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April  2022, 27(4): 2147-2172. doi: 10.3934/dcdsb.2021127

Interactions of time delay and spatial diffusion induce the periodic oscillation of the vegetation system

1. 

School of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan Shanxi 030006, China

2. 

Department of Mathematics, North University of China, Taiyuan Shanxi 030051, China

3. 

Complex Systems Research Center, Shanxi University, Taiyuan Shanxi 030006, China

4. 

Shanxi Key Laboratory of Mathematical Techniques and Big Data, Analysis on Disease Control and Prevention, Taiyuan Shanxi 030006, China

* Corresponding author: Jing Li, Gui-Quan Sun and Zhen Jin

Received  November 2020 Revised  March 2021 Published  April 2022 Early access  April 2021

Empirical data exhibit a common phenomenon that vegetation biomass fluctuates periodically over time in ecosystem, but the corresponding internal driving mechanism is still unclear. Simultaneously, considering that the conversion of soil water absorbed by roots of the vegetation into vegetation biomass needs a period time, we thus introduce the conversion time into Klausmeier model, then a spatiotemporal vegetation model with time delay is established. Through theoretical analysis, we not only give the occurence conditions of stability switches for system without and with diffusion at the vegetation-existence equilibrium, but also derive the existence conditions of saddle-node-Hopf bifurcation of non-spatial system and Hopf bifurcation of spatial system at the coincidence equilibrium. Our results reveal that the conversion delay induces the interaction between the vegetation and soil water in the form of periodic oscillation when conversion delay increases to the critical value. By comparing the results of system without and with diffusion, we find that the critical value decreases with the increases of spatial diffusion factors, which is more conducive to emergence of periodic oscillation phenomenon, while spatial diffusion factors have no effects on the amplitude of periodic oscillation. These results provide a theoretical basis for understanding the spatiotemporal evolution behaviors of vegetation system.

Citation: Jing Li, Gui-Quan Sun, Zhen Jin. Interactions of time delay and spatial diffusion induce the periodic oscillation of the vegetation system. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 2147-2172. doi: 10.3934/dcdsb.2021127
References:
[1]

J. A. BonachelaR. M. PringleE. ShefferT. C. CoverdaleJ. A. GuytonK. K. CaylorS. A. Levin and C. E. Tarnita, Termite mounds can increase the robustness of dryland ecosystems to climatic change, Science, 347 (2015), 651-655.  doi: 10.1126/science.1261487.

[2]

F. BorgognoP. D'OdoricoF. Laio and L. Ridolfi, Mathematical models of vegetation pattern formation in ecohydrology, Rev. Geophys, 47 (2009), 1-36.  doi: 10.1029/2007RG000256.

[3]

X. ChenK.-Y. Lam and Y. Lou, Dynamics of a reaction-diffusion-advection model for two competing species, Discrete Contin. Dyn. Syst., 32 (2012), 3841-3859.  doi: 10.3934/dcds.2012.32.3841.

[4]

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

[5]

M. FazlyM. Lewis and H. Wang, Analysis of propagation for impulsive reaction-diffusion models, SIAM J. Appl. Math., 80 (2020), 521-542.  doi: 10.1137/19M1246481.

[6]

S. Getzin, H. Yizhaq, B. Bell, et al., Discovery of fairy circles in Australia supports self-organization theory, Proc. Natl. Acad. Sci. USA, 113 (2016), 201522130. doi: 10.1073/pnas.1522130113.

[7]

S. GetzinK. WiegandT. WiegandH. von HardenbergJ. Yizhaq and E. Meron, Adopting a spatially explicit perspective to study the mysterious fairy circles of Namibia, Ecography, 38 (2015), 1-11.  doi: 10.1111/ecog.00911.

[8]

E. Gilad, J. von Hardenberg, A. Provenzale, M. Shachak and E. Meron, Ecosystem engineers: from pattern formation to habit creation, Phys. Rev. Lett., 93 (2004), 098105. doi: 10.1103/PhysRevLett.93.098105.

[9]

Z.-G. Guo, G.-Q. Sun, Z. Wang, Z. Jin, L. Li and C. Li, Spatial dynamics of an epidemic model with nonlocal infection, Appl. Math. Comput., 377 (2020), 125158. doi: 10.1016/j.amc.2020.125158.

