Small Parameters in Structured Population Models and the Trotter–Kato Theorem

Abstract

Abstract. The justification of some (often implicit) limit arguments used in the development of structured population models is discussed via two examples. The first example shows how a pair of sink-source terms may transform into a side condition relating the appearance of individuals in the interior of the individual state space to the outflow of individuals at its boundary. The second example considers the usual equation for size-dependent population growth in which it is implicitly assumed that discrete finitely-sized young are produced from infinitesimal contributions by all potential parents. The main mathematical tool for dealing with these examples is the Trotter-Kato theorem for one-parameter semigroups of bounded linear operators.Key words. structured population, limit transition, C0-semigroup, Trotter-Kato theorem AMS(MOS) subject classifications. 92A15, 35A35, 47005 l. Introduction. 1.1. Biological motivation: structured populations, semigroups of operators, and the need for model simplifications. The tenet of the physiologically structured approach to the modeling of the dynamics of populations as set out in Metz and Diekmann (1986) is that, provided all individuals experience the same environmental inputs such as food availability or chance of running into a predator, we may (and should) represent a population as a frequency distribution over a space D of potential states of the individuals comprising the population. (As we frequently need corresponding concepts on the individual and population levels we will, where necessary, use the prefixes iand p-to distinguish the corresponding terms, for example i-state versus p-state, where the latter refers to the frequency distribution.) The main effort in model construction is the determination of an appropriate state representation of i-behavior, where the i-behavior consists of (i) any contributions to population change such as giving birth or dying, and (ii) any quantities relevant to the calculation of the output from the population model, such as the rate at which the individual consumes food. If we make the assumption that the number of individuals is sufficiently large, then for any given course of the environment the present p-state should determine the future p-states in a deterministic and linear fashion. For a constant environment the maps relating subsequent p-states should form a linear semigroup.The transition from i-model to p-model is made through their differential generators. It is here that we leave biology and start doing mathematics: did we really write down a genuine differential generator, and what can be said about the properties of the semigroup so generated?Until now the attention has been mostly restricted to models where the i-state space n is a subset of !Rk, and where the individuals move through n according to the solution of an ordinary differential equation (ODE), possibly alternating with (usually randomly occurring) state jumps, for example, due to an individual losing weight when it splits off a daughter. The reasons ...

Mathematical aspects of physiologically structured populations: the contributions of J. A. J. Metz

“…The Trotter-Kato theorem therefore provides a theoretical framework for a rigorous treatment of the approximation procedures mentioned above. This was demonstrated by Heijmans and Metz [80] for two physiologically structured models, one of which we shall briefly review.…”

“…This is clearly absurd. To justify the model (123)-(124), Heijmans and Metz [80] considered a more complicated model which does not violate any obvious biological constraints. In this model it is assumed that when giving birth to an offspring of size ε the parent loses the same amount of weight.…”

“…Heijmans and Metz [80] used the Trotter-Kato theorem to make these formal manipulations mathematically precise. The limiting model (130)-(131) is the same as (123)-(124) but with the individual growth rate replaced by g(x) − β(x) and the size of newborns equal to zero.…”

“…The opening sentence of the preprint version of Heijmans and Metz[80].Downloaded by [Northern Alberta Inst of Technology] at 04:18 05 February 2015…”

How to lift a model for individual behaviour to the population level?

“…In order to keep a somewhat tractable framework, we in the past fudged the effect of stochasticity in the individual state transitions by having the process of giving birth represented as a continuous rate b (cf. Metz & Diekmann 1986, Remark I.3.2.1; see their Section III.6.3 and Heijmans & Metz 1989 for one possible justification, in the form of a limit argument starting from a more realistic specification). This is also what we did in § §2 and 3 and what we will do below: we shall assume that births are produced at a rate that is proportional to the rate at which energy is channelled towards reproductive activities.…”