| Citation: |
Yongli Song, Heping Jiang, Yuan Yuan. TURING-HOPF BIFURCATION IN THE REACTION-DIFFUSION SYSTEM WITH DELAY AND APPLICATION TO A DIFFUSIVE PREDATOR-PREY MODEL[J]. Journal of Applied Analysis & Computation, 2019, 9(3): 1132-1164. doi: 10.11948/2156-907X.20190015
|
TURING-HOPF BIFURCATION IN THE REACTION-DIFFUSION SYSTEM WITH DELAY AND APPLICATION TO A DIFFUSIVE PREDATOR-PREY MODEL
-
Abstract
The interactions of diffusion-driven Turing instability and delayinduced Hopf bifurcation always give rise to rich spatiotemporal dynamics. In this paper, we first derive the algorithm for the normal forms associated with the Turing-Hopf bifurcation in the reaction-diffusion system with delay, which can be used to investigate the spatiotemporal dynamical classification near the Turing-Hopf bifurcation point in the parameter plane. Then, we consider a diffusive predator-prey model with weak Allee effect and delay. Through investigating the dynamics of the corresponding normal form of Turing-Hopf bifurcation induced by diffusion and delay, the spatiotemporal dynamics near this bifurcation point can be divided into six categories. Especially, stable spatially homogeneous/inhomogeneous periodic solutions and steady states, coexistence of two stable spatially inhomogeneous periodic solutions, coexistence of two stable spaially inhomogeneous steady states and the transition from one kind of spatiotemporal patterns to another are found.-
Keywords:
- Diffusion /
- delay /
- Turing-Hopf bifurcation /
- normal form /
- predatorprey model
-
-
References
[1] S. Busenberg and W. Huang, Stability and Hopf bifurcation for a population delay model with diffusion effects, J. Diff. Eqs., 1996, 124(1), 80-107. doi: 10.1006/jdeq.1996.0003 [2] J. Cao, P. Wang, R. Yuan and Y. Mei, Bogdanov-Takens Bifurcation of a Class of Delayed Reaction-Diffusion System, Internat. J. Bifur. Chaos, 2015, 25(06), 1550082. doi: 10.1142/S0218127415500820 [3] S. Chen and J. Shi, Stability and Hopf bifurcation in a diffusive logistic population model with nonlocal delay effect, J. Diff. Eqs., 2012, 253(12), 3440-3470. doi: 10.1016/j.jde.2012.08.031 [4] S. Chen, J. Shi and J. Wei, Global stability and Hopf bifurcation in a delayed diffusive Leslie-Gower predator-prey system, Internat. J. Bifur. Chaos, 2012, 22(03), 1250061. doi: 10.1142/S0218127412500617 [5] S. Chen and J. Yu, Stability and bifurcation on predator-prey systems with nonlocal prey competition, Discrete Contin. Dyn. Syst., 2018, 38(1), 43-62. doi: 10.3934/dcds.2018002 [6] Y. Dong, S. Li and S. Zhang, Hopf bifurcation in a reaction-diffusion model with degn-harrison reaction scheme, Nonlinear Anal. Real World Appl., 2017, 33, 284-297. doi: 10.1016/j.nonrwa.2016.07.002 [7] L. Du and M. Wang, Hopf bifurcation analysis in the 1-D Lengyel-Epstein reaction-diffusion model, J. Math. Anal. Appl., 2017, 366(2), 473-485. [8] T. Faria, Normal forms and Hopf bifurcation for partial differential equations with delays, Trans. Amer. Math. Soc., 2000, 352(5), 2217-2238. doi: 10.1090/S0002-9947-00-02280-7 [9] T. Faria, Stability and bifurcation for a delayed predator-prey model and the effect of diffusion, J. Math. Anal. Appl., 2001, 254(2), 433-463. doi: 10.1006/jmaa.2000.7182 [10] E. González-Olivares and A. Rojas-Palma, Multiple limit cycles in a Gause type predator-prey model with Holling type Ⅲ functional response and Allee effect on prey, B. Math. Biol., 