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Invasion entire solutions in a time periodic Lotka-Volterra competition system with diffusion
1. | School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China |
This paper is concerned with invasion entire solutions of a monostable time periodic Lotka-Volterra competition-diffusion system. We first give the asymptotic behaviors of time periodic traveling wave solutions at infinity by a dynamical approach coupled with the two-sided Laplace transform. According to these asymptotic behaviors, we then obtain some key estimates which are crucial for the construction of an appropriate pair of sub-super solutions. Finally, using the sub-super solutions method and comparison principle, we establish the existence of invasion entire solutions which behave as two periodic traveling fronts with different speeds propagating from both sides of x-axis. In other words, we formulate a new invasion way of the superior species to the inferior one in a time periodic environment.
References:
[1] |
N. D. Alikakos, P. W. Bates and X. Chen,
Periodic traveling waves and locating oscillating patterns in multidimensional domains, Trans. Amer. Math. Soc., 351 (1999), 2777-2805.
doi: 10.1090/S0002-9947-99-02134-0. |
[2] |
X. Bao, W. T. Li and Z. C. Wang,
Time periodic traveling curved fronts in the periodic Lotka-Volterra competition-diffusion system, J. Dynam. Differential Equations, (2015), 1-36.
doi: 10.1007/s10884-015-9512-4. |
[3] |
X. Bao and Z. C. Wang,
Existence and stability of time periodic traveling waves for a periodic bistable Lotka-Volterra competition system, J. Differential Equations, 255 (2013), 2402-2435.
doi: 10.1016/j.jde.2013.06.024. |
[4] |
P. W. Bates and F. Chen,
Periodic traveling waves for a nonlocal integro-differential model, Electron. J. Differential Equations, 1999 (1999), 1-19.
|
[5] |
H. Berestycki and F. Hamel,
Front propagation in periodic excitable media, Comm. Pure Appl. Math., 55 (2002), 949-1032.
doi: 10.1002/cpa.3022. |
[6] |
Z. H. Bu, Z. C. Wang and N. W. Liu,
Asymptotic behavior of pulsating fronts and entire solutions of reaction-advection-diffusion equations in periodic media, Nonlinear Anal. Real World Appl., 28 (2016), 48-71.
doi: 10.1016/j.nonrwa.2015.09.006. |
[7] |
X. Chen and J. S. Guo,
Existence and uniqueness of entire solutions for a reaction-diffusion equation, J. Differential Equations, 212 (2005), 62-84.
doi: 10.1016/j.jde.2004.10.028. |
[8] |
C. Conley and R. Gardner,
An application of the generalized morse index to travelling wave solutions of a competitive reaction-diffusion model, Indiana Univ. Math. J., 33 (1984), 319-343.
doi: 10.1512/iumj.1984.33.33018. |
[9] |
J. Foldes and P. Poláčik,
On cooperative parabolic systems: Harnack inequalities and asymptotic symmetry, Discrete Cont. Dynam. Syst. Ser. A., 25 (2009), 133-157.
doi: 10.3934/dcds.2009.25.133. |
[10] |
Y. Fukao, Y. Morita and H. Ninomiya,
Some entire solutions of the Allen-Cahn equation, Taiwanese J. Math., 8 (2004), 15-32.
|
[11] |
R. A. Gardner,
Existence and stability of travelling wave solutions of competition models: A degree theoretic approach, J. Differential Equations., 44 (1982), 343-364.
doi: 10.1016/0022-0396(82)90001-8. |
[12] |
J. S. Guo and Y. Morita,
Entire solutions of reaction-diffusion equations and an application to discrete diffusive equations, Discrete Contin. Dyn. Syst. Ser. A., 12 (2005), 193-212.
doi: 10.3934/dcds.2005.12.193. |
[13] |
J. S. Guo and C. H. Wu,
Entire solutions for a two-component competition system in a lattice, Tohoku Math. J., 62 (2010), 17-28.
doi: 10.2748/tmj/1270041024. |
[14] |
F. Hamel,
Qualitative properties of monostable pulsating fronts: Exponential decayed monotonicity, J. Math. Pures Appl., 89 (2008), 355-399.
doi: 10.1016/j.matpur.2007.12.005. |
[15] |
F. Hamel and N. Nadirashvili,
Entire solutions of the KPP equation, Comm. Pure Appl. Math., 52 (1999), 1255-1276.
