doi: 10.3934/dcdsb.2021202
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Spatiotemporal dynamics in a diffusive Holling-Tanner model near codimension-two bifurcations

1. 

Department of Mathematics, Harbin Institute of Technology at Weihai, Weihai 264209, China

2. 

School of Mathematics, Harbin Institute of Technology, Harbin 150001, China

* Corresponding author: Ben Niu

Received  March 2020 Revised  July 2021 Early access August 2021

Fund Project: The authors are supported by National Natural Science Foundation of China (No.11771109), Shandong Provincial Natural Science Foundation (No.ZR2019QA020)

We investigate spatiotemporal patterns near the Turing-Hopf and double Hopf bifurcations in a diffusive Holling-Tanner model on a one- dimensional spatial domain. Local and global stability of the positive constant steady state for the non-delayed system is studied. Introducing the generation time delay in prey growth, we discuss the existence of Turing-Hopf and double Hopf bifurcations and give the explicit dynamical classification near these bifurcation points. Finally, we obtain the complicated dynamics, including periodic oscillations, quasi-periodic oscillations on a three-dimensional torus, the coexistence of two stable nonconstant steady states, the coexistence of two spatially inhomogeneous periodic solutions, and strange attractors.

Citation: Daifeng Duan, Ben Niu, Junjie Wei. Spatiotemporal dynamics in a diffusive Holling-Tanner model near codimension-two bifurcations. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021202
References:
[1]

Q. An and W. Jiang, Spatiotemporal attractors generated by the Turing-Hopf bifurcation in a time-delayed reaction-diffusion system, Discrete Cont. Dyn. Syst. Ser. B., 24 (2019), 487-510.  doi: 10.3934/dcdsb.2018183.

[2]

Q. An and W. Jiang, Bifurcations and spatiotemporal patterns in a ratio-dependent diffusive Holling-Tanner system with time delay, Math. Meth. Appl. Sci., 42 (2019), 440-465.  doi: 10.1002/mma.5299.

[3]

M. A. Aziz-Alaoui and M. Daher Okiye, Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes, Appl. Math. Lett., 16 (2003), 1069-1075.  doi: 10.1016/S0893-9659(03)90096-6.

[4]

M. Banerjee and S. Banerjee, Turing instabilities and spatio-temporal chaos in ratio-dependent Holling-Tanner model, Math. Biosci., 236 (2012), 64-76.  doi: 10.1016/j.mbs.2011.12.005.

[5]

P. M. BattelinoC. GrebogiE. Ott and J. A. Yorke, Chaotic attractors on a $3$-torus, and torus break-up, Physica D., 39 (1989), 299-314.  doi: 10.1016/0167-2789(89)90012-2.

[6]

P. A. Braza, The bifurcation structure of the Holling-Tanner model for predator-prey interaction using two-timing, SIAM J. Appl. Math., 63 (2003), 889-904.  doi: 10.1137/S0036139901393494.

[7]

M. ChenR. WuB. Liu and L. Chen, Spatiotemporal dynamics in a ratio-dependent predator-prey model with time delay near the Turing-Hopf bifurcation point, Commun. Nonlinear Sci. Numer. Simulat., 77 (2019), 141-167.  doi: 10.1016/j.cnsns.2019.04.024.

[8]

S. ChenY. Lou and J. Wei, Hopf bifurcation in a delayed reaction-diffusion-advection population model, J. Differential Equations, 264 (2018), 5333-5359.  doi: 10.1016/j.jde.2018.01.008.

[9]

X. Chen and W. Jiang, Turing-Hopf bifurcation and multi-stable spatio-temporal patterns in the Lengyel-Epstein system, Nonlinear Anal. Real World Appl., 49 (2019), 386-404.  doi: 10.1016/j.nonrwa.2019.03.013.

[10]

J. B. Collings, Bifurcation and stability analysis of a temperature-dependent mite predator-prey interaction model incorporating a prey refuge, Bull. Math. Biol., 57 (1995), 63-76. 

