Numerical bifurcation of a delayed diffusive hematopoiesis model with Dirichlet boundary condition

  • Xueyang Liu School of Mathematics and Statistics, Guangdong University of Technology, Guangzhou, China
  • Qi Wang School of Mathematics and Statistics, Guangdong University of Technology, Guangzhou, China
Keywords: hematopoiesis model, nonstandard finite difference scheme, bifurcation


UDC 517.9

Numerical bifurcation of a delayed diffusive hematopoiesis model with Dirichlet boundary condition is studied by using a nonstandard finite-difference scheme.  We prove that a series of numerical Neimark–Sacker bifurcations appear at the positive equilibrium as the time delay increases. At the same time, the parameter conditions for the existence of numerical Neimark–Sacker bifurcations at positive equilibrium point are presented. Finally, we use several examples to verify the accuracy of the results.


Z. Wang, B. Hu, L. Zhu, J. Lin, M. Xu, D. Wang, Hopf bifurcation analysis for Parkinson oscillation with heterogeneous delays: A theoretical derivation and simulation analysis, Commun. Nonlinear Sci. and Numer. Simul., 114, Article 106614 (2022); DOI: 10.1016/j.cnsns.2022.106614.

A. Neto, A. Secchi, P. Melo, Direct computation of Hopf bifurcation points in differential-algebraic equations, Comput. Chem. Eng., 121, 639–645 (2019); DOI: 10.1016/j.compchemeng.2018.12.008.

J. Wei, X. Zou, Bifurcation analysis of a population model and the resulting SIS epidemic model with delay, J.~Comput. and Appl. Math., 197, № 1, 169–187 (2006); DOI: 10.1016/

A. Suryanto, A nonstandard finite difference scheme for SIS epidemic model with delay: stability and bifurcation analysis, Appl. Math., 3, № 6, 528–534 (2012); DOI: 10.4236/AM.2012.36080.

A. Krawiec, M. Szydlowski, Economic growth cycles driven by investment delay, Econ. Modell., 67, 175–183 (2016); DOI: 10.1016/j.econmod.2016.11.014.

Y. Xiong, W. W. Onyx, The analytical solution for sediment reaction and diffusion equation with generalized initial-boundary conditions, Appl. Math. and Mech., 22, № 4, 404–408 (2001); DOI: 10.1007/BF02438306.

S. Dev, D. Dhar, Electric field of a six-needle array electrode used in drug and DNA delivery in vivo: Analytical versus numerical solution, IEEE. Tran. Biomed. Eng., 50, № 11, 1296–1300 (2003); DOI: 10.1109/TBME.2003.818467.

S. Qamar, J. N. Abbasi, S. Javeed, M. Shah, Analytical solutions and moment analysis of chromatographic models for rectangular pulse injections, J. Chromatogr. A, 1315, 92–106 (2013); DOI: 10.1016/j.chroma.2013.09.031.

H. Fogedby, R. Metzler, A. Svane, Exact solution of a linear molecular motor model driven by two-step fluctuations and subject to protein friction, Phys. Rev. E, 70, № 12, Article 021905 (2004); DOI: 10.1103/PhysRevE.70.021905.

A. Yousef, Stability and further analytical bifurcation behaviors of Moran–Ricker model with delayed density dependent birth rate regulation, J. Comput. and Appl. Math., 355, 143–161 (2019); DOI: 10.1016/

J. Mao, S. Tian, T. Zhang, X. Yan, Lie symmetry analysis, conservation laws and analytical solutions for chiral nonlinear Schrödinger equation in $(2+1)$-dimensions, Nonlinear Anal. Model., 25, № 3, 358–377 (2020); DOI: 10.15388/namc.2020.25.16653.

J. Zhang, Bäcklund transformation and multisoliton-like solutions for $(2+1)$-dimensional dispersive long wave equations, Commun. Theor. Phys., 33, № 4, 577–580 (2000); DOI: 10.1088/0253-6102/33/4/577.

K. Hosseini, S. Salahshour, M. Mirzazadeh, A. Ahmadian, D. Baleanu, A. Khoshrang, The $(2+1)$-dimensional Heisenberg ferromagnetic spin chain equation: its solitons and Jacobi elliptic function solutions, Eur. Phys. J. Plus, 136, № 2, 1–9 (2021); DOI: 10.1140/epjp/s13360-021-01160-1.

Q. Wang, J. Wen, P. Zhang, Oscillation analysis of advertising capital model: Analytical and numerical studies, Appl. Math. and Comput., 354, 365–376 (2019); DOI: 10.1016/j.amc.2019.02.029.

H. Qu, L. Wang, Asymptotical stability and asymptotic periodicity for the Lasota–Wazewska model of fractional order with infinite delays, Math. and Comput. Simulat., 43, № 8, 1091–1107 (2019); DOI: 10.2989/16073606.2019.1600596.

