Coherent Structures and Travelling Waves in Spatial Replicators from a Biased Volterra Lattice

TL;DR

This work extends the Volterra lattice by introducing a bias in the skew-symmetric interaction matrix to yield biased cycle dynamics in spatial replicator systems. It derives the biased Volterra lattice $ \\dot{u}_i = u_i(u_{i-1}-u_{i+1}) - a \\ u_i[C_n(\\mathbf{u}) - u_{i-1}] $ with $ C_n(\\mathbf{u}) = \\sum_{i=0}^{n-1} u_i u_{i+1} $, analyzes fixed points and their stability, and then couples the system to diffusion to obtain one-dimensional travelling-wave ODEs. The main contributions are: (i) explicit demonstration that travelling waves arise in biased $5$- and $6$-cycles for both signs of $a$ via Hopf bifurcations, (ii) absence of travelling waves for the biased $4$-cycle, replaced by stationary 'frozen waves' that form two spatial niches, and (iii) extension of frozen-wave phenomena to higher even cycles (e.g., $6$- and $8$-cycles) and random initial-condition studies showing nontrivial basins of attraction. These results link directed graph structure to novel spatial patterns in replicator dynamics, reveal a new class of stationary ecological niches beyond classical Turing patterns, and open avenues for general conjectures about travelling and frozen waves in biased Volterra lattices with potential ecological implications.

Abstract

The Volterra lattice is a well-known integrable family that is also a special class of replicator dynamics and whose members can be put in one-to-one correspondence with the directed cycle graphs. In this paper, we study a variation of the Volterra lattice by introducing a bias term in the replicator interaction matrix. The resulting system can still be put into one-to-one correspondence with the directed cycles, and the dynamics offer one generalisation of the classic rock-paper-scissors evolutionary game. We study the resulting spatial dynamics of this family, showing that travelling wave solutions are present in those dynamics corresponding to the directed 5- and 6-cycles, but not the 4-cycle. Instead, the 4-cycle exhibits a set of stationary solutions that we call `frozen waves' that are similar to but distinct from Turing patterns. This type of solution is also found in the dynamics generated from the directed 6- and 8-cycles. We discuss how these stationary solutions can represent naturally emergent ecological niches in these systems, and offer generalizing conjectures for the existence of both travelling wave solutions and frozen wave solutions in this family of dynamics as a potential program of future investigation.

Figures (19)

• Figure 1: The visual relationship between the graph structure and interaction matrix shows how a cycle maps to a matrix, which in turn produces the dynamics.
• Figure 2: (Left) Example dynamics of the biased Volterra lattice on four species with $a = \tfrac{1}{2}$. (Right) Example dynamics of the biased Volterra lattice on four species with $a = -\tfrac{1}{2}$.
• Figure 3: (Left) Example dynamics of the biased Volterra lattice on five species with $a = \tfrac{1}{2}$. (Right) Example dynamics of the biased Volterra lattice on five species with $a = -\tfrac{1}{2}$. Note, plots are clipped so that $0 \leq t \leq 200$ at which point the dynamics do not change in character.
• Figure 4: (Left) Example dynamics of the biased Volterra lattice on six species with $a = \tfrac{1}{2}$. (Centre) Example dynamics of the biased Volterra lattice on six species with $a = -\tfrac{1}{2}$, settling at a fixed point corresponding to a non-cycle. (Right) Example dynamics of the biased Volterra lattice on six species with $a = -\tfrac{1}{2}$, settling at a fixed point corresponding to a non-edge. Note, plots are clipped so that $0 \leq t \leq 200$ at which point the dynamics do not change in character.
• Figure 5: (Top) Emergence of a travelling wave with $a = -\tfrac{1}{2}$ and $D = \tfrac{1}{50}$. (Middle) Emergence of a travelling wave with $a = \tfrac{1}{2}$ and $D = \tfrac{1}{50}$. (Bottom) Emergence of a travelling wave with $a = \tfrac{1}{2}$ and $D = \tfrac{1}{30}$. Notice the amplitude of the travelling wave is a function of both $a$ and $D$, which affects that amplitude of the underlying limit cycle. From left to right, the times are given by $t \in \{0, 10, 50,150, 200\}$. Initial conditions are given in \ref{['eqn:TravellingWaveIC']} and periodic boundary conditions are used.

Theorems & Definitions (2)

• Conjecture 1
• Conjecture 2