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January  2020, 19(1): 1-18. doi: 10.3934/cpaa.2020001

Semi-exact solutions and pulsating fronts for Lotka-Volterra systems of two competing species in spatially periodic habitats

Chiun-Chuan Chen 1, , Yin-Liang Huang 2, , Li-Chang Hung 3,, and  Chang-Hong Wu 4,

1. 

Department of Mathematics, National Taiwan University, National Center for Theoretical Sciences, Taipei, Taiwan
2. 

Department of Applied Mathematics, National University of Tainan, Tainan, Taiwan
3. 

Department of Mathematics, National Taiwan University, Taipei, Taiwan
4. 

Department of Applied Mathematics, National Chiao Tung University, Taiwan

* Corresponding author

Received  December 2017 Revised  March 2019 Published  July 2019

Fund Project: The research of C.-C. Chen is partly supported by the grant 102-2115-M-002-011-MY3 of Ministry of Science and Technology, Taiwan. The research of L.-C. Hung is partly supported by the grant 104EFA0101550 of Ministry of Science and Technology, Taiwan. The research of C.-H. Wu is partly supported by the grant MOST 105-2628-M-024-001-MY2 and 107-2636-M-024-001 of Ministry of Science and Technology, Taiwan and National Center for Theoretical Science (NCTS).

We are concerned with the coexistence states of the diffusive Lotka-Volterra system of two competing species when the growth rates of the two species depend periodically on the spacial variable. For the one-dimensional problem, we employ the generalized Jacobi elliptic function method to find semi-exact solutions under certain conditions on the parameters. In addition, we use the sine function to construct a pair of upper and lower solutions and obtain a solution of the above-mentioned system. Next, we provide a sufficient condition for the existence of pulsating fronts connecting two semi-trivial states by applying the abstract theory regarding monotone semiflows. Some numerical simulations are also included.

Keywords: Semi-exact solutions, traveling wave solutions, reaction-diffusion equations.
Mathematics Subject Classification: Primary: 35K57; Secondary: 35C07, 83C15.
Citation: Chiun-Chuan Chen, Yin-Liang Huang, Li-Chang Hung, Chang-Hong Wu. Semi-exact solutions and pulsating fronts for Lotka-Volterra systems of two competing species in spatially periodic habitats. Communications on Pure & Applied Analysis, 2020, 19 (1) : 1-18. doi: 10.3934/cpaa.2020001
References:
[1]

H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model: Ⅰ – Influence of periodic heterogeneous environment on species persistence,, J. Math. Biology, 51 (2005), 75-113.  doi: 10.1007/s00285-004-0313-3.  Google Scholar

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H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model : Ⅱ - Biological invasions and pulsating travelling fronts, J. Math. Pures Appl., 84 (2005), 1101-1146.  doi: 10.1016/j.matpur.2004.10.006.  Google Scholar

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C.-C. Chen, L.-C. Hung, M. Mimura, M. Tohma and D. Ueyama, Semi-exact equilibrium solutions for three-species competition-diffusion systems, Hiroshima Math J., 43 (2013), 176-206.   Google Scholar

[5]

C.-C. Chen, L.-C. Hung, M. Mimura and D. Ueyama, Exact travelling wave solutions of three-species competition-diffusion systems, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2653-2669.  doi: 10.3934/dcdsb.2012.17.2653.  Google Scholar

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E. Fan and J. Zhang, Applications of the Jacobi elliptic function method to special-type nonlinear equations, Physics Letters A, 305 (2002), 383-392.  doi: 10.1016/S0375-9601(02)01516-5.  Google Scholar

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J. Fang and X.-Q. Zhao, Traveling waves for monotone semiflows with weak compactness, SIAM J. Math. Anal., 46 (2014), 3678-3704.  doi: 10.1137/140953939.  Google Scholar

[9]

J. Fang, X. Yu and X.-Q. Zhao, Traveling waves and spreading speeds for time-space periodic monotone systems, J. Funct. Anal., 272 (2017), 4222-4262.  doi: 10.1016/j.jfa.2017.02.028.  Google Scholar

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M. Freidlin and J. Gartner, On the propagation of concentration waves in periodic and random media, Sov. Math. Dokl., 20 (1979), 1282-1286.   Google Scholar

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L. Girardin, Competition in periodic media: Ⅰ-Existence of pulsating fronts, Discrete and Continuous Dynamical Systems-Series B, 22 (2017), 1341-1360.  doi: 10.3934/dcdsb.2017065.  Google Scholar

