness eng
dc.subject Stability eng
dc.subject Structured population model eng
dc.subject Evolutionsoperator ger
dc.subject Stabilität ger
dc.subject Wohlgestelltheit ger
dc.subject Strukturiertes Populationsmodell ger
dc.subject.ddc 510 | Mathematik ger
dc.title Stability of equilibria for a nonlinear population model with age- and spatial-structure eng
dc.type doctoralThesis ger
dc.type Text ger
dc.description.version publishedVersion ger
tib.accessRights frei zug�nglich ger

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