## 2 Materials and Methods
### Pattern Formation  
Pattern formation in mathematics is the process that, by changing a bifurcation parameter, the spatially homogeneous steady states lose stability due to spatially inhomogeneous perturbations, and stable inhomogeneous solutions arise. In this section, we are interested in the pattern formation for the system (3) and (4). We derive here the necessary conditions for diffusion-driven instability (also called Turing instability) of the steady-state and the initiation of spatial pattern for the above-mentioned system. We look for the spatially homogeneous steady states by setting the kinetics on the right-hand side of Equation (3) to be zero:  
\[f(u,v)\ =0,\qquad g(u,v)\ =0. \tag{5}\]  
We use the same technique as [PERSON] [12; 24]. We suppose that \((u_{s},\ v_{s})\) is a positive solution of Equation (5); where \((u_{s},\ v_{s})\) is a steady state of the system  
\[\partial_{t}u=f(u,\ v),\qquad\partial_{t}v=g(u,\ v). \tag{6}\]We set  
\[W=\ \left( \begin{array}{c}u-u_{s}\\ v-v_{s}\end{array} \right)\]  
and the linearized system leads to  
\[\partial_{t}W=AW;\qquad A=\left( \begin{array}{cc}f_{u}&f_{v}\\ g_{u}&g_{v}\end{array} \right)_{u_{s},v_{s}}.\]  
where \(A\) is the stability matrix (i.e., the Jacobian matrix of system (6) at the steady state \((u_{s},\ v_{s})\). From now on, we take the derivatives of \(f\) and \(g\) to be evaluated at the steady state \((u_{s},\ v_{s})\) unless stated otherwise. Therefore, the conditions for which the steady state \((u_{s},\ v_{s})\) is linearly stable; that is, Re \(\lambda<0\), where \(\lambda\) is the eigenvalue of the matrix \(A\), is guaranteed if  
\[\text{tr}(A)\ =f_{u}+g_{v}<0,\qquad|A|\ =f_{u}g_{v}-f_{v}g_{u}>0. \tag{7}\]  
Now, we consider the full chemotaxis model (3). We examine a small perturbation from the homogeneous steady state \((u_{s},\ v_{s})\) of the form  
\[u=us+\varepsilon\widetilde{u}(x,\ t),\qquad v=vs+\varepsilon\widetilde{v}(x, \ t), \tag{8}\]  
where \(\varepsilon\ll 1\) is a small parameter strictly positive. \(f\) is a smooth function, then using an asymptotic expansion we obtain  
\[f(u_{s}+\varepsilon\widetilde{u},\ v_{s}+\varepsilon\widetilde{v})\ =f(u_{s},v_{s})\ +\ \varepsilon \widetilde{u}f_{u}(u_{s},v_{s})\ +\ \varepsilon\widetilde{v}f_{v}(u_{s},v_{s})\ +\ o \Big{(}\varepsilon^{2}\Big{)}\]  
The system (3) transforms into the linearized system in (9) after performing the following steps:  
\[\left\{ \begin{array}{c}\partial_{t}u=d_{1}(q(u_{s})\ -\ q\prime(u_{s})u_{s}) \Delta u\\ -u_{s}q(u_{s})\chi(v_{s})\Delta v+uf_{u}+vf_{v},\\ \partial_{t}u=d_{2}\Delta v+ug_{u}+vg_{v}\end{array} \right. \tag{9}\]  
* Asymptotic expansion for the functions \(g,\ q\), and \(\chi\) around \((u_{s},\ v_{s})\);
* Taking into account Equation (5);
* Equating first-order terms with respect to \(\varepsilon\);
* Neglecting higher-order terms, and dropping the tilde for the convenience. For simplicity, we denote by \(\zeta_{s}=q(u_{s})-q^{\prime}(u_{s})u_{s}\), \(\psi_{s}=u_{s}q(u_{s})\ \chi(v_{s})\) and by \(\chi_{s}=\chi(v_{s})\) with the given functions in (14), one can deduce that \(\zeta_{s}>0\) and \(\psi_{s}<0\). In what follows, we perform the standard argument (e.g., [24], Chapter II). We obtain the dispersion relation associated with system (9)  
\[\lambda^{2}+a\Big{(}k^{2}\Big{)}\lambda+b\Big{(}k^{2}\Big{)}\ =0, \tag{10}\]  
where \(a=a\big{(}k^{2}\big{)}\) and \(b=b\big{(}k^{2}\big{)}\) are two functions given by  
\[a\Big{(}k^{2}\Big{)}\ =\ (s_{s}d_{1}+d_{2})k^{2}-(f_{u}+g_{v});\qquad\qquad b \Big{(}k^{2}\Big{)}\ =\zeta_{s}d_{1}d_{2}k^{4}+(\psi_{s}g_{u}-d_{2}f_{u}-\zeta_{s}d_{1}g_{v})k^{2} +f_{u}\ g_{v}-f_{v}\ g_{u}.\]  
For the steady state to be unstable to spatial perturbations, we require \(\text{Re}(\lambda)>0\) for some \(k\
eq 0\). We remark that the sum of the eigenvalues is strictly negative since the function \(a\big{(}k^{2}\big{)}\) is strictly positive due to the conditions (7).