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
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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 |
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.
Keywords:
Delay,
spatiotemporal dynamics,
Turing-Hopf bifurcation,
double Hopf bifurcation,
attractors.
Mathematics Subject Classification:
Primary: 35K57, 35B10; Secondary: 37G15, 58J55.
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
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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. |
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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. |
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M. Banerjee and S. Banerjee,
Turing instabilities and spatio-temporal chaos in ratio-dependent Holling-Tanner model, Math. Biosci., 236 (2012), 64-76.
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P. M. Battelino, C. Grebogi, E. 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. Chen, R. Wu, B. 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. |
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S. Chen, Y. 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. |
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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. |
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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.
|
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Y. Du, B. Niu, Y. 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. Duan, B. 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. |
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J.-P. Eckmann,
Roads to turbulence in dissipative dynamical systems, Rev. Modern Phys., 53 (1981), 643-654.
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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. |
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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. |
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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. Huang, S. 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. Li, W. 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. Nindjin, M. 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. Song, H. Jiang, Q.-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. Wang, P. 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. Wollkind, J. 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. Yafia, F. 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. Battelino, C. Grebogi, E. 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. Chen, R. Wu, B. 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. Chen, Y. 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. Du, B. Niu, Y. 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. Duan, B. 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. Huang, S. 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. Li, W. 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. Nindjin, M. 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. Song, H. Jiang, Q.-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. Wang, P. 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. Wollkind, J. 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. Yafia, F. 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 2.
The bifurcation set on the $ (\alpha_{1}, \alpha_{2}) $ plane
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 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)
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