[10]

R. HilleRisLambers, M. Rietkerk, F. van den Bosch, H. H. T. Prins and H. de Kroon, Vegetation pattern formation in semi-arid grazing systems, Ecology, 82 (2001), 50-61. https://doi.org/10.1890/0012-9658(2001)082[0050:VPFISA]2.0.CO;2.

[11]

C. A. Klausmeier, Regular and irregular patterns in semiarid vegetation, Science, 284 (1999), 1826-1828.  doi: 10.1126/science.284.5421.1826.

[12]

H. Kobayashi and D. G. Dye, Atmospheric conditions for monitoring the long-term vegetation dynamics in the Amazon using normalized difference vegetation index, Remote Sens. Environ., 97 (2005), 519-525.  doi: 10.1016/j.rse.2005.06.007.

[13]

K.-Y. Lam and W.-M. Ni, Uniqueness and complete dynamics in heterogeneous competition diffusion systems, SIAM J. Appl. Math., 72 (2012), 1695-1712.  doi: 10.1137/120869481.

[14]

R. Lefever and O. Lejeune, On the origin of tiger bush, Bull. Math. Biol., 59 (1997), 263-294.  doi: 10.1007/BF02462004.

[15]

J. LiZ. JinG.-Q. Sun and L.-P. Song, Pattern dynamics of a delayed eco-epidemiological model with disease in the predator, Discrete Cont. Dyn-S, 10 (2017), 1025-1042.  doi: 10.3934/dcdss.2017054.

[16]

L. LiJ. ZhangC. LiuH.-T. ZhangY. Wang and Z. Wang, Analysis of transmission dynamics for Zika virus on networks, Appl. Math. Comput., 347 (2019), 566-577.  doi: 10.1016/j.amc.2018.11.042.

[17]

Q.-X. LiuZ. Jin and B.-L. Li, Numerical investigation of vegetation spatial pattern in a model with feedback function, J. Theor. Biol., 254 (2008), 350-360.  doi: 10.1016/j.jtbi.2008.05.017.

[18]

E. Meron, Modelling dryland landscapes, Math. Model. Nat. Phenom., 6 (2011), 163-187.  doi: 10.1051/mmnp/20116109.

[19]

F. PengW. FanX. Xu and et al., Analysis on temporal-spatial change of vegetation coverage in Hulunbuir Steppe (2000-2014), Acta Scientiarum Naturalium Universitatis Pekinensis, 53 (2017), 563-572. 

[20]

R. M. Pringle and C. E. Tarnita, Spatial self-organization of ecosystems: Integrating multiple mechanisms of regular-pattern formation, Annu. Rev. Entomol., 62 (2017), 359-377.  doi: 10.1146/annurev-ento-031616-035413.

[21]

M. RietkerkS. C. DekkerP. de Ruiter and J. van de Koppel, Self-organized patchiness and catastrophic shifts in ecosystems, Science, 305 (2004), 1926-1929.  doi: 10.1126/science.1101867.

[22]

S. Ruan and J. Wei, On the zeros of a third degree exponential polynomial with applications to a delayed model for the control of testosterone secretion, IMA J. Math. Appl. Med. Biol., 18 (2001), 41-52.  doi: 10.1093/imammb/18.1.41.

[23]

J. A. Sherratt, Pattern solutions of the Klausmeier model for banded vegetation in semiarid environments IV: Slowly moving patterns and their stability, SIAM. J. Appl. Math., 73 (2013), 330-350.  doi: 10.1137/120862648.

[24]

J. A. Sherratt, An analysis of vegetation stripe formation in semi-arid landscapes, J. Math. Biol., 51 (2005), 183-197.  doi: 10.1007/s00285-005-0319-5.

[25]

G.-Q. SunC.-H. Wang and Z.-Y. Wu, Pattern dynamics of a Gierer-Meinhardt model with spatial effects, Nonlinear Dynam., 88 (2017), 1385-1396.  doi: 10.1007/s11071-016-3317-9.

[26]

G.-Q. SunC.-H. WangL.-L. ChangY.-P. WuL. Li and Z. Jin, Effects of feedback regulation on vegetation patterns in semi-arid environments, Appl. Math. Model., 61 (2018), 200-215.  doi: 10.1016/j.apm.2018.04.010.