doi: 10.1002/(SICI)1097-0312(199910)52:10<1255::AID-CPA4>3.0.CO;2-W. |
[16] |
F. Hamel and N. Nadirashvili,
Travelling fronts and entire solutions of the Fisher-KPP equation in $\mathbb R^N$, Arch. Ration. Mech. Anal., 157 (2001), 91-163.
doi: 10.1007/PL00004238. |
[17] |
Y. Hosono,
Singular perturbation analysis of travelling waves of diffusive Lotka-Volterra competition models, Numerical and Applied Mathematics, Part Ⅱ (Paris 1988), (1989), 687-692.
|
[18] |
X. 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. |
[19] |
Y. Kan-On,
Parameter dependence of propagation speed of travelling waves for competition-diffusion equations, SIAM J. Math. Anal., 26 (1995), 340-363.
doi: 10.1137/S0036141093244556. |
[20] |
Y. Kan-On,
Fisher wave fronts for the Lotka-Volterra competition model with diffusion, Nonlinear Anal., 28 (1997), 145-164.
doi: 10.1016/0362-546X(95)00142-I. |
[21] |
W. T. Li, Y. J. Sun and Z. C. Wang,
Entire solutions in the Fisher-KPP equation with nonlocal dispersal, Nonlinear Anal. Real World Appl., 11 (2010), 2302-2313.
doi: 10.1016/j.nonrwa.2009.07.005. |
[22] |
W. T. Li, Z. C. Wang and J. Wu,
Entire solutions in monostable reaction-diffusion equations with delayed nonlinearity, J. Differential Equations, 245 (2008), 102-129.
doi: 10.1016/j.jde.2008.03.023. |
[23] |
W. T. Li, J. B. Wang and L. Zhang,
Entire solutions of nonlocal dispersal equations with monostable nonlinearity in space periodic habitats, J. Differential Equations, 261 (2016), 2472-2501.
doi: 10.1016/j.jde.2016.05.006. |
[24] |
W. T. Li, L. Zhang and G. B. Zhang,
Invasion entire solutions in a competition system with nonlocal dispersal, Discrete Contin. Dyn. Syst. Ser. A., 35 (2015), 1531-1560.
doi: 10.3934/dcds.2015.35.1531. |
[25] |
N. W. Liu, W. T. Li and Z. C. Wang,
Pulsating type entire solutions of monostable reaction-advection-diffusion equations in periodic excitable media, Nonlinear Anal., 75 (2012), 1869-1880.
doi: 10.1016/j.na.2011.09.037. |
[26] |
A. Lunardi,
Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhauser, Boston, 1995. |
[27] |
G. Lv and M. Wang,
Traveling wave front in diffusive and competitive Lotka-Volterra system with delays, Nonlinear Anal. Real World Appl., 11 (2010), 1323-1329.
doi: 10.1016/j.nonrwa.2009.02.020. |
[28] |
Y. Morita and H. Ninomiya,
Entire solutions with merging fronts to reaction-diffusion equations, J. Dynam. Differential Equations, 18 (2006), 841-861.
doi: 10.1007/s10884-006-9046-x. |
[29] |
Y. Morita and K. Tachibana,
An entire solution to the Lotka-Volterra competition-diffusion equations, SIAM J. Math. Anal., 40 (2009), 2217-2240.
doi: 10.1137/080723715. |
[30] |
G. Nadin,
Traveling fronts in space-time periodic media, J. Math. Pures Appl., 92 (2009), 232-262.
doi: 10.1016/j.matpur.2009.04.002. |
[31] |
G. Nadin,
Existence and uniqueness of the solution of a space-time periodic reaction-diffusion equation, J. Differential Equations, 249 (2010), 1288-1304.
doi: 10.1016/j.jde.2010.05.007. |
[32] |
J. Nolen, M. Rudd and J. Xin,
Existence of KPP fronts in spatially-temporally periodic advection and variational principle for propagation speeds, Dyn. Partial Differ. Equ., 2 (2005), 1-24.
doi: 10.4310/DPDE.2005.v2.n1.a1. |
[33] |
W. Shen,
Traveling waves in time periodic lattice differential equations, Nonlinear Anal., 54 (2003), 319-339.
doi: 10.1016/S0362-546X(03)00065-8. |
[34] |
W. J. Sheng and J. B. Wang, Entire solutions of time periodic bistable reaction-advection-diffusion equations in infinite cylinders J. Math. Phys., 56 (2015), 081501, 17 pp.
