American Institute of Mathematical Sciences

Advanced
January  2020, 40(1): 153-187. doi: 10.3934/dcds.2020007

Discrete N-barrier maximum principle for a lattice dynamical system arising in competition models

Chiun-Chuan Chen 1, , Ting-Yang Hsiao 2, and  Li-Chang Hung 2,,

1. 

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

Department of Mathematics, National Taiwan University, Taipei, Taiwan

* Corresponding author

Received  November 2018 Revised  June 2019 Published  October 2019

In the present paper, we show that an analogous N-barrier maximum principle (see [3,7,5]) remains true for lattice systems. This extends the results in [3,7,5] from continuous equations to discrete equations. In order to overcome the difficulty induced by a discretized version of the classical diffusion in the lattice systems, we propose a more delicate construction of the N-barrier which is appropriate for the proof of the N-barrier maximum principle for lattice systems. As an application of the discrete N-barrier maximum principle, we study a coexistence problem of three species arising from biology, and show that the three species cannot coexist under certain conditions.

Keywords: Maximum principle, traveling wave solutions, lattice systems, reaction-diffusion equations, Lotka-Volterra.
Mathematics Subject Classification: Primary: 35B50; Secondary: 35C07, 35K57.
Citation: Chiun-Chuan Chen, Ting-Yang Hsiao, Li-Chang Hung. Discrete N-barrier maximum principle for a lattice dynamical system arising in competition models. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 153-187. doi: 10.3934/dcds.2020007
References:
[1]

R. A. Armstrong and R. McGehee, Competitive exclusion, Amer. Natur., 115 (1980), 151-170.  doi: 10.1086/283553.  Google Scholar

[2]

Cantrell, Ward and Jr., On competition-mediated coexistence, SIAM J. Appl. Math., 57 (1997), 1311-1327.  doi: 10.1137/S0036139995292367.  Google Scholar

[3]

C.-C. Chen and L.-C. Hung, A maximum principle for diffusive lotka-volterra systems of two competing species, J. Differential Equations, 261 (2016), 4573-4592.  doi: 10.1016/j.jde.2016.07.001.  Google Scholar

[4]

C.-C. Chen and L.-C. Hung, Nonexistence of traveling wave solutions, exact and semi-exact traveling wave solutions for diffusive Lotka-Volterra systems of three competing species, Commun. Pure Appl. Anal., 15 (2016), 1451-1469.  doi: 10.3934/cpaa.2016.15.1451.  Google Scholar

[5]

C.-C. Chen and L.-C. Hung, An N-barrier maximum principle for elliptic systems arising from the study of traveling waves in reaction-diffusion systems, Discrete Contin. Dyn. Syst. B, 23 (2018), 1503-1521.  doi: 10.3934/dcdsb.2018054.  Google Scholar

[6]

C.-C. Chen, L.-C. Hung and C.-C. Lai, An N-barrier maximum principle for autonomous systems of n species and its application to problems arising from population dynamics, Commun. Pure Appl. Anal., 18 (2019), 33-50.  doi: 10.3934/cpaa.2019003.  Google Scholar

[7]

C.-C. Chen, L.-C. Hung and H.-F. Liu, N-barrier maximum principle for degenerate elliptic systems and its application, Discrete Contin. Dyn. Syst. A, 38 (2018), 791-821.  doi: 10.3934/dcds.2018034.  Google Scholar

[8]

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), 179-206.  doi: 10.32917/hmj/1372180511.  Google Scholar

[9]

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

[10]

P. de Mottoni, Qualitative analysis for some quasilinear parabolic systems, Institute of Math., Polish Academy Sci., 11 (1979), p190. Google Scholar

[11]

J.-S. Guo and C.-C. Wu, The existence of traveling wave solutions for a bistable three-component lattice dynamical system, J. Differential Equations, 260 (2016), 1445-1455.  doi: 10.1016/j.jde.2015.09.036.  Google Scholar

[12]

J.-S. Guo and C.-H. Wu, Wave propagation for a two-component lattice dynamical system arising in strong competition models, Journal of Differential Equations, 250 (2011), 3504-3533.  doi: 10.1016/j.jde.2010.12.004.  Google Scholar

[13]

J.-S. Guo and C.-H. Wu, Traveling wave front for a two-component lattice dynamical system arising in competition models, Journal of Differential Equations, 252 (2012), 4357-4391.  doi: 10.1016/j.jde.2012.01.009.  Google Scholar

[14]

