### An age-dependent population equation with diffusion and delayed birth process.

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The subject of the paper is reciprocal influence of pure mathematics and applied sciences. We illustrate the idea by giving a review of mathematical results obtained recently, related to the model of stochastic gene expression due to Lipniacki et al. [38]. In this model, featuring mRNA and protein levels, and gene activity, the stochastic part of processes involved in gene expression is distinguished from the part that seems to be mostly deterministic, and the dynamics is expressed by means of a...

Motivated by structured parasite populations in aquaculture we consider a class of size-structured population models, where individuals may be recruited into the population with distributed states at birth. The mathematical model which describes the evolution of such a population is a first-order nonlinear partial integro-differential equation of hyperbolic type. First, we use positive perturbation arguments and utilise results from the spectral...

Let be a locally compact Hausdorff space. Let ${A}_{i}$, i = 0,1,...,N, be generators of Feller semigroups in C₀() with related Feller processes ${X}_{i}={X}_{i}\left(t\right),t\ge 0$ and let ${\alpha}_{i}$, i = 0,...,N, be non-negative continuous functions on with ${\sum}_{i=0}^{N}{\alpha}_{i}=1$. Assume that the closure A of ${\sum}_{k=0}^{N}{\alpha}_{k}{A}_{k}$ defined on ${\bigcap}_{i=0}^{N}\left({A}_{i}\right)$ generates a Feller semigroup T(t), t ≥ 0 in C₀(). A natural interpretation of a related Feller process X = X(t), t ≥ 0 is that it evolves according to the following heuristic rules: conditional on being at a point p ∈ , with probability ${\alpha}_{i}\left(p\right)$, the process...