doi: 10.1063/1.4927712. |
[35] |
Y. J. Sun, W. T. Li and Z. C. Wang,
Entire solutions in nonlocal dispersal equations with bistable nonlinearity, J. Differential Equations, 251 (2011), 551-581.
doi: 10.1016/j.jde.2011.04.020. |
[36] |
M. M. Tang and P. C. Fife,
Propagating fronts for competing species equations with diffusion, Arch. Ration. Mech. Anal., 73 (1980), 69-77.
doi: 10.1007/BF00283257. |
[37] |
J. H. Vuuren,
The existence of traveling plane waves in a general class of competition-diffusion systems, SIMA J. Appl. Math., 55 (1995), 135-148.
doi: 10.1093/imamat/55.2.135. |
[38] |
M. Wang and G. Lv,
Entire solutions of a diffusive and competitive Lotka-Volterra type system with nonlocal delays, Nonlinearity, 23 (2010), 1609-1630.
doi: 10.1088/0951-7715/23/7/005. |
[39] |
Z. C. Wang, W. T. Li and S. Ruan,
Entire solutions in bistable reaction-diffusion equations with nonlocal delayed nonlinearity, Trans. Amer. Math. Soc., 361 (2009), 2047-2084.
doi: 10.1090/S0002-9947-08-04694-1. |
[40] |
Z. C. Wang, W. T. Li and J. Wu,
Entire solutions in delayed lattice differential equations with monostable nonlinearity, SIAM J. Math. Anal., 40 (2009), 2392-2420.
doi: 10.1137/080727312. |
[41] |
H. Yagisita,
Backward global solutions characterizing annihilation dynamics of travelling fronts, Publ. Res. Inst. Math. Sci., 39 (2003), 117-164.
doi: 10.2977/prims/1145476150. |
[42] |
L. Zhang, W. T. Li and S. L. Wu,
Multi-type entire solutions in a nonlocal dispersal epidemic model, J. Dynam. Differential Equations, 28 (2016), 189-224.
doi: 10.1007/s10884-014-9416-8. |
[43] |
G. Zhao and S. Ruan,
Existence, uniqueness and asymptotic stability of time periodic traveling waves for a periodic Lotka-Volterra competition system with diffusion, J. Math. Pures Appl., 95 (2011), 627-671.
doi: 10.1016/j.matpur.2010.11.005. |
[44] |
G. Zhao and S. Ruan,
Time periodic traveling wave solutions for periodic advection-reaction-diffusion systems, J. Differential Equations, 257 (2014), 1078-1147.
doi: 10.1016/j.jde.2014.05.001. |
[45] |
X. Q. Zhao,
Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003.
doi: 10.1007/978-0-387-21761-1. |
show all references
References:
[1] |
N. D. Alikakos, P. W. Bates and X. Chen,
Periodic traveling waves and locating oscillating patterns in multidimensional domains, Trans. Amer. Math. Soc., 351 (1999), 2777-2805.
doi: 10.1090/S0002-9947-99-02134-0. |
[2] |
X. Bao, W. T. Li and Z. C. Wang,
Time periodic traveling curved fronts in the periodic Lotka-Volterra competition-diffusion system, J. Dynam. Differential Equations, (2015), 1-36.
doi: 10.1007/s10884-015-9512-4. |
[3] |
X. Bao and Z. C. Wang,
Existence and stability of time periodic traveling waves for a periodic bistable Lotka-Volterra competition system, J. Differential Equations, 255 (2013), 2402-2435.
doi: 10.1016/j.jde.2013.06.024. |
[4] |
P. W. Bates and F. Chen,
Periodic traveling waves for a nonlocal integro-differential model, Electron. J. Differential Equations, 1999 (1999), 1-19.
|
[5] |
H. Berestycki and F. Hamel,
Front propagation in periodic excitable media, Comm. Pure Appl. Math., 55 (2002), 949-1032.
doi: 10.1002/cpa.3022. |
[6] |
Z. H. Bu, Z. C. Wang and N. W. Liu,
Asymptotic behavior of pulsating fronts and entire solutions of reaction-advection-diffusion equations in periodic media, Nonlinear Anal. Real World Appl., 28 (2016), 48-71.
doi: 10.1016/j.nonrwa.2015.09.006. |
[7] |
X. Chen and J. S. Guo,
Existence and uniqueness of entire solutions for a reaction-diffusion equation, J. Differential Equations, 212 (2005), 62-84.
doi: 10.1016/j.jde.2004.10.028. |
[8] |
C. Conley and R. Gardner,
An application of the generalized morse index to travelling wave solutions of a competitive reaction-diffusion model, Indiana Univ. Math. J., 33 (1984), 319-343.
