Turing patterns induced by cross-diffusion in a predator-prey system with functional response of Holling-II type. (English) Zbl 1537.35054
Summary: In this article, a classical predator-prey system with linear cross-diffusion and Holling-II type functional response and subject to homogeneous Neuamnn boundary condition is considered. The spatially homogeneous Hopf bifurcation curve and Turing bifurcation curve of the constant coexistence equilibrium are established with the help of the linearized analysis. When the bifurcation parameters are restricted to the Turing instability region and near the Turing bifurcation curve, the associated amplitude equations of the original system near the constant coexistence equilibrium are obtained by means of multiple-scale time perturbation analysis. According to the obtained amplitude equations, the stability and classification of spatiotemporal patterns of the original system near the constant coexistence equilibrium are determined. It is shown that the cross-diffusion in the classical predator-prey system plays an important role in formation of spatiotemporal patterns. Also, the theoretical results are verified numerically.
MSC:
| 35B35 | Stability in context of PDEs |
| 35B32 | Bifurcations in context of PDEs |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35K51 | Initial-boundary value problems for second-order parabolic systems |
| 35K57 | Reaction-diffusion equations |
| 92D40 | Ecology |
Keywords:
predator-prey system; cross-diffusion; Turing patterns; amplitude equations; complicated dynamicsReferences:
| [1] | Bentout, S.; Djilali, S.; Atangana, A., Bifurcation analysis of an age-structured prey-predator model with infection developed in prey, Math. Methods Appl. Sci., 45, 1189-1208, 2022 · Zbl 1529.35032 · doi:10.1002/mma.7846 |
| [2] | Bentout, S.; Djilali, S.; Ghanbari, B., Backward, Hopf bifurcation in a heroin epidemic model with treat age, Int. J. Model. Simul. Sci. Comput., 12, 2150018, 2021 · doi:10.1142/S1793962321500185 |
| [3] | Bentout, S.; Djilali, S.; Kuniya, T.; Wang, J-L, Mathematical analysis of a vaccination epidemic model with nonlocal diffusion, Math. Methods Appl. Sci., 46, 10970-10994, 2023 · Zbl 1538.35394 · doi:10.1002/mma.9162 |
| [4] | Boudjema, I.; Djilali, S., Turing-Hopf bifurcation in Gauss-type model with cross diffusion and its application, Nonlinear Stud., 25, 665-687, 2018 · Zbl 1409.35102 |
| [5] | Chen, S-S; Shi, J-P; Wei, J-J, The effect of delay on a diffusive predator-prey system with Holling type-II predator functional response, Commun. Pure Appl. Anal., 12, 481-501, 2013 · Zbl 1264.35120 · doi:10.3934/cpaa.2013.12.481 |
| [6] | Chen, M-X; Wu, R-C; Chen, L-P, Spatiotemporal patterns induced by Turing and Turing-Hopf bifurcations in a predator-prey system, Appl. Math. Comput., 380, 2020 · Zbl 1460.92163 |
| [7] | Cheng, K-S, Uniqueness of a limit cycle for a predator-prey system, SIAM J. Math. Anal., 12, 541-548, 1981 · Zbl 0471.92021 · doi:10.1137/0512047 |
| [8] | Djilali, S., Pattern formation of a diffusive predator-prey model with herd behavior and nonlocal prey competition, Math. Methods Appl. Sci., 43, 2233-2250, 2020 · Zbl 1452.92033 · doi:10.1002/mma.6036 |
| [9] | Djilali, S., Threshold asymptotic dynamics for a spatial age-dependent cell-to-cell transmission model with nonlocal disperse, Discrete Contin. Dyn. Syst. Ser. B, 28, 4108-4143, 2023 · Zbl 1512.35071 · doi:10.3934/dcdsb.2023001 |
| [10] | Djilali, S.; Bentout, S., Pattern formations of a delayed diffusive predator-prey model with predator harvesting and prey social behavior, Math. Methods Appl. Sci., 44, 9128-9142, 2021 · Zbl 1469.92086 · doi:10.1002/mma.7340 |
| [11] | Guin, L-N, Spatial patterns through Turing instability in a reaction-diffusion predator-prey model, Math. Comput. Simul., 109, 174-185, 2015 · Zbl 1540.92113 · doi:10.1016/j.matcom.2014.10.002 |
| [12] | Hsu, S-B, On global stability of a predator-prey system, Math. Biosci., 39, 1-10, 1978 · Zbl 0383.92014 · doi:10.1016/0025-5564(78)90025-1 |
