Monotone traveling wave solutions of Mackey–Glass reaction diffusion equation with delayed diffusion term
-
William Kyle Barker
,
Nguyen Truong Thanh
Abstract
This work investigates the existence of solutions to the Mackey–Glass equation, incorporating delays in both the diffusion and birth functions. This equation, a nonlinear functional differential equation, plays a significant role in modeling phenomena in biology and physiology. Using the monotone iteration method, we construct quasi-upper and quasi-lower solutions to rigorously establish the existence of solutions. The analysis relies on a modified Perron’s theorem.
Acknowledgements
The authors would like to thank the anonymous referee for their thoughtful comments.
References
[1] W. K. Barker, Existence of Traveling waves in a Nicholson blowflies model with delayed diffusion term, Proc. Amer. Math. Soc. (2025), 10.1090/proc/17233. 10.1090/proc/17233Search in Google Scholar
[2] W. K. Barker and N. Van Minh, Traveling waves in reaction-diffusion equations with delay in both diffusion and reaction terms, preprint (2023), https://arxiv.org/abs/2301.11504. Search in Google Scholar
[3] R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley & Sons, Chichester, 2003. 10.1002/0470871296Search in Google Scholar
[4] R. Durrett and S. A. Levin, The importance of being discrete (and spatial), Theoret. Pop. Biol. 46 (1994), 363–394. 10.1006/tpbi.1994.1032Search in Google Scholar
[5] S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Differential Equations 171 (2001), no. 2, 294–314. 10.1006/jdeq.2000.3846Search in Google Scholar
[6] M. C. Mackey and L. Glass, Oscillation and chaos in physiological control system, Science 197 (1977), no. 4300, 287–289. 10.1126/science.267326Search in Google Scholar PubMed
[7] J. Mallet-Paret, The Fredholm alternative for functional-differential equations of mixed type, J. Dynam. Differential Equations 11 (1999), no. 1, 1–47. 10.1023/A:1021889401235Search in Google Scholar
[8] C. Ramirez-Carrasco and J. Molina-Garay, Existence and approximation of traveling wavefronts for the diffusive Mackey–Glass equation, Aust. J. Math. Anal. Appl. 18 (2021), no. 1, 1–12. Search in Google Scholar
[9] K. W. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional-differential equations, Trans. Amer. Math. Soc. 302 (1987), no. 2, 587–615. 10.1090/S0002-9947-1987-0891637-2Search in Google Scholar
[10] J. W.-H. So and X. Zou, Traveling waves for the diffusive Nicholson’s blowflies equation, Appl. Math. Comput. 122 (2001), no. 3, 385–392. 10.1016/S0096-3003(00)00055-2Search in Google Scholar
[11] J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dynam. Differential Equations 13 (2001), no. 3, 651–687. 10.1023/A:1016690424892Search in Google Scholar
© 2025 Walter de Gruyter GmbH, Berlin/Boston
