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June  2020, 40(6): 3683-3714. doi: 10.3934/dcds.2020050

Asymptotic spreading of interacting species with multiple fronts Ⅰ: A geometric optics approach

1. 

Institute for Mathematical Sciences, Renmin University of China, Beijing 100872, China

2. 

Department of Mathematics, Ohio State University, Columbus, OH 43210, USA

* Corresponding author: King-Yeung Lam

Received  April 2019 Revised  September 2019 Published  October 2019

Fund Project: The last author is partially supported by NSF grant DMS-1853561

We establish spreading properties of the Lotka-Volterra competition-diffusion system. When the initial data vanish on a right half-line, we derive the exact spreading speeds and prove the convergence to homogeneous equilibrium states between successive invasion fronts. Our method is inspired by the geometric optics approach for Fisher-KPP equation due to Freidlin, Evans and Souganidis. Our main result settles an open question raised by Shigesada et al. in 1997, and shows that one of the species spreads to the right with a nonlocally pulled front.

Citation: Qian Liu, Shuang Liu, King-Yeung Lam. Asymptotic spreading of interacting species with multiple fronts Ⅰ: A geometric optics approach. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3683-3714. doi: 10.3934/dcds.2020050
References:
[1]

Y. Achdou, G. Barles, H. Ishii and G. L. Litvinov, Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications, Fondazione CIME/CIME Foundation Subseries, Springer, Heidelberg; Fondazione C.I.M.E., Florence, 2013. doi: 10.1007/978-3-642-36433-4.  Google Scholar

[2]

A. Alhasanat and C. H. Ou, On a conjecture raised by Yuzo Hosono, J. Dynam. Differential Equations, 31 (2019), 287-304.  doi: 10.1007/s10884-018-9651-5.  Google Scholar

[3]

A. Alhasanat and C. H. Ou, Minimal-speed selection of traveling waves to the Lotka-Volterra competition model, J. Differential Equations, 266 (2019), 7357-7378.  doi: 10.1016/j.jde.2018.12.003.  Google Scholar

[4]

G. BarlesL. C. Evans and P. E. Souganidis, Wavefront propagation for reaction-diffusion systems of PDE, Duke Math. J., 61 (1990), 835-858.  doi: 10.1215/S0012-7094-90-06132-0.  Google Scholar

[5]

G. Barles and B. Perthame, Discontinuous solutions of deterministic optimal stopping time problems, RAIRO Modél. Math. Anal. Numér., 21 (1987), 557-579.  doi: 10.1051/m2an/1987210405571.  Google Scholar

[6]

G. BarlesH. M. Soner and P. E. Souganidis, Front propagation and phase field theory, SIAM J. Control Optim., 31 (1993), 439-469.  doi: 10.1137/0331021.  Google Scholar

[7]

H. Berestycki and J. Fang, Forced waves of the Fisher-KPP equation in a shifting environment, J. Differential Equations, 264 (2018), 2157-2183.  doi: 10.1016/j.jde.2017.10.016.  Google Scholar

[8]

H. Berestycki and G. Nadin, Spreading speeds for one-dimensional monostable reaction-diffusion equations, J. Math. Phys., 53 (2012), 115619, 23 pp. doi: 10.1063/1.4764932.  Google Scholar

[9]

C. Carrère, Spreading speeds for a two-species competition-diffusion system, J. Differential Equations, 264 (2018), 2133-2156.  doi: 10.1016/j.jde.2017.10.017.  Google Scholar

[10]

M. B. Davis, Quaternary history and stability of forest communities, Forest Succession: Concepts and Applications, Springer-Verlag, New York, (1981), 132–153. doi: 10.1007/978-1-4612-5950-3_10.  Google Scholar

[11]

Y. H. Du and C.-H. Wu, Spreading with two speeds and mass segregation in a diffusive competition system with free boundaries, Calc. Var. Partial Differential Equations, 57 (2018), Art. 52, 36 pp. doi: 10.1007/s00526-018-1339-5.  Google Scholar

[12]

A. Ducrot, T. Giletti and H. Matano, Spreading speeds for multidimensional reaction-diffusion systems of the prey-predator type, Calc. Var. Partial Differential Equations, 58 (2019), Art. 137, 34 pp. doi: 10.1007/s00526-019-1576-2.  Google Scholar

[13]

L. C. Evans and P. E. Souganidis, Differential games and representation formulas for solutions of Hamilton-Jacobi-Isaacs equation, Indiana Univ. Math. J., 33 (1984), 773-797.  doi: 10.1512/iumj.1984.33.33040.  Google Scholar