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X. He and W.-M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system: diffusion and spatial heterogeneity, Ⅰ, Communications on Pure and Applied Mathematics, 69 (2016), 981-1014.  doi: 10.1002/cpa.21596.  Google Scholar

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Y.-L. Huang and C.-H. Wu, Positive steady states of reaction-diffusion-advection competition models in periodic environment, J. Math. Anal. Appl., 453 (2017), 724-745.  doi: 10.1016/j.jmaa.2017.04.026.  Google Scholar

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V. Hutson, Y. Lou and K. Mischaikow, Spatial heterogeneity of resources versus LotkaVolterra dynamics, J. Differential Equations, 185 (2002), 97-136.  doi: 10.1006/jdeq.2001.4157.  Google Scholar

[15]

K.-Y. Lam and W.-M. Ni, Uniqueness and complete dynamics in heterogeneous competition-diffusion systems, SIAM J. Appl. Math., 72 (2012), 1695-1712.  doi: 10.1137/120869481.  Google Scholar

[16]

X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, J. Funct. Anal., 259 (2010), 857-903.  doi: 10.1016/j.jfa.2010.04.018.  Google Scholar

[17]

Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, J. Differential Equations, 223 (2006), 400-426.  doi: 10.1016/j.jde.2005.05.010.  Google Scholar

[18]

Y. Lou, Some challenging mathematical problems in evolution of dispersal and population dynamics, in, Tutorials in Mathematical Biosciences, Ⅳ, Lecture Notes in Math. 1922, Springer, Berlin, 2008, 171–205. doi: 10.1007/978-3-540-74331-6_5.  Google Scholar

[19]

W.-M. Ni, The Mathematics of Diffusion, CBMS-NSF Regional Conf. Ser. in Appl. Math. 82, SIAM, Philadelphia, 2011. doi: 10.1137/1.9781611971972.  Google Scholar

[20]

M. Rodrigo and M. Mimura, Exact solutions of a competition-diffusion system, Hiroshima Math. J., 30 (2000), 257-270.   Google Scholar

[21]

X. Yu and X.-Q. Zhao, Propagation phenomena for a reaction-advection-diffusion competition model in a periodic habitat, J. Dynamics and Differential Equations, 29 (2017), 41-66.  doi: 10.1007/s10884-015-9426-1.  Google Scholar

[22]

H. Zhang, Extended Jacobi elliptic function expansion method and its applications, Communications in Nonlinear Science and Numerical Simulation, 12, 627–635. doi: 10.1016/j.cnsns.2005.08.003.  Google Scholar

show all references

References:
[1]

H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model: Ⅰ – Influence of periodic heterogeneous environment on species persistence,, J. Math. Biology, 51 (2005), 75-113.  doi: 10.1007/s00285-004-0313-3.  Google Scholar

[2]

H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model : Ⅱ - Biological invasions and pulsating travelling fronts, J. Math. Pures Appl., 84 (2005), 1101-1146.  doi: 10.1016/j.matpur.2004.10.006.  Google Scholar

[3]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-diffusion Equations, Chichester, UK: Wiley, 2003. doi: 10.1002/0470871296.  Google Scholar

[4]

C.-C. Chen, L.-C. Hung, M. Mimura, M. Tohma and D. Ueyama, Semi-exact equilibrium solutions for three-species competition-diffusion systems, Hiroshima Math J., 43 (2013), 176-206.   Google Scholar

[5]

C.-C. Chen, L.-C. Hung, M. Mimura and D. Ueyama, Exact travelling wave solutions of three-species competition-diffusion systems, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2653-2669.  doi: 10.3934/dcdsb.2012.17.2653.  Google Scholar

[6]

J. Dockery, V. Hutson, K. Mischaikow and M. Pernarowski, The evolution of slow dispersal rates: a reaction diffusion model, J. Math. Biol., 37 (1998), 61-83.  doi: 10.1007/s002850050120.  Google Scholar

[7]

E. Fan and J. Zhang, Applications of the Jacobi elliptic function method to special-type nonlinear equations, Physics Letters A, 305 (2002), 383-392.  doi: 10.1016/S0375-9601(02)01516-5.  Google Scholar

[8]

J. Fang and X.-Q. Zhao, Traveling waves for monotone semiflows with weak compactness, SIAM J. Math. Anal., 46 (2014), 3678-3704.  doi: 10.1137/140953939.  Google Scholar

[9]

J. Fang, X. Yu and X.-Q. Zhao, Traveling waves and spreading speeds for time-space periodic monotone systems, J. Funct. Anal., 272 (2017), 4222-4262.  doi: 10.1016/j.jfa.2017.02.028.  Google Scholar