C. Huang, H. Li, J. Cao, A novel strategy of bifurcation control for a delayed fractional predator–prey model, Appl. Math. and Comput., 347, 808–838 (2019); DOI: 10.1016/j.amc.2018.11.031.

L. Berezansky, E. Braverman, On stability of delay equations with positive and negative coefficients with applications, Z. Anal. und Anwend., 38, № 2, 157–189 (2019); DOI: 10.4171/ZAA/1633.

L. Li, M. Wang, Global existence and blow-up of solutions of nonlocal diffusion problems with free boundaries, Nonlinear Anal. Real World Appl., 58, Article 103231 (2021); DOI: 10.1016/j.nonrwa.2020.103231.

Y. Qu, J. Wei, Global Hopf bifurcation analysis for a time-delayed model of asset prices, Discrete Dyn. Nat. and Soc., 2010, Article 432821 (2010); DOI: 10.1155/2010/432821.

H. Alfifi, Stability and Hopf bifurcation analysis for the diffusive delay logistic population model with spatially heterogeneous environment, Appl. Math. and Comput., 408, Article 126362 (2021); DOI: 10.1016/j.amc.2021.126362.

R. Zhang, X. Liu, C. Wei, Stability and Hopf bifurcation of a delayed mutualistic system, Int. J. Bifurcat. and Chaos, 31, № 14, Article 2150212 (2021); DOI: 10.1142/S0218127421502126.

Q. Shi, J. Shi, Y. Song, Hopf bifurcation in a reaction-diffusion equation with distributed delay and Dirichlet boundary condition, J. Different. Equat., 263, № 10, 6537–6575 (2017); DOI: 10.1016/j.jde.2017.07.024.

Y. Wang, X. Ding, Numerical bifurcation of a delayed diffusive food-limited model with Dirichlet boundary condition, Math. Methods Appl. Sci., 38, № 13, 2888–2900 (2015); DOI: 10.1002/mma.3513.

X. Ding, D. Fan, M. Liu, Stability and bifurcation of a numerical discretization Mackey–Gass system, Chaos. Solitons Fractals, 34, № 2, 383–393 (2007); DOI: 10.1016/j.chaos.2006.03.053.

Q. Wang, X. Wang, Runge–Kutta methods for systems of differential equation with piecewise continuous arguments: convergence and stability, Numer. Funct. Anal. and Optim., 39, № 7, 784–799 (2017); DOI: 10.1080/ 01630563.2017.1421554.

X. Ding, H. Su, Dynamics of a discretization physiological control system, Discrete Dyn. Nat. and Soc., 2007, Article 51406 (2007); DOI: 10.1155/2007/51406.

H. Su, X. Ding, W. Li, Numerical bifurcation control of Mackey–Glass system, Appl. Math. Model., 35, № 7, 3460–3472 (2011); DOI: 10.1016/j.apm.2011.01.009.

J. Yao, Q. Wang, Numerical dynamics of nonstandard finite difference method for Mackey–Glass system, J. Math., 42, № 1, 63–72 (2022); DOI: 10.13548/j.sxzz.2022.01.006.

J. Wei, Bifurcation analysis in a scalar delay differential equation, Nonlinearity, 20, № 11, 2483–2498 (2007); DOI: 10.1088/0951-7715/20/11/002.

Y. Wang, X. Ding, Dynamics of numerical discretization in a delayed diffusive Nicholson's blowflies equation, Appl. Math. and Comput., 222, 589–603 (2013); DOI: 10.1016/j.amc.2013.07.082.

X. Zhuang, Q. Wang, J. Wen, Numerical dynamics of nonstandard finite difference method for nonlinear delay differential equation, Int. J. Bifurcat. and Chaos, 28, № 11, Article 1850133 (2018); DOI: 10.1142/S021812741850133X.

X. Pan, H. Shu, L. Wang, X. Wang, Dirichlet problem for a delayed diffusive hematopoiesis model, Nonlinear Anal. Real., 48, 493–516 (2019); DOI: 10.1016/j.nonrwa.2019.01.008.

S. Ruan, J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dyn. Contin. Discrete Impuls. Syst. Ser. A, 10, № 6, 863–874 (2003); DOI: 10.1093/imammb/18.1.41.

R. E. Mickens, A nonstandard finite-difference scheme for the Lotka–Volterra system, Appl. Numer. Math., 45, № 2-3, 309–314 (2003); DOI: 10.1016/S0168-9274(02)00223-4.

K. C. Patidar, On the use of nonstandard finite difference methods, J. Different. Equat. and Appl., 11, № 8, 735–758 (2005); DOI: 10.1080/10236190500127471.

How to Cite
Liu, X., and Q. Wang. “Numerical Bifurcation of a Delayed Diffusive Hematopoiesis Model With Dirichlet Boundary Condition”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 76, no. 1, Feb. 2024, pp. 147 -56, doi:10.3842/umzh.v76i1.7295.
Research articles