[11]

Y. DuB. NiuY. Guo and J. Wei, Double Hopf bifurcation in delayed reaction-diffusion systems, J. Dyn. Differ. Equ., 32 (2020), 313-358.  doi: 10.1007/s10884-018-9725-4.

[12]

Y. Du and S.-B. Hsu, A diffusive predator-prey model in heterogeneous environment, J. Differential Equations, 203 (2004), 331-364.  doi: 10.1016/j.jde.2004.05.010.

[13]

D. DuanB. Niu and J. Wei, Hopf-Hopf bifurcation and chaotic attractors in a delayed diffusive predator-prey model with fear effect, Chaos, Solitons, Fractals, 123 (2019), 206-216.  doi: 10.1016/j.chaos.2019.04.012.

[14]

J.-P. Eckmann, Roads to turbulence in dissipative dynamical systems, Rev. Modern Phys., 53 (1981), 643-654.  doi: 10.1103/RevModPhys.53.643.

[15]

T. Faria, Normal forms and Hopf bifurcation for partial differential equations with delays, Trans. Amer. Math. Soc., 352 (2000), 2217-2238.  doi: 10.1090/S0002-9947-00-02280-7.

[16]

T. Faria, Stability and bifurcation for a delayed predator-prey model and the effect of diffusion, J. Math. Anal. Appl., 254 (2001), 433-463.  doi: 10.1006/jmaa.2000.7182.

[17]

T. Faria and W. Huang, Stability of periodic solutions arising from Hopf bifurcation for a reaction-diffusion equation with time delay, Fields Inst. Comm., 31 (2002), 125-141. 

[18]

T. Faria and L. T. Magalhães, Normal forms for retarded functional differential equations and applications to Bogdanov-Takens singularity, J. Differential Equations., 122 (1995), 201-224.  doi: 10.1006/jdeq.1995.1145.

[19]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York, 1983. doi: 10.1007/978-1-4612-1140-2.

[20]

C. S. Holling, The functional response of predator to prey density and its role in mimicry and population regulation, Mem. Ent. Soc. Can., 97 (1965), 5-60.  doi: 10.4039/entm9745fv.

[21]

S. B. Hsu and T. W. Huang, Global stability for a class of predator-prey system, SIAM J. Appl. Math., 55 (1995), 763-783. doi: 10.1137/S0036139993253201.

[22]

J. HuangS. Ruan and J. Song, Bifurcations in a predator-prey system of Leslie type with generalized Holling type III functional response, J. Differential Equations, 257 (2014), 1721-1752.  doi: 10.1016/j.jde.2014.04.024.

[23]

G. E. Hutchinson, Circular causal systems in ecology, Ann. N. Y. Acad. Sci., 50 (1948), 221-246.  doi: 10.1111/j.1749-6632.1948.tb39854.x.

[24]

W. Ko and K. Ryu, Non-constant positive steady-states of a diffusive predator-prey system in homogeneous environment, J. Math. Anal. Appl., 327 (2007), 539-549.  doi: 10.1016/j.jmaa.2006.04.077.

[25]

P. H. Leslie and J. C. Gower, The properties of a stochastic model for the predator-prey type of interaction between two species, Biometrika, 47 (1960), 219-234.  doi: 10.1093/biomet/47.3-4.219.

[26]

J. Li and W. Gao, A strongly coupled predator-prey system with modified Holling-Tanner functional response, Comput. Math. Appl., 60 (2010), 1908-1916.  doi: 10.1016/j.camwa.2009.03.124.

[27]

X. LiW. Jiang and J. Shi, Hopf bifurcation and Turing instability in the reaction-diffusion Holling-Tanner predator-prey model, IMA J. Appl. Math., 78 (2013), 287-306.  doi: 10.1093/imamat/hxr050.

[28]

Z.-P. Ma and W.-T. Li, Bifurcation analysis on a diffusive Holling-Tanner predator-prey model, Appl. Math. Model., 37 (2013), 4371-4384.  doi: 10.1016/j.apm.2012.09.036.

