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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:

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  • 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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