Numerical continuation of spiral waves in heteroclinic networks of cyclic dominance

Abstract

Heteroclinic-induced spiral waves may arise in systems of partial differential equations that exhibit robust heteroclinic cycles between spatially uniform equilibria. Robust heteroclinic cycles arise naturally in systems with invariant subspaces, and their robustness is considered with respect to perturbations that preserve these invariances. We make use of particular symmetries in the system to formulate a relatively low-dimensional spatial two-point boundary-value problem in Fourier space that can be solved efficiently in conjunction with numerical continuation. The standard numerical set-up is formulated on an annulus with small inner radius, and Neumann boundary conditions are used on both inner and outer radial boundaries. We derive and implement alternative boundary conditions that allow for continuing the inner radius to zero and so compute spiral waves on a full disk. As our primary example, we investigate the formation of heteroclinic-induced spiral waves in a reaction–diffusion model that describes the spatiotemporal evolution of three competing populations in a 2D spatial domain—much like the Rock–Paper–Scissors game. We further illustrate the efficiency of our method with the computation of spiral waves in a larger network of cyclic dominance between five competing species, which describes the so-called Rock–Paper–Scissors–Lizard–Spock game.

Spatiotemporal stability of periodic travelling waves in a heteroclinic-cycle model

“…Dodson and Sandstede [6] used a continuation scheme to analyze the spectral properties of spiral waves and investigate the underlying mechanisms for instabilities. Performing continuation-based techniques to study the existence and stability of spiral wave solutions in system (2) is ongoing work [11].…”

“…System (11) needs to be solved together with integral conditions (9) and this computation is usually done in parallel with solving system (10) together with appropriate periodicity and phase conditions.…”

“…Vertical and horizontal axes correspond to the real and imaginary parts of the temporal eigenvalue λ, respectively. The essential spectrum is computed by continuation of solutions to the boundary value problem ( 9)- (11), starting from the origin λ = 0 (red dot) with φ = 0 and initial data obtained by substituting the known periodic TW solution W into the right-hand side of system (10); here, both λ and φ are allowed to vary. The black curve in figure 3 is the spectrum of the periodic TW and the blue dots are selected λ-values that correspond to various phases φ.…”

“…The formulation of the corresponding boundary value problem is described in [21], where the required second derivative is found via implicit differentiation of the eigenvalue problem with respect to ν = iφ. We adapt the set-up to our system (11), which is written in logarithmic coordinates. We use the notation V φ , V φφ and V zφ , V zφφ for the first and second derivatives with respect to iφ of V and V z at λ = φ = 0, respectively.…”