Numerical solution of structured population models (original) (raw)
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Considering the environmental condition as a given function of time, we formulate a physiologically structured population model as a linear non-autonomous integral equation for the, in general distributed, population level birth rate. We take this renewal equation as the starting point for addressing the following question: When does a physiologically structured population model allow reduction to an ODE without loss of relevant information? We formulate a precise condition for models in which the state of individuals changes deterministically, that is, according to an ODE. Specialising to a one-dimensional individual state, like size, we present various sufficient conditions in terms of individual growth-, death-, and reproduction rates, giving special attention to cell fission into two equal parts and to the catalogue derived in an other paper of ours (submitted). We also show how to derive an ODE system describing the asymptotic large time behaviour of the population when growth,...
Numerical solutions of a linear age-structured population model
We study numerical solutions of an age-structured population model. The model is a linear partial differential equation (PDE) with nonlocal boundary condition. With this model, we aim to understand how the populations of individuals change in both time and age. We use finite difference operators to replace the partial derivatives and composite trapezoidal method to compute the integral at the boundary. By numerically solving the PDE in Matlab, we generate preliminary results about age-structured populations, and explore convergence of the numerical scheme.
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We consider an age-structured population model achieved by modifying the classical Sharpe-Lotka-McKendrick model, incorporating an overcrowding effect or competition for resources term. This term depends on the whole population rather than on any specific age group, in the case of overcrowding or limitation of resources. We investigate the solutions for arbitrary initial conditions. We consider the existence of a steady age distribution and its stability and are able to determine this for a simple illustrative case. If the non-trivial steady age distribution is unstable, there is a critical initial population size beyond which the population explodes. This watershed is independent of the shape of the initial age distribution.
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Growth equations are established for a population of individuals that have fixed age dependent reproduction and mortality rates. Equations are obtained both for the population density and for the numerical size of the population in a fixed age group. Age and time dependent migration is taken into consideration. The usual integral equation of renewal type for these variables is shown to be equivalent to a functional differential equation of retarded type; these differential equations are of main interest in this work. The role of initial data in characterizing a unique solution of the functional differential equation is examined in detail. Finally, some special cases for the reproduction and mortality rates are considered where the functional differential equations take a reasonably simple form.
A quasilinear parabolic model for population evolution
Differential Equations & Applications, 2012
A quasilinear parabolic problem is investigated. It models the evolution of a single population species with a nonlinear diffusion and a logistic reaction function. We present a new treatment combining standard theory of monotone operators in L 2 (Ω) with some orderpreserving properties of the evolutionary equation. The advantage of our approach is that we are able to obtain the existence and long-time asymptotic behavior of a weak solution almost simultaneously. We do not employ any uniqueness results; we rely on the uniqueness of the minimal and maximal solutions instead. At last, we answer the question of (long-time) survival of the population in terms of a critical value of a spectral parameter.