| [13] | Hu, G-P; Li, X-L, Turing patterns of a predator-prey model with a modified Leslie-Gower term and cross-diffusions, Int. J. Biomath., 5, 1250060, 2012 · Zbl 1297.92063 · doi:10.1142/S179352451250060X |
| [14] | Iida, M.; Mimura, M.; Ninomiya, H., Diffusion, cross-diffusion and competitive interaction, J. Math. Biol., 53, 617-641, 2006 · Zbl 1113.92064 · doi:10.1007/s00285-006-0013-2 |
| [15] | Li, Y-X; Liu, H.; Wei, Y-M; Ma, M., Turing pattern of a reaction-diffusion predator-prey model with weak Allee effect and delay, J. Phys. Conf. Ser., 1707, 2020 · doi:10.1088/1742-6596/1707/1/012025 |
| [16] | Mezouaghi, A.; Djilali, S.; Bentout, S.; Biroud, K., Bifurcation analysis of a diffusive predator-prey model with prey social behavior and predator harvesting, Math. Methods Appl. Sci., 45, 718-731, 2022 · Zbl 1530.35024 · doi:10.1002/mma.7807 |
| [17] | Murray, J-D, Mathematical Biology II, 1993, Heidelberg: Springer, Heidelberg · doi:10.1007/978-3-662-08542-4 |
| [18] | Ouyang, Q.; Gunaratne, GH; Swinney, HL, Rhombic patterns: broken hexagonal symmetry, Chaos, 3, 707-711, 1993 · doi:10.1063/1.165931 |
| [19] | Ou, Y-X, Nonlinear Science and the Pattern Dynamics Introduction, 2010, Beijing: Peking University Press, Beijing |
| [20] | Peng, R.; Shi, J-P, Non-existence of non-constant positive steady states of two Holling type-II predator-prey systems: Strong interaction case, J. Differ. Equ., 247, 866-886, 2009 · Zbl 1169.35328 · doi:10.1016/j.jde.2009.03.008 |
| [21] | Song, Y-L; Tang, X-S, Stability, steady-state bifurcations, and Turing patterns in a predator-prey model with herd behavior and prey-taxis, Stud. Appl. Math., 139, 371-404, 2017 · Zbl 1373.35324 · doi:10.1111/sapm.12165 |
| [22] | Tang, X-S; Song, Y-L, Cross-diffusion induced spatiotemporal patterns in a predator-prey model with herd behavior, Nonlinear Anal. Real World Appl., 24, 36-49, 2015 · Zbl 1336.92073 · doi:10.1016/j.nonrwa.2014.12.006 |
| [23] | Turing, A-M, The chemical basis of morphogenesis, Bull. Math. Biol., 237, 37-72, 1952 · Zbl 1403.92034 |
| [24] | Vanag, V.; Epstein, I., Cross-diffusion and pattern formation in reaction-diffusion systems, Phys. Chem. Chem. Phys., 11, 897-912, 2009 · doi:10.1039/B813825G |
| [25] | Wang, Q-F; Peng, Y-H, Turing instability and pattern induced by cross-diffusion in a predator-prey system, J. Shanghai Normal Univ. (Nat. Sci.), 47, 331-337, 2018 · Zbl 1424.35043 |
| [26] | Wu, J-H, Theory and Applications of Partial Functional Differential Equations, 1996, New York: Springer, New York · Zbl 0870.35116 · doi:10.1007/978-1-4612-4050-1 |
| [27] | Yan, X-P; Zhang, C-H, Stability and Turing instability in a diffusive predator-prey system with Beddington-DeAngelis functional response, Nonlinear Anal. Real World Appl., 20, 1-13, 2014 · Zbl 1295.35071 · doi:10.1016/j.nonrwa.2014.04.001 |
| [28] | Yang, B., Pattern formation in a diffusive ratio-dependent Holling-Tanner predator-prey model with Smith growth, Discrete Dyn. Nat. Soc., 1, 87-118, 2013 · Zbl 1264.34169 |
| [29] | Yi, F-Q; Wei, J-J; Shi, J-P, Bifurcation and spatio-temporal patterns in a homogeneous diffusive predator-prey system, J. Differ. Equ., 246, 1944-1977, 2009 · Zbl 1203.35030 · doi:10.1016/j.jde.2008.10.024 |
| [30] | Yuan, S-L; Xu, C-Q; Zhang, T-H, Spatial dynamics in a predator-prey model with herd behavior, Chaos, 23, 33102-33102, 2013 · Zbl 1323.92197 · doi:10.1063/1.4812724 |
| [31] | Zhang, J-F; Li, W-T; Yan, X-P, Hopf bifurcation and Turing instability in spatial homogeneous and inhomogeneous predator-prey models, Appl. Math. Comput., 218, 1883-1893, 2011 · Zbl 1228.92082 |
| [32] | Zhou, Y.; Yan, X-P; Zhang, C-H, Turing patterns induced by self-diffusion in a predator-prey model with schooling behavior in predator and prey, Nonlinear Dyn., 105, 3731-3747, 2021 · Zbl 1537.35059 · doi:10.1007/s11071-021-06743-2 |
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