Abstract
In this paper, the stability and the bifurcations of the nutrient-microorganism model with chemotaxis are analyzed, subject to no-flux boundary conditions. By choosing the chemotaxis coefficient as the control parameter, it is found that the steady state bifurcation, the Hopf–Turing bifurcation, can happen in the model. The induced spatially homogeneous periodic solution, the non-constant steady state, and the spatially inhomogeneous periodic solution are exhibited. The results suggest that chemotaxis assimilated into the model could give rise to rich spatiotemporal dynamical behaviors.
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 11971032
Funding source: China Postdoctoral Science Foundation
Award Identifier / Grant number: 2021M701118
-
Author contributions: Mengxin Chen: Formal analysis, Writing-original draft, Project administration. Ranchao Wu: Methodology, Writing-review & editing, Project administration.
-
Research funding: This work was supported by the National Natural Science Foundation of China (No. 11971032) and China Postdoctoral Science Foundation (No. 2021M701118).
-
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
References
[1] X. B. Zhang and H. Y. Zhao, “Dynamics analysis of a delayed reaction-diffusion predator-prey system with non-continuous threshold harvesting,” Math. Biosci., vol. 289, pp. 130–141, 2017. https://doi.org/10.1016/j.mbs.2017.05.007.Search in Google Scholar PubMed
[2] J. L. Yue and Z. P. Ma, “Stability and Hopf bifurcation of a reaction-diffusion system with weak Allee effect,” Int. J. Bifurcation Chaos, vol. 31, no. 7, p. 2150098, 2021. https://doi.org/10.1142/s021812742150098x.Search in Google Scholar
[3] Q. Hu and J. W. Shen, “Turing instability of the modified reaction-diffusion Holling-Tanner model in random network,” Int. J. Bifurcation Chaos, vol. 32, p. 2250049, 2022.10.1142/S0218127422500493Search in Google Scholar
[4] D. M. Luo, “Global bifurcation for a reaction-diffusion predator-prey model with Holling-II functional response and prey-taxis,” Chaos, Solit. Fractals, vol. 147, p. 110975, 2021. https://doi.org/10.1016/j.chaos.2021.110975.Search in Google Scholar
[5] L. Q. Pu and Z. G. Lin, “Effects of depth and evolving rate on phytoplankton growth in a periodically evolving environment,” J. Math. Anal. Appl., vol. 493, p. 124502, 2021. https://doi.org/10.1016/j.jmaa.2020.124502.Search in Google Scholar
[6] C. Y. Xu, Q. Li, T. H. Zhang, and S. L. Yuan, “Stability and Hopf bifurcation for a delayed diffusive competition model with saturation effect,” Math. Biosci. Eng., vol. 17, no. 6, pp. 8037–8051, 2020. https://doi.org/10.3934/mbe.2020407.Search in Google Scholar PubMed
[7] Q. Hu and J. W. Shen, “Delay-induced self-organization dynamics in a predator-prey network with diffusion,” Nonlinear Dynam., vol. 108, pp. 4499–4510, 2022. https://doi.org/10.1007/s11071-022-07431-5.Search in Google Scholar
[8] M. Baurmann and U. Feudel, “Turing patterns in a simple model of a nutrient-microorganism system in the sediment,” Ecol. Complex., vol. 1, pp. 77–94, 2004. https://doi.org/10.1016/j.ecocom.2004.01.001.Search in Google Scholar
[9] M. X. Chen, R. C. Wu, B. Liu, and L. P. Chen, “Hopf-Hopf bifurcation in the delayed nutrient-microorganism model,” Appl. Math. Model., vol. 86, pp. 460–483, 2020. https://doi.org/10.1016/j.apm.2020.05.024.Search in Google Scholar
[10] M. X. Chen and R. C. Wu, “Dynamics of diffusive nutrient-microorganism model with spatially heterogeneous environment,” J. Math. Anal. Appl., vol. 511, p. 126078, 2022. https://doi.org/10.1016/j.jmaa.2022.126078.Search in Google Scholar
[11] M. X. Chen, Q. Q. Zheng, R. C. Wu, and L. P. Chen, “Hopf bifurcation in delayed nutrient-microorganism model with network structure,” J. Biol. Dynam., vol. 16, no. 1, pp. 1–13, 2022. https://doi.org/10.1080/17513758.2021.2020915.Search in Google Scholar PubMed
[12] Q. Cao and J. H. Wu, “Pattern formation of reaction-diffusion system with chemotaxis terms,” Chaos, vol. 31, p. 113118, 2021. https://doi.org/10.1063/5.0054708.Search in Google Scholar PubMed
[13] K. Qu, C. J. Li, and F. Y. Zhang, “Asymptotic and stability analysis of solutions for a Keller Segel chemotaxis model,” Concurr. Eng. Res. Appl., vol. 29, pp. 75–81, 2021. https://doi.org/10.1177/1063293x21998087.Search in Google Scholar
[14] B. X. Dai and G. X. Sun, “Turing-Hopf bifurcation of a delayed diffusive predator-prey system with chemotaxis and fear effect,” Appl. Math. Lett., vol. 111, p. 106644, 2020. https://doi.org/10.1016/j.aml.2020.106644.Search in Google Scholar
[15] R. Celinski and A. Raczynski, “Asymtotic profile of solutions to a certain chemotaxis system,” Commun. Pure Appl. Anal., vol. 19, no. 2, pp. 911–922, 2020. https://doi.org/10.3934/cpaa.2020041.Search in Google Scholar
[16] Y. Sugiyama, Y. Tsutsui, and J. J. L. Velázquez, “Global solutions to a chemotaxis system with non-diffusive memory,” J. Math. Anal. Appl., vol. 410, no. 2, pp. 908–917, 2014. https://doi.org/10.1016/j.jmaa.2013.08.065.Search in Google Scholar
[17] H. Hattori and A. Lagha, “Existence of global solutions to chemotaxis fluid system with logistic source,” Electron. J. Qual. Theory Differ. Equ., vol. 2021, no. 53, pp. 1–27, 2021. https://doi.org/10.14232/ejqtde.2021.1.53.Search in Google Scholar
[18] P. Y. H. Pang, Y. F. Wang, and J. X. Yin, “Asymptotic profile of a two-dimensional chemotaxis-Navier-Stokes system with singular sensitivity and logistic source,” Math. Model Methods Appl. Sci., vol. 31, no. 03, pp. 577–618, 2021. https://doi.org/10.1142/s0218202521500135.Search in Google Scholar
[19] J. P. Gao, S. J. Guo, and L. Ma, “Global existence and spatiotemporal pattern formation of a nutrient-microorganism model with nutrient-taxis in the sediment,” Nonlinear Dynam., vol. 108, pp. 4207–4229, 2022. https://doi.org/10.1007/s11071-022-07355-0.Search in Google Scholar
[20] M. X. Chen and Q. Q. Zheng, “Steady state bifurcation of a population model with chemotaxis,” Physica A, vol. 609, p. 128381, 2023. https://doi.org/10.1016/j.physa.2022.128381.Search in Google Scholar
[21] H. Amann, “Dynamic theory of quasilinear parabolic equations II,” Differ. Integral Equ., vol. 3, no. 1, pp. 13–75, 1990.10.57262/die/1371586185Search in Google Scholar
[22] D. Horstmann and M. Winkler, “Boundedness vs. blow-up in a chemotaxis system,” J. Differ. Equ., vol. 215, pp. 52–107, 2005. https://doi.org/10.1016/j.jde.2004.10.022.Search in Google Scholar
© 2023 Walter de Gruyter GmbH, Berlin/Boston
