Macroscopic Dominance from Microscopic Extremes: Symmetry Breaking in Spatial Competition

Abstract

How do competing populations convert a spatial advantage into macroscopic dominance? We introduce a stochastic model for resource competition that decouples the transient discovery phase from monopolization. Initial symmetry breaking is governed by extreme value statistics of first-passage times: a linear spatial disadvantage requires an exponentially larger population to overcome. However, transient superiority cannot stabilize dominance. A non-reciprocal interaction bias is strictly necessary to arrest local fluctuations and drive the system into a robust absorbing state.

Figures (3)

• Figure 1: Competition outcomes under symmetry breaking. Pie charts show the final-state probability of the absorbing outcome: colony 1 dominance, colony 2 dominance, or coexistence (“both”). (A) Symmetric conditions ($N_1 = N_2$, $d_1 = d_2$, $\beta_1 = \beta_2$) produce a nonuniform distribution of outcomes; (B–D) Breaking symmetry through (B) population bias, (C) spatial bias, and (D) interaction bias shifts the outcome probabilities. Outcomes summarize 1000 trials. Rightmost column shows representative lattice equilibria in each case.
• Figure 2: The scaling parameter $\chi$ completely determines symmetry. (A) Phase diagram of $\mathbb{P}(T_2 < T_1)$ based on the full lattice simulations. The dashed white line denotes the asymptotic balance condition ($\chi = 1$), where an exponential population advantage perfectly cancels a linear spatial disadvantage. Each point in the heatmap emerged from averaging over 10000 trials. (B) Probability that colony 2 dominates the food source. (C) Pie chart demonstrating the reemergence of symmetry with $\chi = 1$ in the absorbing outcomes despite unequal parameters. Simulation parameters: lattice size $M = 50 \times 50$, $D = 10$, $\gamma = 2$, $N_1 8^{-d_1} = 0.43$.
• Figure 3: Asymmetric interaction bias controls the sharpness of the competitive phase transition. (A) The probability of Colony 2 establishing macroscopic dominance as a function of the relative distance ratio $d_2/d_1$, given a strict numerical disadvantage ($N_1 > N_2$) and a fixed competitor bias ($\beta_1 = 55$). (B) The probability of dominance as a function of the population ratio $N_2/N_1$, given a strict spatial disadvantage ($d_1 < d_2$) and constant $\beta_1 = 55$. As indicated by the arrows, increasing Colony 2's interaction bias ($\beta_2$) shifts the competitive outcome from a noise-dominated, probabilistic regime into a strictly polarized state. For large $\beta_2$, the symmetry-breaking boundary steepens into a sharp phase transition, enabling Colony 2 to overcome structural deficits to secure the resource and drive the system into the absorbing state of competitive exclusion.