Finite dimensional state representation of physiologically structured populations

Area of Science:

• Mathematical Biology
• Theoretical Ecology
• Population Dynamics

Background:

• Physiologically structured population models (PSPMs) characterize individuals by continuous variables (i-state) and environmental conditions.
• PSPMs integrate submodels for i-state dynamics, survival, reproduction, and output functions, with rates dependent on i-state and environment.
• Density dependence and inter-population interactions are modeled via feedback through a shared environment.

Purpose of the Study:

• To develop a test for determining if a PSPM is reducible to a finite-dimensional ordinary differential equation (ODE).
• To provide a catalogue of ODE-reducible PSPMs under specific restrictions (deterministic i-state dynamics, one-dimensional i-state space, specific birth rate structure).

Main Methods:

• The study defines ODE-reducibility for PSPMs where infinite-dimensional models can be replaced by finite-dimensional ODEs without information loss.
• A specific test is proposed to identify ODE-reducible PSPMs.
• A catalogue of ODE-reducible models is presented based on defined restrictions.

Main Results:

• A test for ODE-reducibility of PSPMs is established.
• Under the specified restrictions, the conditions for ODE-reducibility are shown to be both sufficient and necessary.
• The restrictions ensure that population trajectories are fully determined by the ODE solution, providing a complete dynamic picture.

Conclusions:

• The methodology enables the formulation of community models by coupling state-linear population models.
• ODE-reducibility simplifies the analysis of complex population dynamics.
• The findings offer a systematic approach to understanding population and community dynamics through reduced mathematical frameworks.