# Waves in a Hyperbolic Predator-Prey System
## 4 The Travelling Shock Waves for the Predator-Independent Diffusivity  
Let \(d/dt\) stand for the total derivative along a characteristic. From system (18) it follows that:  
\[\frac{dq}{dt}+\lambda\frac{dp}{dt}=\kappa\chi(s)ps_{x}-\
u q+cF(p,s), \tag{28}\]  
where the notation of \(\lambda=\pm\sqrt{\mu(s)}\) stands for the corresponding characteristic speed. If this characteristic supports a shock, then the same equations hold on the shores of the shock with the limit values of every quantity involved. Subtracting them and eliminating the variable \(q\) with the use of the Rankine-Hugoniot conditions (19) lead to the following equation on the shock  
\[\frac{d(\lambda_{*}[p])}{dt}+\lambda_{*}\frac{d[p]}{dt}=\kappa\chi_{*}[ps_{x} ]-\
u\lambda_{*}[p]+c[F],\quad q=\lambda_{*}[p], \tag{29}\]  
where subscript \({}_{*}\) indicates the quantities, which depend on the values of \(s_{*}\) only. Furthermore, from the continuity of the prey's density, \(s\), it follows that the gap of \(\
abla s\) across the shock is normal to it everywhere--that is,  
\[[s_{t}]+\lambda_{*}[s_{x}]=0. \tag{30}\]  
So, we are discussing the travelling shock waves. By definition, such a wave propagates at a constant speed \(c\), and the corresponding solution depends on only one variable,  
\[\xi=t-x/c. \tag{31}\]  
Then the characteristics at which the shocks occur are the parallel lines determined by equations \(\xi=\xi_{*}\) for the values of \(\xi_{*}\) varying over some set, \(\Sigma\), which we assume to be finite. This set gathers all the discontinuities of the solution in variable \(\xi\). On the complement of the singular set, the system (18) reduces to an ODE system. Namely,  
\[(c^{2}-\mu(s))p_{\xi}=c^{2}(F(p,s)-\
u(p-r))-\kappa\chi(s)pG(p,s),\;r_{\xi}=F( p,s),\;s_{\xi}=G(p,s), \tag{32}\]  
where \(r=p-q/c\), \(\xi\
ot\in\Sigma\). The conditions for matching the solutions are defined on the adjacent intervals separated by a discontinuity following from Equations (29) and (30). The latter holds trivially for every function in only variable \(\xi\), while the former turns to a system of functional equations as follows:  
\[c^{2}([F]-\
u[p])-\kappa\chi_{*}[pG]=0,\quad[s]=0,\;[r]=0. \tag{33}\]  
The last equation in this set is equivalent to \([q]=c[p]\). The following constraint is for the wave speed, \(c\), to obey:  
\[\exists\,c:\;\forall\;\xi_{*}\in\Sigma\quad\lambda_{*}=\pm\sqrt{\mu(s_{*})}=c,\;s_{*}=s(\xi_{*}). \tag{34}\]Alluding to the values of the unknown, \(s\), taken on \(\Sigma\) is correct by the continuity that is consistent with the matching condition (33). The same is true regarding another unknown, \(r\). The first equation in system (32) degenerates when the independent variable, \(\xi\), is approaching a discontinuity. Let us restrict ourselves within the solutions obeying the following condition:  
\[p_{\xi}^{\pm}=o\Big{(}(\xi-\xi_{*})^{-1}\Big{)},\quad\xi\to\xi_{*}, \tag{35}\]  
where the superscripts, \({}^{+}\) and \({}^{-}\), are to distinguish the solutions settled on the right and left intervals adjacent to the discontinuity point, \(\xi_{*}\). This estimate entails one more condition for matching the solutions at the discontinuity. Namely, the following equations are to obey:  
\[c^{2}\big{(}F(p_{*}^{\pm},s_{*})-\
u(p_{*}^{\pm}-r_{*})\big{)}-\kappa\chi_{*}p_ {*}^{\pm}G(p_{*}^{\pm},s_{*})=0, \tag{36}\]  
where the notations of \(p_{*}^{\pm}\), \(s_{*}\) and \(r_{*}\) stand for the unilateral limits of variable \(p\) and the bilateral limits of variables \(s\) and \(r\) correspondingly. It is worth noting that subtracting the equalities (36) gives the first of matching condition (33). At first, let us address the waves which allow a single shock only. Then there must be only one discontinuity on the \(\xi\)-axis, so let us place the origin there, and put \(\xi_{*}=0\).