# American Institute of Mathematical Sciences

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

 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.

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 and Continuous Dynamical Systems, 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. [2] Cantrell, Ward and Jr., On competition-mediated coexistence, SIAM J. Appl. Math., 57 (1997), 1311-1327.  doi: 10.1137/S0036139995292367. [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. [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. [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. [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. [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. [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. [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. [10] P. de Mottoni, Qualitative analysis for some quasilinear parabolic systems, Institute of Math., Polish Academy Sci., 11 (1979), p190. [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. [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. [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. [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. [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. [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. [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. [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. [19] J. Kastendiek, Competitor-mediated coexistence: Interactions among three species of benthic macroalgae, Journal of Experimental Marine Biology and Ecology, 62 (1982), 201-210. [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. [21] M. Mimura and M. Tohma, Dynamic coexistence in a three-species competition–diffusion system, Ecological Complexity, 21 (2015), 215-232. [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.

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##### References:
 [1] R. A. Armstrong and R. McGehee, Competitive exclusion, Amer. Natur., 115 (1980), 151-170.  doi: 10.1086/283553. [2] Cantrell, Ward and Jr., On competition-mediated coexistence, SIAM J. Appl. Math., 57 (1997), 1311-1327.  doi: 10.1137/S0036139995292367. [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. [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. [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. [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. [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. [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. [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. [10] P. de Mottoni, Qualitative analysis for some quasilinear parabolic systems, Institute of Math., Polish Academy Sci., 11 (1979), p190. [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. [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. [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. [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. [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. [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. [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. [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. [19] J. Kastendiek, Competitor-mediated coexistence: Interactions among three species of benthic macroalgae, Journal of Experimental Marine Biology and Ecology, 62 (1982), 201-210. [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. [21] M. Mimura and M. Tohma, Dynamic coexistence in a three-species competition–diffusion system, Ecological Complexity, 21 (2015), 215-232. [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.
$\phi(x)$ in (12) and $\psi(x) = \frac{\phi(x)}{h^2}$ in Definition 2.1 $(iv)$
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)
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)
Graph of $\Psi (x)$ given by (49)
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)
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 $\lambda_1 = \eta_1\,\min\left\{1,d\right\}$
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