S.-B. Hsu and T.-H. Hsu, Competitive exclusion of microbial species for a single nutrient with internal storage, SIAM J. Appl. Math., 68 (2008), 1600-1617.  doi: 10.1137/070700784.  Google Scholar

[15]

S. B. Hsu, H. L. Smith and P. Waltman, Competitive exclusion and coexistence for competitive systems on ordered Banach spaces, Trans. Amer. Math. Soc., 348 (1996), 4083-4094.  doi: 10.1090/S0002-9947-96-01724-2.  Google Scholar

[16]

L.-C. Hung, Exact traveling wave solutions for diffusive Lotka-Volterra systems of two competing species, Jpn. J. Ind. Appl. Math., 29 (2012), 237-251.  doi: 10.1007/s13160-012-0056-2.  Google Scholar

[17]

S. R.-J. Jang, Competitive exclusion and coexistence in a Leslie-Gower competition model with Allee effects, Appl. Anal., 92 (2013), 1527-1540.  doi: 10.1080/00036811.2012.692365.  Google Scholar

[18]

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.  Google Scholar

[19]

J. Kastendiek, Competitor-mediated coexistence: Interactions among three species of benthic macroalgae, Journal of Experimental Marine Biology and Ecology, 62 (1982), 201-210.   Google Scholar

[20]

R. McGehee and R. A. Armstrong, Some mathematical problems concerning the ecological principle of competitive exclusion, J. Differential Equations, 23 (1977), 30-52.  doi: 10.1016/0022-0396(77)90135-8.  Google Scholar

[21]

M. Mimura and M. Tohma, Dynamic coexistence in a three-species competition–diffusion system, Ecological Complexity, 21 (2015), 215-232.   Google Scholar

[22]

H. L. Smith and P. Waltman, Competition for a single limiting resource in continuous culture: the variable-yield model, SIAM J. Appl. Math., 54 (1994), 1113-1131.  doi: 10.1137/S0036139993245344.  Google Scholar

show all references

References:
[1]

R. A. Armstrong and R. McGehee, Competitive exclusion, Amer. Natur., 115 (1980), 151-170.  doi: 10.1086/283553.  Google Scholar

[2]

Cantrell, Ward and Jr., On competition-mediated coexistence, SIAM J. Appl. Math., 57 (1997), 1311-1327.  doi: 10.1137/S0036139995292367.  Google Scholar

[3]

C.-C. Chen and L.-C. Hung, A maximum principle for diffusive lotka-volterra systems of two competing species, J. Differential Equations, 261 (2016), 4573-4592.  doi: 10.1016/j.jde.2016.07.001.  Google Scholar

[4]

C.-C. Chen and L.-C. Hung, Nonexistence of traveling wave solutions, exact and semi-exact traveling wave solutions for diffusive Lotka-Volterra systems of three competing species, Commun. Pure Appl. Anal., 15 (2016), 1451-1469.  doi: 10.3934/cpaa.2016.15.1451.  Google Scholar

[5]

C.-C. Chen and L.-C. Hung, An N-barrier maximum principle for elliptic systems arising from the study of traveling waves in reaction-diffusion systems, Discrete Contin. Dyn. Syst. B, 23 (2018), 1503-1521.  doi: 10.3934/dcdsb.2018054.  Google Scholar

[6]

C.-C. Chen, L.-C. Hung and C.-C. Lai, An N-barrier maximum principle for autonomous systems of n species and its application to problems arising from population dynamics, Commun. Pure Appl. Anal., 18 (2019), 33-50.  doi: 10.3934/cpaa.2019003.  Google Scholar

[7]

C.-C. Chen, L.-C. Hung and H.-F. Liu, N-barrier maximum principle for degenerate elliptic systems and its application, Discrete Contin. Dyn. Syst. A, 38 (2018), 791-821.  doi: 10.3934/dcds.2018034.  Google Scholar

[8]

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), 179-206.  doi: 10.32917/hmj/1372180511.  Google Scholar

[9]

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

[10]

P. de Mottoni, Qualitative analysis for some quasilinear parabolic systems, Institute of Math., Polish Academy Sci., 11 (1979), p190. Google Scholar

[11]

J.-S. Guo and C.-C. Wu, The existence of traveling wave solutions for a bistable three-component lattice dynamical system, J. Differential Equations, 260 (2016), 1445-1455.  doi: 10.1016/j.jde.2015.09.036.  Google Scholar

[12]

