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Monotone dynamical systems: Reflections on new advances & applications
Stationary and time-periodic patterns of two-predator and one-prey systems with prey-taxis
Department of Mathematics, Southwestern University of Finance and Economics, 555 Liutai Ave, Wenjiang, Chengdu, Sichuan 611130, China |
This paper concerns pattern formation in a class of reaction-advection-diffusion systems modeling the population dynamics of two predators and one prey. We consider the biological situation that both predators forage along the population density gradient of the preys which can defend themselves as a group. We prove the global existence and uniform boundedness of positive classical solutions for the fully parabolic system over a bounded domain with space dimension $ N=1,2 $ and for the parabolic-parabolic-elliptic system over higher space dimensions. Linearized stability analysis shows that prey-taxis stabilizes the positive constant equilibrium if there is no group defense while it destabilizes the equilibrium otherwise. Then we obtain stationary and time-periodic nontrivial solutions of the system that bifurcate from the positive constant equilibrium. Moreover, the stability of these solutions is also analyzed in detail which provides a wave mode selection mechanism of nontrivial patterns for this strongly coupled system. Finally, we perform numerical simulations to illustrate and support our theoretical results.
References:
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P. Abrams and H. Matsuda,
Effects of adaptive predatory and anti-predator behaviour in a two-prey-one-predator system, Evolutionary Ecology, 7 (1993), 312-326.
doi: 10.1007/BF01237749. |
[2] |
B. E. Ainseba, M. Bendahmane and A. Noussair,
A reaction-diffusion system modeling predator-prey with prey-taxis, Nonlinear Anal. Real World Appl., 9 (2008), 2086-2105.
doi: 10.1016/j.nonrwa.2007.06.017. |
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N. Alikakos,
$ L^p $ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868.
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H. Amann,
Dynamic theory of quasilinear parabolic equations. Ⅱ. Reaction-diffusion systems, Differential Integral Equations, 3 (1990), 13-75.
|
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H. Amann,
Hopf bifurcation in quasilinear reaction-diffusion systems, Delay Differential Equations and Dynamical Systems, Lecture Notes in Mathematics, 1475 (1991), 53-63.
doi: 10.1007/BFb0083479. |
[6] |
H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value
problems, Function Spaces, differential operators and nonlinear Analysis, Teubner, Stuttgart,
Leipzig, 133 (1993), 9-126.
doi: 10.1007/978-3-663-11336-2_1. |
[7] |
A. Chakraborty, M. Singh, D. Lucy and P. Ridland,
Predator-prey model with prey-taxis and diffusion, Math. Comput. Modelling, 46 (2007), 482-498.
doi: 10.1016/j.mcm.2006.10.010. |
[8] |
A. Chertock, A. Kurganov, X. Wang and Y. Wu,
On a chemotaxis model with saturated chemotactic flux, Kinet. Relat. Models, 5 (2012), 51-95.
doi: 10.3934/krm.2012.5.51. |
[9] |
M. G. Crandall and P. H. Rabinowitz,
Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340.
doi: 10.1016/0022-1236(71)90015-2. |
[10] |
M. G. Crandall and P. H. Rabinowitz,
Bifurcation, perturbation of simple eigenvalues, and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180.
doi: 10.1007/BF00282325. |
[11] |
M. G. Crandall and P. H. Rabinowitz,
The Hopf bifurcation theorem in infinite dimensions, Arch. Rational Mech. Anal., 67 (1977), 53-72.
doi: 10.1007/BF00280827. |
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T. Czaran, Spatiotemporal Models of Population and Community Dynamics, Chapman and Hall, London, 1998. |
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The wave of advance of advantageous genes, Annals of Eugenics, 7 (1937), 355-369.
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Geometry for the selfish herd, J. Theoret. Biol., 31 (1971), 295-311.
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Global boundedness of solutions in a reaction-diffusion system of predator-prey model with prey-taxis, Appl. Math. Lett., 49 (2015), 73-77.
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Generalizing the Keller-Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species, J. Nonlinear Sci., 21 (2011), 231-270.