2011, 73(6), 1378-1397. doi: 10.1007/s11538-010-9577-5 [11] S. Guo, Stability and bifurcation in a reaction-diffusion model with nonlocal delay effect, J. Diff. Eqs., 2015, 259(4), 1409-1448. doi: 10.1016/j.jde.2015.03.006 [12] S. Guo and L. Ma, Stability and bifurcation in a delayed reaction-diffusion equation with Dirichlet boundary condition, J. Nonlinear Sci., 2016, 26(2), 545- 580. doi: 10.1007/s00332-016-9285-x [13] K. P. Hadeler and S. Ruan, Interaction of diffusion and delay, Discrete Contin. Dyn. Syst. Ser. B, 2007, 8(1), 95-105. doi: 10.3934/dcdsb [14] M. Haragus and G. Iooss, Local bifurcations, center manifolds, and normal forms in infinite-dimensional dynamical systems, Springer, London, 2010. [15] R. Hu and Y. Yuan, Spatially nonhomogeneous equilibrium in a reactiondiffusion system with distributed delay, J. Diff. Eqs., 2011, 250(6), 2779-2806. doi: 10.1016/j.jde.2011.01.011 [16] R. Hu and Y. Yuan, Stability and Hopf bifurcation analysis for Nicholson's blowflies equation with non-local delay, European J. Appl. Math., 2012, 23(6), 777-796. doi: 10.1017/S0956792512000265 [17] W. Jiang, Q. An and J. Shi, Formulation of the normal forms of Turing-Hopf bifurcation in reaction-diffusion systems with time delay. https://arxiv.org/abs/1802.10286 [18] J. Jin, J. Shi, J. Wei and F.Yi, Bifurcations of patterned solutions in the diffusive Lengyel-Epstein system of CIMA chemical reactions, Rocky Mountain J. Math., 2013, 43(5), 1637-1674. doi: 10.1216/RMJ-2013-43-5-1637 [19] W. Just, M. Bose, S. Bose, H. Engel and E.Schöll, Spatiotemporal dynamics near a supercritical Turing-Hopf bifurcation in a two-dimensional reactiondiffusion system, Physical Rev. E, 2001, 64, 026219. doi: 10.1103/PhysRevE.64.026219 [20] H. Kidachi, On mode interactions in reaction-diffusion equation with nearly degenerate bifurcations, Prog. Theoret. Phy., 1980, 63, 1152-1169. doi: 10.1143/PTP.63.1152 [21] S. Kondo and T. Miura, Reaction-Diffusion Model as a Framework for Understanding Biological Pattern Formation, Science, 2010, 329(5999), 1616-1620. doi: 10.1126/science.1179047 [22] Y. Lv, Y. Pei and R. Yuan, Hopf bifurcation and global stability of a diffusive Gause-type predator-prey models, Comput. Math. Appl., 2016, 72(10), 2620-2635. doi: 10.1016/j.camwa.2016.09.022 [23] Z. Mei, Numerical Bifurcation Analysis for Reaction-Diffusion Equations, Springer-Verlag, Berlin, 2000. [24] M. C. Memory, Bifurcation and asymptotic behavior of solutions of a delaydifferential equation with diffusion, SIAM J. Math. Anal., 1989, 20(3), 533-546. doi: 10.1137/0520037 [25] P. J. Pal, T. Saha, M. Sen and M. Banerjee, A delayed predator-prey model with strong allee effect in prey population growth, Nonlinear Dynam., 2012, 68(1), 23-42. [26] J. E. Pearson, Complex patterns in a simple system, Science, 1993, 261(5118), 189-192. doi: 10.1126/science.261.5118.189 [27] Y. Peng and H. Ling, Pattern formation in a ratio-dependent predator-prey model with cross-diffusion, Appl. Math. Comput., 2018, 331, 307-318. [28] Y. Peng and T. Zhang, Turing instability and pattern induced by cross-diffusion in a predator-prey system with Allee effect, Appl. Math. Comput., 2016, 275, 1-12. [29] H. Shi and S. Ruan, Spatial, temporal and spatiotemporal patterns of diffusive predator-prey models with mutual interference, IMA J. Appl. Math., 2015, 80(5), 1534-1568. doi: 10.1093/imamat/hxv006 [30] Q. Shi, J. Shi and