[14]

L. C. Evans and P. E. Souganidis, A PDE approach to geometric optics for certain semilinear parabolic equations, Indiana Univ. Math. J., 38 (1989), 141-172.  doi: 10.1512/iumj.1989.38.38007.  Google Scholar

[15]

J. FangY. J. Lou and J. H. Wu, Can pathogen spread keep pace with its host invasion?, SIAM J. Appl. Math., 74 (2016), 1633-1657.  doi: 10.1137/15M1029564.  Google Scholar

[16]

R. A. Fisher, The wave of advance of advantageous genes, Ann. Hum. Genet., 7 (1937), 355-369.  doi: 10.1111/j.1469-1809.1937.tb02153.x.  Google Scholar

[17]

M. Freidlin, Limit theorems for large deviations and reaction-diffusion equation, Ann. Probab., 13 (1985), 639-675.  doi: 10.1214/aop/1176992901.  Google Scholar

[18]

L. Girardin and K.-Y. Lam, Invasion of an empty habitat by two competitors: Spreading properties of monostable two-species competition-diffusion systems, P. Lond. Math. Soc., 119 (2019), 1279-1335.  doi: 10.1112/plms.12270.  Google Scholar

[19]

J.-S. Guo and C.-H. Wu, Dynamics for a two-species competition-diffusion model with two free boundaries, Nonlinearity, 28 (2015), 1-27.  doi: 10.1088/0951-7715/28/1/1.  Google Scholar

[20]

F. Hamel and G. Nadin, Spreading properties and complex dynamics for monostable reaction-diffusion equations, Comm. Partial Differential Equations, 37 (2012), 511-537.  doi: 10.1080/03605302.2011.647198.  Google Scholar

[21]

M. Holzer and A. Scheel, Accelerated fronts in a two stage invasion process, SIAM J. Math. Anal., 46 (2014), 397-427.  doi: 10.1137/120887746.  Google Scholar

[22]

X. J. Hou and A. W. Leung, Traveling wave solutions for a competitive reaction-diffusion system and their asymptotics, Nonlinear Anal. Real World Appl., 9 (2008), 2196-2213.  doi: 10.1016/j.nonrwa.2007.07.007.  Google Scholar

[23]

W. Z. Huang, Problem on minimum wave speed for a Lotka-Volterra reaction-diffusion competition model, J. Dynam. Differential Equations, 22 (2010), 285-297.  doi: 10.1007/s10884-010-9159-0.  Google Scholar

[24]

W. Z. Huang and M. Han, Non-linear determinacy of minimum wave speed for a Lotka-Volterra competition model, J. Differential Equations, 251 (2011), 1549-1561.  doi: 10.1016/j.jde.2011.05.012.  Google Scholar

[25]

M. IidaR. Lui and H. Ninomiya, Stacked fronts for cooperative systems with equal diffusion coefficients, SIAM J. Math. Anal., 43 (2011), 1369-1389.  doi: 10.1137/100792846.  Google Scholar

[26]

A. N. KolmogorovI. G. Petrovsky and N. S. Piskunov, Étude de léquation de la diffusion avec croissance de la quantité de matiére et son application à un probléme biologique, Bull. Univ. Moscow, Ser. Internat., Sec. A, 1 (1937), 1-26.   Google Scholar

[27]

M. A. LewisB. T. Li and H. F. Weinberger, Spreading speed and linear determinacy for two-species competition models, J. Math. Biol., 45 (2002), 219-233.  doi: 10.1007/s002850200144.  Google Scholar

[28]

B. T. Li, Multiple invasion speeds in a two-species integro-difference competition model, J. Math. Biol., 76 (2018), 1975-2009.  doi: 10.1007/s00285-017-1200-z.  Google Scholar

[29]

B. T. LiH. F. Weinberger and M. A. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, J. Math. Bios., 196 (2005), 82-98.  doi: 10.1016/j.mbs.2005.03.008.  Google Scholar

[30]

X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40.  doi: 10.1002/cpa.20154.  Google Scholar

[31]

G. Lin and W.-T. Li, Asymptotic spreading of competition diffusion systems: The role of interspecific competitions, European J. Appl. Math., 23 (2012), 669-689.  doi: 10.1017/S0956792512000198.  Google Scholar

[32]

S. Y. LiuH. M. Huang and M. X. Wang, Asymptotic spreading of a diffusive competition model with different free boundaries, J. Differential Equations, 266 (2019), 4769-4799.  doi: 10.1016/j.jde.2018.10.009.  Google Scholar

