Phase transitions in multiplicative competitive processes

We introduce a discrete multiplicative process as a generic model of competition. Players with different abilities successively join the game and compete for finite resources. Emergence of dominant players and evolutionary development occur as a phase transition. The competitive dynamics underlying this transition is understood from a formal analogy to statistical mechanics. The theory is applicable to bacterial competition, predicting novel population dynamics near criticality.

Figure 1

(a) Normalized cumulative gain $n_t\left({ϵ}_i\right)$ for $m=5$ and $T<T_c$ (left), $T\approx T_c$ (center), and $T>T_c$ (right). The solid line is the Bose distribution with $\alpha$ from Eq. (11). The plus sign indicates a pioneer. Plots of the last ten entrants were excluded. (b) Numerical calculation of $\mid \alpha \mid$ with $\alpha$ from Eq. (11), averaged over 500 trials. Symbols +, 엯, and × indicate $m=5$, 10, and 20. Change in the sign of $\alpha$ indicates the transition. Triangles on the abscissa are the transition temperatures from Eq. (17); $T_c\approx 3.33$, 6.67, and 13.3. Shown in inset is the dependence of $T_c$ on exponent $\sigma$ for $m=20$ (dots) and the analytical solution from Eq. (17), solid line. (c) Cumulative occupation by the most capable player was calculated from ${\phi }_t\left({ϵ}_{\min }\right)∕{\sum }_j{\phi }_t\left({ϵ}_j\right)$ at $t=200$. Symbols as in (b). Dashed line, occupation by pioneer for $m=20$. In all simulations, $y_0=1$, $l=1$, $\sigma =3$, and ${ϵ}_{\max }=1$.

Figure 2

A stochastic branching process with mutation and selection described in the text approaches a critical state by natural selection. Average total population is $\tilde{m}=100$. Simulation started with 100 unique strains whose fitness is drawn from a fitness distribution with $T=1$, $\sigma =3∕2$, and ${ϵ}_{\max }=1$. The $i\text{th}$ strain produces mutants (mutation rate $\eta =0.01$) whose fitness is drawn from a fitness distribution with $T_i$ given by Eq. (19) $(\gamma =2)$. (a) Average temperature of the system stays near the transition point $T_c=33$. In early stages, the average temperature increases, indicating initial adaptive evolution. (b) Number of existing strains at each steps (diversity) decreases during initial adaptive evolution. However, further dominance by a few species is suppressed as the system approaches near criticality.