[27]

C. E. TarnitaJ. A. BonachelaE. ShefferJ. A. GuytonT. C. CoverdaleR. A. Long and R. M. Pringle, A theoretical foundation for multi-scale regular vegetation patterns, Nature, 541 (2017), 398-401.  doi: 10.1038/nature20801.

[28]

C. TianQ. ShiX. CuiJ. GuoZ. Yang and J. Shi, Spatiotemporal dynamics of a reaction-diffusion model of pollen tube tip growth, J. Math. Biol., 79 (2019), 1319-1355.  doi: 10.1007/s00285-019-01396-7.

[29]

J. von Hardenberg, E. Meron, M. Shachak and Y. Zarmi, Diversity of vegetation patterns and desertification, Phys. Rev. Lett., 87 (2001), 19810. doi: 10.1103/PhysRevLett.87.198101.

[30]

X. WangW. Wang and G. Zhang, Vegetation pattern formation of a water-biomass model, Commun. Nonlinear. Sci. Numer. Simulat., 42 (2017), 571-584.  doi: 10.1016/j.cnsns.2016.06.008.

show all references

References:
[1]

J. A. BonachelaR. M. PringleE. ShefferT. C. CoverdaleJ. A. GuytonK. K. CaylorS. A. Levin and C. E. Tarnita, Termite mounds can increase the robustness of dryland ecosystems to climatic change, Science, 347 (2015), 651-655.  doi: 10.1126/science.1261487.

[2]

F. BorgognoP. D'OdoricoF. Laio and L. Ridolfi, Mathematical models of vegetation pattern formation in ecohydrology, Rev. Geophys, 47 (2009), 1-36.  doi: 10.1029/2007RG000256.

[3]

X. ChenK.-Y. Lam and Y. Lou, Dynamics of a reaction-diffusion-advection model for two competing species, Discrete Contin. Dyn. Syst., 32 (2012), 3841-3859.  doi: 10.3934/dcds.2012.32.3841.

[4]

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

[5]

M. FazlyM. Lewis and H. Wang, Analysis of propagation for impulsive reaction-diffusion models, SIAM J. Appl. Math., 80 (2020), 521-542.  doi: 10.1137/19M1246481.

[6]

S. Getzin, H. Yizhaq, B. Bell, et al., Discovery of fairy circles in Australia supports self-organization theory, Proc. Natl. Acad. Sci. USA, 113 (2016), 201522130. doi: 10.1073/pnas.1522130113.

[7]

S. GetzinK. WiegandT. WiegandH. von HardenbergJ. Yizhaq and E. Meron, Adopting a spatially explicit perspective to study the mysterious fairy circles of Namibia, Ecography, 38 (2015), 1-11.  doi: 10.1111/ecog.00911.

[8]

E. Gilad, J. von Hardenberg, A. Provenzale, M. Shachak and E. Meron, Ecosystem engineers: from pattern formation to habit creation, Phys. Rev. Lett., 93 (2004), 098105. doi: 10.1103/PhysRevLett.93.098105.

[9]

Z.-G. Guo, G.-Q. Sun, Z. Wang, Z. Jin, L. Li and C. Li, Spatial dynamics of an epidemic model with nonlocal infection, Appl. Math. Comput., 377 (2020), 125158. doi: 10.1016/j.amc.2020.125158.

[10]

R. HilleRisLambers, M. Rietkerk, F. van den Bosch, H. H. T. Prins and H. de Kroon, Vegetation pattern formation in semi-arid grazing systems, Ecology, 82 (2001), 50-61. https://doi.org/10.1890/0012-9658(2001)082[0050:VPFISA]2.0.CO;2.

[11]

C. A. Klausmeier, Regular and irregular patterns in semiarid vegetation, Science, 284 (1999), 1826-1828.  doi: 10.1126/science.284.5421.1826.

[12]

H. Kobayashi and D. G. Dye, Atmospheric conditions for monitoring the long-term vegetation dynamics in the Amazon using normalized difference vegetation index, Remote Sens. Environ., 97 (2005), 519-525.  doi: 10.1016/j.rse.2005.06.007.