doi: 10.1512/iumj.1984.33.33018. |
[9] |
J. Foldes and P. Poláčik,
On cooperative parabolic systems: Harnack inequalities and asymptotic symmetry, Discrete Cont. Dynam. Syst. Ser. A., 25 (2009), 133-157.
doi: 10.3934/dcds.2009.25.133. |
[10] |
Y. Fukao, Y. Morita and H. Ninomiya,
Some entire solutions of the Allen-Cahn equation, Taiwanese J. Math., 8 (2004), 15-32.
|
[11] |
R. A. Gardner,
Existence and stability of travelling wave solutions of competition models: A degree theoretic approach, J. Differential Equations., 44 (1982), 343-364.
doi: 10.1016/0022-0396(82)90001-8. |
[12] |
J. S. Guo and Y. Morita,
Entire solutions of reaction-diffusion equations and an application to discrete diffusive equations, Discrete Contin. Dyn. Syst. Ser. A., 12 (2005), 193-212.
doi: 10.3934/dcds.2005.12.193. |
[13] |
J. S. Guo and C. H. Wu,
Entire solutions for a two-component competition system in a lattice, Tohoku Math. J., 62 (2010), 17-28.
doi: 10.2748/tmj/1270041024. |
[14] |
F. Hamel,
Qualitative properties of monostable pulsating fronts: Exponential decayed monotonicity, J. Math. Pures Appl., 89 (2008), 355-399.
doi: 10.1016/j.matpur.2007.12.005. |
[15] |
F. Hamel and N. Nadirashvili,
Entire solutions of the KPP equation, Comm. Pure Appl. Math., 52 (1999), 1255-1276.
doi: 10.1002/(SICI)1097-0312(199910)52:10<1255::AID-CPA4>3.0.CO;2-W. |
[16] |
F. Hamel and N. Nadirashvili,
Travelling fronts and entire solutions of the Fisher-KPP equation in $\mathbb R^N$, Arch. Ration. Mech. Anal., 157 (2001), 91-163.
doi: 10.1007/PL00004238. |
[17] |
Y. Hosono,
Singular perturbation analysis of travelling waves of diffusive Lotka-Volterra competition models, Numerical and Applied Mathematics, Part Ⅱ (Paris 1988), (1989), 687-692.
|
[18] |
X. 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. |
[19] |
Y. Kan-On,
Parameter dependence of propagation speed of travelling waves for competition-diffusion equations, SIAM J. Math. Anal., 26 (1995), 340-363.
doi: 10.1137/S0036141093244556. |
[20] |
Y. Kan-On,
Fisher wave fronts for the Lotka-Volterra competition model with diffusion, Nonlinear Anal., 28 (1997), 145-164.
doi: 10.1016/0362-546X(95)00142-I. |
[21] |
W. T. Li, Y. J. Sun and Z. C. Wang,
Entire solutions in the Fisher-KPP equation with nonlocal dispersal, Nonlinear Anal. Real World Appl., 11 (2010), 2302-2313.
doi: 10.1016/j.nonrwa.2009.07.005. |
[22] |
W. T. Li, Z. C. Wang and J. Wu,
Entire solutions in monostable reaction-diffusion equations with delayed nonlinearity, J. Differential Equations, 245 (2008), 102-129.
doi: 10.1016/j.jde.2008.03.023. |
[23] |
W. T. Li, J. B. Wang and L. Zhang,
Entire solutions of nonlocal dispersal equations with monostable nonlinearity in space periodic habitats, J. Differential Equations, 261 (2016), 2472-2501.
doi: 10.1016/j.jde.2016.05.006. |
[24] |
W. T. Li, L. Zhang and G. B. Zhang,
Invasion entire solutions in a competition system with nonlocal dispersal, Discrete Contin. Dyn. Syst. Ser. A., 35 (2015), 1531-1560.
doi: 10.3934/dcds.2015.35.1531. |
[25] |
N. W. Liu, W. T. Li and Z. C. Wang,
Pulsating type entire solutions of monostable reaction-advection-diffusion equations in periodic excitable media, Nonlinear Anal., 75 (2012), 1869-1880.
doi: 10.1016/j.na.2011.09.037. |
[26] |
A. Lunardi,
Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhauser, Boston, 1995. |
[27] |
G. Lv and M. Wang,
Traveling wave front in diffusive and competitive Lotka-Volterra system with delays, Nonlinear Anal. Real World Appl., 11 (2010), 1323-1329.