Y. Song, Hopf bifurcation in a reaction-diffusion equation with distributed delay and Dirichlet boundary condition, J. Diff. Eqs., 2017, 263(10), 6537-6575. doi: 10.1016/j.jde.2017.07.024 [31] J. Shi, Z. Xie and K. Little, Cross-diffusion induced instability and stability in reaction-diffusion systems, J. Appl. Anal. Comput., 2011, 1(1), 95-119. [32] Y. Song, H. Jiang, Q. Liu and Y. Yuan, Spatiotemporal Dynamics of the Diffusive Mussel-Algae Model Near Turing-Hopf Bifurcation, SIAM J. Appl. Dyn. Syst., 2017, 16(4), 2030-2062. doi: 10.1137/16M1097560 [33] Y. Song and J. Jiang, Hopf and steady-state-Hopf bifurcations in delay differential equations with applications to a damped harmonic oscillator with delay feedback, Internat. J. Bifur. Chaos, 2012, 22(12), 1250286. doi: 10.1142/S0218127412502860 [34] Y. Song, T. Zhang and Y. Peng, Turing-Hopf bifurcation in the reactiondiffusion equations and its applications, Commun. Nonlinear Sci. Numer. Simul., 2016, 33, 229-258. doi: 10.1016/j.cnsns.2015.10.002 [35] Y. Su, J. Wei and J. Shi, Hopf bifurcations in a reaction-diffusion population model with delay effect, J. Diff. Eqs., 2009, 247(4), 1156-1184. doi: 10.1016/j.jde.2009.04.017 [36] Y. Su and X. Zou, Transient oscillatory patterns in the diffusive non-local blowfly equation with delay under the zero-flux boundary condition, Nonlinearity, 2014, 27(1), 87-104. [37] X. Tang and Y. Song, Stability, Hopf bifurcations and spatial patterns in a delayed diffusive predator-prey model with herd behavior, Appl. Math. Comput., 2015, 254, 375-391. [38] A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London Ser. B, 1952, 237(641), 37-72. doi: 10.1098/rstb.1952.0012 [39] V. K. Vanag and I. R. Epstein, Pattern formation mechanisms in reactiondiffusion systems, Int. J. Dev. Biol., 2009, 53(5-6), 673-681. doi: 10.1387/ijdb.072484vv [40] A. I. Volpert, Vitaly Volpert and V. A. Volpert, Traveling wave solutions of parabolic systems, vol. 140, American Mathematical Soc., 1994. [41] W. Wang, X. Gao, Y. Cai, H. Shi and S. Fu, Turing patterns in a diffusive epidemic model with saturated infection force, J. Franklin. Inst., 2018, 355, 7226-7245. doi: 10.1016/j.jfranklin.2018.07.014 [42] J. Wang, J. Liang, Y. Liu and J. Wang, Zero singularities of codimension two in a delayed predator-prey diffusion system, Neurocomputing, 2017, 227, 10-17. doi: 10.1016/j.neucom.2016.07.060 [43] J. Wang, J. Shi and J. Wei, Predator-prey system with strong allee effect in prey, J. Math. Biol., 2011, 62(3), 291-331. doi: 10.1007/s00285-010-0332-1 [44] S. Wu and Y. Song, Stability and spatiotemporal dynamics in a diffusive predator-prey model with nonlocal prey competition, Nonlinear Anal. Real World Appl., 2019, 48, 12-39. doi: 10.1016/j.nonrwa.2019.01.004 [45] H. Wu and X. Wu, Bogdanov-Takens singularity for a system of reactiondiffusion equations, J. Math. Chem., 2016, 54(1), 120-136. doi: 10.1007/s10910-015-0553-z [46] J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with Delay, J. Dynam. Differential Equations, 2001, 13(3), 651-687. doi: 10.1023/A:1016690424892 [47] R. Yang and Y. Song, Spatial resonance and Turing-Hopf bifurcations in the Gierer-Meinhardt model, Nonlinear Anal. Real World Appl., 2016, 31, 356-387. doi: 10.1016/j.nonrwa.2016.02.006 [48] T. Zhang, X. Liu, X. Meng and T. Zhang, Spatio-temporal dynamics near the steady state of a planktonic system, Comput. Math. Appl., 2018, 75(12), 4490- 4504. doi: 10.1016/j.camwa.2018.03.044 [49] F. Yi, J. Wei and J. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Diff. Eqs., 2009, 246(5), 1944-1977. doi: 10.1016/j.jde.2008.10.024 [50] W. Zuo and J. Wei, Stability and Hopf bifurcation in a diffusive predator-prey system with delay effect, Nonlinear Anal. Real World Appl., 2011, 12(4), 1998-2011. doi: 10.1016/j.nonrwa.2010.12.016 -