J.-S. Guo and C.-H. Wu, Wave propagation for a two-component lattice dynamical system arising in strong competition models, Journal of Differential Equations, 250 (2011), 3504-3533.  doi: 10.1016/j.jde.2010.12.004.  Google Scholar

[13]

J.-S. Guo and C.-H. Wu, Traveling wave front for a two-component lattice dynamical system arising in competition models, Journal of Differential Equations, 252 (2012), 4357-4391.  doi: 10.1016/j.jde.2012.01.009.  Google Scholar

[14]

S.-B. Hsu and T.-H. Hsu, Competitive exclusion of microbial species for a single nutrient with internal storage, SIAM J. Appl. Math., 68 (2008), 1600-1617.  doi: 10.1137/070700784.  Google Scholar

[15]

S. B. Hsu, H. L. Smith and P. Waltman, Competitive exclusion and coexistence for competitive systems on ordered Banach spaces, Trans. Amer. Math. Soc., 348 (1996), 4083-4094.  doi: 10.1090/S0002-9947-96-01724-2.  Google Scholar

[16]

L.-C. Hung, Exact traveling wave solutions for diffusive Lotka-Volterra systems of two competing species, Jpn. J. Ind. Appl. Math., 29 (2012), 237-251.  doi: 10.1007/s13160-012-0056-2.  Google Scholar

[17]

S. R.-J. Jang, Competitive exclusion and coexistence in a Leslie-Gower competition model with Allee effects, Appl. Anal., 92 (2013), 1527-1540.  doi: 10.1080/00036811.2012.692365.  Google Scholar

[18]

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.  Google Scholar

[19]

J. Kastendiek, Competitor-mediated coexistence: Interactions among three species of benthic macroalgae, Journal of Experimental Marine Biology and Ecology, 62 (1982), 201-210.   Google Scholar

[20]

R. McGehee and R. A. Armstrong, Some mathematical problems concerning the ecological principle of competitive exclusion, J. Differential Equations, 23 (1977), 30-52.  doi: 10.1016/0022-0396(77)90135-8.  Google Scholar

[21]

M. Mimura and M. Tohma, Dynamic coexistence in a three-species competition–diffusion system, Ecological Complexity, 21 (2015), 215-232.   Google Scholar

[22]

H. L. Smith and P. Waltman, Competition for a single limiting resource in continuous culture: the variable-yield model, SIAM J. Appl. Math., 54 (1994), 1113-1131.  doi: 10.1137/S0036139993245344.  Google Scholar

Figure 1.  $ \phi(x) $ in (12) and $ \psi(x) = \frac{\phi(x)}{h^2} $ in Definition 2.1 $ (iv) $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Integral domain of $ \int_{z_2}^{z_1} \left((v\ast\psi)''(x)\,v'(x)-(v\ast\psi)'(x)\,v''(x)\right)\,dx $ in (44)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Domain of the integral $ \int_{s_1}^{s_2}\int_{\bar{s}_1}^{\bar{s}_2}I_2(z_2)\,dz_2\,dz_1 $ in (48)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Graph of $ \Psi (x) $ given by (49)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  The distance $ L $ between the two lines $ \frac{S_u}{\hat{u}}+\frac{S_v}{\hat{v}} = 1 $ and $ \frac{S_u}{\underline{u}}+\frac{S_v}{\underline{v}} = 1 $ given by (87)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  N-barrier for the case $ a_1 $, $ a_2>1 $ in the $ S_uS_v $-plane. Dashed black curves: the solution $ (u(x),v(x)) $ of (BVP*); black lines: $ 1-u-a_1\,v = 0 $ and $ 1-a_2\,u-v = 0 $; green curve: $ F(u,v) = 0 $; magenta line (above): $ \frac{u}{\underline{u}}+\frac{v}{\underline{v}} = 1 $, where $ \underline{u} $ and $ \underline{v} $ are given by (83); magenta line (below): $ \frac{u}{\hat{u}}+\frac{v}{\hat{v}} = 1 $, where $ \hat{u} $ and $ \hat{v} $ are given by (84) and (85); blue line (above): $ \alpha\,S_u+\beta\,d\,S_v = \lambda_2 $, where $ \lambda_2 = \min\left\{\alpha\,\hat{u},\beta\,d\,\hat{v}\right\} $; red lines: $ \alpha\,S_u+\beta\,S_v = \eta_2 $ (above), where $ \eta_2 = \lambda_2\,\min\left\{1,1/d\right\} $, and $ \alpha\,S_u+\beta\,S_v = \eta_1 $ (below), where $ \eta_1 $ satisfies (99); blue line (below): $ \alpha\,S_u+\beta\,d\,S_v = \lambda_1 $, where $ \lambda_1 = \eta_1\,\min\left\{1,d\right\} $
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Cheng-Hsiung Hsu, Ting-Hui Yang. Traveling plane wave solutions of delayed lattice differential systems in competitive Lotka-Volterra type. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 111-128. doi: 10.3934/dcdsb.2010.14.111
[2]

