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Simple models for complex systems: exploiting the relationship between local and global densities.

Mercedes Pascual¹, Manojit Roy¹, Karina Laneri²

¹Department of Ecology and Evolutionary Biology, University of Michigan, Ann Arbor, MI USA ; HHMI, and Department of Ecology and Evolutionary, University of Michigan, Ann Arbor, MI 48109-1048 USA.

Theoretical Ecology

|December 26, 2014

Summary

This summary is machine-generated.

Spatial structure in predator-prey models can justify simple temporal models by establishing a bi-power law relationship between global densities and pair densities. This finding explains why simplified population dynamics can accurately reflect complex spatial interactions.

Keywords:

Criticality From individuals to populations Implicit space in ecological models Modified mean-field equations Moment closure Scaling

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Area of Science:

Ecology
Mathematical Biology
Theoretical Ecology

Background:

Traditional mean-field models simplify ecological interactions by assuming individuals are well-mixed, ignoring spatial heterogeneity.
Evidence suggests spatial structure and distributed interactions are crucial across various scales in ecological systems.
Mean-field models are often considered overly simplified due to their neglect of spatial dynamics.

Purpose of the Study:

To investigate why simplified temporal population models can be effective despite violating mass-action assumptions.
To explore the relationship between spatial structure and global densities in ecological models.
To demonstrate a novel mathematical relationship that reconciles spatial complexity with simplified population dynamics.

Main Methods:

Developed a spatial and stochastic individual-based predator-prey model.
Used the classic Lotka-Volterra model as the mean-field counterpart for comparison.
Analyzed the stationary and transient dynamics of global densities and pair densities (spatial covariances).

Main Results:

Identified a 'bi-power law' relationship between global densities and pair densities in the stationary and transient states.
Demonstrated that this bi-power law allows global density dynamics to be described solely by global densities.
Observed remarkable robustness in the bi-power law exponents for predation rates across parameter variations.

Conclusions:

Spatial structure, when exhibiting a specific relationship with global densities, can validate the use of simplified temporal models.
The bi-power law offers a mathematical bridge between complex spatial dynamics and simpler mean-field descriptions.
Findings suggest potential applications for modified mean-field equations in other ecological systems exhibiting critical phase transitions.