[33]

A. J. Majda and P. E. Souganidis, Large-scale front dynamics for turbulent reaction-diffusion equations with separated velocity scales, Nonlinearity, 7 (1994), 1-30.  doi: 10.1088/0951-7715/7/1/001.  Google Scholar

[34]

V. Méndez, J. Fort, H. G. Rotstein and S. Fedotov, Speed of reaction-diffusion fronts in spatially heterogeneous media, Phys. Rev. E, 68 (2003), 041105, 11 pp. doi: 10.1103/PhysRevE.68.041105.  Google Scholar

[35]

L. RoquesY. HosonoO. Bonnefon and T. Boivin, The effect of competition on the neutral intraspecific diversity of invasive species, J. Math. Biol., 71 (2015), 465-489.  doi: 10.1007/s00285-014-0825-4.  Google Scholar

[36] N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford University Press, Oxford, 1997.   Google Scholar
[37]

P. E. Souganidis, Front propagation: Theory and applications, Viscosity solutions and applications, Lecture Notes in Math., Fond. CIME/CIME Found. Subser., Springer, Berlin, 1660 (1997), 186-242.  doi: 10.1007/BFb0094298.  Google Scholar

[38]

M. M. Tang and P. C. Fife, Propagating fronts for competing species equations with diffusion, Arch. Rational Mech. Anal., 73 (1980), 69-77.  doi: 10.1007/BF00283257.  Google Scholar

[39]

A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems, Translations of Mathematical Monographs, 140. American Mathematical Society, Providence, RI, 1994.  Google Scholar

[40]

M. X. Wang and Y. Zhang, Note on a two-species competition-diffusion model with two free boundaries, Nonlinear Anal., 159 (2017), 458-467.  doi: 10.1016/j.na.2017.01.005.  Google Scholar

[41]

M. X. Wang and Y. Zhang, Dynamics for a diffusive prey-predator model with different free boundaries, J. Differential Equations, 264 (2018), 3527-3558.  doi: 10.1016/j.jde.2017.11.027.  Google Scholar

[42]

C.-H. Wu, The minimal habitat size for spreading in a weak competition system with two free boundaries, J. Differential Equations, 259 (2015), 873-897.  doi: 10.1016/j.jde.2015.02.021.  Google Scholar

[43]

J. Xin, Front propagation in heterogeneous media, SIAM Rev., 42 (2000), 161-230.  doi: 10.1137/S0036144599364296.  Google Scholar

show all references

References:
[1]

Y. Achdou, G. Barles, H. Ishii and G. L. Litvinov, Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications, Fondazione CIME/CIME Foundation Subseries, Springer, Heidelberg; Fondazione C.I.M.E., Florence, 2013. doi: 10.1007/978-3-642-36433-4.  Google Scholar

[2]

A. Alhasanat and C. H. Ou, On a conjecture raised by Yuzo Hosono, J. Dynam. Differential Equations, 31 (2019), 287-304.  doi: 10.1007/s10884-018-9651-5.  Google Scholar

[3]

A. Alhasanat and C. H. Ou, Minimal-speed selection of traveling waves to the Lotka-Volterra competition model, J. Differential Equations, 266 (2019), 7357-7378.  doi: 10.1016/j.jde.2018.12.003.  Google Scholar

[4]

G. BarlesL. C. Evans and P. E. Souganidis, Wavefront propagation for reaction-diffusion systems of PDE, Duke Math. J., 61 (1990), 835-858.  doi: 10.1215/S0012-7094-90-06132-0.  Google Scholar

[5]

G. Barles and B. Perthame, Discontinuous solutions of deterministic optimal stopping time problems, RAIRO Modél. Math. Anal. Numér., 21 (1987), 557-579.  doi: 10.1051/m2an/1987210405571.  Google Scholar

[6]

G. BarlesH. M. Soner and P. E. Souganidis, Front propagation and phase field theory, SIAM J. Control Optim., 31 (1993), 439-469.  doi: 10.1137/0331021.  Google Scholar

[7]

H. Berestycki and J. Fang, Forced waves of the Fisher-KPP equation in a shifting environment, J. Differential Equations, 264 (2018), 2157-2183.  doi: 10.1016/j.jde.2017.10.016.  Google Scholar

[8]

H. Berestycki and G. Nadin, Spreading speeds for one-dimensional monostable reaction-diffusion equations, J. Math. Phys., 53 (2012), 115619, 23 pp. doi: 10.1063/1.4764932.  Google Scholar

[9]