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D. Horstmann and M. Winkler,
Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.
doi: 10.1016/j.jde.2004.10.022. |
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L. Hsiao and P. de Mottoni,
Persistence in reacting-diffusing systems: Interaction of two predators and one prey, Nonlinear Anal., 11 (1987), 877-891.
doi: 10.1016/0362-546X(87)90058-7. |
[23] |
L. Jin, Q. Wang and Z. Zhang, Qualitative Studies of Advective Competition System with
Beddington-DeAngelis Functional Response, preprint, http://arxiv.org/abs/1412.3371 |
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D. D. Joseph and D. Nield,
Stability of bifurcating time-periodic and steady solutions of arbitrary amplitude, Arch. Rational Mech. Anal., 58 (1975), 369-380.
doi: 10.1007/BF00250296. |
[25] |
D. D. Joseph and D. H. Sattinger,
Bifurcating time periodic solutions and their stability, Arch. Rational Mech. Anal., 45 (1972), 75-109.
doi: 10.1007/BF00253039. |
[26] |
P. Kareiva and G. Odell,
Swarms of predators exhibit "preytaxis" if individual predators use area-restricted search, The American Naturalist, 130 (1987), 233-270.
doi: 10.1086/284707. |
[27] |
T. Kato,
Functional Analysis, Springer Classics in Mathematics, 1995.
doi: 10.1007/978-3-642-61859-8. |
[28] |
E. F. Keller and L. A. Segel,
Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[29] |
K. Kuto,
Stability of steady-state solutions to a prey-predator system with cross-diffusion, J. Differential Equations, 197 (2004), 293-314.
doi: 10.1016/j.jde.2003.10.016. |
[30] |
K. Kuto and Y. Yamada,
Multiple coexistence states for a prey-predator system with cross-diffusion, J. Differential Equations, 197 (2004), 315-348.
doi: 10.1016/j.jde.2003.08.003. |
[31] |
O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva,
Linear and Quasi-Linear Equations of Parabolic Type, American Mathematical Society, 1968,648 pages. |
[32] |
J. M. Lee, T. Hilllen and M. A. Lewis,
Continuous traveling waves for prey-taxis, Bull. Math. Biol., 70 (2008), 654-676.
doi: 10.1007/s11538-007-9271-4. |
[33] |
J. M. Lee, T. Hilllen and M. A. Lewis,
Pattern formation in prey-taxis systems, J. Biol. Dyn., 3 (2009), 551-573.
doi: 10.1080/17513750802716112. |
[34] |
C. Li, X. Wang and Y. Shao,
Steady states of a predator-prey model with prey-taxis, Nonlinear Anal., 97 (2014), 155-168.
doi: 10.1016/j.na.2013.11.022. |
[35] |
J.-J. Lin, W. Wang, C. Zhao and T.-H. Yang,
Global dynamics and traveling wave solutions of two predators-one prey models, Discrete Contin. Dyn. Syst-Series B, 20 (2015), 1135-1154.
doi: 10.3934/dcdsb.2015.20.1135. |
[36] |
P. Liu, J. Shi and Z.-A. Wang,
Pattern formation of the attraction-repulsion Keller-Segel system, Discrete Contin. Dyn. Syst-Series B, 18 (2013), 2597-2625.
doi: 10.3934/dcdsb.2013.18.2597. |
[37] |
I. Loladze, Y. Kuang, J.-J. Elser and W.-F. Fagan,
Competition and stoichiometry: Coexistence of two predators on one prey, Theoret. Pop. Biol., 65 (2004), 1-15.
doi: 10.1016/S0040-5809(03)00105-9. |
[38] |
Z. Maciej Gliwicz, P. Maszczyk, J. Jabłoński and D. Wrzosek,
Patch exploitation by planktivorous fish and the concept of aggregation as an antipredation defense in zooplankton, Limnology and Oceanography, 58 (2013), 1621-1639.
doi: 10.4319/lo.2013.58.5.1621. |
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M. Mimura and K. Kawasaki,
Spatial segregation in competitive interaction-diffusion equations, J. Math. Biol., 9 (1980), 49-64.
doi: 10.1007/BF00276035. |
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J. D. Murray,
Mathematical Biology, Springer, New York, 1993.
doi: 10.1007/b98869. |
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T. Nagai, T. Senba and K. Yoshida,
Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433.
doi: 10.1155/AAA/2006/23061. |
[42] |
W. Nagata and S.-M. Merchant,
Wave train selection behind invasion fronts in reaction-diffusion predator-prey models, Phys. D, 239 (2010), 1670-1680.
doi: 10.1016/j.physd.2010.04.014. |
[43] |
K. Nakashima and Y. Yamada,
Positive steady states for prey-predator models with cross-diffusion, Adv. Differential Equations, 1 (1996), 1099-1122.
|
[44] |
A. Okubo and S. A. Levin,
Diffusion and Ecological Problems, Modern Perspectives 2nd Edition, Springer-Verlag, New York, 2001.
doi: 10.1007/978-1-4757-4978-6. |
[45] |
P. Pang and M. Wang,
Strategy and stationary pattern in a three-species predator-prey model, J. Differential Equations, 200 (2004), 245-273.
