Daphnias: from the individual based model to the large population equation

The class of deterministic 'Daphnia' models treated by Diekmann et al. (J Math Biol 61: 277-318, 2010) has a long history going back to Nisbet and Gurney (Theor Pop Biol 23: 114-135, 1983) and Diekmann et al. (Nieuw Archief voor Wiskunde 4: 82-109, 1984). In this note, we formulate the individual based models (IBM) supposedly underlying those deterministic models. The models treat the interaction between a general size-structured consumer population ('Daphnia') and an unstructured resource ('algae'). The discrete, size and age-structured Daphnia population changes through births and deaths of its individuals and throught their aging and growth. The birth and death rates depend on the sizes of the individuals and on the concentration of the algae. The latter is supposed to be a continuous variable with a deterministic dynamics that depends on the Daphnia population. In this model setting we prove that when the Daphnia population is large, the stochastic differential equation describing the IBM can be approximated by the delay equation featured in (Diekmann et al., l.c.)

When Does Evolution Optimise?

Goal: Elucidating the role of the eco-evolutionary feedback loop in determining evolutionarily stable life histories, with particular reference to the methodological status of the optimisation procedures of classical evolutionary ecology. Conclusion: A pure optimisation approach holds water only when the eco-evolutionary feedbacks are of a particularly simple kind

When Does Evolution Optimize? On the Relation Between Types of Density Dependence and Evolutionarily Stable Life History Parameters

In this paper we (i) put forward a simple notational device clarifying the, undeniable but generally ignored, role of density dependence in determining evolutionarily stable life histories, (ii) use this device to derive necessary and sufficient conditions for (a) the existence of an evolutionary extremization principle, and (b) the reduction of such a principle to straight r- or RO-maximization, (iii) use the latter results to analyze a simple concrete example showing that the details of the population dynamical embedding may influence our evolutionary predictions to an unexpected extent

On the Formulation and Analysis of General Deterministic Structured Population Models

We define a linear physiologically structured population model by two rules, one for reproduction and one for "movement" and survival. We use these ingredients to give a constructive definition of next-population-state operators. For the autonomous case we define the basic reproduction ratio R0 and the Malthusian parameter r and we compute the resolvent in terms of the Laplace transform of the ingredients. A key feature of our approach is that unbounded operators are avoided throughout. This will facilitate the treatment of nonlinear models as a next step

On the Formulation and Analysis of General Deterministic Structured Population Models. I. Linear Theory

We define a linear physiologically structured population model by two rules, one for reproduction and one for "movement" and survival. We use these ingredients to give a constructive definition of next-population-state operators. For the autonomous case we define the basic reproduction ratio Ro and the Malthusian parameter r and we compute the resolvent in terms of the Laplace transform of the ingredients. A key feature of our approach is that unbounded operators are avoided throughout. This will facilitate the treatment of nonlinear models as a next step

Population growth in discrete time: a renewal equation oriented survey

Traditionally, population models distinguish individuals on the basis of their current state. Given a distribution, a discrete time model then specifies (precisely in deterministic models, probabilistically in stochastic models) the population distribution at the next time point. The renewal equation alternative concentrates on newborn individuals and the model specifies the production of offspring as a function of age. This has two advantages: (i) as a rule, there are far fewer birth states than individual states in general, so the dimension is often low; (ii) it relates seamlessly to the next-generation matrix and the basic reproduction number. Here we start from the renewal equation for the births and use results of Feller and Thieme to characterise the asymptotic large time behaviour. Next we explicitly elaborate the relationship between the two bookkeeping schemes. This allows us to transfer the characterisation of the large time behaviour to traditional structured-population models

The 'cumulative' formulation of (physiologically) structured population models

bibliographical data to be processed -- Evolution equations, control theory, and biomathematics (Han sur Lesse, 1991) Pages: 145--154 Series: Lecture Notes in Pure and Appl. Math. Vol: 155 -- Dekker (New York) --