C. Carrère, Spreading speeds for a two-species competition-diffusion system, J. Differential Equations, 264 (2018), 2133-2156.  doi: 10.1016/j.jde.2017.10.017.  Google Scholar

[10]

M. B. Davis, Quaternary history and stability of forest communities, Forest Succession: Concepts and Applications, Springer-Verlag, New York, (1981), 132–153. doi: 10.1007/978-1-4612-5950-3_10.  Google Scholar

[11]

Y. H. Du and C.-H. Wu, Spreading with two speeds and mass segregation in a diffusive competition system with free boundaries, Calc. Var. Partial Differential Equations, 57 (2018), Art. 52, 36 pp. doi: 10.1007/s00526-018-1339-5.  Google Scholar

[12]

A. Ducrot, T. Giletti and H. Matano, Spreading speeds for multidimensional reaction-diffusion systems of the prey-predator type, Calc. Var. Partial Differential Equations, 58 (2019), Art. 137, 34 pp. doi: 10.1007/s00526-019-1576-2.  Google Scholar

[13]

L. C. Evans and P. E. Souganidis, Differential games and representation formulas for solutions of Hamilton-Jacobi-Isaacs equation, Indiana Univ. Math. J., 33 (1984), 773-797.  doi: 10.1512/iumj.1984.33.33040.  Google Scholar

[14]

L. C. Evans and P. E. Souganidis, A PDE approach to geometric optics for certain semilinear parabolic equations, Indiana Univ. Math. J., 38 (1989), 141-172.  doi: 10.1512/iumj.1989.38.38007.  Google Scholar

[15]

J. FangY. J. Lou and J. H. Wu, Can pathogen spread keep pace with its host invasion?, SIAM J. Appl. Math., 74 (2016), 1633-1657.  doi: 10.1137/15M1029564.  Google Scholar

[16]

R. A. Fisher, The wave of advance of advantageous genes, Ann. Hum. Genet., 7 (1937), 355-369.  doi: 10.1111/j.1469-1809.1937.tb02153.x.  Google Scholar

[17]

M. Freidlin, Limit theorems for large deviations and reaction-diffusion equation, Ann. Probab., 13 (1985), 639-675.  doi: 10.1214/aop/1176992901.  Google Scholar

[18]

L. Girardin and K.-Y. Lam, Invasion of an empty habitat by two competitors: Spreading properties of monostable two-species competition-diffusion systems, P. Lond. Math. Soc., 119 (2019), 1279-1335.  doi: 10.1112/plms.12270.  Google Scholar

[19]

J.-S. Guo and C.-H. Wu, Dynamics for a two-species competition-diffusion model with two free boundaries, Nonlinearity, 28 (2015), 1-27.  doi: 10.1088/0951-7715/28/1/1.  Google Scholar

[20]

F. Hamel and G. Nadin, Spreading properties and complex dynamics for monostable reaction-diffusion equations, Comm. Partial Differential Equations, 37 (2012), 511-537.  doi: 10.1080/03605302.2011.647198.  Google Scholar

[21]

M. Holzer and A. Scheel, Accelerated fronts in a two stage invasion process, SIAM J. Math. Anal., 46 (2014), 397-427.  doi: 10.1137/120887746.  Google Scholar

[22]

X. J. Hou and A. W. Leung, Traveling wave solutions for a competitive reaction-diffusion system and their asymptotics, Nonlinear Anal. Real World Appl., 9 (2008), 2196-2213.  doi: 10.1016/j.nonrwa.2007.07.007.  Google Scholar

[23]

W. Z. Huang, Problem on minimum wave speed for a Lotka-Volterra reaction-diffusion competition model, J. Dynam. Differential Equations, 22 (2010), 285-297.  doi: 10.1007/s10884-010-9159-0.  Google Scholar

[24]

W. Z. Huang and M. Han, Non-linear determinacy of minimum wave speed for a Lotka-Volterra competition model, J. Differential Equations, 251 (2011), 1549-1561.  doi: 10.1016/j.jde.2011.05.012.  Google Scholar

[25]

M. IidaR. Lui and H. Ninomiya, Stacked fronts for cooperative systems with equal diffusion coefficients, SIAM J. Math. Anal., 43 (2011), 1369-1389.  doi: 10.1137/100792846.  Google Scholar

[26]

A. N. KolmogorovI. G. Petrovsky and N. S. Piskunov, Étude de léquation de la diffusion avec croissance de la quantité de matiére et son application à un probléme biologique, Bull. Univ. Moscow, Ser. Internat., Sec. A, 1 (1937), 1-26.   Google Scholar