[29] R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, 1973. 
[30]

A. F. NindjinM. A. Aziz-Alaoui and M. Cadivel, Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with time delay, Nonlinear Anal. Real World Appl., 7 (2006), 1104-1118.  doi: 10.1016/j.nonrwa.2005.10.003.

[31]

R. Peng and M. Wang, Positive steady-states of the Holling-Tanner prey-predator model with diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 149-164.  doi: 10.1017/S0308210500003814.

[32]

S. Ruan and J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dyn. Contin. Discrete Impuls. Syst. Ser. A: Math. Anal., 10 (2003), 863-874. 

[33]

D. Ruelle and F. Takens, On the nature of turbulence, Commun. Math. Phys., 20 (1971), 167-192.  doi: 10.1007/BF01646553.

[34]

Z. Shen and J. Wei, Spatiotemporal patterns near the Turing-Hopf bifurcation in a delay-diffusion mussel-algae model, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 29 (2019), 1950164, 25 pp. doi: 10.1142/S0218127419501645.

[35]

H.-B. Shi and S. Ruan, Spatial, temporal and spatiotemporal patterns of diffusive predator-prey models with mutual interference, IMA J. Appl. Math., 80 (2015), 1534-1568.  doi: 10.1093/imamat/hxv006.

[36]

Y. SongH. JiangQ.-X. Liu and Y. Yuan, Spatiotemporal dynamics of the diffusive Mussel-Algae model near Turing-Hopf bifurcation, SIAM J. Appl. Dyn. Syst., 16 (2017), 2030-2062.  doi: 10.1137/16M1097560.

[37]

J. T. Tanner, The stability and the intrinsic growth rates of prey and predator populations, Ecology, 56 (1975), 855-867.  doi: 10.2307/1936296.

[38]

R. K. Upadhyay and S. R. K. Iyengar, Effect of seasonality on the dynamics of 2 and 3 species prey-predator system, Nonlinear Anal. Real World Appl., 6 (2005), 509-530.  doi: 10.1016/j.nonrwa.2004.11.001.

[39]

R. K. Upadhyay and V. Rai, Crisis-limited chaotic dynamics in ecological systems, Chaos Solitons Fractals, 12 (2001), 205-218.  doi: 10.1016/S0960-0779(00)00141-7.

[40]

M. WangP. Y. H. Pang and W. Chen, Sharp spatial patterns of the diffusive Holling-Tanner prey-predator model in heterogeneous environment, IMA. J. Appl. Math., 73 (2008), 815-835.  doi: 10.1093/imamat/hxn016.

[41]

D. J. WollkindJ. B. Collings and J. A. Logan, Metastability in a temperature-dependent model system for predator-prey mite outbreak interactions on fruit flies, Bull. Math. Biol., 50 (1988), 379-409. 

[42]

X. Xu and J. Wei, Turing-Hopf bifurcation of a class of modified Leslie-Gower model with diffusion, Discrete Cont. Dyn. Syst. Ser. B., 23 (2018), 765-783.  doi: 10.3934/dcdsb.2018042.

[43]

R. YafiaF. El Adnani and H. T. Alaoui, Limit cycle and numerical similations for small and large delays in a predator-prey model with modified Leslie-Gower and Holling-type II schemes, Nonlinear Anal. Real World Appl., 9 (2008), 2055-2067.  doi: 10.1016/j.nonrwa.2006.12.017.

[44]

R. Yang and C. Zhang, Dynamics in a diffusive modified Leslie-Gower predator-prey model with time delay and prey harvesting, Nonlinear Dyn., 87 (2017), 863-878.  doi: 10.1007/s11071-016-3084-7.

show all references

References:
[1]

Q. An and W. Jiang, Spatiotemporal attractors generated by the Turing-Hopf bifurcation in a time-delayed reaction-diffusion system, Discrete Cont. Dyn. Syst. Ser. B., 24 (2019), 487-510.  doi: 10.3934/dcdsb.2018183.

[2]

Q. An and W. Jiang, Bifurcations and spatiotemporal patterns in a ratio-dependent diffusive Holling-Tanner system with time delay, Math. Meth. Appl. Sci., 42 (2019), 440-465.  doi: 10.1002/mma.5299.

