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Traveling waves with dispersive variability and time delay
1College of Engineering and Applied Sciences, State University of New York, Stony Brook, New York 11794, USA. Stewart.Harris@sunysb.edu
Summary
Few dispersers can drive population expansion waves similarly to the Fisher equation. A generalized Cook model with time delays accurately models historical population spread, like the Neolithic transition.
Area of Science:
- Mathematical Biology
- Population Dynamics
- Ecological Modeling
Background:
- The Fisher equation and Cook model are used to study population expansion waves.
- Previous linearized wavefront analysis suggested few dispersers can significantly impact wave speed.
- The Cook model, like the Fisher equation, has limitations in describing historical population spread.
Purpose of the Study:
- To determine an approximate traveling wave profile for the Cook model with a small number of dispersers.
- To generalize the Cook model by incorporating time-delay effects.
- To assess the generalized Cook model's accuracy in describing historical population spread, specifically the Neolithic transition.
Main Methods:
- Approximate traveling wave profile determination for the Cook model with few dispersers.
- Linearized wavefront analysis for comparison.
- Generalization of the Cook model to include time-delay effects.
- Comparison of model predictions with historical data.
Main Results:
- The approximate traveling wave profile for the Cook model with few dispersers is determined.
- Results align with linearized analysis, showing few dispersers can drive expansion waves with speeds comparable to single-disperser populations.
- The dimensionless wavespeed is found to be close to unity.
- The generalized Cook model with time delays shows close agreement with the historical record of the Neolithic transition.
Conclusions:
- Relatively few dispersers can significantly influence population expansion wave speed, contrary to initial intuition.
- The generalized Cook model, incorporating time delays, offers a more accurate representation of historical population spread compared to the basic Cook model or Fisher equation.
- This work provides a refined mathematical tool for understanding past and potentially future population dynamics.
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