Net reproduction functions for nonlinear structured population models

Abstract

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...

Boundary perturbations and steady states of structured populations

“…To this end, for every fixed u ∈ Y 0 + we define the net reproduction number as Note that naturally R : Y 0 + → R + can be considered as a functional, and so we may call it the net reproduction functional, see also [21] for more details in this direction. Also note that for the specific model we considered here we had to formally define R at the extinction steady state (zero population density) separately, since F 1 is not defined at 0 in general.…”

“…Indeed, net reproduction numbers, functions and functionals play an important role when studying the existence and the local asymptotic stability of equilibria of physiologically structured population dynamical models, for more details see e.g. the recent papers [4,5,15,19,21,22,23,35,36].…”

Theory of optimal harvesting for a size structured model of fish

“…As a result, modelling population dynamics, it is natural to assume that the vital rates such as fertility, mortality, and growth rates of individuals depended on its body size (see [7,9]). In [6], the authors investigated the net reproduction functions for nonlinear structured population models. Moreover, construction of a net reproduction function in structured populations is crucial for understanding stability of their stationary states (see [8][9][10]).…”

Modelling evolution of virulence in populations with a distributed parasite load

“…nutrition), age, sex, space and other factors (Dwyer et al 1997;Keith and Mitchell-Olds 2013;Sorci et al 2013) and it was shown to strongly affect the course of evolution, in particular resulting in evolutionary branching (Pugliese 2011;Lion and Metz 2018). The importance of considering the heterogeneity of host is closely related to the current trend in modelling in population dynamics which highlights the necessity of studying physiologically structured populations (Metz and Diekmann 1986;Calsina and Farkas 2014;Farkas 2018;Farkas and Hagen 2007;Magal et al 2010;de Roos and Persson 2013).…”