Steady-state thermodynamics for population growth in fluctuating environments

We report that population dynamics in fluctuating environments is characterized by a mathematically equivalent structure to steady-state thermodynamics. By employing the structure, population growth in fluctuating environments is decomposed into housekeeping and excess parts. The housekeeping part represents the integral of the stationary growth rate for each condition during a history of the environmental change. The excess part accounts for the excess growth induced by environmental fluctuations. Focusing on the excess growth, we obtain a Clausius inequality, which gives the upper bound of the excess growth. The equality is shown to be achieved in quasistatic environmental changes. We also clarify that this bound can be evaluated by the “lineage fitness”, which is an experimentally observable quantity.

Figure 1

Types of individuals are represented by colors. In experiments, we can calculate the lineage fitness by the following steps: (i) we observe the fraction of individuals with type $x$ at initial time, that is the occupation probability $v_y\left(x\right)$; (ii) we culture the population in a fixed environmental condition $y$ for a sufficiently long time; (iii) we finally observe the fraction of the offspring of the individuals with type $x$ at initial time, which is denoted as $P_{R_y}^{\mathrm{s}\mathrm{t}}\left(x\right)$; (iv) we obtain the lineage fitness of type $x,u_y\left(x\right)$, by the ratio, $u_y\left(x\right)=P_{R_y}^{\mathrm{s}\mathrm{t}}\left(x\right)/v_y\left(x\right)$.

Figure 2

Three maximum excess growths.