---

Journal of Applied Nonlinear Dynamics 14(4) (2025) 887--898 | DOI:10.5890/JAND.2025.12.010

M.R. Ricard

Facultad de Matem'atica y Computaci'on, Universidad de La Habana, San L'azaro y L, La Habana, CP 10400, Cuba

Download Full Text PDF

 

Abstract

In this paper are being studied Turing and Turing--Hopf instabilities of solutions for reaction--diffusion systems under a certain type of nonselfadjoint boundary conditions. The root subspaces of the Laplace operator provided with such boundary conditions form a Riesz basis of subspaces of $L_2(\Omega)$, that allows the standard procedure for the study of spatial or spatiotemporal pattern formation by considering the unstable Fourier normal modes of the linearized perturbations.

Acknowledgments

The author wishes to thank the valuable observations of the anonymous referees, made regarding the content and presentation of the paper.

References

1. [1]	Murray, J.D. (2001), Mathematical Biology, I. An Introduction, Third Edition, Springer.

2. [2]	Murray, J.D. (2003), Mathematical Biology, II, Third Edition, Springer.

3. [3]	Perthame, B. (2007), Transport Equations in Biology, Birkh\"{a}user Verlag.

4. [4]	Perthame, B. (2015), Parabolic Equations in Biology: Growth, Reaction, Movement and Diffusion, Lecture Notes on Mathematical Modelling in the Life Sciences, Springer International Publishing.

5. [5]	Turing, A.M. (1952), The chemical basis of morphogenesis, Philosophical Transactions of the Royal Society of London, Series B, 237, 37-72.

6. [6]	Ricard, M.R. and Mischler, S. (2009), Turing instabilities at Hopf bifurcation, Journal of Nonlinear Science, 19(5), 467-496.

7. [7]	Baurmann, M., Gross, T., and Feudel, U. (2007), Instabilities in spatially extended predator-prey systems: Spatio-temporal patterns in the neighborhood of Turing-Hopf bifurcations, Journal of Theoretical Biology, 245(2), 220-229.

8. [8]	Woolley, T.E., Baker, R.E., and Maini, P.K. (2017), Turing’s Theory of Morphogenesis: Where We Started, Where We Are and Where We Want to Go, in Cooper, S. and Soskova, M. (Eds.), The Incomputable: Journeys Beyond the Turing Barrier, Springer. https://doi.org/10.1007/978-3-319-43669-2

9. [9]	Maini, P.K., Painter, K.J., and Chau, H.N.P. (1997), Spatial Pattern Formation in chemical and biological systems, Journal of the Chemical Society, Faraday Transactions, 93(20), 3601-3610.

10. [10]	Just, W., Bose, M., Bose, S., Engel, H., and Sch\"{o}ll, E. (2001), Spatiotemporal dynamics near a supercritical Turing-Hopf bifurcation in a two-dimensional reaction-diffusion system, Physical Review E, 64, 026219, 1-12.

11. [11]	Meixner, M., De Wit, A., Bose, S., and Sch\"{o}ll, E. (1997), Generic spatiotemporal dynamics near codimension-two Turing-Hopf bifurcations, Physical Review E, 55(6), 6690-6697.

12. [12]	Garzon-Alvarado, D.A., Garcia-Aznar, J.M., and Doblare, M. (2009a), Appearance and location of secondary ossification centres may be explained by a reaction-diffusion mechanism, Computers in Biology and Medicine, 39, 554-561.

13. [13]	Garzon-Alvarado, D.A., Garcia-Aznar, J.M., and Doblare, M. (2009b), A reaction-diffusion model for long bones growth, Biomechanics and Modeling in Mechanobiology, 8, 381-395. https://doi.org/10.1007/s10237-008-0144-z

14. [14]	Yang, L. and Epstein, I.R. (2003), Oscillatory Turing Patterns in Reaction-Diffusion Systems with Two Coupled Layers, Physical Review Letters, 90(17), 178303, 1-4.

15. [15]	Sarria-Gonzalez, J., Sgura, I., and Ricard, M.R. (2021), Bifurcations in twinkling patterns for the Lengyel-Epstein reaction-diffusion model, International Journal of Bifurcation and Chaos, 31(11), 2150164.

16. [16]	Lacitignola, D., Sgura, I., and Bozzini, B. (2021), Turing-Hopf patterns in a morphochemical model for electrodeposition with cross-diffusion, Applications in Engineering Science, 5, 100034.

17. [17]	Settanni, G. and Sgura, I. (2015), Devising efficient numerical methods for oscillating patterns in reaction-diffusion systems, Journal of Computational and Applied Mathematics, https://doi.org/10.1016/j.cam.2015.04.044

18. [18]	Dillon, R., Maini, P.K., and Oltmer, H.G. (1994), Pattern formation in generalized Turing systems I. Steady-state patterns in systems with mixed boundary conditions, Journal of Mathematical Biology, 32, 345-393.

19. [19]	Gohberg, I.C. and Krein, M.G. (1969), Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Space, Translations of Mathematical Monographs, Vol.18, American Mathematical Society, Providence, Rhode Island.

20. [20]	Naimark, M.A. (1968), Linear Differential Operators, Vols. 1-2, Harrap (Translated from Russian).

21. [21]	Golubitsky, M., Knobloch, E., and Stewart, I. (2000), Target Patterns and Spirals in Planar Reaction-Diffusion Systems, Journal of Nonlinear Science, 10, 333-354.

22. [22]	Henry, D. (1981), Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, New York.

23. [23]	Kuznetsov, Yu.A. (2004), Elements of Applied Bifurcation Theory, Third Edition, Applied Mathematical Sciences, Vol.112, Springer Verlag, New York.

24. [24]	Bogoliubov, N.N. and Mitropolski, Y. (1961), Asymptotic Methods in the Theory of Nonlinear Oscillations, Gordon and Breach.

25. [25]	Schnakenberg, J. (1979), Simple chemical reactions with limit cycle behaviour, Journal of Theoretical Biology, 81, 389-400.

26. [26]	Edelstein-Keshet, L. (2005), Mathematical Models in Biology, SIAM, Vol.46, Classics in Applied Mathematics.

27. [27]	Sanders, J.A., Verhulst, F., and Murdock, J. (2007), Averaging Methods in Nonlinear Dynamical Systems, Springer, Vol.59.