[13]

K.-Y. Lam and W.-M. Ni, Uniqueness and complete dynamics in heterogeneous competition diffusion systems, SIAM J. Appl. Math., 72 (2012), 1695-1712.  doi: 10.1137/120869481.

[14]

R. Lefever and O. Lejeune, On the origin of tiger bush, Bull. Math. Biol., 59 (1997), 263-294.  doi: 10.1007/BF02462004.

[15]

J. LiZ. JinG.-Q. Sun and L.-P. Song, Pattern dynamics of a delayed eco-epidemiological model with disease in the predator, Discrete Cont. Dyn-S, 10 (2017), 1025-1042.  doi: 10.3934/dcdss.2017054.

[16]

L. LiJ. ZhangC. LiuH.-T. ZhangY. Wang and Z. Wang, Analysis of transmission dynamics for Zika virus on networks, Appl. Math. Comput., 347 (2019), 566-577.  doi: 10.1016/j.amc.2018.11.042.

[17]

Q.-X. LiuZ. Jin and B.-L. Li, Numerical investigation of vegetation spatial pattern in a model with feedback function, J. Theor. Biol., 254 (2008), 350-360.  doi: 10.1016/j.jtbi.2008.05.017.

[18]

E. Meron, Modelling dryland landscapes, Math. Model. Nat. Phenom., 6 (2011), 163-187.  doi: 10.1051/mmnp/20116109.

[19]

F. PengW. FanX. Xu and et al., Analysis on temporal-spatial change of vegetation coverage in Hulunbuir Steppe (2000-2014), Acta Scientiarum Naturalium Universitatis Pekinensis, 53 (2017), 563-572. 

[20]

R. M. Pringle and C. E. Tarnita, Spatial self-organization of ecosystems: Integrating multiple mechanisms of regular-pattern formation, Annu. Rev. Entomol., 62 (2017), 359-377.  doi: 10.1146/annurev-ento-031616-035413.

[21]

M. RietkerkS. C. DekkerP. de Ruiter and J. van de Koppel, Self-organized patchiness and catastrophic shifts in ecosystems, Science, 305 (2004), 1926-1929.  doi: 10.1126/science.1101867.

[22]

S. Ruan and J. Wei, On the zeros of a third degree exponential polynomial with applications to a delayed model for the control of testosterone secretion, IMA J. Math. Appl. Med. Biol., 18 (2001), 41-52.  doi: 10.1093/imammb/18.1.41.

[23]

J. A. Sherratt, Pattern solutions of the Klausmeier model for banded vegetation in semiarid environments IV: Slowly moving patterns and their stability, SIAM. J. Appl. Math., 73 (2013), 330-350.  doi: 10.1137/120862648.

[24]

J. A. Sherratt, An analysis of vegetation stripe formation in semi-arid landscapes, J. Math. Biol., 51 (2005), 183-197.  doi: 10.1007/s00285-005-0319-5.

[25]

G.-Q. SunC.-H. Wang and Z.-Y. Wu, Pattern dynamics of a Gierer-Meinhardt model with spatial effects, Nonlinear Dynam., 88 (2017), 1385-1396.  doi: 10.1007/s11071-016-3317-9.

[26]

G.-Q. SunC.-H. WangL.-L. ChangY.-P. WuL. Li and Z. Jin, Effects of feedback regulation on vegetation patterns in semi-arid environments, Appl. Math. Model., 61 (2018), 200-215.  doi: 10.1016/j.apm.2018.04.010.

[27]

C. E. TarnitaJ. A. BonachelaE. ShefferJ. A. GuytonT. C. CoverdaleR. A. Long and R. M. Pringle, A theoretical foundation for multi-scale regular vegetation patterns, Nature, 541 (2017), 398-401.  doi: 10.1038/nature20801.

[28]

C. TianQ. ShiX. CuiJ. GuoZ. Yang and J. Shi, Spatiotemporal dynamics of a reaction-diffusion model of pollen tube tip growth, J. Math. Biol., 79 (2019), 1319-1355.  doi: 10.1007/s00285-019-01396-7.

[29]

J. von Hardenberg, E. Meron, M. Shachak and Y. Zarmi, Diversity of vegetation patterns and desertification, Phys. Rev. Lett., 87 (2001), 19810. doi: 10.1103/PhysRevLett.87.198101.

