In this paper we investigate bifurcation phenomena in a single-species reaction-diffusion model with spatiotemporal delay under the conditions of the weak and strong kernel functions. We have found that when the weak kernel function is introduced there is Hopf bifurcation but no Turing bifurcation and wave bifurcation to occur, but when the strong kernel function is introduced there exist Hopf bifurcation and wave bifurcation but no Turing bifurcation to occur. Especially, taking the inverse of the average time delay as a bifurcation parameter, we investigate influences of the time delay on the formation of spatiotemporal patterns through the numerical method. Some spatiotemporal patterns induced by Hopf bifurcation and wave bifurcation are respectively shown to illustrate the mechanism of the complexity of spatiotemporal dynamics.
Citation: Gaoxiang Yang, Xiaoyu Li. Bifurcation phenomena in a single-species reaction-diffusion model with spatiotemporal delay[J]. AIMS Mathematics, 2021, 6(7): 6687-6698. doi: 10.3934/math.2021392
In this paper we investigate bifurcation phenomena in a single-species reaction-diffusion model with spatiotemporal delay under the conditions of the weak and strong kernel functions. We have found that when the weak kernel function is introduced there is Hopf bifurcation but no Turing bifurcation and wave bifurcation to occur, but when the strong kernel function is introduced there exist Hopf bifurcation and wave bifurcation but no Turing bifurcation to occur. Especially, taking the inverse of the average time delay as a bifurcation parameter, we investigate influences of the time delay on the formation of spatiotemporal patterns through the numerical method. Some spatiotemporal patterns induced by Hopf bifurcation and wave bifurcation are respectively shown to illustrate the mechanism of the complexity of spatiotemporal dynamics.
| [1] |
N. F. Britton, Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model, SIAM J. Appl. Math., 50 (1990), 1663–1688. doi: 10.1137/0150099
|
| [2] |
N. F. Britton, Aggregation and the competitive exclusion principle, J. Theor. Biol., 136 (1989), 57–66. doi: 10.1016/S0022-5193(89)80189-4
|
| [3] |
S. A. Gourley, M. A. Chaplain, F. A. Davidson, Spatio-temporal pattern formation in a nonlocal reaction-diffusion equation, Dyn. Syst., 16 (2001), 173–192. doi: 10.1080/14689360116914
|
| [4] | J. Billingham, Dynamics of a strongly nonlocal reaction-diffusion population model, Nonlinearity, 17 (2004), 313–346. |
| [5] |
S. Merchant, W. Nagata, Instabilities and spatiotemporal patterns behind predator invasions with nonlocal prey competition, Theor. Popul. Biol., 80 (2011), 289–297. doi: 10.1016/j.tpb.2011.10.001
|
| [6] |
W. Zuo, Y. Song, Stability and bifurcation analysis of a reaction-diffusion equation with spatio-temporal delay, J. Math. Anal. Appl., 430 (2015), 243–261. doi: 10.1016/j.jmaa.2015.04.089
|
| [7] |
B. Han, Z. Wang, Turing patterns of a Lotka-Volterra competitive system with nonlocal delay, Int. J. Bifurcat. Chaos, 28 (2018), 1830021. doi: 10.1142/S0218127418300215
|
| [8] |
M. Banerjee, V. Volpert, Spatio-temporal pattern formation in Rosenzweig-MacArthur model: Effect of nonlocal interactions, Ecol. Complex., 30 (2017), 2–10. doi: 10.1016/j.ecocom.2016.12.002
|
| [9] |
S. Wu, Y. Song, Stability and spatiotemporal dynamics in a diffusive predator-prey model with nonlocal prey competition, Nonlinear Anal.-Real World Appl., 48 (2019), 12–39. doi: 10.1016/j.nonrwa.2019.01.004
|
| [10] |
H. Jiang, Turing bifurcation in a diffusive predator-prey model with schooling behavior, Appl. Math. Lett., 96 (2019), 230–235. doi: 10.1016/j.aml.2019.05.010
|
| [11] |
M. Baurmann, T. Gross, U. Feudel, Instabilities in spatially extended predator-prey systems: Spatio-temporal patterns in the neighborhood of Turing-Hopf bifurcations, J. Theor. Biol., 245 (2007), 220–229. doi: 10.1016/j.jtbi.2006.09.036
|
| [12] | Y. Song, H. Jiang, Y. Yuan, Turing-Hopf bifurcation in the reaction-diffusion system with delay and application to a diffusive predator-prey model, J. Appl. Anal. Comput., 9 (2019), 1132–1146. |
| [13] |
Y. Song, T. Zhang, Y. Peng, Turing-Hopf bifurcation in the reaction-diffusion equations and its applications, Commun. Nonlinear Sci., 33 (2016), 229–258. doi: 10.1016/j.cnsns.2015.10.002
|
| [14] |
Y. Song, S. Wu, H. Wang, Spatiotemporal dynamics in the single population model with memory-based diffusion and nonlocal effect, J. Differ. Equations, 267 (2019), 6316–6351. doi: 10.1016/j.jde.2019.06.025
|
| [15] |
L. Yang, M. Dolnik, A. M. Zhabotinsky, I. R. Epstein, Pattern formation arising from interactions between Turing and wave instabilities, J. Chem. Phys., 117 (2002), 7259–7265. doi: 10.1063/1.1507110
|
| [16] | B. D. Hassard, N. D. Kazarinoff, Y. H. Wan, Theory and applications of Hopf bifurcation, Cambridge University Press, 1981. |
| [17] | J. D. Murray, Mathematical biology I: An introduction (3 Eds.), Springer-Verlag, 2002. |
| [18] | S. A. Gourley, J. W. H. So, Dynamics of a food-limited population model incorporating nonlocal delays on a finite domain, J. Math. Biol., 77 (2002), 49–78. |
| [19] |
S. Pal, S. Ghorai, M. Banerjee, Effect of kernels on spatio-temporal patterns of a non-local prey-predator model, Math. Biosci., 310 (2019), 96–107. doi: 10.1016/j.mbs.2019.01.011
|
Figures(2)
Gaoxiang Yang, Xiaoyu Li. Bifurcation phenomena in a single-species reaction-diffusion model with spatiotemporal delay[J]. AIMS Mathematics, 2021, 6(7): 6687-6698. doi: 10.3934/math.2021392
DownLoad: