On the formulation and analysis of general deterministic structured population models. II. Nonlinear theory.

Area of Science:

• Mathematical Biology
• Population Dynamics
• Ecological Modeling

Background:

• Physiologically structured population models are essential for understanding population dynamics.
• Existing models often struggle to incorporate complex environmental feedback loops.
• A need exists for a robust methodology to construct nonlinear population models.

Purpose of the Study:

• To present a novel two-step modeling methodology for defining future population states.
• To introduce a constructive approach for solving nonlinear population models.
• To demonstrate the application of the contraction mapping principle in population dynamics.

Main Methods:

• Developing a linear physiologically structured population model based on reproduction and development rules.
• Introducing environmental conditions as a function of time, ensuring individual independence for linearity.
• Incorporating a feedback law linking environmental conditions to population size and composition.
• Solving the resulting fixed-point problem using the contraction mapping principle.

Main Results:

• Constructive definition of future population state operators from individual behavior and environmental conditions.
• Successful formulation and solution of nonlinear population models via a fixed-point problem.
• Generation of a population semiflow by applying the fixed-point solution to the linear model.

Conclusions:

• The proposed two-step methodology provides a constructive solution for nonlinear physiologically structured population models.
• This approach effectively integrates individual behavior, environmental conditions, and population-level feedback.
• The method offers a powerful tool for predicting future population states under dynamic environmental influences.