[3]

M. A. Aziz-Alaoui and M. Daher Okiye, Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes, Appl. Math. Lett., 16 (2003), 1069-1075.  doi: 10.1016/S0893-9659(03)90096-6.

[4]

M. Banerjee and S. Banerjee, Turing instabilities and spatio-temporal chaos in ratio-dependent Holling-Tanner model, Math. Biosci., 236 (2012), 64-76.  doi: 10.1016/j.mbs.2011.12.005.

[5]

P. M. BattelinoC. GrebogiE. Ott and J. A. Yorke, Chaotic attractors on a $3$-torus, and torus break-up, Physica D., 39 (1989), 299-314.  doi: 10.1016/0167-2789(89)90012-2.

[6]

P. A. Braza, The bifurcation structure of the Holling-Tanner model for predator-prey interaction using two-timing, SIAM J. Appl. Math., 63 (2003), 889-904.  doi: 10.1137/S0036139901393494.

[7]

M. ChenR. WuB. Liu and L. Chen, Spatiotemporal dynamics in a ratio-dependent predator-prey model with time delay near the Turing-Hopf bifurcation point, Commun. Nonlinear Sci. Numer. Simulat., 77 (2019), 141-167.  doi: 10.1016/j.cnsns.2019.04.024.

[8]

S. ChenY. Lou and J. Wei, Hopf bifurcation in a delayed reaction-diffusion-advection population model, J. Differential Equations, 264 (2018), 5333-5359.  doi: 10.1016/j.jde.2018.01.008.

[9]

X. Chen and W. Jiang, Turing-Hopf bifurcation and multi-stable spatio-temporal patterns in the Lengyel-Epstein system, Nonlinear Anal. Real World Appl., 49 (2019), 386-404.  doi: 10.1016/j.nonrwa.2019.03.013.

[10]

J. B. Collings, Bifurcation and stability analysis of a temperature-dependent mite predator-prey interaction model incorporating a prey refuge, Bull. Math. Biol., 57 (1995), 63-76. 

[11]

Y. DuB. NiuY. Guo and J. Wei, Double Hopf bifurcation in delayed reaction-diffusion systems, J. Dyn. Differ. Equ., 32 (2020), 313-358.  doi: 10.1007/s10884-018-9725-4.

[12]

Y. Du and S.-B. Hsu, A diffusive predator-prey model in heterogeneous environment, J. Differential Equations, 203 (2004), 331-364.  doi: 10.1016/j.jde.2004.05.010.

[13]

D. DuanB. Niu and J. Wei, Hopf-Hopf bifurcation and chaotic attractors in a delayed diffusive predator-prey model with fear effect, Chaos, Solitons, Fractals, 123 (2019), 206-216.  doi: 10.1016/j.chaos.2019.04.012.

[14]

J.-P. Eckmann, Roads to turbulence in dissipative dynamical systems, Rev. Modern Phys., 53 (1981), 643-654.  doi: 10.1103/RevModPhys.53.643.

[15]

T. Faria, Normal forms and Hopf bifurcation for partial differential equations with delays, Trans. Amer. Math. Soc., 352 (2000), 2217-2238.  doi: 10.1090/S0002-9947-00-02280-7.

[16]

T. Faria, Stability and bifurcation for a delayed predator-prey model and the effect of diffusion, J. Math. Anal. Appl., 254 (2001), 433-463.  doi: 10.1006/jmaa.2000.7182.

[17]

T. Faria and W. Huang, Stability of periodic solutions arising from Hopf bifurcation for a reaction-diffusion equation with time delay, Fields Inst. Comm., 31 (2002), 125-141. 

[18]

T. Faria and L. T. Magalhães, Normal forms for retarded functional differential equations and applications to Bogdanov-Takens singularity, J. Differential Equations., 122 (1995), 201-224.  doi: 10.1006/jdeq.1995.1145.

[19]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York, 1983. doi: 10.1007/978-1-4612-1140-2.