-
Figures(10) / Tables(1)
Export File
Citation
Yongli Song, Heping Jiang, Yuan Yuan. TURING-HOPF BIFURCATION IN THE REACTION-DIFFUSION SYSTEM WITH DELAY AND APPLICATION TO A DIFFUSIVE PREDATOR-PREY MODEL[J]. Journal of Applied Analysis & Computation, 2019, 9(3): 1132-1164. doi: 10.11948/2156-907X.20190015
Format
Content
DownLoad:
-
Figure 1. Stability and Turing instability regions for system (3.9). (A): when
$ 0<d_2\leq d_1 $ , there is no Turing instability and the shaded region$ R_1 $ is the stability region of the positive equilibrium$ E^* $ ; (B): when$ 0<d_1<d_2 $ , Turing instability region is the white region$ R_{12} $ and the shaded region$ R_{11} $ is the stability region of the positive equilibrium$ E^* $ . Here$ d_1 = 0.0125, d_2 = 0.125 $ -
Figure 2. Bifurcation diagrams (A) and dynamical classification (B) near the Turing-Hopf point.
$ \mu_1 $ and$ \mu_2 $ are the perturbation for the parameters$ \delta $ and$ \tau $ , respectively, at$ (\delta_*, \tau_*) = (0.25,2.5278) $ -
Figure 3. When
$ (\mu_1,\mu_2) = (0.01,-0.12)\in D_1 $ , the positive constant equilibrium$ E^* $ is asymptotically stable -
Figure 4. When
$ (\mu_1,\mu_2) = (0.01,0.5)\in D_2 $ , the positive constant equilibrium$ E^* $ is unstable and there is a stable spatially homogeneous periodic solution -
Figure 5. When
$ (\mu_1,\mu_2) = (-0.01,0.01)\in D_3 $ , there are unstable spatially inhomogeneous steady states, stable spatially homogeneous periodic solution and there exists a orbit connecting these two states -
Figure 6. When
$ (\mu_1,\mu_2) = (-0.01,-0.095)\in D_4 $ , there are unstable spatially inhomogeneous steady states and stable spatially inhomogeneous periodic solutions. For the initial condtion$ u(x, 0) = 0.3-0.01\cos (2x), v(x,0)) = 0.21+0.002\cos (2x) $ , there is a transition from unstable spatially inhomogeneous steady state to stable spatially inhomogeneous periodic solution -
Figure 7. When
$ (\mu_1,\mu_2) = (-0.01,-0.095)\in D_4 $ , there are unstable spatially inhomogeneous steady states and stable spatially inhomogeneous periodic solutions. For the initial condtion$ u(x, 0) = 0.3-0.01, v(x,0)) = 0.21+0.001\cos (2x) $ , there is a transition from unstable spatially homogeneous periodic solution to stable spatially inhomogeneous periodic solution -
Figure 8. When
$ (\mu_1,\mu_2) = (-0.01,-0.095)\in D_4 $ , there are unstable spatially inhomogeneous steady states and stable spatially inhomogeneous periodic solutions. For the initial condtion$ u(x, 0) = 0.3-0.01, v(x,0)) = 0.21-0.001\cos (2x) $ , there is another stable spatially inhomogeneous periodic solution, which is different from that shown in Fig. 7 -
Figure 9. When
$ (\mu_1,\mu_2) = (-0.01,-0.1)\in D_5 $ , there are unstable spatially inhomogeneous steady states and stable spatially inhomogeneous periodic solutions -
Figure 10. When
$ (\mu_1,\mu_2) = (-0.01,-0.5)\in D_6 $ , the positive constant equilibrium is unstable and there is stable spatially inhomogeneous steady states with$ \cos(2x)-\mathrm{like} $ shape