On the net reproduction rate of continuous structured populations with distributed states at birth

We consider a nonlinear structured population model with a distributed recruitment term. The question of the existence of non-trivial steady states can be treated (at least!) in three different ways. One approach is to study spectral properties of a parametrized family of unbounded operators. The alternative approach, on which we focus here, is based on the reformulation of the problem as an integral equation. In this context we introduce a density dependent net reproduction rate and discuss its relationship to a biologically meaningful quantity. Finally, we briefly discuss a third approach, which is based on the finite rank approximation of the recruitment operator.Comment: To appear in Computers and Mathematics with Application

Stability Analysis for the Gurtin-MacCamy’s Age-Structured Population Dynamics Model

The stability of the Gurtin-MacCamy’s age-structured population dynamics model is investigated. We determine the steady states and study their stability. The results in this paper generalize previous results

A numerical method for nonlinear age-structured population models with finite maximum age

AbstractWe propose a new numerical method for the approximation of solutions to a non-autonomous form of the classical Gurtin–MacCamy population model with a mortality rate that is the sum of an intrinsic age-dependent rate that becomes unbounded as the age approaches its maximum value, plus a non-local, non-autonomous, bounded rate that depends on some weighted population size. We prove that our new quadrature based method converges to second-order and we show the results of several numerical simulations

Numerical integration of an age-structured population model with infinite life span

Producción CientíficaThe choice of age as a physiological parameter to structure a population and to describe its dynamics involves the election of the life-span. The analysis of an unbounded life-span age-structured population model is motivated because, not only new models continue to appear in this framework, but also it is required by the study of the asymptotic behaviour of its dynamics. The numerical integration of the corresponding model is usually performed in bounded domains through the truncation of the age life-span. Here, we propose a new numerical method that avoids the truncation of the unbounded age domain. It is completely analyzed and second order of convergence is established. We report some experiments to exhibit numerically the theoretical results and the behaviour of the problem in the simulation of the evolution of the Nicholson’s blowflies model.Ministerio de Economía, Industria y Competitividad - Fondo Europeo de Desarrollo Regional (project MTM2017-85476-C2-1-P)Ministerio de Ciencia, Innovación y Universidades - Agencia Estatal de Investigación (grants PID2020-113554GB-I00/AEI/10.13039/501100011033 and RED2018-102650-T)Junta de Castilla y Leon - Fondo Europeo de Desarrollo Regional (grant VA193P20)Junta de Castilla y León (grant VA138G18

Size-structured populations: immigration, (bi)stability and the net growth rate

We consider a class of physiologically structured population models, a first order nonlinear partial differential equation equipped with a nonlocal boundary condition, with a constant external inflow of individuals. We prove that the linearised system is governed by a quasicontraction semigroup. We also establish that linear stability of equilibrium solutions is governed by a generalized net reproduction function. In a special case of the model ingredients we discuss the nonlinear dynamics of the system when the spectral bound of the linearised operator equals zero, i.e. when linearisation does not decide stability. This allows us to demonstrate, through a concrete example, how immigration might be beneficial to the population. In particular, we show that from a nonlinearly unstable positive equilibrium a linearly stable and unstable pair of equilibria bifurcates. In fact, the linearised system exhibits bistability, for a certain range of values of the external inflow, induced potentially by All\'{e}e-effect.Comment: to appear in Journal of Applied Mathematics and Computin

Neutral delay equations from and for population dynamics

For a certain class of neutral differential equations it is shown that these equations can serve as population models in the sense that they can be interpreted as special cases or caricatures of the standard Gurtin-MacCamy model for a population structured by age with birth and death rate depending on the total adult population. The delayed logistic equation does not belong to this class but the blowfly equation does. These neutral delay equations can be written as forward systems of an ordinary differential equation and a shift map. There are several quite distinct ways to perform the transformation to a system, either following a method of Hale or following more closely the renewal process. Similarly to the delayed logistic equation, the neutral equation (and the blowfly equation as a special case) exhibit periodic solutions, although only for a restricted range of parameters