Nonlinear lattices from the physics of ecosystems: The Lefever-Lejeune nonlinear lattice in $\mathbb{Z}^2$

Abstract

We argue that the spatial discretization of the strongly nonlinear Lefever-Lejeune partial differential equation defines a nonlinear lattice that is physically relevant in the context of the nonlinear physics of ecosystems, modelling the dynamics of vegetation densities in dry lands. We study the system in the lattice $\mathbb{Z}^2$, which is especially relevant because of its natural dimension for the emergence of pattern formation. Theoretical results identify parametric regimes for the system that distinguish between extinction and potential convergence to non-trivial states. Importantly, we analytically identify conditions for Turing instability, detecting thresholds on the discretization parameter for the manifestation of this mechanism. Numerical simulations reveal the sharpness of the analytical conditions for instability and illustrate the rich potential for pattern formation even in the strongly discrete regime, emphasizing the importance of the interplay between higher dimensionality and discreteness.

Figures (10)

• Figure 1: Implementation of the Dirichlet boundary conditions \ref{['DBC']} leads to anti-symmetric conditions for the virtual nearest neighbors of the set of the boundary points $\partial\Omega$. Details in the text.
• Figure 2: Dynamics of the initial condition \ref{['eq:tent']} for various values of $A$, varying the parameter $\gamma_3$. Rest of parameters: $\alpha=\beta=0.1$, $\gamma_1=0.125$, $\gamma_2=0.5$, $h=2$, $L=20$. Blue (solid) curve: $A=0.001$, $\gamma_3=8$. Dotted-Dashed (orange) curve: $A=0.1$, $\gamma_3=9$. Dotted (green) curve: $A=0.001$, $\gamma_3=9$. Dashed (red) curve: $A=0.1$, $\gamma_3=10$. $||U^0||_{\ell^2}$ is the value of the corresponding $\ell^2$-norm of the initial condition \ref{['eq:tent']}, for each case of $A$. Details in the text.
• Figure 3: Left panel: Graphs of the polynomial $\Lambda(x)$\ref{['polyn']}, varying $h$ for the $\gamma_1=0.125$, $\gamma_2=0.5$, $\gamma_3=0.5$. $\alpha=0.02$ and $\beta=0.1$. Right panel: The graph of the function $x(k_1,k_2)$\ref{['pfk']} in the fundamental period $\left(\frac{2\pi}{h},\frac{2\pi}{h}\right)$, for $h=1$.
• Figure 4: Top row: three-dimensional plots (light (blue) color) of the eigenvalue function $\lambda(k_1,k_2)$ for different values of the discretization parameter $h$. First panel for $h=0.5\in (0,h_{+})$. Second panel for $h=1.8\in (h_{+},h_1)$. Third panel for $h=3\in (h_1,h_c)$ and fourth panel for $h=3.3>h_c$. The critical values are $h_+=1.63$, $h_1=2.05$ and $h_c=3.16$. Rest of lattice parameters: $\gamma_{1}=0.125, \gamma_{2}= 0.5, \gamma_{3}= 0.005, \alpha=0.02, \beta=0.1$. The dark (red) colored plane is the plane $\lambda=0$. Bottom row: Each graph depicts the corresponding cross-sections of the above plots of $\lambda(k_1,k_2)$ for each $h$ along the line $k=k_1$. of the $(k_1,k_2)$-plane.
• Figure 5: Instability sets $J_{i,j}$ in the $(k_1,k_2)$-plane when $i=0,1$ and $j=0,1$ for the set of parameters $\alpha=0.02$, $\beta=0.1$, $\gamma_1=0.125$, $\gamma_2=0.5$, $\gamma_3=0.005$ and $h=3$.

Theorems & Definitions (8)

• Theorem \oldthetheorem
• Proposition 3.1
• Lemma 3.1
• Theorem \oldthetheorem
• Proposition 4.1
• Theorem \oldthetheorem
• Lemma 2.1
• Lemma 2.2