### Structured population dynamics, a modelling perspective

Kooi, B. W. and Kelpin, F. D. L. 2003.
Structured population dynamics, a modelling perspective.
* Comments on Theoretical Biology*,
to appear
#### Abstract

In classical population dynamic models individuals are assumed to be
identical. The resulting mathematical model consists of an
ordinary differential equations (ode) describing the
temporal change of the state, biomass or number of individuals, of
the population. Subsequently the dynamics of ecosystems where these
populations interplay with each other and with the abiotic
environment is generally described by a system of ordinary
differential equations. These models are sometimes called
unstructured population models, to the contrary of structured
population models, where differences of the individuals are taken
into account.
Now the development of each individual from birth till death is
often described by an ode where the change in time of the state of
the individual is its growth rate. Complemented with birth and death
models, the state of a population is not described by a single
number but by some distribution over the individual state, such as
age or size. When the number of individuals is large, this
distribution is taken continuous instead of discrete. When such a
distribution is smooth the resulting population model, derived
straightforwardly from the individual model using balance laws, is a
partial differential equation (pde).

Initially age was used to describe the state of the
individuals. However, for many organisms not age, or not age alone,
characterizes its state but also size or energy reserves are
important. In addition, the life cycle of the individuals consists
of a number of life-stages, such as egg, juvenile and adult or the
distribution of the number of individuals with respect to the state
variable is irregular. This lead to physiologically structured
population models psp. The introduction of these two extra
components, namely several life-stages and non-smooth distributions,
gave mathematical difficulties which were difficult to overcome with
the pde formulation. Therefore, recently a cumulative formulation
was proposed where the governing formulation is in terms of renewal
integral equations.