## 3 Results  
### Numerical Scheme and Simulations  
In this section, we use a finite volume scheme to discretize system (3). Following [19], assuming that \(\blacksquare\) is an open bounded polygonal domain for which we consider an admissible mesh consisting of a triplet \((\mathcal{T}\,,\,\mathcal{E}\,,\,\mathcal{P}\,)\), where \(T\) is a family of disjoint open polygonal convex subsets of \(\blacksquare\) called control volumes, \(\mathcal{E}\) is a family of subsets of \(\overline{\Omega}\) containing the edges of the control volumes, and \(\mathcal{P}\) is a family of points of \(\blacksquare\) satisfying the orthogonality condition; that is, the line segment joining the centers of two neighboring control volumes is orthogonal to the interface between these control volumes (see, e.g., [25]). We give some notations on the mesh:  
The set of neighbors of \(K\) is denoted by \(\mathcal{N}(K)\), which is \(\mathcal{N}(K)=\{L\in\,\mathcal{T}\,|\,\exists\sigma\in\,\mathcal{E}_{K},\, \,\overline{\sigma}=\overline{K}\cap\overline{L}\}\), where \(\mathcal{E}_{K}\) is the subset of \(\mathcal{E}\) containing the edges of \(K\). We denote by \(\eta_{K,L}\) the unit normal vector to \(\sigma_{K,L}=\partial\overline{K}\cap\partial\overline{L}\) outward to \(K\). Integrating the equations of system (3) over each control volume \(K\), and using the Green-Gauss formula, one can deduce the following discretization for system (3) for all \(K\in\mathcal{T}\):  
\[U_{K}^{0}=\frac{1}{|K|}\,\,\int_{K}u_{0}(x)dx,\,\,\,\,V_{K}^{0}=\frac{1}{|K|} \,\,\int_{K}v_{0}(x)dx, \tag{17}\]  
for all \(K\in\mathcal{T}\) and \(n\in\,[0,\ldots,N]\):  
\[ \begin{split}|K|\frac{U_{K}^{n+1}-U_{K}^{n}}{t}-\sum\limits_{L\in \mathcal{N}(K)}\frac{|\sigma_{K,L}|}{d_{K,L}}\Big{(}A\Big{(}U_{L}^{n+1}\Big{)} -A\Big{(}U_{K}^{n+1}\Big{)}\Big{)}+\sum\limits_{L\in\mathcal{N}(K)}G\Big{(}U_ {K}^{n+1};U_{L}^{n+1};dV_{K,L}^{n+1}\Big{)}\\ =|K|f\Big{(}U_{K}^{n+1}\Big{)}\\ |K|\frac{V_{K}^{n+1}-V_{K}^{n}}{\Delta t}-d_{2}\sum\limits_{L\in \mathcal{N}(K)}\frac{|\sigma_{K,L}|}{d_{K,L}}\Big{(}V_{L}^{n+1}-V_{K}^{n+1} \Big{)}\,\,=K\Big{(}\
u U_{K}^{n}-\delta V_{K}^{n+1}\Big{)}\end{split}  \tag{18}\]  
Figure 1: Plot of the real part of the eigenvalue \(\lambda(k^{2})\). varies as a function of \(k^{2}\) defined by Equation (10). When \(\chi>\chi_{c}\), there is a range of unstable wave numbers \(k_{1}^{2}<k^{2}<k_{2}^{2}\) for which the homogeneous steady state is linearly unstable. With the discrete unknowns \(U=\left(U_{K}^{n+1}\right)_{K\in\mathcal{T}}\) and \(V=\left(V_{K}^{n+1}\right)_{K\in\mathcal{T}},\ n\in 0,\ldots,N\). \(dV_{L,K}^{n+1}:=\frac{|\sigma_{K,\parallel}}{d_{K,\perp}}\left(U_{L}^{n+1}-U_ {K}^{n+1}\right)\). The function \(A\) stands to a differentiable function defined by \(A(u):=\int_{0}^{u}D(r)dr\), while the function \(G\) is a numerical convection flux function defined by  
\[G(a,b;c)=c^{+}\big{(}\Phi_{\downarrow}(a)+\Phi_{\downarrow}(b)\big{)}-c^{-} \big{(}\Phi_{\downarrow}(a)+\Phi_{\downarrow}(b)\big{)}, \tag{19}\]  
where \(c^{+}=\max(c,0)\), \(c^{-}=\max(-c,0)\), and \(\Phi=\chi\phi(u)\). \(\Phi_{\downarrow}\) represents the nonincreasing part of \(\Phi\), and \(\Phi_{\uparrow}\) represents the nondecreasing part of \(\Phi\). We refer to [19] for details about the convergence analysis of the discretized problem (17) and (18). In the next section, we show the numerical simulations of the model (3) in two-dimensional space to illustrate our experimental results and to demonstrate the patterns generated by the system. The [PERSON] algorithm is used to approach the solution of the nonlinear system defined by the first Equation of (18). This algorithm is coupled with a bigradient method to solve linear systems arising from the Newton algorithm process. The simulation of the model is applied in two-dimensional domain \(\blacksquare=(0,\ 10)\ \times\ (0,\ 10)\) for which a non-uniform admissible grid is considered.