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Modelling population growth with delayed nonlocal reaction in 2-dimensions
Dong Liang1, Jianhong Wu, Fan Zhang
1Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada M3J 1P3. dliang@math.yorku.ca.
Mathematical Biosciences and Engineering : MBE
|April 8, 2010
Summary
This study models single-species population growth using reaction-diffusion equations with nonlocal delayed effects in 2D spaces. Numerical analysis reveals stable steady states and periodic waves, influenced by diffusion and maturation delays.
Area of Science:
- Mathematical Biology
- Computational Science
- Ecology
Background:
- Population dynamics are crucial in ecology and are often modeled using reaction-diffusion equations.
- Incorporating spatial aspects and time delays (maturation) adds complexity to these models.
- Understanding these complex dynamics is key to predicting species distribution and stability.
Purpose of the Study:
- To develop novel reaction-diffusion equation models for single-species population growth in 2D bounded domains.
- To analyze the interplay between diffusion, nonlocal reactions, and maturation delays.
- To numerically investigate the emergence of stable states and periodic waves.
Main Methods:
- Development of new reaction-diffusion models incorporating nonlocal delayed reactions.
- Numerical analysis of population dynamics using established birth functions.
- Investigation of asymptotically stable steady states and periodic wave solutions.
Main Results:
- The models successfully capture the combined effects of diffusion and delayed nonlocal maturation.
- Numerical simulations show the occurrence of both asymptotically stable steady states and periodic waves.
- Parameter variations influence the characteristics of periodic waves and steady states.
Conclusions:
- Reaction-diffusion models with nonlocal delayed reactions provide a robust framework for studying spatial population dynamics.
- The interplay of diffusion and maturation delays significantly impacts population distribution and stability.
- Numerical methods are effective in exploring complex behaviors like periodic waves and stable states in ecological models.
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