Research article

Taylor approximation of the solution of age-dependent stochastic delay population equations with Ornstein-Uhlenbeck process and Poisson jumps

  • Received: 04 December 2019 Accepted: 18 February 2020 Published: 09 March 2020
  • Numerical approximation is a vital method to investigate the properties of stochastic age-dependent population systems, since most stochastic age-dependent population systems cannot be solved explicitly. In this paper, a Taylor approximation scheme for a class of age-dependent stochastic delay population equations with mean-reverting Ornstein-Uhlenbeck (OU) process and Poisson jumps is presented. In case that the coefficients of drift and diffusion are Taylor approximations, we prove that the numerical solutions converge to the exact solutions for these equations. Moreover, the convergence order of the numerical scheme is given. Finally, some numerical simulations are discussed to illustrate the theoretical results.

    Citation: Wenrui Li, Qimin Zhang, Meyer-Baese Anke, Ming Ye, Yan Li. Taylor approximation of the solution of age-dependent stochastic delay population equations with Ornstein-Uhlenbeck process and Poisson jumps[J]. Mathematical Biosciences and Engineering, 2020, 17(3): 2650-2675. doi: 10.3934/mbe.2020145

    Related Papers:

  • Numerical approximation is a vital method to investigate the properties of stochastic age-dependent population systems, since most stochastic age-dependent population systems cannot be solved explicitly. In this paper, a Taylor approximation scheme for a class of age-dependent stochastic delay population equations with mean-reverting Ornstein-Uhlenbeck (OU) process and Poisson jumps is presented. In case that the coefficients of drift and diffusion are Taylor approximations, we prove that the numerical solutions converge to the exact solutions for these equations. Moreover, the convergence order of the numerical scheme is given. Finally, some numerical simulations are discussed to illustrate the theoretical results.


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    [1] R. Li, W. Pang, Q. Wang, Numerical analysis for stochastic age-dependent population equations with Poisson jumps, J. Math. Anal. Appl., 327 (2007), 1214-1224.
    [2] W. Ma, Q. Zhang, J. Gao, The existence and uniqueness of solutions to stochastic age-dependent population equations with jumps, Chin. J. Appl. Probab. Stat.,32 (2016), 441-451.
    [3] L. Wang, X. Wang, Convergence of the semi-implicit Euler method for stochastic age-dependent population equations with Poisson jumps, Appl. Math. Model., 34 (2010), 2034-2043.
    [4] A. Rathinasamy, B. Yin, B. Yasodha, Numerical analysis for stochastic age-dependent population equations with Poisson jump and phase semi-Markovian switching, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 350-362.
    [5] J. Tan, A. Rathinasamy, Y. Pei, Convergence of the split-step method for stochastic age-dependent population equations with Poisson jumps, Appl. Math. Comput., 254 (2015), 305-317.
    [6] H. Chen, Existence and uniqueness, attraction for stochastic age-structured population systems with diffusion and Poisson jump, J. Math. Phys., 54 (2013), 082701.
    [7] M. Wei, Exponential stability of numerical solutions to stochastic age-dependent population equations with Poisson jumps, Engin. Tech., 55 (2011), 1103-1108.
    [8] Y. Pei, H. Yang, Q. Zhang, F. Shen, Asymptotic mean-square boundedness of the numerical solutions of stochastic age-dependent population equations with Poisson jumps, Appl. Math. Comput., 320 (2018), 524-534.
    [9] W. Ma, Q. Zhang, Convergence of numerical solutions to stochastic age-dependent population equations with Poisson jumps and Markovian switching, J. Shanxi Univ., 2011 (2011).
    [10] S. Zhao, S. Yuan, H. Wang, Threshold behavior in a stochastic algal growth model with stoichiometric constraints and seasonal variation, J. Differ. Equ., 2019 (2019).
    [11] B. Wang, J. Sun, B. Huang, The model analysis for stochastic age-dependent population with Poisson Jumps, Int. Inst. Appl. Stat. Stud., 2008 (2008).
    [12] W. Li, Q. Zhang, Construction of positivity-preserving numerical method for stochastic SIVS epidemic model, Adv. Differ. Equ., 2019 (2019).
    [13] Y. Li, M. Ye, Q. Zhang, Strong convergence of the partially truncated Euler Maruyama scheme for a stochastic age-structured SIR epidemic model, Appl. Math. Comput., 362 (2019), 124519.
    [14] D. Duffie, Dynamic asset pricing theory, Princeton University Press, (2010.)
    [15] E. Allen, Environmental variability and mean-reverting processes, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 2073.
    [16] Y. Zhao, S. Yuan, J. Ma, Survival and stationary distribution analysis of a stochastic competitive model of three species in a polluted environment, Bull. Math. Biol., 77 (2015), 1285-1326.
    [17] W. Wang, Y. Cai, Z. Ding, Z. Gui, A stochastic differential equation SIS epidemic model incorporating Ornstein-Uhlenbeck process, Phys. A, 509 (2018), 921-936.
    [18] Y. Cai, J. Jiao, Z. Gui, Y. Liu, W. Wang, Environmental variability in a stochastic epidemic model, Appl. Math. Comput., 329 (2018), 210-226.
    [19] S. Deng, W. Fei, Y. Liang, X. Mao, Convergence of the split-step-method for stochastic age-dependent population equations with Markovian switching and variable delay, Appl. Numer. Math., 139 (2019), 15-37.
    [20] Q. Li, Q. Zhang, B. Cao, Mean-square stability of stochastic age-dependent delay population systems with jumps, Acta Math. Appl. Sin., 34 (2018), 145-154.
    [21] J. Tan, W. Men, Y. Pei, Y. Guo, Construction of positivity preserving numerical method for stochastic age-dependent population equations, Appl. Math. Comput., 293 (2017), 57-64.
    [22] F. Jiang, Y. Shen, L. Liu, Taylor approximation of the solutions of stochastic differential delay equations with Poisson jump, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 798-804.
    [23] X. Yu, S, Yuan, T. Zhang, Asymptotic properties of stochastic nutrient-plankton food chain models with nutrient recycling, Nonlinear Anal. Hybrid Syst., 34 (2019), 209-225.
    [24] S. Jankovic, D. Ilic, An analytic approximation of solutions of stochastic differential equations, Comput. Math. Appl., 47 (2004), 903-912.
    [25] A. Ivanov, Y. Kazmerchuk, A. Swishchuk, Theory, stochastic stability and applications of stochastic delay differential equations: A survey of results, Differ. Equ. Dyna. Syst., 11 (2003), 55-115.
    [26] Q. Zhang, W. Liu, Z. Nie, Existence, uniqueness and exponential stability for stochastic age-dependent population, Appl. Math. Comput., 154 (2004), 183-201.
    [27] R. Li, H. Meng, Y. Dai, Convergence of numerical solutions to stochastic delay differential equations with jumps, Appl. Math. Comput., 172 (2006), 584-602.
    [28] L. Wang, C. Mei, H. Xue, The semi-implicit Euler method for stochastic differential delay equation with jumps, Appl. Math. Comput., 192 (2007), 567-578.
    [29] S. Anita, Analysis and control of age-dependent population dynamics, Springer Science & Business Media, (2000).
    [30] D. Higham, P. Kloeden, Strong convergence rates for backward Euler on a class of nonlinear jump-diffusion problems, J. Comput. Appl. Math., 205 (2007), 949-956.
    [31] I. Bihari, A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations, Acta Math. Hungarica, 7 (1956), 81-94.
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