[20]

C. S. Holling, The functional response of predator to prey density and its role in mimicry and population regulation, Mem. Ent. Soc. Can., 97 (1965), 5-60.  doi: 10.4039/entm9745fv.

[21]

S. B. Hsu and T. W. Huang, Global stability for a class of predator-prey system, SIAM J. Appl. Math., 55 (1995), 763-783. doi: 10.1137/S0036139993253201.

[22]

J. HuangS. Ruan and J. Song, Bifurcations in a predator-prey system of Leslie type with generalized Holling type III functional response, J. Differential Equations, 257 (2014), 1721-1752.  doi: 10.1016/j.jde.2014.04.024.

[23]

G. E. Hutchinson, Circular causal systems in ecology, Ann. N. Y. Acad. Sci., 50 (1948), 221-246.  doi: 10.1111/j.1749-6632.1948.tb39854.x.

[24]

W. Ko and K. Ryu, Non-constant positive steady-states of a diffusive predator-prey system in homogeneous environment, J. Math. Anal. Appl., 327 (2007), 539-549.  doi: 10.1016/j.jmaa.2006.04.077.

[25]

P. H. Leslie and J. C. Gower, The properties of a stochastic model for the predator-prey type of interaction between two species, Biometrika, 47 (1960), 219-234.  doi: 10.1093/biomet/47.3-4.219.

[26]

J. Li and W. Gao, A strongly coupled predator-prey system with modified Holling-Tanner functional response, Comput. Math. Appl., 60 (2010), 1908-1916.  doi: 10.1016/j.camwa.2009.03.124.

[27]

X. LiW. Jiang and J. Shi, Hopf bifurcation and Turing instability in the reaction-diffusion Holling-Tanner predator-prey model, IMA J. Appl. Math., 78 (2013), 287-306.  doi: 10.1093/imamat/hxr050.

[28]

Z.-P. Ma and W.-T. Li, Bifurcation analysis on a diffusive Holling-Tanner predator-prey model, Appl. Math. Model., 37 (2013), 4371-4384.  doi: 10.1016/j.apm.2012.09.036.

[29] R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, 1973. 
[30]

A. F. NindjinM. A. Aziz-Alaoui and M. Cadivel, Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with time delay, Nonlinear Anal. Real World Appl., 7 (2006), 1104-1118.  doi: 10.1016/j.nonrwa.2005.10.003.

[31]

R. Peng and M. Wang, Positive steady-states of the Holling-Tanner prey-predator model with diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 149-164.  doi: 10.1017/S0308210500003814.

[32]

S. Ruan and J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dyn. Contin. Discrete Impuls. Syst. Ser. A: Math. Anal., 10 (2003), 863-874. 

[33]

D. Ruelle and F. Takens, On the nature of turbulence, Commun. Math. Phys., 20 (1971), 167-192.  doi: 10.1007/BF01646553.

[34]

Z. Shen and J. Wei, Spatiotemporal patterns near the Turing-Hopf bifurcation in a delay-diffusion mussel-algae model, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 29 (2019), 1950164, 25 pp. doi: 10.1142/S0218127419501645.

[35]

H.-B. Shi and S. Ruan, Spatial, temporal and spatiotemporal patterns of diffusive predator-prey models with mutual interference, IMA J. Appl. Math., 80 (2015), 1534-1568.  doi: 10.1093/imamat/hxv006.

[36]

Y. SongH. JiangQ.-X. Liu and Y. Yuan, Spatiotemporal dynamics of the diffusive Mussel-Algae model near Turing-Hopf bifurcation, SIAM J. Appl. Dyn. Syst., 16 (2017), 2030-2062.  doi: 10.1137/16M1097560.

[37]

J. T. Tanner, The stability and the intrinsic growth rates of prey and predator populations, Ecology, 56 (1975), 855-867.  doi: 10.2307/1936296.

[38]

R. K. Upadhyay and S. R. K. Iyengar, Effect of seasonality on the dynamics of 2 and 3 species prey-predator system, Nonlinear Anal. Real World Appl., 6 (2005), 509-530.  doi: 10.1016/j.nonrwa.2004.11.001.

