Numerical Oscillation Analysis for Gompertz Equation with One Delay

Abstract

This paper concerns with the oscillation of numerical solutions of a kind of nonlinear delay differential equation proposed by Benjamin Gompertz, this equation usually be used to describe the population dynamics and tumour growth. We obtained some conditions under which the numerical solutions are oscillatory. The non-oscillatory behaviors of numerical solutions are also analyzed. Numerical examples are given to test our theoretical results.

Keywords

• Asymptotic behavior,Gompertz equation,Oscillation,Theta-method

References

1. [1] J. Dzurina, I. Jadlovska, Oscillation theorems for fourth-order delay differential equations with a negative middle term, Math. Meth. Appl. Sci., 40 (2017), 7830-7842.
2. [2] K. M. Chudinov, On exact sufficient oscillation conditions for solutions of linear differential and difference equations of the first order with after effects, Russian Math., 62 (2018), 79-84.
3. [3] J. F. Gao, M. F. Song, M. Z. Liu, Oscillation analysis of numerical solutions for nonlinear delay differential equations of population dynamics, Math. Model. Anal., 16 (2011), 365-375.
4. [4] J. F. Gao, M. F. Song, Oscillation analysis of numerical solutions for nonlinear delay differential equations of hematopoiesis with unimodal production rate, Appl. Math. Comput., 264 (2015), 72-84.
5. [5] Q. Wang, Oscillation analysis of q-methods for the Nicholson’s blowflies model, Math. Meth. Appl. Sci., 39 (2016), 941-948.
6. [6] Y. Z. Wang, J. F. Gao, Oscillation analysis of numerical solutions for delay differential equations with real coefficients, J. Comput. Appl. Math., 337 (2018), 73-86.
7. [7] M. Bodnar, U. Foryss, Three types of simple DDE’s describing tumor growth, J. Biol. Syst., 15 (2007), 453-471.
8. [8] L. E. B. Cabrales, A. R. Aguilera, R. P. Jiméenéz, et al., Mathematical modeling of tumor growth in mice following low-level direct electric current, Math. Comput. Simulat., 78 (2008), 112-120.

9. [9] B. Gompertz, On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies, Philos. T. R. Soc. B, 115 (1825), 513-583.
10. [10] C. P. Winsor, The Gompertz Curve as a Growth Curve, Proc. Natl Acad. Sci., 18 (1932), 1-8.
11. [11] L. Ferrante, S. Bompadre, L. P. Leone, Parameter estimation in a Gompertzian stochastic model for tumor growth, Biometrics, 56 (2000), 1076-1081.
12. [12] M. J. Piotrowska, U. Forys, The nature of Hopf bifurcation for the Gompertz model with delays, Math. Comput. Model., 54 (2011), 2183-2198.
13. [13] M. J. Piotrowska, U. Forys, Analysis of the Hopf bifurcation for the family of angiogenesis models. J. Math. Anal. Appl., 382 (2011), 180-203.
14. [14] M. Bodnar, M. J. Piotrowska, U. Forys, Gompertz model with delays and treatment: Mathematical analysis, Math. Biosci. Eng., 10 (2013), 551-563.
15. [15] I. Gyori, G. Ladas, Oscillation Theory of Delay Differential Equations: with Applications, Oxford University Press, 1991.
16. [16] M. H. Song, Z. W. Yang, M. Z. Liu, Stability of q-methods for advanced differential equations with piecewise continuous arguments, Comput. Math. Appl., 49 (2005), 1295-1301.