[30]

X. WangW. Wang and G. Zhang, Vegetation pattern formation of a water-biomass model, Commun. Nonlinear. Sci. Numer. Simulat., 42 (2017), 571-584.  doi: 10.1016/j.cnsns.2016.06.008.

Figure 1.  Time series of vegetation coverage in Gansu Province from January 2000 to December 2018
Figure 2.  Stability of equilibria of system (5) with respect to parameter $ a $ for m = 0.8, the solid line and dotted line respectively indicate that the equilibria are stable and unstable, except for pink dotted line $ a = 2m $
Figure 3.  The time series (a)-(c) and phase plane (d)-(f) diagrams of system (4) without diffusion when $ \tau>\tau_{00}^{0} $ are plotted: (a)(d) $ m = 1.4, \tau = 4.5>\tau_{00}^{0} = 4.2650; $ (b)(e) $ m = 1.6, \tau = 3.3>\tau_{00}^{0} = 2.9887; $ (c)(f) $ m = 1.8, \tau = 2.7>\tau_{00}^{0} = 2.3657. $ Red dot, blue dot and green closed loop denote the coincidence equilibrium $ E_{**} $, initial value and stable limit cycle, respectively
Figure 4.  (a) $ \tau = 0.9 <\tau_{01}^{0} $, $ E^{*} $ is asymptotically stable; (b) $ \tau = 2.8>\tau_{01}^{0} $, $ E^{*} $ becomes unstable, and Hopf bifurcation is bifurcated from $ E^{*} $. (c) intercepts a part of (b) from $ t = 3450 $ to $ t = 3500 $. Parameters are $ a = 3.61 $, $ m = 1.8 $. The initial conditions are $ n(t) = 1.2 $ and $ s(t) = 1.8 $, $ t \in[-\tau, 0] $
Figure 5.  The evolutions of vegetation biomass and the density of soil water with time and space: (a) $ \tau = 2<\tau_{2}^{0}; $ (b) $ \tau = 4.2>\tau_{2}^{0} $. Other parameters are $ a = 2.8 $, $ m = 1.4 $, $ \beta = 0.0 $, $ d_{1} = 0.02 $, $ d_{2} = 0.2 $. The initial conditions are $ n(x, t) = 1.2 $ and $ s(x, t) = 1.6 $, $ (x, t)\in[0, \pi]\times[-\tau, 0] $
Figure 6.  The spatiotemporal evolutions of $ n(x, t) $ and $ w(x, t) $ with respect to $ \tau $: (a)(d) $ \tau = 2<\tau_{2}^{0}; $ (b)(c)(e)(f) $ \tau = 4.2>\tau_{2}^{0} $. Other parameters are $ a = 2.8 $, $ m = 1.4 $, $ \beta = 0.0 $, $ d_{1} = 0.02 $, $ d_{2} = 0.2 $. The initial conditions are $ n(x, t) = 1.2 $ and $ s(x, t) = 1.6 $, $ (x, t)\in[0, \pi]\times[-\tau, 0] $
Figure 7.  The variation of the value of $ \tau_{c} $ for three groups $ a $ and $ m $: (a) with $ \beta $ when $ d_{1} = 0.02 $, $ d_{2} = 0.2 $; (b) with $ d_{2} $ when $ \beta = 0.0 $, $ d_{1} = 0.02 $. (c) The variation of the critical bifurcation parameter $ \tau_{c} $, the amplitude and the period of the spatially homogeneous periodic solutions with respect to $ m $. Other parameters as $ d_{1} = 0.02 $, $ d_{2} = 0.2 $, $ \beta = 0.0 $, $ a = 2m $ and $ \tau = 4.2 $
