When does cyclic dominance lead to stable spiral waves?

In these notes, we provide some supplementary material to the manuscript [SMR]. The notes include a schematic description of the model and movies allowing the visualization of the results. [skip to movies]

The generic model of cyclic competition is defined as a square periodic lattice of L^{2} patches (L being the linear size) labeled by a vector **l** = (l_{1}, l_{2}). Each patch has a limited carrying capacity, accommodating at most N individuals, and consists of a well-mixed population of N_{S1} individuals of species S_{1}, N_{S2} of type S_{2}, N_{S3} of type S_{3} and N_{Ø} = N - N_{S1} - N_{S2} - N_{S3} empty spaces denoted Ø.

Within each patch, the population composition evolves according to the following reactions:

Birth at rate β:S

S

S

Selection (dominance-removal) at rate σ:

S

S

S

Zero-sum (dominance-replacement) at rate ζ:

S

S

S

Mutation at rate μ:

S

S

S

S

S

S

Here, μ is a small mutation rate which allows for the mathematical treatment around the onset of the Hopf bifurcation. Inspired by nonlinear biological movement, we *divorce* pair-exchanges (rate δ_{E}) from hopping (rate δ_{D}) between nearest-neighbor patches **l** and **l'**, according to:

Pair-exchange at rate δ

Here, "**[** **]** **[** **]**" symbolises neighbouring patches **l** and **l'** which lie in 4-neighborhood.

**Mov. 1:** Upper: Reactive steady states in stochastic simulations in the system's 4 phases (see Fig. 1 of [SMR]). Here, L^{2} = 128^{2} and N = 64 while the parameters are β = σ = δ_{D} = δ_{E} = 1, μ = 0.02 < μ_{H} = 0.042 (ε = 0.25) and, from left to right, ζ = (1.8, 1.2, 0.6, 0) respectively. Each pixel describes a single metapopulation with normalized RGB representation (red, green, blue) = (N_{S1}, N_{S2}, N_{S3})/N. The right-most panel shows an oscillatory homogeneous state in which each of the species dominates the population in turn (no species goes extinct). Lower: Typical simulation of the partial differential equation (PDE) (5) of [SMR] in each phase AI, EI, BS, SA from left to right, same parameters used (see Fig. 3 of [SMR]).

**Mov. 2:** Stochastic simulations of a system with low carrying capacity N. While the effects of the demographic noise are visible, the predictions of our theory are still valid provided that the increased mobility facilitates mixing of the individual in metapopulations. Here, L^{2} = 256^{2}, N = 2, and δ_{D} = δ_{E} = 4 while the other parameters are as in Mov. 1. Please see [SMR] for the related discussion.

**Mov. 3:** Influence of nonlinear mobility on spiraling patterns in lattice simulations for (δ_{D}, δ_{E}) = (0.05, 0.05), (0.20, 0.05) from left to right respectively. When δ_{D} ≠ δ_{E}, nonlinear mobility typically causes far-field break-up of the spiral waves (see Fig. 4 of [SMR]). Other parameters are L^{2} = 128^{2}, N = 1024, β = σ = 1, ζ = 0.1, μ = 10^{-6}.

**Mov. 4:** With the mutation μ = 0.05 > μ_{H} = 0.042, the reactive fixed point remains stable and no coherent patterns are observed. The fluctuations due to demographic noise diminish as the carrying capacity of the metapopulations in increased. In this setup, N = (64, 256, 1024) left to right with , L^{2} = 128^{2}, β = σ = 1, ζ = 0, δ_{D} = δ_{E} = 1. Related discussion can be found in [SMR].

**Mov. 5:** Comparison of the stochastic simulations with carrying capacity N = (4, 16, 64, 256, 1024) left to right and the PDE (right most panel). The parameters are β = σ = 1, ζ = 0.6, μ = 0.02, δ_{D} = δ_{E} = 1. The metapopulation lattice size was L^{2} = 128^{2} while the PDE was solved on a grid with 128^{2} points. Please see [SMR] for details.