Cheng-Hsiung Hsu, Jian-Jhong Lin. Stability analysis of traveling wave solutions for lattice reaction-diffusion equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1757-1774. doi: 10.3934/dcdsb.2020001
[3]

Chiun-Chuan Chen, Li-Chang Hung. An N-barrier maximum principle for elliptic systems arising from the study of traveling waves in reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1503-1521. doi: 10.3934/dcdsb.2018054
[4]

Anthony W. Leung, Xiaojie Hou, Wei Feng. Traveling wave solutions for Lotka-Volterra system re-visited. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 171-196. doi: 10.3934/dcdsb.2011.15.171
[5]

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 & Applied Analysis, 2016, 15 (4) : 1451-1469. doi: 10.3934/cpaa.2016.15.1451
[6]

Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189
[7]

Zhi-Cheng Wang, Hui-Ling Niu, Shigui Ruan. On the existence of axisymmetric traveling fronts in Lotka-Volterra competition-diffusion systems in ℝ3. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 1111-1144. doi: 10.3934/dcdsb.2017055
[8]

Jian Fang, Jianhong Wu. Monotone traveling waves for delayed Lotka-Volterra competition systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (9) : 3043-3058. doi: 10.3934/dcds.2012.32.3043
[9]

Bang-Sheng Han, Zhi-Cheng Wang, Zengji Du. Traveling waves for nonlocal Lotka-Volterra competition systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1959-1983. doi: 10.3934/dcdsb.2020011
[10]

Jong-Shenq Guo, Ying-Chih Lin. The sign of the wave speed for the Lotka-Volterra competition-diffusion system. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2083-2090. doi: 10.3934/cpaa.2013.12.2083
[11]

Jiang Liu, Xiaohui Shang, Zengji Du. Traveling wave solutions of a reaction-diffusion predator-prey model. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1063-1078. doi: 10.3934/dcdss.2017057
[12]

Bang-Sheng Han, Zhi-Cheng Wang. Traveling wave solutions in a nonlocal reaction-diffusion population model. Communications on Pure & Applied Analysis, 2016, 15 (3) : 1057-1076. doi: 10.3934/cpaa.2016.15.1057
[13]

Xiaojie Hou, Yi Li. Local stability of traveling-wave solutions of nonlinear reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 681-701. doi: 10.3934/dcds.2006.15.681
[14]

Tongren Ding, Hai Huang, Fabio Zanolin. A priori bounds and periodic solutions for a class of planar systems with applications to Lotka-Volterra equations. Discrete & Continuous Dynamical Systems - A, 1995, 1 (1) : 103-117. doi: 10.3934/dcds.1995.1.103
[15]

Shaohua Chen, Runzhang Xu, Hongtao Yang. Global and blowup solutions for general Lotka-Volterra systems. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1757-1768. doi: 10.3934/cpaa.2016012
[16]

Yuzo Hosono. Traveling waves for the Lotka-Volterra predator-prey system without diffusion of the predator. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 161-171. doi: 10.3934/dcdsb.2015.20.161
[17]

Yang Wang, Xiong Li. Uniqueness of traveling front solutions for the Lotka-Volterra system in the weak competition case. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3067-3075. doi: 10.3934/dcdsb.2018300
[18]

Zhi-Xian Yu, Rong Yuan. Traveling wave fronts in reaction-diffusion systems with spatio-temporal delay and applications. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 709-728. doi: 10.3934/dcdsb.2010.13.709
[19]

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 & Continuous Dynamical Systems - B, 2014, 19 (10) : 3267-3281. doi: 10.3934/dcdsb.2014.19.3267
[20]

Kota Ikeda, Masayasu Mimura. Traveling wave solutions of a 3-component reaction-diffusion model in smoldering combustion. Communications on Pure & Applied Analysis, 2012, 11 (1) : 275-305. doi: 10.3934/cpaa.2012.11.275

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (121)
  • HTML views (107)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences