Stochastic models in population biology and their deterministic analogs

Area of Science:

• Ecology
• Theoretical Biology
• Mathematical Biology

Background:

• Stochastic population models are crucial for understanding ecological dynamics.
• Mean-field theories often simplify complex spatial interactions.
• Patch dynamics offer a framework to bridge spatial and non-spatial ecological models.

Purpose of the Study:

• Introduce a novel class of stochastic population models based on patch dynamics.
• Quantify deviations of these models from mean-field theories by varying patch size.
• Analyze two-species competition within this framework.

Main Methods:

• Developed a mathematical analysis of the patch dynamics model.
• Derived precise competition mean-field equations and first-order corrections for the non-spatial case.
• Conducted simulation studies to validate theoretical predictions.

Main Results:

• Derived competition mean-field equations that differ from phenomenological formulations.
• Demonstrated that mean-field theory is more robust in spatial models compared to single isolated patches.
• Identified deviations from mean-field theory due to discrete effects in modest-sized patches.

Conclusions:

• Spatial models with interpatch diffusion enhance the robustness of mean-field theory by diluting stochastic effects.
• Discrete effects in finite-sized patches can lead to substantial departures from mean-field predictions, even in spatial contexts.
• The patch dynamics framework provides a flexible tool for studying a wide range of biological processes.