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A stochastic Gilpin-Ayala mutualism model driven by mean-reverting OU process with Lévy jumps
1School of Science, Northeast Forestry University, Harbin 150040, China.
Mathematical Biosciences and Engineering : MBE
|March 29, 2024
Summary
This study introduces a stochastic Gilpin-Ayala mutualism model with Ornstein-Uhlenbeck (OU) process and Lévy jumps. It analyzes the model
Area of Science:
- Mathematical Biology
- Stochastic Processes
- Ecology
Background:
- The Gilpin-Ayala model describes mutualistic interactions.
- Environmental randomness and sudden disturbances (jump noise) are crucial in ecological dynamics.
- Stochastic differential equations are vital for modeling such complex systems.
Purpose of the Study:
- To establish and analyze a stochastic Gilpin-Ayala mutualism model incorporating a mean-reverting Ornstein-Uhlenbeck (OU) process and Lévy jumps.
- To investigate the asymptotic behaviors of this novel stochastic ecological model.
Main Methods:
- Utilizing the Ornstein-Uhlenbeck (OU) process to simulate environmental noise.
- Incorporating Lévy jumps to model sudden disturbances.
- Employing Lyapunov functions to prove the existence of global solutions.
- Analyzing moment boundedness and the existence of stationary distributions.
- Proving the extinction of the model.
Main Results:
- Existence and uniqueness of the global solution for the stochastic Gilpin-Ayala model were proven.
- Moment boundedness of the model's solutions was established.
- The existence of a stationary distribution for the model was demonstrated.
- Conditions for the extinction of the model were derived.
Conclusions:
- The established stochastic Gilpin-Ayala model with OU process and Lévy jumps provides a robust framework for ecological dynamics.
- Theoretical findings on solution existence, boundedness, stationary distribution, and extinction were validated through numerical simulations.
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