Supplemental material to "Evolution of a Fluctuating Population in Randomly Switching Environment"

These are videos to supplement the arguments in <i>Evolution of a Fluctuating Population in Randomly Switching Environment</i> by K. Wienand, E. Frey, and M. Mobilia.<br><br><b>Abstract<br></b>Environmental conditions play a fundamental role in the competition for resources, and therefore on evolution. Here, we study a well-mixed finite population consisting of slow-growing and fast-growing individuals competing for limited resources in a stochastic environment under two scenarios (pure competition and public good). We consider that resources randomly switch between states of abundance and scarcity leading to growth and decay of the population under coupled external and internal noise. By analytical and computational means, we investigate the interplay between these sources of noise and their impact of the population’s fixation properties and its size distribution. We show that demographic and environmental noise can significantly enhance the slow type’s fixation probability, and find when the population size distribution is unimodal, bimodal or multimodal and undergoes noise-induced transition. We also unveil the subtle influence on fixation and population size distribution of random switching and by the coupling of the population composition to its size.<br><br><b>Viewing tips:</b> Downloading all files and opening "index.html" in a browser will display all videos, with their description. Videos can also be played singularly, and are described in the following.<br><div><b><br></b><div><b>The pure competition scenario</b><br>Videos 1-5 illustrate the population dynamics in the pure competition (b=0) scenario for the parameters (s, K₊,K_-,x₀)=(0.02,450,50,0.5).</div><div><br><b> Videos 1-3</b> present sample paths of population size N and composition x for nu=20 and 0.01, respectively. They illustrate the timescale separation between the two dynamics. Furthermore, <b>Video 2</b> shows that, even after fixation, the population size continues its behavior, jumping between K₊ and K_- . <b>Video 3</b> (in which nu=0.0001) shows that, when nu is extremely small, populations are indeed locked in one environmental state until fixation.</div><div><br><b> Video 4 and 5</b> present the development of the histograms of N and x for nu=0.2 and 20, respectively. In both cases, the histogram for N rapidly reaches its steady-state form, while fixations slowly build at the margins of the x histogram. More specifically, in the high-nu regime, the N-distribution becomes unimodal, centered around the harmonic mean of K₊ and K_-; in the low-nu regime, instead, the distribution becomes bimodal, with peaks at K₊ and K_-.</div><div><br><b>The public good scenario</b></div><div>Videos 6-10 illustrate the population dynamics in the public good (b=2) scenario for the parameters (s, K₊,K_-,x₀)=(0.02,450,50,0.5)</div><div><br><b> Videos 6 and 7</b> present sample paths of population size N and composition x for nu=20 and 1.2, respectively. In the b>0 case there is no timescale separation between N and x, but N remains the fast variable, slaved to the slow dynamics of x. Indeed, populations with higher x have higher N as well.</div><div>In particular, for nu=20 (<b>Video 6</b>), the populations eventually split into a group with x=1, with N fluctuating around relatively high values, and one with x=0, whose N fluctuates at lower values.</div><div>For nu=1.2 (<b>Video 7</b>) it is also possible to see a different behavior of N: abruptly jumping (bimodal regime) when x is high, fluctuating about a single point (unimodal regime) for low x.</div><div><br><b> Video 8-10</b> present the time development of the histograms of N and x for nu=20, 1.2 and 0.2, respectively. Unlike the pure competition case, the histogram of N cannot settle to a steady state until fixations occur. Moreover, the steady-state distributions have more peaks, because populations fixated to x=0 or x=1 reach different stationary states.</div><div>In the nu=20 case (<b>Video 8</b>), the steady state for x=0 and x=1 is unimodal, but peaking at different values, resulting in a bimodal distribution.</div><div>For nu=1.2 (<b>Video 9</b>), populations fixating to x=0 have a unimodal equilibrium, whereas those with x=1 are bimodal, so the stationary histogram has three peaks.<i><br></i></div></div><div>For nu=0.2 (<b>Video 10</b>), all populations reach a bimodal stationary state, but with different peaks, creating a 4-peaked distribution for N