2017, 14(5-6): 1187-1213. doi: 10.3934/mbe.2017061

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

2. 

School of Mathematics and Statistics, Lanzhou University Lanzhou, Gansu 730000, China

3. 

School of Mathematics and Statistics, Lanzhou University Lanzhou, Gansu 730000, China

 

Received  March 2016 Revised  October 2016 Accepted  October 2016 Published  May 2017

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.

Citation: Li-Jun Du, Wan-Tong Li, Jia-Bing Wang. Invasion entire solutions in a time periodic Lotka-Volterra competition system with diffusion. Mathematical Biosciences & Engineering, 2017, 14 (5-6) : 1187-1213. doi: 10.3934/mbe.2017061
References:
[1]

N. D. AlikakosP. 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. BaoW. 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. BuZ. 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. Polá\v{c}ik, 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. FukaoY. 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. LiY. 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. LiZ. 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. LiJ. 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. LiL. 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. LiuW. 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. NolenM. 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.

[35]

Y. J. SunW. 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. WangW. 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. WangW. 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. ZhangW. 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.

show all references

References:
[1]

N. D. AlikakosP. 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. BaoW. 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. BuZ. 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. Polá\v{c}ik, 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. FukaoY. 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. LiY. 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. LiZ. 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. LiJ. 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. LiL. 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. LiuW. 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. NolenM. 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.

[35]

Y. J. SunW. 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. WangW. 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. WangW. 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. ZhangW. 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.

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