[39]

R. K. Upadhyay and V. Rai, Crisis-limited chaotic dynamics in ecological systems, Chaos Solitons Fractals, 12 (2001), 205-218.  doi: 10.1016/S0960-0779(00)00141-7.

[40]

M. WangP. Y. H. Pang and W. Chen, Sharp spatial patterns of the diffusive Holling-Tanner prey-predator model in heterogeneous environment, IMA. J. Appl. Math., 73 (2008), 815-835.  doi: 10.1093/imamat/hxn016.

[41]

D. J. WollkindJ. B. Collings and J. A. Logan, Metastability in a temperature-dependent model system for predator-prey mite outbreak interactions on fruit flies, Bull. Math. Biol., 50 (1988), 379-409. 

[42]

X. Xu and J. Wei, Turing-Hopf bifurcation of a class of modified Leslie-Gower model with diffusion, Discrete Cont. Dyn. Syst. Ser. B., 23 (2018), 765-783.  doi: 10.3934/dcdsb.2018042.

[43]

R. YafiaF. El Adnani and H. T. Alaoui, Limit cycle and numerical similations for small and large delays in a predator-prey model with modified Leslie-Gower and Holling-type II schemes, Nonlinear Anal. Real World Appl., 9 (2008), 2055-2067.  doi: 10.1016/j.nonrwa.2006.12.017.

[44]

R. Yang and C. Zhang, Dynamics in a diffusive modified Leslie-Gower predator-prey model with time delay and prey harvesting, Nonlinear Dyn., 87 (2017), 863-878.  doi: 10.1007/s11071-016-3084-7.