doi: 10.1016/j.jde.2004.01.004. |
[46] |
C. S. Patlak,
Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338.
doi: 10.1007/BF02476407. |
[47] |
P. Rabinowitz,
Some global results for nonlinear eigenvalue problems, J. Functional Analysis, 7 (1971), 487-513.
doi: 10.1016/0022-1236(71)90030-9. |
[48] |
K. Ryu and I. Ahn,
Positive steady-states for two interacting species models with linear self-cross diffusions, Discrete Contin. Dyn. Syst., 9 (2003), 1049-1061.
doi: 10.3934/dcds.2003.9.1049. |
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The role of prey taxis in biological control: A spatial theoretical model, The American Naturalist, 162 (2003), 61-76.
doi: 10.1086/375297. |
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On global bifurcation for quasilinear elliptic systems on bounded domains, J. Differential Equations, 246 (2009), 2788-2812.
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show all references
References:
[1] |
P. Abrams and H. Matsuda,
Effects of adaptive predatory and anti-predator behaviour in a two-prey-one-predator system, Evolutionary Ecology, 7 (1993), 312-326.
doi: 10.1007/BF01237749. |
[2] |
B. E. Ainseba, M. Bendahmane and A. Noussair,
A reaction-diffusion system modeling predator-prey with prey-taxis, Nonlinear Anal. Real World Appl., 9 (2008), 2086-2105.
doi: 10.1016/j.nonrwa.2007.06.017. |
[3] |
N. Alikakos,
$ L^p $ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868.
doi: 10.1080/03605307908820113. |
[4] |
H. Amann,
Dynamic theory of quasilinear parabolic equations. Ⅱ. Reaction-diffusion systems, Differential Integral Equations, 3 (1990), 13-75.
|
[5] |
H. Amann,
Hopf bifurcation in quasilinear reaction-diffusion systems, Delay Differential Equations and Dynamical Systems, Lecture Notes in Mathematics, 1475 (1991), 53-63.
doi: 10.1007/BFb0083479. |
[6] |
H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value
problems, Function Spaces, differential operators and nonlinear Analysis, Teubner, Stuttgart,
Leipzig, 133 (1993), 9-126.
doi: 10.1007/978-3-663-11336-2_1. |
[7] |
A. Chakraborty, M. Singh, D. Lucy and P. Ridland,
Predator-prey model with prey-taxis and diffusion, Math. Comput. Modelling, 46 (2007), 482-498.
doi: 10.1016/j.mcm.2006.10.010. |
[8] |
A. Chertock, A. Kurganov, X. Wang and Y. Wu,
On a chemotaxis model with saturated chemotactic flux, Kinet. Relat. Models, 5 (2012), 51-95.
doi: 10.3934/krm.2012.5.51. |
[9] |
M. G. Crandall and P. H. Rabinowitz,
Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340.
doi: 10.1016/0022-1236(71)90015-2. |
[10] |
M. G. Crandall and P. H. Rabinowitz,
Bifurcation, perturbation of simple eigenvalues, and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180.
doi: 10.1007/BF00282325. |
[11] |
M. G. Crandall and P. H. Rabinowitz,
The Hopf bifurcation theorem in infinite dimensions, Arch. Rational Mech. Anal., 67 (1977), 53-72.
doi: 10.1007/BF00280827. |
[12] |
T. Czaran, Spatiotemporal Models of Population and Community Dynamics, Chapman and Hall, London, 1998. |
[13] |
R. A. Fisher,
The wave of advance of advantageous genes, Annals of Eugenics, 7 (1937), 355-369.
doi: 10.1111/j.1469-1809.1937.tb02153.x. |
[14] |
D. Grünbaum,
Advection-diffusion equations for generalized tactic searching behaviours, J. Math. Biol., 38 (1999), 169-194.
doi: 10.1007/s002850050145. |
[15] |
W. D. Hamilton,
Geometry for the selfish herd, J. Theoret. Biol., 31 (1971), 295-311.
doi: 10.1016/0022-5193(71)90189-5. |
[16] |
X. He and S. Zheng,
Global boundedness of solutions in a reaction-diffusion system of predator-prey model with prey-taxis, Appl. Math. Lett., 49 (2015), 73-77.
doi: 10.1016/j.aml.2015.04.017. |
[17] |
D. Henry,
Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag-Berlin-New York, 1981.
doi: 10.1007/BFb0089647. |
[18] |
T. Hillen and K. J. Painter,
A user's guidence to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[19] |
D. Horstmann,
1970 until present: The Keller-Segel model in chemotaxis and its consequences. Ⅰ, Jahresber DMV, 105 (2003), 103-165.
