Interaction of planarly stable trivial and non-trivial travelling waves 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 two planarly stable travelling wave solutions. In the movies here presented, the result of the interaction of these two waves in two spatial dimensions is reported for several value of r₃. The species u, v and w are denoted by the red, green and blue colours respectively. The yellow line marks the interface between the species u and v. As the value of the free parameter decreases, we observe a transition from a regular spiral pattern, to a breathing spiral and finally to a complex spatio-temporal pattern born from the break-up of the spiral. This complex pattern may be either periodic or chaotic in the long run, as can be seen in the movies for longer time intervals T.