Nonlinear Physiologically Structured Population Models with Two Internal Variables

Abstract

First-order hyperbolic partial differential equations with two internal variables have been used to model biological and epidemiological problems with two physiological structures, such as chronological age and infection age in epidemic models, age and another physiological character (maturation, size, stage) in population models, and cell-age and molecular content (cyclin content, maturity level, plasmid copies, telomere length) in cell population models. In this paper, we study nonlinear double physiologically structured population models with two internal variables by applying integrated semigroup theory and non-densely defined operators. We consider first a semilinear model and then a nonlinear model, use the method of characteristic lines to find the resolvent of the infinitesimal generator and the variation of constant formula, apply Krasnoselskii’s fixed point theorem to obtain the existence of a steady state, and study the stability of the steady state by estimating the essential growth bound of the semigroup. Finally, we generalize the techniques to investigate a nonlinear age-size structured model with size-dependent growth rate.

A Appendix: Positive Operators

In this “Appendix”, we recall some definitions and results of positive operator theory on ordered Banach spaces from Inaba (2006). For more complete exposition, we refer to Heijmans (1986); Marek (1970), and Sawashima (1964).

Let E be a real or complex Banach space and $E^*$ be its dual (the space of all linear functionals on E). Write the value of $f\in E^*$ at $\psi \in E$ as $\langle f, \psi \rangle$. A nonempty closed subset $E_+$ is called a cone if the following hold: (i) $E_++E_+\subset E_+$, (ii) $\lambda E_+\subset E_+$ for $\lambda \ge 0$, (iii) $E_+\cap (-E_+)=\{0\}$. Define the order in E such that $x\le y$ if and only if $y-x\in E_+$ and $x<y$ if and only if $y-x\in E_+{\setminus }\{0\}$. The cone $E_+$ is called total if the set $\{\psi -\phi : \psi , \phi \in E_+\}$ is dense in E. The dual cone $E^*_+$ is the subset of $E^*$ consisting of all positive linear functionals on E; that is, $f\in E^*_+$ if and only if $\langle f, \psi \rangle \ge 0$ for all $\psi \in E_+$. $\psi \in E_+$ is called a quasi-interior point if $\langle f, \psi \rangle >0$ for all $f\in E^*_+{\setminus }\{0\}$. $f\in E^*_+$ is said to be strictly positive if $\langle f, \psi \rangle >0$ for all $\psi \in E_+{\setminus }\{0\}$. The cone $E_+$ is called generating if $E=E_+-E_+$ and is called normal if $E^*=E^*_+-E^*_+$.

An ordered Banach space $(E, \le )$ is called a Banach lattice if (i) any two elements $x, y \in E$ have a supremum $x \vee y=\sup \{x, y\}$ and an infimum $x \wedge y=\inf \{x, y\}$ in E;  and (ii) $|x| \le |y|$ implies $\Vert x\Vert \le \Vert y\Vert$ for $x, y \in E,$ where the modulus of x is defined by $|x|=x \vee (-x).$

Let B(E) be the set of bounded linear operators from E to E. $T\in B(E)$ is said to be positive if $T(E_+)\subset E_+$. For $T, S\in B(E)$, we say $T\ge S$ if $(T-S)(E_+)\subset E_+$. A positive operator $T\in B(E)$ is called semi-nonsupporting if for every pair $\psi \in E_+{\setminus }\{0\}, f\in E^*_+{\setminus }\{0\}$, there exists a positive integer $p=p(\psi , f)$ such that $\langle f, T^p\psi \rangle >0$. A positive operator $T\in B(E)$ is called nonsupporting if for every pair $\psi \in E_+{\setminus }\{0\}, f\in E^*_+{\setminus }\{0\}$, there exists a positive integer $p=p(\psi , f)$ such that $\langle f, T^n\psi \rangle >0$ for all $n\ge p$. The spectral radius of $T\in B(E)$ is denoted by r(T). $\sigma (T)$ denotes the spectrum of T and $\sigma _P(T)$ denotes the point spectrum of T. If there exists a nonzero $x\in E$ which satisfies $Tx=\lambda x$, $\lambda$ is called a proper value and x a proper vector corresponding to $\lambda$.

From results in Sawashima (1964) and Inaba (2006), we state the following propositions.

Proposition A.1

Let E be a Banach space, and let $T\in B(E)$ be compact and semi-nonsupporting. Then, the following statements hold:

• (i) $r(T)\in \sigma _P(T){\setminus }\{0\}$ and r(T) is a simple pole of the resolvent $\lambda I-T;$ that is, r(T) is an algebraically simple eigenvalue of T;

• (ii) The eigenspace of T corresponding to r(T) is one-dimensional and the corresponding eigenvector $\psi \in E_+$ is a quasi-interior point. The relation $T\phi =\mu \phi$ with $\phi \in E_+$ implies that $\phi =c\psi$ for some constant c;

• (iii) The eigenspace of $T^*$ corresponding to r(T) is also a one-dimensional subspace of $E^*$ spanned by a strictly positive functional $f\in E^*_+$.

Proposition A.2

Let E be a Banach space with positive cone $E_+$ which is total. Let $T\in B(E)$ be positive and have the resolvent $\lambda I-T$ with the point r(T) as its pole. Then, T is a semi-nonsupporting operator if and only if $r(T)>0$ and T satisfies (A), where

• (A) Every proper eigenvector corresponding to the proper eigenvalue r(T) lying in $E_+$ is a quasi-interior point of $E_+$ and every proper eigenvector corresponding to r(T) lying in $E_+^*$ is strictly positive.