|
[20] |
D. Horstmann,
Generalizing the Keller-Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species, J. Nonlinear Sci., 21 (2011), 231-270.
doi: 10.1007/s00332-010-9082-x. |
[21] |
D. Horstmann and M. Winkler,
Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.
doi: 10.1016/j.jde.2004.10.022. |
[22] |
L. Hsiao and P. de Mottoni,
Persistence in reacting-diffusing systems: Interaction of two predators and one prey, Nonlinear Anal., 11 (1987), 877-891.
doi: 10.1016/0362-546X(87)90058-7. |
[23] |
L. Jin, Q. Wang and Z. Zhang, Qualitative Studies of Advective Competition System with
Beddington-DeAngelis Functional Response, preprint, http://arxiv.org/abs/1412.3371 |
[24] |
D. D. Joseph and D. Nield,
Stability of bifurcating time-periodic and steady solutions of arbitrary amplitude, Arch. Rational Mech. Anal., 58 (1975), 369-380.
doi: 10.1007/BF00250296. |
[25] |
D. D. Joseph and D. H. Sattinger,
Bifurcating time periodic solutions and their stability, Arch. Rational Mech. Anal., 45 (1972), 75-109.
doi: 10.1007/BF00253039. |
[26] |
P. Kareiva and G. Odell,
Swarms of predators exhibit "preytaxis" if individual predators use area-restricted search, The American Naturalist, 130 (1987), 233-270.
doi: 10.1086/284707. |
[27] |
T. Kato,
Functional Analysis, Springer Classics in Mathematics, 1995.
doi: 10.1007/978-3-642-61859-8. |
[28] |
E. F. Keller and L. A. Segel,
Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[29] |
K. Kuto,
Stability of steady-state solutions to a prey-predator system with cross-diffusion, J. Differential Equations, 197 (2004), 293-314.
doi: 10.1016/j.jde.2003.10.016. |
[30] |
K. Kuto and Y. Yamada,
Multiple coexistence states for a prey-predator system with cross-diffusion, J. Differential Equations, 197 (2004), 315-348.
doi: 10.1016/j.jde.2003.08.003. |
[31] |
O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva,
Linear and Quasi-Linear Equations of Parabolic Type, American Mathematical Society, 1968,648 pages. |
[32] |
J. M. Lee, T. Hilllen and M. A. Lewis,
Continuous traveling waves for prey-taxis, Bull. Math. Biol., 70 (2008), 654-676.
doi: 10.1007/s11538-007-9271-4. |
[33] |
J. M. Lee, T. Hilllen and M. A. Lewis,
Pattern formation in prey-taxis systems, J. Biol. Dyn., 3 (2009), 551-573.
doi: 10.1080/17513750802716112. |
[34] |
C. Li, X. Wang and Y. Shao,
Steady states of a predator-prey model with prey-taxis, Nonlinear Anal., 97 (2014), 155-168.
doi: 10.1016/j.na.2013.11.022. |
[35] |
J.-J. Lin, W. Wang, C. Zhao and T.-H. Yang,
Global dynamics and traveling wave solutions of two predators-one prey models, Discrete Contin. Dyn. Syst-Series B, 20 (2015), 1135-1154.
doi: 10.3934/dcdsb.2015.20.1135. |
[36] |
P. Liu, J. Shi and Z.-A. Wang,
Pattern formation of the attraction-repulsion Keller-Segel system, Discrete Contin. Dyn. Syst-Series B, 18 (2013), 2597-2625.
doi: 10.3934/dcdsb.2013.18.2597. |
[37] |
I. Loladze, Y. Kuang, J.-J. Elser and W.-F. Fagan,
Competition and stoichiometry: Coexistence of two predators on one prey, Theoret. Pop. Biol., 65 (2004), 1-15.
doi: 10.1016/S0040-5809(03)00105-9. |
[38] |
Z. Maciej Gliwicz, P. Maszczyk, J. Jabłoński and D. Wrzosek,
Patch exploitation by planktivorous fish and the concept of aggregation as an antipredation defense in zooplankton, Limnology and Oceanography, 58 (2013), 1621-1639.
doi: 10.4319/lo.2013.58.5.1621. |
[39] |
M. Mimura and K. Kawasaki,
Spatial segregation in competitive interaction-diffusion equations, J. Math. Biol., 9 (1980), 49-64.
doi: 10.1007/BF00276035. |
[40] |
J. D. Murray,
Mathematical Biology, Springer, New York, 1993.
doi: 10.1007/b98869. |
[41] |
T. Nagai, T. Senba and K. Yoshida,
Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433.
