Entropy Production of Cyclic Population Dynamics

Entropy serves as a central observable in equilibrium thermodynamics. However, many biological and ecological systems operate far from thermal equilibrium. Here we show that entropy production can characterize the behavior of such nonequilibrium systems. To this end we calculate the entropy production for a population model that displays nonequilibrium behavior resulting from cyclic competition. At a critical point the dynamics exhibits a transition from large, limit-cycle-like oscillations to small, erratic oscillations. We show that the entropy production peaks very close to the critical point and tends to zero upon deviating from it. We further provide analytical methods for computing the entropy production which agree excellently with numerical simulations.

Figure 1

Probability distributions ($k=1$, $N=100$). (a) For a mutation rate $m=0.003$ smaller than $m_c=k/(2N)$ the probability distribution is concentrated near the edges and particularly near the corners of the phase space. (b) A mutation rate $m=0.1$ larger than $m_c$ leads to a Gaussian distribution around the center.

Figure 2

Entropy production in the steady state for different system sizes (□, $N=2$; $■$, $N=5$; ○, $N=10$; ●, $N=20$; $△$, $N=50$; ▲, $N=100$; $▽$, $N=200$; ▼, $N=400$). The entropy production vanishes for very high and very low mutation rates and exhibits a maximum at an intermediate value $m_{\max }$. The value $m_{\max }$ is near the critical mutation rate as shown in the inset where the black line indicates $m_c+0.001$, and red circles represent data obtained from simulations.

Figure 3

Entropy production in the limiting cases $m\ll m_c$ (a) and $m\gg m_c$ (b). Analytical results (black lines) agree excellently with simulations (○, $N=30$; $△$, $N=100$; □, $N=200$; $◇$, $N=400$). The data confirm that the entropy production is proportional to the squared system size $N^2$ for small mutation rates and proportional to $N$ for large mutation rates. The simulation results further confirm that the entropy production decays as $m$ for $m\to 0$ and as $1/m$ for $m\to \infty$.