Structured populations with distributed recruitment: from PDE to delay formulation

Calsina, Àngel; Diekmann, Odo; Farkas, Jozsef Zoltan

doi:10.1002/mma.3898

Cited by 15 publications

(11 citation statements)

References 39 publications

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“…Structured population models involve book-keeping considerations and there are multiple ways to organise these. Accordingly, there are as well multiple ways of associating a dynamical system at the population level with the model (Barril et al in preparation;Calsina et al 2016). We favour book-keeping based on an individual's state-at-birth and age, which leads to an abstract renewal equation for the population level birth rate .…”

Section: Discussionmentioning

confidence: 99%

On models of physiologically structured populations and their reduction to ordinary differential equations

2019

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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, death and reproduction all depend on the environmental condition through a common factor (so for a very strict form of physiological age).

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Section: Discussionmentioning

confidence: 99%

On models of physiologically structured populations and their reduction to ordinary differential equations

2019

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“…Linear and nonlinear structured population models with distributed recruitment processes, but formulated on Lebesgue spaces were introduced and studied recently for example in [1,13,24]. However, it is important to note that in contrast to [1,13,24] here we do not impose a finite maximal value for the structuring variable x. For some applications this might be a more natural assumption; but at the same time this poses additional challenges in the spectral analysis of model ( 2).…”

Section: Problem Formulation and Existence Of Solutionsmentioning

confidence: 99%

Asymptotic behaviour of a structured population model on a space of measures

Farkas¹,

Gwiazda²,

Marciniak-Czochra³

2019

Preprint

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In this paper we consider a physiologically structured population model with distributed states at birth, formulated on the space of non-negative Radon measures. Using a characterisation of the pre-dual space of bounded Lipschitz functions, we show how to apply the theory of strongly continuous positive semigroups to such a model. In particular, we establish the exponential convergence of solutions to a one-dimensional global attractor.

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“…Also note that nonlinearities, in this case induced by scramble competition effects, are introduced via density dependence through individual mortality µ, fertility β, and growth/development γ rates. (We do not specify here the necessary regularity assumptions on the model ingredients, but instead refer the interested reader to [1,9,28]. ) In model (2.8), the environment individuals are experiencing is simply determined by the total population size (scramble competition), and therefore we set…”

Section: (Re)formulation Of a Nonlinear Problemmentioning

confidence: 99%

Net reproduction functions for nonlinear structured population models

Farkas

2018

Math. Model. Nat. Phenom.

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The goal of this note is to present a general approach to define the net reproduction function for a large class of nonlinear physiologically structured population models. In particular, we are going to show that this can be achieved in a natural way by reformulating a nonlinear problem as a family of linear ones; each of the linear problems describing the evolution of the population in a different, but constant environment. The reformulation of a nonlinear population model as a family of linear ones is a new approach, and provides an elegant way to study qualitative questions, for example the existence of positive steady states. To define the net reproduction number for any fixed (constant) environment, i.e. for the linear models, we use a fairly recent spectral theoretic result, which characterizes the connection between the spectral bound of an unbounded operator and the spectral radius of a corresponding bounded operator. For nonlinear models, varying the environment naturally leads to a net reproduction function. ProloguePhysiologically structured population models have been developed and studied extensively in the past decades, see for example the monographs [18,35,38,45] for an in-depth introduction and review of the topic. Our general research agenda, to relate the stability of steady states of physiologically structured population models to biologically meaningful quantities, allowed us to obtain a number of interesting results, see e.g. [27,29,30,31,32]. In particular, as we have seen for example in [1,27,29,30,31,32], the existence and stability of steady states of nonlinear structured population models can often be characterised using an appropriately defined net reproduction function. In fact we note that stability questions of nonlinear matrix population models were already investigated in the same spirit, see for example the paper [46]. However, we would like to emphasize that previously net reproduction functions were defined for concrete nonlinear models on an 'ad hoc' basis, but typically via analysing the corresponding steady state equations, see e.g. [1,27,28,29,30,31,32] for more details. It is our goal in this paper to present a general framework, which is applicable to different classes of nonlinear models.The existence of positive steady states is an important and interesting question, when studying nonlinear population models; and to establish the existence of non-trivial steady states is often challenging. This is especially the case for models formulated as infinite dimensional dynamical systems, for example delay equations, partial differential equations or integro-differential equations, see e.g. [33,34]. To overcome some of the difficulties, we have developed a very general framework to treat steady state problems of some classes of nonlinear partial differential or partial integro-differential equations, see [10,11,12]. The power of the method we developed becomes apparent for models with so-called infinite dimensional nonlinearities, when in fact the existence 1991 Mathematics Subj...

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Structured populations with distributed recruitment: from PDE to delay formulation

Cited by 15 publications

References 39 publications

On models of physiologically structured populations and their reduction to ordinary differential equations

On models of physiologically structured populations and their reduction to ordinary differential equations

Asymptotic behaviour of a structured population model on a space of measures

Net reproduction functions for nonlinear structured population models

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