Control and Optimization in Applied Mathematics

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An Adaptive Time-Stepping Algorithm to Solve a Stochastic Lotka-Volterra Competition System with Time-Variable Delays

Document Type : Research Article

Authors

1 Department of Computer Sciences‎, ‎University of Mazandaran‎, ‎Babolsar‎, ‎Iran

2 Department of Applied Mathematics‎, ‎University of Mazandaran‎, ‎Babolsar‎, ‎Iran‎.

10.30473/coam.2024.67940.1237

Abstract

‎This paper introduces ‎a variable step size strategy for a stochastic time-delays Lotka-Volterra competition system‎. ‎This adaptive strategy utilizes the Milstein method for numerical solutions. It employs two local error estimates, corresponding to the diffusion and drift components of the model, to select and control the step sizes‎. ‎The algorithm is described in detail‎, ‎and numerical experiments are conducted to demonstrate the efficiency of the proposed method‎. ‎The primary objective of this research is to propose a dynamic strategy for generating and controlling the step sizes in the finite difference algorithm employed. ‎This adaptive approach accelerates the numerical procedure and improves efficiency compared to a constant-size scheme‎. ‎ As an analytical solution for the model is unavailable‎, ‎a numerical estimation with a small fixed step size is considered a reference solution‎. ‎The numerical results demonstrate the superior accuracy of the proposed strategy compared to a reference solution‎.

Keywords

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References
[1] Bahar, A., Mao, X. (2004). “Stochastic delay Lotka-Volterra model”, Journal of Mathematical Analysis and Applications, 292(2), 364-380.
[2] Cai, Y., Guo, Q., Mao, X. (2023). “Positivity preserving truncated scheme for the stochastic Lotka-Volterra model with small moment convergence”, Calcolo, 60(2), 24.
 [3] Hairer, E., Wanner, G., Nørsett, S.P. (2009). “Solving ordinary differential equations. Part I. Non-stiff problems”, Springer.
[4] Higham, D.J. (2001). “An algorithmic introduction to numerical simulation of stochastic differential equations”, SIAM Review, 43(3), 525-546.
[5] Jiang, D., Ji, C., Li, X., O’Regan, D. (2012). “Analysis of autonomous Lotka-Volterra competition systems with random perturbation”, Journal of Mathematical Analysis and Applications, 390(2), 582-595.
[6] Kloeden, P.E., Platen, E. (1999). “Numerical solution of stochastic differential equations”, Springer, Berlin.
[7] Lamba, H. (1997). “Step size control for the Milstein scheme using first-exit-times”, Preprint, available at http://math.gmu.edu/~harbir/newtsm102307.pdf. Accessed, 8(08), 2013.
[8] Li, Y., Cao, W.(2023).“ A positivity preserving Lamperti transformed Euler–Maruyama method for solving the stochastic Lotka-Volterra competition model”, Communications in Nonlinear Science and Numerical Simulation, 122, 107260.
[9] Liu, Q. (2015). “The effects of time-dependent delays on global stability of stochastic Lotka-Volterra competitive model”, Physica A: Statistical Mechanics and its Applications, 420, 108-115.
[10] Lotka, A.J. (1925). “Elements of physical biology”, Williams & Wilkins.
[11] Mao, X., Wei, F., Wiriyakraikul, T. (2021). “Positivity preserving truncated Euler–Maruyama Method for stochastic Lotka-Volterra competition model”, Journal of Computational and Applied Mathematics, 394, 113566.
[12] Mauthner, S. (1998). “Step size control in the numerical solution of Stochastic Differential Equations”, J. Comput. Appl. Math. 100, 93-109.
[13] Milshtein, G.N. (1986). “Weak approximation of solutions of systems of stochastic differential equations”, Theory of Probability and Its Applications, 30(4), 750-766.
[14] Nguyen, D.H., Yin, G. (2017). “Coexistence and exclusion of stochastic competitive Lotka-Volterra models”, Journal of Differential Equations, 262(3), 1192-1225.
[15] Shekarabi, H.F., Khodabin M. (2016). “Numerical solutions of stochastic Lotka-Volterra equations via operational matrices”‎, Journal of Interpolation and Approximation in Scientific Computing, 37-42.
[16] Tran, K., Yin, G. (2015). “Optimal harvesting strategies for stochastic competitive Lotka-Volterra ecosystems”, Automatica 55, 236-246.
[17] Vadillo, F. (2019). “Comparing stochastic Lotka-Volterra predator-prey models”, Applied Mathematics and Computation, 360, 181-189.
[18] Valinejad, A., Hosseini, S.M. (2010). “A variable step-size control algorithm for the weak approximation of stochastic differential equations”, Numerical Algorithms, 55, 429-446.
[19] Volterra, V.(1926).“ Variazioni e fluttuazioni del numero d'individui in specie animali conviventi”, Società anonima tipografica Leonardo da Vinci.
[20] Wang, H., Dong, L. (2021). “Analysis of a periodic stochastic three-species Lotka-Volterra model with distributed delay and dispersal”, Mathematical Methods in the Applied Sciences. DOI: 10.1002/mma.7338.
[21] Wei, C. (2021). “Parameter estimation for stochastic Lotka-Volterra model driven by small Lévy noises from discrete observations”, Communications in Statistics-Theory and Methods, 50(24), 6014-6023.
[22] Zhang, C., Xie, Y. (2019). “Backward Euler-Maruyama method applied to nonlinear hybrid stochastic differential equations with time-variable delay”, Science China Mathematics, 62, 597-616.
[23] Zhang, M., Tian, J., Zou, K. (2023). “Asymptotic stability of a stochastic age-structured cooperative Lotka-Volterra system with Poisson jumps”, Electronic Journal of Differential Equations, 2, 1-18.
Control and Optimization in Applied Mathematics

Articles in Press, Accepted Manuscript
Available Online from 10 May 2024
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