doi: 10.1155/AAA/2006/23061. |
[42] |
W. Nagata and S.-M. Merchant,
Wave train selection behind invasion fronts in reaction-diffusion predator-prey models, Phys. D, 239 (2010), 1670-1680.
doi: 10.1016/j.physd.2010.04.014. |
[43] |
K. Nakashima and Y. Yamada,
Positive steady states for prey-predator models with cross-diffusion, Adv. Differential Equations, 1 (1996), 1099-1122.
|
[44] |
A. Okubo and S. A. Levin,
Diffusion and Ecological Problems, Modern Perspectives 2nd Edition, Springer-Verlag, New York, 2001.
doi: 10.1007/978-1-4757-4978-6. |
[45] |
P. Pang and M. Wang,
Strategy and stationary pattern in a three-species predator-prey model, J. Differential Equations, 200 (2004), 245-273.
doi: 10.1016/j.jde.2004.01.004. |
[46] |
C. S. Patlak,
Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338.
doi: 10.1007/BF02476407. |
[47] |
P. Rabinowitz,
Some global results for nonlinear eigenvalue problems, J. Functional Analysis, 7 (1971), 487-513.
doi: 10.1016/0022-1236(71)90030-9. |
[48] |
K. Ryu and I. Ahn,
Positive steady-states for two interacting species models with linear self-cross diffusions, Discrete Contin. Dyn. Syst., 9 (2003), 1049-1061.
doi: 10.3934/dcds.2003.9.1049. |
[49] |
N. Sapoukhina, Y. Tyutyunov and R. Arditi,
The role of prey taxis in biological control: A spatial theoretical model, The American Naturalist, 162 (2003), 61-76.
doi: 10.1086/375297. |
[50] |
J. Shi and X. Wang,
On global bifurcation for quasilinear elliptic systems on bounded domains, J. Differential Equations, 246 (2009), 2788-2812.
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1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |
| 66.98 | 18.98 | 10.30 | 7.49 | 6.41 | 6.05 | 6.09 | 6.36 | 6.80 |
1204.20 | 550.84 | 504.20 | 575.80 | 705.50 | 878.48 | 1089.70 | 1336.90 | 1619.19 |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |
| 66.98 | 18.98 | 10.30 | 7.49 | 6.41 | 6.05 | 6.09 | 6.36 | 6.80 |
1204.20 | 550.84 | 504.20 | 575.80 | 705.50 | 878.48 | 1089.70 | 1336.90 | 1619.19 |
Interval length | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| 1 | 2 | 3 | 4 | 5 | 5 | 6 | 7 |
6.09 | 6.09 | 6.09 | 6.09 | 6.09 | 6.08 | 6.05 | 6.04 | |
Interval length | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |
8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | |
| 6.04 | 6.03 | 6.04 | 6.04 | 6.04 | 6.04 | 6.04 | 6.04 |
Interval length | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| 1 | 2 | 3 | 4 | 5 | 5 | 6 | 7 |
6.09 | 6.09 | 6.09 | 6.09 | 6.09 | 6.08 | 6.05 | 6.04 | |
Interval length | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |
8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | |
| 6.04 | 6.03 | 6.04 | 6.04 | 6.04 | 6.04 | 6.04 | 6.04 |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |
106.4 | 98.63 | 107.32 | 122.53 | 143.15 | 169.05 | 200.24 | 236.80 | 278.76 | |
186.37 | 96.73 | 92.57 | 105.46 | 127.03 | 155.24 | 189.40 | 229.24 | 274.64 |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |
106.4 | 98.63 | 107.32 | 122.53 | 143.15 | 169.05 | 200.24 | 236.80 | 278.76 | |
186.37 | 96.73 | 92.57 | 105.46 | 127.03 | 155.24 | 189.40 | 229.24 | 274.64 |
Interval length | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
1 | 2 | 3 | 4 | 5 | 5 | 6 | 7 | |
97.68 | 92.13 | 97.68 | 91.49 | 92.13 | 92.57 | 91.15 | 92.13 | |
Interval length | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 |
8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | |
| 91.5 | 91.2 | 91.13 | 91.21 | 91.30 | 91.49 | 91.15 | 91.40 |
Interval length | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
1 | 2 | 3 | 4 | 5 | 5 | 6 | 7 | |
97.68 | 92.13 | 97.68 | 91.49 | 92.13 | 92.57 | 91.15 | 92.13 | |
Interval length | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 |
8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | |
| 91.5 | 91.2 | 91.13 | 91.21 | 91.30 | 91.49 | 91.15 | 91.40 |
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