# Waves in a Hyperbolic Predator-Prey System
## 3 Sedentary Prey and Hyperbolicity  
Throughout this section, we consider the system (7)-(9) with \(\delta=0\). Then it becomes a first order quasi-linear PDE system, which we put into the form:  
\[z_{t}+A(z)z_{x}=b(z),\quad\text{where} \tag{13}\] \[z=\left( \begin{array}{c}p\\ q\\ s\end{array} \right),\;\;b:\left( \begin{array}{c}p\\ q\\ s\end{array} \right)\mapsto\left( \begin{array}{c}F(p,s)\\ -vq\\ G(p,s)\end{array} \right),\;A:\left( \begin{array}{c}p\\ q\\ s\end{array} \right)\mapsto\left( \begin{array}{ccc}0&1&0\\ \mu(p,s)&0&-\kappa p\chi(p,s)\\ 0&0&0\end{array} \right). \tag{14}\]  
The eigenvalues of matrix \(A\) are \(\pm\sqrt{\mu(p,s)},0\). They are real and distinct one from another as long as \(\mu(p,s)>0\). It is always true by assumption. Hence the system of Equations (13) and (14) is strictly hyperbolic. The above triple of eigenvalues determines the triple of _characteristic speeds_--that is, every characteristic of system (13) allows a parametrization by the mapping \(t\mapsto(t,X(t))\) that satisfies an equation \(\dot{X}(t)=\lambda(p(X(t),t),s(X(t),t))\), where \(\lambda\in\{0,\pm\sqrt{\mu(p,s)}\}\). A question to ask about a first-order hyperbolic system is whether or not it allows diagonalizing by a pointwise transform \(\rho=\mathcal{R}(z)\). Such an ansatz generally does not exist for a system that includes more than two equations. The diagonalizing is feasible, provided that the system matrix, \(A=A(z)\), satisfies an integrability criterion, which allows a straight algebraic formulation. Another formulation requires satisfying the Frobenius condition \(d\omega_{i}\wedge\omega_{i}=0\) with every 1-form \(\omega_{i}=\ell_{i}\cdot dz\) generated by the vectors of the dual eigenbasis \(\{\ell_{1},\ell_{2},...\}\) of the matrix \(A\) (this is the eigenbasis of the transposed matrix). Then for every \(i\) there exists a factor \(\alpha_{i}=\alpha_{i}(z)\) such that \(\alpha_{i}\omega_{i}=d\rho_{i}\), and \(\rho_{i}=\mathcal{R}_{i}\). The last criterion is handy to use as long as there is a handy dual eigenbasis, as in the case of matrix (14), for example. Then a routine calculation reduces the diagonalization criterion to the following condition:  
\[\mu_{s}(p,s)=\kappa p\chi(p,s)5\bigg{(}\ln\frac{\mu(p,s)}{p^{2}\chi^{2}(p,s)} \bigg{)}_{p}. \tag{15}\]  
Under condition (12) the obtained criterion simplifies to:  
\[\mu(p,s)=pc(s)\chi(p,s), \tag{16}\]  
where \(c\) stands for an arbitrary function. Thus, assuming the diagonalization entails restrictions that are too artificial even in the simplified form. That is why we will not be considering this option in this study anymore. Let the condition (12) hold throughout all subsequent considerations. Then there exists a single-valued function \(\varphi=\varphi(p,s)\) such that:  
\[d\varphi=\mu(p,s)dp-\kappa p\chi(p,s)ds. \tag{17}\]  
Hence, the system (13) and (14) consists of conservation laws, namely:  
\[p_{t}+q_{x}=F(p,s),\quad q_{t}+\varphi_{x}=-vq,\quad s_{t}=G(p,s). \tag{18}\]  
This feature makes feasible the generalized solutions, e.g., the shock waves. Consider now a shock wave with discontinuities at some curve \(x=X(t)\). Then the velocity of the shock, \(\dot{X}\), has to satisfy the Rankine-Hugoniot conditions entailed by the conservations laws (18). They read as:  
\[\dot{X}[p]=[q],\quad\dot{X}[q]=[\varphi],\quad\dot{X}[s]=0. \tag{19}\]Here the notation of a dependent variable put in the square brackets stands for the jump of this variable across the discontinuity--that is, the difference between the limit values evaluated for \((x,t)\rightarrow(X(t)+0,t)\) and \((x,t)\rightarrow(X(t)-0,t)\). At that,  
\[[\varphi]=\int\limits_{(p^{-}s^{-})}^{(p^{+},s^{+})}\mu(p,s)dp-\kappa p\chi(p,s )ds, \tag{20}\]  
where the superscripts \(+\) or \(-\) stand for the unilateral limit values at the left and right shores of the discontinuity (when the observer looks forward alongside the time axis). There are two possibilities.