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A Non-local Cross-Diffusion Model of Population Dynamics I: Emergent Spatial and Spatiotemporal Patterns
Nick P Taylor1, Hyunyeon Kim2, Andrew L Krause2
1Department of Plant Sciences, University of Cambridge, Downing Street, Cambridge, CB2 3EA, UK.
Bulletin of Mathematical Biology
|August 12, 2020
Summary
This study shows that multi-species aggregation models with directed motion can form colonies. Colony formation requires at least three species for directed motion along fecundity gradients, but can occur with two species under different dispersal strategies.
Area of Science:
- Mathematical Biology
- Ecological Modeling
- Theoretical Ecology
Background:
- Existing models explore species aggregation and colony formation through cross-diffusion and directed motion.
- Turing instability is a known mechanism for pattern formation in biological systems.
- Previous work suggested limitations for colony formation in two-species systems under specific directed motion conditions.
Purpose of the Study:
- To extend spatially non-local cross-diffusion models to include multiple species and general dispersal.
- To investigate the conditions for colony formation and persistence in multi-species aggregation models.
- To analyze the impact of directed motion, dispersal types, and resource heterogeneity on emergent spatial patterns.
Main Methods:
- Development and analysis of a generalized spatially non-local cross-diffusion model.
- Investigation of diffusive instabilities and Turing instabilities.
- Numerical simulations to illustrate pattern formation under various conditions.
- Analysis of the interplay between pattern formation and spatial resource heterogeneity.
Main Results:
- Colony formation and persistence via Turing instability require at least three species when motion is along fecundity gradients.
- Two-species colony formation is possible when directed motion deviates from fecundity gradients or with generalized dispersal.
- Advection-dominated cases yield broad, global patterns, while diffusion-dominated cases produce localized colonies.
- Spatial heterogeneity in resources modifies Turing patterns, especially with large variations.
Conclusions:
- The extended model provides a more comprehensive framework for understanding species aggregation and colony formation.
- Directed motion strategies and dispersal mechanisms significantly influence emergent spatial structures.
- Resource heterogeneity plays a crucial role in shaping ecological patterns, deviating from predictions on homogeneous environments.
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