Abstract
We consider a linear size-structured population model with diffusion in the size-space. Individuals are recruited into the population at arbitrary sizes. We equip the model with generalized Wentzell-Robin (or dynamic) boundary conditions. This approach allows the modelling of populations in which individuals may have distinguished physiological states. We establish existence and positivity of solutions by showing that solutions are governed by a positive quasicontractive semigroup of linear operators on the biologically relevant state space. These results are obtained by establishing dissipativity of a suitably perturbed semigroup generator. We also show that solutions of the model exhibit balanced exponential growth, that is, our model admits a finite-dimensional global attractor. In case of strictly positive fertility we are able to establish that solutions in fact exhibit asynchronous exponential growth.
Highlights
A significant amount of interest has been devoted to the analysis of mathematical models arising in structured population dynamics
In this note we introduced a linear structured population model with diffusion in the size space
Diffusion amounts to adding noise in a deterministic fashion
Summary
A significant amount of interest has been devoted to the analysis of mathematical models arising in structured population dynamics (see e.g. [21, 27] for references). [21, 27] for references) Such models often assume spatial homogeneity of the population in a given habitat and only focus on the dynamics of the population arising from differences between individuals with respect to some physiological structure. In this context, reproduction, death and growth characterize individual behavior which may be affected by competition, for example for available resources. The main question addressed in [18] is what type of boundary conditions are necessary for a biologically plausible and mathematically sound model In this context some special cases of a general Robin boundary condition were considered. When the structuring variable represents a parasite load as
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