# American Institute of Mathematical Sciences

December  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

Received  March 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. 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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##### 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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