
-
Previous Article
Entire subsolutions of Monge-Ampère type equations
- CPAA Home
- This Issue
- Next Article
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. |






[1] |
Chiun-Chuan Chen, Li-Chang Hung. Nonexistence of traveling wave solutions, exact and semi-exact traveling wave solutions for diffusive Lotka-Volterra systems of three competing species. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1451-1469. doi: 10.3934/cpaa.2016.15.1451 |
[2] |
Cheng-Hsiung Hsu, Jian-Jhong Lin. Stability analysis of traveling wave solutions for lattice reaction-diffusion equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (5) : 1757-1774. doi: 10.3934/dcdsb.2020001 |
[3] |
Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189 |
[4] |
Jiang Liu, Xiaohui Shang, Zengji Du. Traveling wave solutions of a reaction-diffusion predator-prey model. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 1063-1078. doi: 10.3934/dcdss.2017057 |
[5] |
Bang-Sheng Han, Zhi-Cheng Wang. Traveling wave solutions in a nonlocal reaction-diffusion population model. Communications on Pure and Applied Analysis, 2016, 15 (3) : 1057-1076. doi: 10.3934/cpaa.2016.15.1057 |
[6] |
Xiaojie Hou, Yi Li. Local stability of traveling-wave solutions of nonlinear reaction-diffusion equations. Discrete and Continuous Dynamical Systems, 2006, 15 (2) : 681-701. doi: 10.3934/dcds.2006.15.681 |
[7] |
Bingtuan Li, William F. Fagan, Garrett Otto, Chunwei Wang. Spreading speeds and traveling wave solutions in a competitive reaction-diffusion model for species persistence in a stream. Discrete and Continuous Dynamical Systems - B, 2014, 19 (10) : 3267-3281. doi: 10.3934/dcdsb.2014.19.3267 |
[8] |
Kota Ikeda, Masayasu Mimura. Traveling wave solutions of a 3-component reaction-diffusion model in smoldering combustion. Communications on Pure and Applied Analysis, 2012, 11 (1) : 275-305. doi: 10.3934/cpaa.2012.11.275 |
[9] |
Guo Lin, Haiyan Wang. Traveling wave solutions of a reaction-diffusion equation with state-dependent delay. Communications on Pure and Applied Analysis, 2016, 15 (2) : 319-334. doi: 10.3934/cpaa.2016.15.319 |
[10] |
Joaquin Riviera, Yi Li. Existence of traveling wave solutions for a nonlocal reaction-diffusion model of influenza a drift. Discrete and Continuous Dynamical Systems - B, 2010, 13 (1) : 157-174. doi: 10.3934/dcdsb.2010.13.157 |
[11] |
Xiaojie Hou, Yi Li, Kenneth R. Meyer. Traveling wave solutions for a reaction diffusion equation with double degenerate nonlinearities. Discrete and Continuous Dynamical Systems, 2010, 26 (1) : 265-290. doi: 10.3934/dcds.2010.26.265 |
[12] |
Jong-Shenq Guo, Yoshihisa Morita. Entire solutions of reaction-diffusion equations and an application to discrete diffusive equations. Discrete and Continuous Dynamical Systems, 2005, 12 (2) : 193-212. doi: 10.3934/dcds.2005.12.193 |
[13] |
Michele V. Bartuccelli, K. B. Blyuss, Y. N. Kyrychko. Length scales and positivity of solutions of a class of reaction-diffusion equations. Communications on Pure and Applied Analysis, 2004, 3 (1) : 25-40. doi: 10.3934/cpaa.2004.3.25 |
[14] |
Peter Poláčik, Eiji Yanagida. Stable subharmonic solutions of reaction-diffusion equations on an arbitrary domain. Discrete and Continuous Dynamical Systems, 2002, 8 (1) : 209-218. doi: 10.3934/dcds.2002.8.209 |
[15] |
Shi-Liang Wu, Yu-Juan Sun, San-Yang Liu. Traveling fronts and entire solutions in partially degenerate reaction-diffusion systems with monostable nonlinearity. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 921-946. doi: 10.3934/dcds.2013.33.921 |
[16] |
Ming Mei. Stability of traveling wavefronts for time-delayed reaction-diffusion equations. Conference Publications, 2009, 2009 (Special) : 526-535. doi: 10.3934/proc.2009.2009.526 |
[17] |
Masaharu Taniguchi. Multi-dimensional traveling fronts in bistable reaction-diffusion equations. Discrete and Continuous Dynamical Systems, 2012, 32 (3) : 1011-1046. doi: 10.3934/dcds.2012.32.1011 |
[18] |
Masaharu Taniguchi. Axisymmetric traveling fronts in balanced bistable reaction-diffusion equations. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3981-3995. doi: 10.3934/dcds.2020126 |
[19] |
Masaharu Taniguchi. Traveling fronts in perturbed multistable reaction-diffusion equations. Conference Publications, 2011, 2011 (Special) : 1368-1377. doi: 10.3934/proc.2011.2011.1368 |
[20] |
Henri Berestycki, Guillemette Chapuisat. Traveling fronts guided by the environment for reaction-diffusion equations. Networks and Heterogeneous Media, 2013, 8 (1) : 79-114. doi: 10.3934/nhm.2013.8.79 |
2020 Impact Factor: 1.916
Tools
Metrics
Other articles
by authors
[Back to Top]