doi: 10.1016/j.nonrwa.2009.02.020. |
[28] |
Y. Morita and H. Ninomiya,
Entire solutions with merging fronts to reaction-diffusion equations, J. Dynam. Differential Equations, 18 (2006), 841-861.
doi: 10.1007/s10884-006-9046-x. |
[29] |
Y. Morita and K. Tachibana,
An entire solution to the Lotka-Volterra competition-diffusion equations, SIAM J. Math. Anal., 40 (2009), 2217-2240.
doi: 10.1137/080723715. |
[30] |
G. Nadin,
Traveling fronts in space-time periodic media, J. Math. Pures Appl., 92 (2009), 232-262.
doi: 10.1016/j.matpur.2009.04.002. |
[31] |
G. Nadin,
Existence and uniqueness of the solution of a space-time periodic reaction-diffusion equation, J. Differential Equations, 249 (2010), 1288-1304.
doi: 10.1016/j.jde.2010.05.007. |
[32] |
J. Nolen, M. Rudd and J. Xin,
Existence of KPP fronts in spatially-temporally periodic advection and variational principle for propagation speeds, Dyn. Partial Differ. Equ., 2 (2005), 1-24.
doi: 10.4310/DPDE.2005.v2.n1.a1. |
[33] |
W. Shen,
Traveling waves in time periodic lattice differential equations, Nonlinear Anal., 54 (2003), 319-339.
doi: 10.1016/S0362-546X(03)00065-8. |
[34] |
W. J. Sheng and J. B. Wang, Entire solutions of time periodic bistable reaction-advection-diffusion equations in infinite cylinders J. Math. Phys., 56 (2015), 081501, 17 pp.
doi: 10.1063/1.4927712. |
[35] |
Y. J. Sun, W. T. Li and Z. C. Wang,
Entire solutions in nonlocal dispersal equations with bistable nonlinearity, J. Differential Equations, 251 (2011), 551-581.
doi: 10.1016/j.jde.2011.04.020. |
[36] |
M. M. Tang and P. C. Fife,
Propagating fronts for competing species equations with diffusion, Arch. Ration. Mech. Anal., 73 (1980), 69-77.
doi: 10.1007/BF00283257. |
[37] |
J. H. Vuuren,
The existence of traveling plane waves in a general class of competition-diffusion systems, SIMA J. Appl. Math., 55 (1995), 135-148.
doi: 10.1093/imamat/55.2.135. |
[38] |
M. Wang and G. Lv,
Entire solutions of a diffusive and competitive Lotka-Volterra type system with nonlocal delays, Nonlinearity, 23 (2010), 1609-1630.
doi: 10.1088/0951-7715/23/7/005. |
[39] |
Z. C. Wang, W. T. Li and S. Ruan,
Entire solutions in bistable reaction-diffusion equations with nonlocal delayed nonlinearity, Trans. Amer. Math. Soc., 361 (2009), 2047-2084.
doi: 10.1090/S0002-9947-08-04694-1. |
[40] |
Z. C. Wang, W. T. Li and J. Wu,
Entire solutions in delayed lattice differential equations with monostable nonlinearity, SIAM J. Math. Anal., 40 (2009), 2392-2420.
doi: 10.1137/080727312. |
[41] |
H. Yagisita,
Backward global solutions characterizing annihilation dynamics of travelling fronts, Publ. Res. Inst. Math. Sci., 39 (2003), 117-164.
doi: 10.2977/prims/1145476150. |
[42] |
L. Zhang, W. T. Li and S. L. Wu,
Multi-type entire solutions in a nonlocal dispersal epidemic model, J. Dynam. Differential Equations, 28 (2016), 189-224.
doi: 10.1007/s10884-014-9416-8. |
[43] |
G. Zhao and S. Ruan,
Existence, uniqueness and asymptotic stability of time periodic traveling waves for a periodic Lotka-Volterra competition system with diffusion, J. Math. Pures Appl., 95 (2011), 627-671.
doi: 10.1016/j.matpur.2010.11.005. |
[44] |
G. Zhao and S. Ruan,
Time periodic traveling wave solutions for periodic advection-reaction-diffusion systems, J. Differential Equations, 257 (2014), 1078-1147.
doi: 10.1016/j.jde.2014.05.001. |
[45] |
X. Q. Zhao,
Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003.
doi: 10.1007/978-0-387-21761-1. |
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