Persistent distribution functions for the dispersion of structured populations
This paper is concerned with the large-population dynamics of individuals that are characterized by a varying state vector \b{X}, whose birth intensity depends only on parental age and their state at birth, whose mortality function is a sum of an age-dependent component and a component depending on absolute time and the evolution of the population density, and which diffuse in a crowd-avoiding manner. More specifically, the state \b{X} of individuals is supposed to vary randomly throughout the individual's lifetime. This variation is age dependent: The state \b{X} of an individual aged a changes to \b{X}' by \(a+\Delta a\) with probability W(\b{X}\(\to \underline X',a)\Delta a\). Apart from this the author's model is a standard one. Thus if \(\rho\) (a,\b{X},ṟ,t) is the population density of individuals aged a, having state \b{X} and situated at position ṟ, all at time t, then \[ (\partial_ t+\partial_ u)\rho +div_{\underline r}\Phi +\mu \rho =\int W(\underline X'\to \underline X,a)\rho (a,\underline X,\underline r,t)d\underline X'-\int W(\underline X\to \underline X',a)\rho (a,\underline X,\underline r,t)d\underline X' \] \[ and\quad \rho (a=0,\underline X,\underline r,t)=\int \int^{\infty}_{0}\lambda (a',\underline X',\underline X,...)\rho (a',\underline X',\underline r,t)da'd\underline X', \] where \(\lambda\) (\(\cdot)\) and \(\mu\) (\(\cdot)\) are the birth and death intensities, respectively, and \(\Phi\) is the diffusion flux. Under the above conditions on \(\lambda\), \(\mu\) and \(\Phi\), the author seeks formal product solutions of this system along lines explored by \textit{M. E. Gurtin} and \textit{R. C. MacCamy}, ibid. 62, 157-167 (1982; Zbl 0505.92019). He then calculates several demographic averages from these solutions. As a particular example he considers the case where there is a single state variable, body size, and he computes expressions for body size and energy production rate averages. Finally, the author considers a generalization where the individual state process is not Markovian.
persistent age-stage distribution functions
scientific article; zbMATH DE number 4055439
scientific article