Research ArticleNo Access

Theoretical mechanism of boundary-driven instability of the reaction–diffusion population system

Yong-Li Song

https://orcid.org/0000-0002-2209-2458

School of Mathematics and Statistics, Ankang University, Ankang, Shaanxi 725000, P. R. China

School of Mathematics, Hangzhou Normal University, Hangzhou 311121, P. R. China

E-mail Address: syl.mail@163.com

Search for more papers by this author

and

Gao-Xiang Yang

https://orcid.org/0000-0002-8070-6593

School of Mathematics and Statistics, Ankang University, Ankang, Shaanxi 725000, P. R. China

Institute of Mathematics and Applied Mathematics, Ankang University, Ankang, Shaanxi 725000, P. R. China

E-mail Address: gaoxiang7661@126.com

Corresponding author.

Search for more papers by this author

https://doi.org/10.1142/S1793524523500912Cited by:0 (Source: Crossref)

Abstract

In this paper, we study the stability of a constant equilibrium solution of the reaction–diffusion population equation under different boundary conditions through analysis of its characteristic equation. In a scalar reaction–diffusion equation, we have found that the stability of a constant equilibrium solution is different when the scalar reaction–diffusion equation is subject to Neumann boundary conditions, Dirichlet boundary conditions and the mixed type boundary conditions, respectively. Similarly, the more complex results are found in the two reaction–diffusion equations with all different kinds boundary conditions. The relevant numerical calculation results are carried out to demonstrate the validity of theoretical analysis.

Keywords:

AMSC: 35K51

Remember to check out the Most Cited Articles in IJB!

Check out new Biomathematics books in our Mathematics 2018 catalogue!
Featuring author Frederic Y M Wan and more!

References

1. M. Belabbas, A. Ouahab and F. Souna , Rich dynamics in a stochastic predator-prey model with protection zone for the prey and multiplicative noise applied on both species, Nonlinear Dyn. 106 (2021) 2761–2780. Crossref, Web of Science, Google Scholar
2. S. Chen and J. Yu , Stability and bifurcation on predator-prey systems with nonlocal prey competition, Discrete Contin. Dyn. Syst. 38 (2018) 43–62. Crossref, Web of Science, Google Scholar
3. S. Djilali , Spatiotemporal patterns induced by cross-diffusion in predator–prey model with prey herd shape effect, Int. J. Biomath. 13 (2020) 2050030. Link, Web of Science, Google Scholar
4. L. N. Guin, S. Pal, S. Chakravarty and S. Djilali , Pattern dynamics of a reaction–diffusion predator-prey system with both refuge and harvesting, Int. J. Biomath. 14 (2021) 2050084. Link, Web of Science, Google Scholar
5. M. Kucera and M. Vath , Bifurcation for a reaction–diffusion system with unilateral and Neumann boundary conditions, J. Differ. Equ. 252 (2012) 2951–2982. Crossref, Web of Science, Google Scholar
6. P. K. Maini and M. R. Myerscough , Boundary-driven instability, Appl. Math. Lett. 10 (1997) 1–4. Crossref, Web of Science, Google Scholar
7. Y. Peng and G. Zhang , Dynamics analysis of a predator-prey model with herd behavior and nonlocal prey competition, Math. Comput. Simul. 170 (2020) 366–378. Crossref, Web of Science, Google Scholar
8. L. A. Segel and J. L. Jachson , Dissipative structure: An explanation and an ecological example, J. Theor. Biol. 37 (1972) 545–559. Crossref, Web of Science, Google Scholar
9. Q. Shi and Y. Song , Spatially nonhomogeneous periodic patterns in a delayed predator–prey model with predator-taxis diffusion, Appl. Math. Lett. 131 (2022) 108062. Crossref, Web of Science, Google Scholar
10. Y. Song, T. Zhang and Y. Peng , Turing-Hopf bifurcation in the reaction–diffusion equations and its applications, Commun. Nonlinear Sci. Numer. Simul. 33 (2016) 229–258. Crossref, Web of Science, Google Scholar
11. F. Souna, M. Belabbas and Y. Menacer , Complex pattern formations induced by the presence of cross-diffusion in a generalized predator–prey model incorporating the Holling type functional response and generalization of habitat complexity effect, Math. Comput. Simul. 204 (2023) 597–618. Crossref, Web of Science, Google Scholar
12. F. Souna, S. Djilali and F. Charif , Mathematical analysis of a diffusive predator-prey model with herd behavior and prey escaping, Math. Model. Nat. Phenom. 15 (2020) 23. Crossref, Web of Science, Google Scholar
13. F. Souna and A. Lakmeche , Spatiotemporal patterns in a diffusive predator–prey system with Leslie–Gower term and social behavior for the prey, Math. Methods Appl. Sci. 44 (2021) 13920–13944. Crossref, Web of Science, Google Scholar
14. F. Souna, A. Lakmeche and S. Djilali , Spatiotemporal patterns in a diffusive predator-prey model with protection zone and predator harvesting, Chaos Solitons Fractals 140 (2020) 110180. Crossref, Web of Science, Google Scholar
15. F. Souna, P. K. Tiwari, M. Belabbas and Y. Menacer , A predator–prey system with prey social behavior and generalized Holling III functional response: Role of predator-taxis on spatial patterns, Math. Methods Appl. Sci. 46 (2023) 13991–14006. Crossref, Web of Science, Google Scholar
16. A. Turing , The chemical basis of morphogenesis, Phil. Trans. R. Soc. Lond. B 237 (1952) 37–72. Crossref, Web of Science, Google Scholar
17. G. Yang and X. Li , Bifurcation phenomena in a single-species reaction–diffusion model with spatiotemporal delay, AIMS Math. 6 (2021) 6687–6698. Crossref, Web of Science, Google Scholar
18. G. X. Yang and X. Y. Li , Effect of delay on pattern formation of a Rosenzweig-MacArthur type reaction–diffusion model with spatiotemporal delay, Int. J. Biomath. 15 (2022) 2250008. Link, Web of Science, Google Scholar
19. G. Yang and X. Tang , Dynamics analysis of three-species reaction–diffusion system via the multiple scale perturbation method, J. Appl. Anal. Comput. 12 (2022) 206–229. Web of Science, Google Scholar
20. C. Zhu and Y. Peng , Stability and bifurcation analysis in a nonlocal diffusive predator–prey model with hunting cooperation, J. Nonlinear Model. Anal. 5 (2023) 95–107. Google Scholar
21. N. Zhu and S. Yuan , Spatial dynamics of a diffusive prey–predator model with stage structure and fear effect, J. Nonlinear Model. Anal. 4 (2022) 392–408. Google Scholar