Figure 8.  Region divisions of parameter space $ a-m $: (a) for system (4) without diffusion; (b) for system (4) with parameters as $ \beta = 0.1 $, $ d_{1} = 0.02 $, $ d_{2} = 0.2 $
Figure 9.  The variation of the sign of expression $ P(k) $ with $ k $: (a) under different $ a = 4.0, \ 3.8 $ and $ 3.61 $. Other parameter as $ m = 1.8 $, $ d_{1} = 0.02 $, $ d_{2} = 0.2 $; (b) under different $ d_{2} = 1, \ 0.6 $ and $ 0.2 $. Other parameter as $ a = 3.61 $, $ m = 1.8 $, $ d_{1} = 0.02 $; (c) under different $ m = 1.8, \ 1.6 $ and $ 1.4 $. Other parameter as $ a = 3.61 $, $ d_{1} = 0.02 $, $ d_{2} = 0.2 $
Figure 10.  The variation of the sign of expression $ Q_{2}(k) $ with $ k $: (a) under different $ \beta = 0.2, \ 0.1 $ and $ 0.0 $. Other parameter as $ a = 3.2 $, $ m = 1.6 $, $ d_{1} = 0.02 $, $ d_{2} = 0.2 $; (b) under different $ d_{2} = 1, \ 0.6 $ and $ 0.2 $. Other parameter as $ \beta = 0.1 $, $ a = 3.2 $, $ m = 1.6 $, $ d_{1} = 0.02 $; (c) under different $ m = 1.8, \ 1.6 $ and $ 1.4 $. Other parameter as $ \beta = 0.1 $, $ a = 2m $, $ d_{1} = 0.02 $, $ d_{2} = 0.2 $
Figure 11.  The evolutions of vegetation biomass and the density of soil water with time and space: (a)(d) vegetation biomass, (b)(e) the density of soil water. The evolutions of vegetation biomass and the density of soil water: (c) with space at $ t = 700 $, (f) with time at $ x = 20 $. The initial conditions are $ n(x, t) = u(i, j) = 1.2+0.1\mathrm{cos}(2i) $ and $ s(x, t) = v(i, j) = 1.6-0.2\mathrm{cos}(2i) $, $ (x, t)\in[0, \pi]\times[-\tau, 0] $. Parameters: $ a = 2.8 $, $ m = 1.4 $, $ \beta = 0.0 $, $ d_{1} = 0.02 $, $ d_{2} = 0.2 $, $ \tau = 2 $
Figure 12.  (a) The influence of $ \beta $ on the amplitudes corresponding to the vegetation and soil water under different $ m $. Other parameters are $ d_{1} = 0.02 $, $ d_{2} = 0.2 $, $ \tau = 2 $; (b) The influence of $ d_{2} $ on the amplitudes corresponding to the vegetation and soil water under different $ m $. Other parameters are $ \beta = 0.0 $, $ d_{1} = 0.02 $, $ \tau = 2 $; (c) The influence of $ m $ on the amplitudes corresponding to the vegetation and soil water under different $ \beta $. Other parameters are $ d_{1} = 0.02 $, $ d_{2} = 0.2 $, $ \tau = 2 $
Table 1.  Comparison of $ \omega_{00} $, $ \tau_{00}^{0} $, $ \omega_{2} $ and $ \tau_{2}^{0} $ of system (4) without and with diffusion
Order $ m $ $ \omega_{00} $ $ \tau_{00}^{0} $ $ \omega_{2}^{0} $ $ \tau_{2}^{0} $
