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Mathematical Study of a Resource-Based Diffusion Model with Gilpin-Ayala Growth and Harvesting
Ishrat Zahan1, Md Kamrujjaman2, Saleh Tanveer3
1Department of Mathematics, Bangladesh University of Engineering and Technology, Dhaka, 1000, Bangladesh.
Bulletin of Mathematical Biology
|September 15, 2022
Summary
This study examines a population model with spatial diffusion, revealing that enhanced diffusion can prevent species survival even with sufficient resources. This highlights diffusion
Area of Science:
- Mathematical Biology
- Ecology
- Population Dynamics
Background:
- Investigates single-species population distribution using a Gilpin-Ayala growth model.
- Incorporates spatial diffusion and Neumann boundary conditions for a heterogeneous environment.
- Assumes population spread is proportional to the gradient of population per unit resource.
Purpose of the Study:
- To analyze the global well-posedness of the mathematical model.
- To determine conditions for the existence and stability of non-trivial equilibrium states under harvesting.
- To examine conditions leading to species extinction and the existence of time-periodic solutions.
Main Methods:
- Mathematical modeling of population dynamics with spatial diffusion.
- Analysis of global well-posedness and stability of equilibrium states.
- Investigation of time-periodic solutions for time-periodic parameters.
- Numerical simulations to explore equilibrium states and parameter dependence.
Main Results:
- Established conditions for the existence and global stability of non-trivial and trivial equilibrium states based on harvesting rates.
- Proved the existence of time-periodic states when growth, resource, capacity, and harvesting functions are time-periodic.
- Demonstrated enhanced diffusion effects, particularly for small Gilpin-Ayala parameters, which can preclude non-trivial states.
Conclusions:
- The Gilpin-Ayala model with specific diffusion dynamics can lead to species extinction under certain harvesting conditions.
- Enhanced diffusion can prevent the existence of non-trivial population states, even when local growth rates exceed harvesting.
- Numerical results highlight the influence of resource, capacity, and the Gilpin-Ayala parameter on population distribution.
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