# Waves in a Hyperbolic Predator-Prey System
## 5 The Inertial Limit for the Sedentary Prey  
At this point, we need more details regarding the reaction terms, \(F=F(p,s)\) and \(G=G(p,s)\). Henceforth, we will be assuming the following:  
\[F=pf(p,s),\,f|_{s=0}<0,\qquad G=sg(p,s),\,g(0,0)>0,\,g_{p}|_{p=0}=-1,\,g_{s}| _{s=0}=-1. \tag{39}\]  
The last two assumptions are equivalent to inequalities \(g_{p}|_{p=0}<0\), \(g_{s}|_{s=0}<0\) modulo the scaling of variables \(p,s\) since the characteristic scales of these variables, \(P\) and \(S\), are indefinite still. For example, the Lotka-Volterra kinetics while being normalized in accord with (39) reads as  
\[F=p(\gamma s-\beta),\,\,G=s(\alpha-s-p),\quad\alpha,\beta,\gamma=\text{const} >0. \tag{40}\]  
Thus, parameter \(\alpha\) plays the part of the carrying capacity given the scaling adopted here. Consider the Cauchy problem  
\[S_{\tau}=G(0,S),\quad S|_{\tau=0}=a. \tag{41}\]  
Assume there exists a number \(a_{0}>0\) such that for every \(a\in[0,a_{0}]\) problem (41) has a solution \(S=S(\tau,a)\) defined on \(\mathbb{R}\), and such that the mapping \(a\to S(\tau,a)\) sends the segment \([0,a_{0}]\) to itself for every \(\tau\in\mathbb{R}\). Consider also the following system of functional equations:  
\[g(p,s)=0,\quad f(p,s)=0, \tag{42}\]  
and assume there exists a positive solution \(s=s_{e}>0,p=p_{e}>0\). The last assumption entails the existence of a strictly positive equilibrium for general system (7)-(9). By equilibrium, we mean a particular solution specified as follows: \(q=0,p=\text{const},s=\text{const}\), so that both species distribute themselves homogeneously. A nonnegative equilibrium not being strictly positive can make sense too. For example, the Lotka-Volterra kinetics (40) allows an equilibrium with \(p=0,s=\alpha\) for every parameter setting, and there exists the strictly positive equilibrium \(s_{e}=\beta/\gamma,p_{e}=\alpha-s_{e}\) provided that \(\beta<\alpha\gamma\). Throughout the rest of this section, we will be considering the inertial limit and sedentary prey. So, we put \(v=\delta=0\). We also assume all the listed assertions on the kinetics to be true. The Lotka-Volterra kinetics defined in (40) meet such an assumption for \(\beta<\alpha\gamma\); at that, the ODE involved in the Cauchy problem (41) reads as \(S_{\tau}=S(\alpha-S)\). Given the assumption made, there are at least two solutions, \(x=P_{i}(y,z),\,i=1,2\) to Equation (37) defined in some vicinity of every point \((y,s_{e})\). At that, \(P_{1}\equiv 0\), while function \(P_{2}\) (in fact, in one variable, \(z\)) is that defined implicitly by equations \(\mu(z)f(x,z)=\kappa\chi(z)g(x,z)=0,\,P_{i}(s_{e})=p_{e}\). However, we do not need much manipulating with the values \(y,z\) here, since the choice is evident; namely, \(x_{1}=0,\,x_{2}=p_{e},\,z=s_{e}\), and the values of \(y\) are arbitrary. Then there are two pairs of the travelling shock waves, one pair propagates at speed \(c=\sqrt{\mu(s_{e})}\), and the other one propagates at the opposite speed. The formulae for both are the same, and they read as:  
\[p=0,\,r=y,\,s=S(\xi,s_{e}),\,\xi\in(-\infty,0), p=p_{e},\,r=y,\,s=s_{e},\,\xi\in(0,\infty), \tag{43}\] \[p=0,\,r=y,\,s=S(\xi,s_{e}),\,\xi\in(0,\infty), p=p_{e},\,r=y,\,s=s_{e},\,\xi\in(-\infty,0), \tag{44}\]  
where \(y\) is an arbitrary constant. Waves (43) and (44) are conjugated in the sense explained in the previous section but not mirroring one by the other one. Although both species in both waves take the same equilibrium values within the areas settled by predators, outside them, prey spreads itself differently. Indeed, the solutions to the Cauchy problem (41) generically behave differently for \(\pm\tau>0\) (except for the equilibria). In Figure 1, the left and middle frames show profiles of waves (43) and (44) correspondingly. For both waves, the front of the predator's invasion moves towards the smaller concentration of prey. This feature is not as paradoxical as it can seem given the locality of the predator-prey interactions.