These are videos to supplement the arguments in Evolution of a Fluctuating Population in Randomly Switching Environment by K. Wienand, E. Frey, and M. Mobilia.
Abstract 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.
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The pure competition scenario 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).
Videos 1-3 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, Video 2 shows that, even after fixation, the population size continues its behavior, jumping between K+ and K- . Video 3 (in which nu=0.0001) shows that, when nu is extremely small, populations are indeed locked in one environmental state until fixation.
Video 4 and 5 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-.
The public good scenario
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)
Videos 6 and 7 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.
In particular, for nu=20 (Video 6), 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.
For nu=1.2 (Video 7) 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.
Video 8-10 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.
In the nu=20 case (Video 8), the steady state for x=0 and x=1 is unimodal, but peaking at different values, resulting in a bimodal distribution.
For nu=1.2 (Video 9), populations fixating to x=0 have a unimodal equilibrium, whereas those with x=1 are bimodal, so the stationary histogram has three peaks.
For nu=0.2 (Video 10), all populations reach a bimodal stationary state, but with different peaks, creating a 4-peaked distribution for N