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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.

Matthew Cassell
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Iesha Patterson
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  • Just let $U' = A, V' = B$ to get \begin{align} U' &= A \ D A' &= c A - U (1 - U - V) \ V' &= B \ B' &= cB - a V (U - b) \end{align} and then do a qualitative analysis as stated below, considering the cases $D = 0, D \ne 0$ separately. The books 'Mathematical Biology', volumes I and II, by Murray give a good treatment of this. – Matthew Cassell Apr 24 '23 at 11:24
  • @MatthewCassell Thank you for your reply. I have conducted the analysis of the eigenvalues to show existence of TWS. However, I was looking at the books along with the external references and it appears no one explicitly derived TWS as a function of U(z). I have not seen how to deal with TWS of a system of 2 or even 3 odes do you have any references to analytically do so? – Iesha Patterson Apr 25 '23 at 20:15

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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.

whpowell96
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  • Thats what I was thinking but I also had a small voice telling me that there might be analytical solutions. I will take a look at the pasted literature. Do you have any idea of an explanation as to why analytical solutions do not exist? – Iesha Patterson Apr 24 '23 at 05:35
  • They may exist but it may not be worth the effort to find them – whpowell96 Apr 24 '23 at 12:04