Perturbing the travelling pulse in a three-species competition-diffusion system

We consider the situation where an exotic species w invades an ecosystem inhabited by two native species u and v. All species are competing for the same limited resource. Supposing that u and v are not able to coexist in the absence of the invader, we want to determine whether a successful invasion by w may allow all species to coexist (competitor-mediated coexistence). Mathematically, this problem can be modelled by the following three-species competition-diffusion system
\( \left\{ \begin{alignedat}{6} u_t &= d_1 \, \Delta u &&+ (r_1 &&- u &&- b_{12} \, v &&- b_{13} \, w &&)\,u, \\ v_t &= d_2 \, \Delta v &&+ (r_2 &&- v &&- b_{21} \, u &&- b_{23} \, w &&)\,v, \\ w_t &= d_3 \, \Delta w &&+ (r_3 &&- w &&- b_{31} \, u &&- b_{32} \, v &&)\,w, \end{alignedat} \right.\)
where all parameters are positive constants.

We are interested in the case in which the invading species is weaker than the native ones, i.e., it is not able to survive in the diffusion-free system obtained by setting d₁ = d₂ = d₃ = 0.
We fix all parameters as
\( \begin{aligned} & d_1 = d_2 = d_3 = 1, \\ & r_1 = r_2 = 28, \\ & \begin{aligned} b_{12} &= 22/21, & b_{13} &= 4, \\ b_{21} &= 1.87, & b_{23} &= 3/4, \\ b_{31} &= 26/21, & b_{32} &= 22/21, \\ \end{aligned} \end{aligned}\)
and leave r₃, which measures the strength of the exotic species, as a free parameter. Depending on the value of r₃, the invasion can be either successful or not and competitor-mediated coexistence may or may not occur.

It turns out that if r₃ lies in a certain range of values, the three-species competition-diffusion system admits several types of travelling wave solutions. In particular, there exists a travelling pulse which is stable for relatively high values of r₃ and then becomes unstable when r₃ decreases. In the movie here presented, we show the outcome of perturbing the unstable travelling pulse for r₃ = 26.75. In this case, the pulse splits in two three-species waves moving in opposite directions.