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Semi-exact solutions and pulsating fronts for Lotka-Volterra systems of two competing species in spatially periodic habitats
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 |
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.
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. |
[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. |
[3] |
R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-diffusion Equations, Chichester, UK: Wiley, 2003.
doi: 10.1002/0470871296. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[10] |
M. Freidlin and J. Gartner,
On the propagation of concentration waves in periodic and random media, Sov. Math. Dokl., 20 (1979), 1282-1286.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[20] |
M. Rodrigo and M. Mimura,
Exact solutions of a competition-diffusion system, Hiroshima Math. J., 30 (2000), 257-270.
|
[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. |
[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. |
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. |
[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. |
[3] |
R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-diffusion Equations, Chichester, UK: Wiley, 2003.
doi: 10.1002/0470871296. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[10] |
M. Freidlin and J. Gartner,
On the propagation of concentration waves in periodic and random media, Sov. Math. Dokl., 20 (1979), 1282-1286.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[20] |
M. Rodrigo and M. Mimura,
Exact solutions of a competition-diffusion system, Hiroshima Math. J., 30 (2000), 257-270.
|
[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. |
[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. |






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