Supplementary material:
When does cyclic dominance lead to stable spiral waves?

Bartosz Szczesny, Mauro Mobilia and Alastair M. Rucklidge

Department of Applied Mathematics, School of Mathematics
University of Leeds, Leeds LS2 9JT, U.K.

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 L2 patches (L being the linear size) labeled by a vector l = (l1, l2). Each patch has a limited carrying capacity, accommodating at most N individuals, and consists of a well-mixed population of NS1 individuals of species S1, NS2 of type S2, NS3 of type S3 and NØ = N - NS1 - NS2 - NS3 empty spaces denoted Ø.

(click image to enlarge)

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

Birth at rate β:

S1 + Ø → 2S1
S2 + Ø → 2S2
S3 + Ø → 2S3

Selection (dominance-removal) at rate σ:

S1 + S2S1 + Ø
S2 + S3S2 + Ø
S3 + S1S3 + Ø

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

S1 + S2 → 2S1
S2 + S3 → 2S2
S3 + S1 → 2S3

Mutation at rate μ:


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 δE) from hopping (rate δD) between nearest-neighbor patches l and l', according to:

Hopping at rate δD:

[S1] [Ø][Ø] [S1]
[S2] [Ø][Ø] [S2]
[S3] [Ø][Ø] [S3]

Pair-exchange at rate δE:

[S1] [S2][S2] [S1]
[S1] [S3][S3] [S1]
[S2] [S3][S3] [S2]
[S2] [S1][S1] [S2]
[S3] [S1][S1] [S3]
[S3] [S2][S2] [S3]

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, L2 = 1282 and N = 64 while the parameters are β = σ = δD = δE = 1, μ = 0.02 < μH = 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) = (NS1, NS2, NS3)/N. The right-most panel shows an oscillatory homogeneous state in which each of the species dominates the habitat in turn (no species goes extinct). Lower: Typical simulation of the PDE 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 small metapopulation size. 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, L2 = 2562, N = 2, and δD = δE = 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 (δD, δE) = (0.05, 0.05), (0.20, 0.05) from left to right respectively. When δD ≠ δE, nonlinear mobility typically causes far-field break-up of the spiral waves (see Fig. 4 of [SMR]). Other parameters are L2 = 1282, β = σ = 1, ζ = 0.1, μ = 10-6.

Mov. 4: With the mutation μ = 0.05 > μH = 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 , L2 = 1282, β = σ = 1, ζ = 0, δD = δE = 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, δD = δE = 1. The metapopulation lattice size was L2 = 1282 while the PDE was solved on a grid with 1282 points. Please see [SMR] for details.