I am looking for traveling wave solutions of
\begin{align} \frac{\partial U}{\partial t} &= AU\left(1-\frac{U}{K}\right)-BUV+D_{1}\nabla^{2}U \\ \frac{\partial V}{\partial t} &= CUV-DV+D_{2}\nabla^{2}V \end{align}
Where $A,B,C,D,K,D_{1}$,and $D_{2}$ are constants.
After nondimensionalising I arrived at the following system of equations
\begin{align} \frac{\partial u}{\partial t} &= u(1-u-v)+D\frac{\partial^{2}u}{\partial x^{2}} \\ \frac{\partial v}{\partial t} &= av(u-b)+\frac{\partial^{2}v}{\partial x^{2}} \end{align}
I then substituted $u(x,t)=U(z), v(x,t)=V(z), z=x+ct$ to get
\begin{align} cU' &= U(1-U-V)+DU'' \\ cV' &= aV(U-b)+V'' \end{align}
The considering the case where $D = 0$ and letting $V' = W$ I get the following system of first order ODEs
\begin{align} U' &= \frac{1}{c}U(1-U-V) \\ V' &= W \\ W' &= cW-aV(U-b) \end{align}
I am now stuck here I am unsure how to handle this system and get a traveling wave solution. I would appreciate any tips or suggestions on methods of solutions.
Finding explicit solutions of these equations is almost certainly impossible. However, there has been some work on analysis of traveling wave solutions to this equation (Lotka-Volterra with logistic prey growth). In general, once you start getting systems of nonlinear ODEs, it may be more productive to start doing qualitative analysis such as stability of critical points, existence of periodic/heteroclinic orbits, invariant sets, etc. as opposed to attempting to find explicit solutions.
Dunbar, Steven R., Traveling wave solutions of diffusive Lotka-Volterra equations: A heteroclinic connection in (R^ 4), Trans. Am. Math. Soc. 286, 557-594 (1984). ZBL0556.35078.