1. $ 1.40 $ $ 1.3711 $ $ 4.2650 $ $ 1.4055 $ $ 4.0708 $
2. $ 1.43 $ $ 1.4611 $ $ 3.9880 $ $ 1.4570 $ $ 3.9233 $
3. $ 1.46 $ $ 1.5475 $ $ 3.7526 $ $ 1.5095 $ $ 3.7835 $
4. $ 1.49 $ $ 1.6310 $ $ 3.5494 $ $ 1.5630 $ $ 3.6507 $
5. $ 1.52 $ $ 1.7121 $ $ 3.3717 $ $ 1.6173 $ $ 3.5247 $
6. $ 1.55 $ $ 1.7909 $ $ 3.2145 $ $ 1.6725 $ $ 3.4051 $
7. $ 1.58 $ $ 1.8679 $ $ 3.0742 $ $ 1.7285 $ $ 3.2916 $
8. $ 1.61 $ $ 1.9433 $ $ 2.9480 $ $ 1.7852 $ $ 3.1839 $
9. $ 1.64 $ $ 2.0171 $ $ 2.8337 $ $ 1.8426 $ $ 3.0817 $
10. $ 1.67 $ $ 2.0897 $ $ 2.7295 $ $ 1.9006 $ $ 2.9847 $
11. $ 1.70 $ $ 2.1610 $ $ 2.6341 $ $ 1.9591 $ $ 2.8927 $
12. $ 1.73 $ $ 2.2313 $ $ 2.5462 $ $ 2.0181 $ $ 2.8053 $
13. $ 1.76 $ $ 2.3006 $ $ 2.4650 $ $ 2.0775 $ $ 2.7223 $
14. $ 1.79 $ $ 2.3690 $ $ 2.3897 $ $ 2.1373 $ $ 2.6435 $
15. $ 1.82 $ $ 2.4366 $ $ 2.3195 $ $ 2.1974 $ $ 2.5687 $
16. $ 1.85 $ $ 2.5035 $ $ 2.2540 $ $ 2.2578 $ $ 2.4975 $
17. $ 1.88 $ $ 2.5697 $ $ 2.1926 $ $ 2.3184 $ $ 2.4299 $
18. $ 1.91 $ $ 2.6352 $ $ 2.1350 $ $ 2.3791 $ $ 2.3655 $
19. $ 1.94 $ $ 2.7001 $ $ 2.0807 $ $ 2.4400 $ $ 2.3042 $
20. $ 1.97 $ $ 2.7645 $ $ 2.0295 $ $ 2.5010 $ $ 2.2460 $
21. $ 2.00 $ $ 2.8284 $ $ 1.9811 $ $ 2.5620 $ $ 2.1903 $
Order $ m $ $ \omega_{00} $ $ \tau_{00}^{0} $ $ \omega_{2}^{0} $ $ \tau_{2}^{0} $
1. $ 1.40 $ $ 1.3711 $ $ 4.2650 $ $ 1.4055 $ $ 4.0708 $
2. $ 1.43 $ $ 1.4611 $ $ 3.9880 $ $ 1.4570 $ $ 3.9233 $
3. $ 1.46 $ $ 1.5475 $ $ 3.7526 $ $ 1.5095 $ $ 3.7835 $
4. $ 1.49 $ $ 1.6310 $ $ 3.5494 $ $ 1.5630 $ $ 3.6507 $
5. $ 1.52 $ $ 1.7121 $ $ 3.3717 $ $ 1.6173 $ $ 3.5247 $
6. $ 1.55 $ $ 1.7909 $ $ 3.2145 $ $ 1.6725 $ $ 3.4051 $
7. $ 1.58 $ $ 1.8679 $ $ 3.0742 $ $ 1.7285 $ $ 3.2916 $
8. $ 1.61 $ $ 1.9433 $ $ 2.9480 $ $ 1.7852 $ $ 3.1839 $
9. $ 1.64 $ $ 2.0171 $ $ 2.8337 $ $ 1.8426 $ $ 3.0817 $
10. $ 1.67 $ $ 2.0897 $ $ 2.7295 $ $ 1.9006 $ $ 2.9847 $
11. $ 1.70 $ $ 2.1610 $ $ 2.6341 $ $ 1.9591 $ $ 2.8927 $
12. $ 1.73 $ $ 2.2313 $ $ 2.5462 $ $ 2.0181 $ $ 2.8053 $
13. $ 1.76 $ $ 2.3006 $ $ 2.4650 $ $ 2.0775 $ $ 2.7223 $
14. $ 1.79 $ $ 2.3690 $ $ 2.3897 $ $ 2.1373 $ $ 2.6435 $
15. $ 1.82 $ $ 2.4366 $ $ 2.3195 $ $ 2.1974 $ $ 2.5687 $
16. $ 1.85 $ $ 2.5035 $ $ 2.2540 $ $ 2.2578 $ $ 2.4975 $
17. $ 1.88 $ $ 2.5697 $ $ 2.1926 $ $ 2.3184 $ $ 2.4299 $
18. $ 1.91 $ $ 2.6352 $ $ 2.1350 $ $ 2.3791 $ $ 2.3655 $
19. $ 1.94 $ $ 2.7001 $ $ 2.0807 $ $ 2.4400 $ $ 2.3042 $
20. $ 1.97 $ $ 2.7645 $ $ 2.0295 $ $ 2.5010 $ $ 2.2460 $
21. $ 2.00 $ $ 2.8284 $ $ 1.9811 $ $ 2.5620 $ $ 2.1903 $
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