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

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• 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.

Mathematics Subject Classification: Primary: 35B50; Secondary: 35C07, 35K57.

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

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

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 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 $\lambda_1 = \eta_1\,\min\left\{1,d\right\}$

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