Stationary Distribution of Stochastic Age-Dependent Population–Toxicant Model with Markov Switching
Abstract
:1. Introduction
- A stochastic age-dependent population–toxicant model with Markov switching is established. The ergodicity of the invariant measure for this model is obtained, applying stochastic techniques such as Gronwall inequality, Young inequality, and so on.
- Under certain suitable conditions, the explicit EM semi-discrete method is used for the time variables, and the convergence of the numerical invariant measure is analyzed.
2. Model and Preliminaries
2.1. Model Formulation
2.2. Preliminaries
- Let be a complete probability space with as the natural filtration generated by Brownian motion (), which means augmented with all P-null sets of .
- stands for the expectation corresponding to .
- C denotes a positive constant whose value may change in different occurrences.
- , where represents generalized partial derivatives, and is a Sobolev space.
- denotes the duality product between V and , and is the scalar product in H.
3. Invariant Measure
3.1. Invariant Measure of Exact Solution
3.2. Numerical Invariant Measure
4. Numerical Example
5. Concluding Remarks
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Parameter | Biological Meaning |
---|---|
The density of the population | |
The depuration rate of the toxicant | |
Toxic substances in organisms | |
Toxic substances in the environment | |
The net organismal excretion rate of the toxicant | |
The mortality rate function of the population | |
The fertility rate function of the population | |
The total loss rate of the toxicant from the environment | |
The net organismal uptake rate of the toxicant from the environment | |
The diffusion coefficient dependent on a, t, and | |
The total density of the population at time t | |
The exogenous total toxicant input into the environment at time t |
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Du, Y.; Wang, Z. Stationary Distribution of Stochastic Age-Dependent Population–Toxicant Model with Markov Switching. Mathematics 2024, 12, 1212. https://doi.org/10.3390/math12081212
Du Y, Wang Z. Stationary Distribution of Stochastic Age-Dependent Population–Toxicant Model with Markov Switching. Mathematics. 2024; 12(8):1212. https://doi.org/10.3390/math12081212
Chicago/Turabian StyleDu, Yanyan, and Zong Wang. 2024. "Stationary Distribution of Stochastic Age-Dependent Population–Toxicant Model with Markov Switching" Mathematics 12, no. 8: 1212. https://doi.org/10.3390/math12081212
APA StyleDu, Y., & Wang, Z. (2024). Stationary Distribution of Stochastic Age-Dependent Population–Toxicant Model with Markov Switching. Mathematics, 12(8), 1212. https://doi.org/10.3390/math12081212