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An N-barrier maximum principle for elliptic systems arising from the study of traveling waves in reaction-diffusion systems

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  • By employing the N-barrier method developed in C.-C. Chen and L.-C. Hung, 2016 ([6]), we establish a new N-barrier maximum principle for diffusive Lotka-Volterra systems of two competing species. To this end, this gives rise to the N-barrier maximum principle for a second-order elliptic equation involving two distinct unknown functions and a quadratic nonlinearity. An immediate consequence of the N-barrier maximum principle is an a priori estimate for the total populations of the two species. As an application of this maximum principle, we show under certain conditions the existence and nonexistence of traveling waves solutions for systems of three competing species. In addition, new $(1, 0, 0)$-$(u^{*}, v^{*}, 0)$ waves are given in terms of the tanh function, provided that the system's parameters satisfy certain conditions.

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

    Citation:

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  • Figure 1.  Red line: $\sigma_1-c_{11}\,u-c_{12}\,v = 0$; blue line: $\sigma_2-c_{21}\,u-c_{22}\,v = 0$; green curve: $\alpha\,u\,(\sigma_1-c_{11}\,u-c_{12}\,v)+\beta\,v\,(\sigma_2-c_{21}\,u-c_{22}\,v) = 0$. $\sigma_1 = \sigma_2 = c_{11} = c_{22} = 1,c_{12} = \frac{1}{2},c_{21} = \frac{2}{3}$. (a) $\alpha = \frac{1}{2}$, $\beta = 4$ (hyperbola). (b) $\alpha = 2$, $\beta = \frac{3}{20}$ (hyperbola). (c) $\alpha = 2$, $\beta = \frac{15}{2}+3 \sqrt{6}\approx14.8485$ (parabola). (d) $\alpha = 2$, $\beta = \frac{15}{2}-3 \sqrt{6}\approx0.1515$ (parabola). (e) $\alpha = 2$, $\beta = 3$ (ellipse). (f) zooming out of (a).

    Figure 2.  Red line: $\sigma_1-c_{11}\,u-c_{12}\,v = 0$; blue line: $\sigma_2-c_{21}\,u-c_{22}\,v = 0$; green curve: $\alpha\,u\,(\sigma_1-c_{11}\,u-c_{12}\,v)+\beta\,v\,(\sigma_2-c_{21}\,u-c_{22}\,v) = 0$; magenta line (above): $\alpha\,d_1\,u+\beta\,d_2\,v = \lambda_2$; magenta line (below): $\alpha\,d_1\,u+\beta\,d_2\,v = \lambda_1$; yellow line: $\alpha\,u+\beta\,v = \eta$; dashed curve: $(u(x),v(x))$. $d_1 = \sigma_1 = \sigma_2 = c_{11} = c_{22} = 1$. (a) $c_{12} = 2$, $c_{21} = 3$, $\alpha = 17$, $\beta = 18$, and $d_2 = 2$ give $\lambda_1 = \frac{17}{6}$, $\lambda_2 = \frac{17}{3}$, and $\eta = \frac{17}{6}$. (b) $c_{12} = 2$, $c_{21} = 3$, $\alpha = 17$, $\beta = 5$, and $d_2 = 2$ give $\lambda_1 = \frac{5}{2}$, $\lambda_2 = 5$, and $\eta = \frac{5}{2}$. (c) $c_{12} = 2$, $c_{21} = 3$, $\alpha = 17$, $\beta = 18$, and $d_2 = \frac{2}{3}$ give $\lambda_1 = \frac{34}{9}$, $\lambda_2 = \frac{17}{3}$, and $\eta = \frac{17}{3}$. (d) $c_{12} = 2$, $c_{21} = 3$, $\alpha = 17$, $\beta = 18$, and $d_2 = \frac{1}{2}$ give $\lambda_1 = \frac{9}{4}$, $\lambda_2 = \frac{9}{2}$, and $\eta = \frac{9}{2}$.

    Figure 3.  Red line: $\sigma_1-c_{11}\,u-c_{12}\,v = 0$; blue line: $\sigma_2-c_{21}\,u-c_{22}\,v = 0$; green curve: $\alpha\,u\,(\sigma_1-c_{11}\,u-c_{12}\,v)+\beta\,v\,(\sigma_2-c_{21}\,u-c_{22}\,v) = 0$; magenta line (below): $\alpha\,d_1\,u+\beta\,d_2\,v = \lambda_2$; magenta line (above): $\alpha\,d_1\,u+\beta\,d_2\,v = \lambda_1$; yellow line: $\alpha\,u+\beta\,v = \eta$; dashed curve: $(u(x),v(x))$. $d_1 = \sigma_1 = \sigma_2 = c_{11} = c_{22} = 1$. (a) $c_{12} = 2$, $c_{21} = 3$, $\alpha = 17$, $\beta = 18$, and $d_2 = 2$ give $\lambda_1 = 72$, $\lambda_2 = 36$, and $\eta = 36$. (b) $c_{12} = 2$, $c_{21} = 3$, $\alpha = 17$, $\beta = 5$, and $d_2 = 2$ give $\lambda_1 = 34$, $\lambda_{2} = 17$, and $\eta = 17$. (c) $c_{12} = 2$, $c_{21} = 3$, $\alpha = 17$, $\beta = 33$, and $d_2 = \frac{2}{3}$ give $\lambda_1 = 33$, $\lambda_2 = 22$, and $\eta = 33$. (d) $c_{12} = 2$, $c_{21} = 3$, $\alpha = 17$, $\beta = 18$, and $d_2 = \frac{1}{2}$ give $\lambda_1 = 34$, $\lambda_2 = 17$, and $\eta = 34$.

    Figure 4.  Profiles of the solution $(u(x),v(x),w(x))$.

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