 Odo Diekmann Mathematical Institute, Utrecht University, The Netherlands

On the formulation and analysis of physiologically structured population models

Abstract:

The aim of physiologically structured population models (PSPM), is to enable the theoretical analysis of the relationship between mechanisms at the i-level (i for individual) and phenomena at the p-level (p for population). In such models, individuals are characterised by (discrete and) continuous variables, like age and size, collectively called their i-state. The world in which these individuals live is characterised by another set of variables, collectively called the environmental condition. The model consists of submodels for (i) the dynamics of the i-state, e.g. growth and maturation, (ii) survival, (iii) reproduction, with the relevant rates described as a function of (i-state, environmental condition), (iv) functions of (i-state, environmental condition), like biomass or feeding rate, that integrated over the i-state distribution together produce the output of the population model. When the environmental condition is treated as a given function of time (input), the population model becomes linear in the state. Density dependence and interaction with other populations is captured by feedback via a shared environment, i.e., by letting the environmental condition be influenced by the populations' outputs. This yields a systematic methodology for formulating community models by coupling nonlinear input-output relations defined by state-linear population models.

The first step to analyse such models mathematically, consists of associating a dynamical system with the model. In order to do so, we lift the concept of state from the i- to the p-level by describing a population by a measure (often represented by a density) over the i-state space. The dynamical system updates this measure.

Even though this is the conceptually designated way to define a dynamical system, alternatively we can use the history of both the birth rate and the environmental condition to code, a bit implicitly, the current state. The model specifies the rules for extending these variables into the future (such rules take the form of renewal equations and delay differential equations). The dynamical system updates the histories, so is defined by shifting along the extended functions.

To analyse the models, we need

• a qualitative stability and bifurcation theory for the corresponding dynamical systems (for instance, the linearized, around a steady state, version of the delay equations yields a characteristic equation, and we want to draw conclusions about the local behaviour of solutions of the nonlinear equations from information about the position in the complex plane of the roots of the characteristic equation)
• numerical bifurcation tools

After a general introduction, the lecture will focus on the state-of-the-art of these two topics and speculate about imminent progress.