Figure 1.  (a) The diagram of $ d_{2}(n^{2}) $ on the $ d_{2} $-$ n $ plane. In region "R", $ B_{n}+C_{n}<0 $. (b) The bifurcation curves on the $ \tau $-$ d_{2} $ plane, where "TH" is the Turing-Hopf bifurcation point
Figure 2.  The bifurcation set on the $ (\alpha_{1}, \alpha_{2}) $ plane
Figure 3.  The dynamical classifications in region $ \Re_{1} $-$ \Re_{6} $
Figure 4.  Choosing $ (\alpha_{1}, \alpha_{2}) = (-0.3936, -0.0858) $ at the red dot in region $ \Re_{1} $, the positive equilibrium $ E_{\ast}(u_{\ast}, v_{\ast}) $ is asymptotically stable. The initial functions are $ u(x, 0) = 1.2+0.02\cos2x $, $ v(x, 0) = 13.8-0.02\cos2x $
Figure 5.  Choosing $ (\alpha_{1}, \alpha_{2}) = (-0.3936, 0.0742) $ at the red dot in region $ \Re_{2} $, two nonconstant steady-states are stable. (a) The initial functions are $ u(x, 0) = 1.2+0.02\cos2x $, $ v(x, 0) = 13.8-0.02\cos2x $; (b) The initial functions are $ u(x, 0) = 1.2-0.02\cos2x $, $ v(x, 0) = 13.8+0.02\cos2x $
Figure 6.  Choosing $ (\alpha_{1}, \alpha_{2}) = (-0.0936, 0.0742) $ at the red dot in region $ \Re_{3} $, two spatially inhomogeneous periodic solutions are stable. The initial functions of (a) and (b) are the same as those in Figure 5(a), (b), respectively
Figure 7.  Choosing $ (\alpha_{1}, \alpha_{2}) = (0.0064, 0.0742) $ at the red dot in region $ \Re_{4} $, the spatially homogeneous periodic solution is unstable (transient state) and two spatially inhomogeneous periodic solutions are stable. The initial functions of (a) and (b) are $ u(x, 0) = 1.2+0.02\cos2x $, $ v(x, 0) = 13.8-0.02\cos2x $; The initial functions of (c) and (d) are $ u(x, 0) = 1.2-0.02\cos2x $, $ v(x, 0) = 13.8+0.02\cos2x $
Figure 8.  The partial bifurcation set on the $ \tau $-$ r_{2} $ plane, where the colored curves stand for Hopf bifurcation curves
Figure 9.  Complete bifurcation sets near the double Hopf point "HH" of system (2)
Figure 10.  The dynamical classifications in region $ \textbf{D1} $-$ \textbf{D8} $
Figure 11.  When $ (\tau, r_{2}) = (6.25, 0.23) $ (see P1 in Figure 9), the constant steady state $ (u_{\ast}, v_{\ast}) $ of system (2) is asymptotically stable with the initial functions $ u_{0}(x) = 2.18+0.02\cos2x $, $ v_{0}(x) = 9.24-0.02\cos2x $
Figure 12.  When $ (\tau, r_{2}) = (32.43, 0.23) $ (see P2 in Figure 9), the spatially homogeneous periodic solutions of system (2) are stable with the initial functions $ u_{0}(x) = 2.18+0.02\cos2x $, $ v_{0}(x) = 9.24-0.02\cos2x $
Figure 13.  When $ (\tau, r_{2}) = (33.58, 0.23) $ (see P3 in Figure 9), the spatially homogeneous periodic or quasi-periodic solutions of system (2) are unstable with the initial functions $ u_{0}(x) = 2.18+0.02\cos2x $, $ v_{0}(x) = 9.24-0.02\cos2x $. (a): the periodic solutions; (b): the quasi-periodic solutions
Figure 14.  The corresponding Poincaré map on the $ u(0, t-2\tau)-v(0, t) $. (a) $ \tau = 33.58 $; (b) $ \tau = 34.2 $; (c) $ \tau = 35.41 $ (see P3, P4, P5 in Figure 9). The parameter $ r_{2} = 0.23 $ is fixed with the initial functions $ u_{0}(x) = 2.18+0.02\cos2x $, $ v_{0}(x) = 9.24-0.02\cos2x $
Figure 15.  The local maximum map of $ u $. (a) $ (\tau, r_{2}) = (34.2, 0.23) $; (b) $ (\tau, r_{2}) = (35.41, 0.23) $
Table 1.  The twelve unfoldings of system (20)
$ \mathrm{Case} $ $ \mathrm{Ia} $ $ \mathrm{Ib} $ $ \mathrm{II} $ $ \mathbf{III} $ $ \mathrm{IVa} $ $ \mathrm{IVb} $ $ \mathrm{V} $ $ \mathbf{VIa} $ $ \mathrm{VIb} $ $ \mathrm{VIIa} $ $ \mathrm{VIIb} $ $ \mathrm{VIII} $
$ d $ + + + + + + $ {-} $ $ {-} $ $ {-} $ $ {-} $ $ {-} $ $ {-} $
$ b $ + + + $ {-} $ $ {-} $ $ {-} $ + + + $ {-} $ $ {-} $ $ {-} $
$ c $ + + $ {-} $ + $ {-} $ $ {-} $ + $ {-} $ $ {-} $ + + $ {-} $
$ d-bc $ + $ {-} $ + + + $ {-} $ $ {-} $ + $ {-} $ + $ {-} $ $ {-} $
$ \mathrm{Case} $ $ \mathrm{Ia} $ $ \mathrm{Ib} $ $ \mathrm{II} $ $ \mathbf{III} $ $ \mathrm{IVa} $ $ \mathrm{IVb} $ $ \mathrm{V} $ $ \mathbf{VIa} $ $ \mathrm{VIb} $ $ \mathrm{VIIa} $ $ \mathrm{VIIb} $ $ \mathrm{VIII} $
$ d $ + + + + + + $ {-} $ $ {-} $ $ {-} $ $ {-} $ $ {-} $ $ {-} $
$ b $ + + + $ {-} $ $ {-} $ $ {-} $ + + + $ {-} $ $ {-} $ $ {-} $
$ c $ + + $ {-} $ + $ {-} $ $ {-} $ + $ {-} $ $ {-} $ + + $ {-} $
$ d-bc $ + $ {-} $ + + + $ {-} $ $ {-} $ + $ {-} $ + $ {-} $ $ {-} $
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