[10]

M. Freidlin and J. Gartner, On the propagation of concentration waves in periodic and random media, Sov. Math. Dokl., 20 (1979), 1282-1286.   Google Scholar

[11]

L. Girardin, Competition in periodic media: Ⅰ-Existence of pulsating fronts, Discrete and Continuous Dynamical Systems-Series B, 22 (2017), 1341-1360.  doi: 10.3934/dcdsb.2017065.  Google Scholar

[12]

X. He and W.-M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system: diffusion and spatial heterogeneity, Ⅰ, Communications on Pure and Applied Mathematics, 69 (2016), 981-1014.  doi: 10.1002/cpa.21596.  Google Scholar

[13]

Y.-L. Huang and C.-H. Wu, Positive steady states of reaction-diffusion-advection competition models in periodic environment, J. Math. Anal. Appl., 453 (2017), 724-745.  doi: 10.1016/j.jmaa.2017.04.026.  Google Scholar

[14]

V. Hutson, Y. Lou and K. Mischaikow, Spatial heterogeneity of resources versus LotkaVolterra dynamics, J. Differential Equations, 185 (2002), 97-136.  doi: 10.1006/jdeq.2001.4157.  Google Scholar

[15]

K.-Y. Lam and W.-M. Ni, Uniqueness and complete dynamics in heterogeneous competition-diffusion systems, SIAM J. Appl. Math., 72 (2012), 1695-1712.  doi: 10.1137/120869481.  Google Scholar

[16]

X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, J. Funct. Anal., 259 (2010), 857-903.  doi: 10.1016/j.jfa.2010.04.018.  Google Scholar

[17]

Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, J. Differential Equations, 223 (2006), 400-426.  doi: 10.1016/j.jde.2005.05.010.  Google Scholar

[18]

Y. Lou, Some challenging mathematical problems in evolution of dispersal and population dynamics, in, Tutorials in Mathematical Biosciences, Ⅳ, Lecture Notes in Math. 1922, Springer, Berlin, 2008, 171–205. doi: 10.1007/978-3-540-74331-6_5.  Google Scholar

[19]

W.-M. Ni, The Mathematics of Diffusion, CBMS-NSF Regional Conf. Ser. in Appl. Math. 82, SIAM, Philadelphia, 2011. doi: 10.1137/1.9781611971972.  Google Scholar

[20]

M. Rodrigo and M. Mimura, Exact solutions of a competition-diffusion system, Hiroshima Math. J., 30 (2000), 257-270.   Google Scholar

[21]

X. Yu and X.-Q. Zhao, Propagation phenomena for a reaction-advection-diffusion competition model in a periodic habitat, J. Dynamics and Differential Equations, 29 (2017), 41-66.  doi: 10.1007/s10884-015-9426-1.  Google Scholar

[22]

H. Zhang, Extended Jacobi elliptic function expansion method and its applications, Communications in Nonlinear Science and Numerical Simulation, 12, 627–635. doi: 10.1016/j.cnsns.2005.08.003.  Google Scholar

Figure 1.  The profile of $ \phi(x) $
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Figure 2.  The profiles of $ u $ (red), $ v $ (green), $ m_1 $ (blue) and $ m_2 $ (cyan)
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Figure 3.  The profiles of $ u $ (red), $ v $ (green), $ m_1 $ (blue) and $ m_2 $ (cyan)
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Figure 4.  $ \frac{c_{22}}{c_{12}} = \frac{1}{3} $ (red), $ \frac{m_2(x)}{m_1(x)} $ (green) and $ \frac{c_{21}}{c_{11}} = 2 $ (blue), where $ \frac{m_2(x)}{m_1(x)} = \frac{18 \left(20+3 \sqrt{3}\right) \phi (x)+8 \left(26+9\sqrt{17}\right)}{27 \left(24+\sqrt{3}+2 \sqrt{51}\right) \phi(x)+36 \left(18+\sqrt{17}\right)} $ and $ \phi = \phi(x) $ is the solution of (17) with $ \phi'(0) = \frac{2\sqrt{2}}{9} $
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Figure 5.  The profile of the long time behavior of the solution with initial data in (42)
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Figure 6.  The profile of the long time behavior of the solution with initial data in (43)
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Figure 7.  The spreading of $ u $ occurs
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Figure 8.  The profile of the long time behavior of Example 2 shows $ u $ invades $ v $ eventually
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Figure 9.  The profile of the long time behavior of Example 3 with $ h = 0.01 $
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