[27]

M. A. LewisB. T. Li and H. F. Weinberger, Spreading speed and linear determinacy for two-species competition models, J. Math. Biol., 45 (2002), 219-233.  doi: 10.1007/s002850200144.  Google Scholar

[28]

B. T. Li, Multiple invasion speeds in a two-species integro-difference competition model, J. Math. Biol., 76 (2018), 1975-2009.  doi: 10.1007/s00285-017-1200-z.  Google Scholar

[29]

B. T. LiH. F. Weinberger and M. A. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, J. Math. Bios., 196 (2005), 82-98.  doi: 10.1016/j.mbs.2005.03.008.  Google Scholar

[30]

X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40.  doi: 10.1002/cpa.20154.  Google Scholar

[31]

G. Lin and W.-T. Li, Asymptotic spreading of competition diffusion systems: The role of interspecific competitions, European J. Appl. Math., 23 (2012), 669-689.  doi: 10.1017/S0956792512000198.  Google Scholar

[32]

S. Y. LiuH. M. Huang and M. X. Wang, Asymptotic spreading of a diffusive competition model with different free boundaries, J. Differential Equations, 266 (2019), 4769-4799.  doi: 10.1016/j.jde.2018.10.009.  Google Scholar

[33]

A. J. Majda and P. E. Souganidis, Large-scale front dynamics for turbulent reaction-diffusion equations with separated velocity scales, Nonlinearity, 7 (1994), 1-30.  doi: 10.1088/0951-7715/7/1/001.  Google Scholar

[34]

V. Méndez, J. Fort, H. G. Rotstein and S. Fedotov, Speed of reaction-diffusion fronts in spatially heterogeneous media, Phys. Rev. E, 68 (2003), 041105, 11 pp. doi: 10.1103/PhysRevE.68.041105.  Google Scholar

[35]

L. RoquesY. HosonoO. Bonnefon and T. Boivin, The effect of competition on the neutral intraspecific diversity of invasive species, J. Math. Biol., 71 (2015), 465-489.  doi: 10.1007/s00285-014-0825-4.  Google Scholar

[36] N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford University Press, Oxford, 1997.   Google Scholar
[37]

P. E. Souganidis, Front propagation: Theory and applications, Viscosity solutions and applications, Lecture Notes in Math., Fond. CIME/CIME Found. Subser., Springer, Berlin, 1660 (1997), 186-242.  doi: 10.1007/BFb0094298.  Google Scholar

[38]

M. M. Tang and P. C. Fife, Propagating fronts for competing species equations with diffusion, Arch. Rational Mech. Anal., 73 (1980), 69-77.  doi: 10.1007/BF00283257.  Google Scholar

[39]

A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems, Translations of Mathematical Monographs, 140. American Mathematical Society, Providence, RI, 1994.  Google Scholar

[40]

M. X. Wang and Y. Zhang, Note on a two-species competition-diffusion model with two free boundaries, Nonlinear Anal., 159 (2017), 458-467.  doi: 10.1016/j.na.2017.01.005.  Google Scholar

[41]

M. X. Wang and Y. Zhang, Dynamics for a diffusive prey-predator model with different free boundaries, J. Differential Equations, 264 (2018), 3527-3558.  doi: 10.1016/j.jde.2017.11.027.  Google Scholar

[42]

C.-H. Wu, The minimal habitat size for spreading in a weak competition system with two free boundaries, J. Differential Equations, 259 (2015), 873-897.  doi: 10.1016/j.jde.2015.02.021.  Google Scholar

[43]

J. Xin, Front propagation in heterogeneous media, SIAM Rev., 42 (2000), 161-230.  doi: 10.1137/S0036144599364296.  Google Scholar

Figure 1.  Asymptotic behaviors of the solutions to (1) with $ a = 0.6, \, b = 0.5, \, r = 1 $, and $ d = 1.5 $ in $ \rm(a) $, $ d = 1 $ in $ \rm(b) $, $ d = 0.5 $ in $ \rm(c) $, where the initial data are chosen as $ u(0, x) = \chi_{[-1000, 0]} $ and $ v(0, x) = \chi_{[-20, 0]} $
Figure 2.  The graphs of $ x_i(t)/t $ ($ i = 1, 2, 3 $) with $ a = 0.6, \, b = 0.5, \, r = 1 $ and $ d = 1.5 $ where the initial data are chosen as $ u(0, x) = \chi_{[-1000, 0]} $ and $ v(0, x) = \chi_{[-20, 0]} $
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