Resource allocation in a PDE ecosystem model

Area of Science:

• Ecology
• Mathematical Biology
• Systems Biology

Background:

• Habitat heterogeneity significantly influences population dynamics and species distribution.
• Resource availability and its spatiotemporal dynamics are critical factors in ecosystem stability.
• Reaction-diffusion models are essential for understanding population spread and interaction.

Purpose of the Study:

• To investigate the effects of habitat heterogeneity on a diffusing population.
• To analyze resource allocation strategies in an ecosystem using a reaction-diffusion system.
• To formulate and solve an optimal control problem for maximizing species abundance.

Main Methods:

• Formulation of a reaction-diffusion system of partial differential equations.
• A priori estimates to prove the existence of state solutions.
• Development of an optimal control problem with cost minimization for resource allocation.
• Analysis of optimal control existence, uniqueness, and characterization.
• Numerical simulations in 1D and 2D with various boundary conditions.

Main Results:

• Demonstrated the existence of state solutions under control.
• Established the existence, uniqueness, and characterization of the optimal control.
• Identified an optimal intermediate diffusion rate for population spread.
• Illustrated the model's behavior through numerical simulations.

Conclusions:

• Habitat heterogeneity and resource allocation strategies critically affect diffusing populations.
• Optimal control provides a framework for managing ecosystems to maximize species abundance.
• The study confirms the existence of an optimal diffusion rate balancing spread and resource use.