# "Evolution of a Fluctuating Population in a Randomly Switching Environment''

Karl Wienand (LMU Munich, Karl.Wienand@physik.uni-muenchen.de), Erwin Frey (LMU Munich, frey@lmu.de), and Mauro Mobilia (University of Leeds, M.Mobilia@leeds.ac.uk)

Paper published in Physical Review Letters on 13/10/2017: [Phys. Rev. Lett. Vol.119, 158301:1-6 (2017)]
Supplementary Material to the paper: [Switching-K SM]
ArXiv e-print: [Switching-K on ArXiv]

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.

In all Videos, the parameters are set to $(s,K_+, K_-, x_0)=(0.02,450,50,0.5)$.

### The pure competition scenario ($b=0$)

Video 1: Sample paths of $N(t)$ (left) and $x(t)$ (right) for $b=0$ and $\nu=20$. Note that the paths for $N(t)$ reach their quasi-stationary behavior at a much faster timescale than those of $x(t)$, which keep drifting until they reach fixation.

Video 2: Sample paths of $N(t)$ (left) and $x(t)$ (right) for $b=0$ and $\nu=0.01$. Besides the timescale separation highlighted above, this shows that the population size continues jumping between $K_+$ and $K_-$ even after fixation. The time courses of $N$ and $x$ are, in fact, independent in the pure competition scenario.

Video 3: Sample paths of $N(t)$ (left) and $x(t)$ (right) for $b=0$ and $\nu=0.0001$. This video shows the dynamics when $\nu\ll 1$: populations remain stuck in a constant-$K$ environment well beyond fixation.

Video 4: Histograms of the population size $N$ (left) and of the fraction of slow strains $x$ (right) for $b=0$ and $\nu=0.2$. The histogram for $N$ rapidly reaches its quasi-stationary form, while fixations slowly build at the margins of the $x$ histogram. Since population sizes jump endlessly between $K_+$ and $K_-$ (see Video 2), the distribution takes a bimodal shape.

Video 5: Histograms of the population size $N$ (left) and of the fraction of slow strains $x$ (right) for $b=0$ and $\nu=20$. At this high value for $\nu$, population sizes fluctuate around an equilibrium value (the harmonic mean of $K_+$ and $K_-$), resulting in a unimodal distribution.

### The public good scenario ($b>0$)

Video 6: Sample paths of $N(t)$ (left) and $x(t)$ (right) for $b=2$ and $\nu=20$. In the $b>0$ case, there is no timescale separation between $N$ and $x$, although $N$ remains the fast variable, slaved to the slow dynamics of $x$. In this case, the population eventually splits into a group with $x=1$, with $N$ fluctuating around relatively high values, and one with $x=0$, whose $N$ fluctuates at lower values.

Video 7: Sample paths of $N(t)$ (left) and $x(t)$ (right) for $b=2$ and $\nu=1.2$. Here it is also possible to see a further consequence of the coupled dynamics. High $x$ speeds up the $N$ dynamics, resulting in abrupt jumping size (bimodal regime), whereas low $x$ results in $N$ fluctuating about a single point (unimodal regime).

Video 8: Histograms of the population size $N$ (left) and of the fraction of slow strains $x$ (right) for $b=2$ and $\nu=20$. Unlike the pure competition case, the histogram of $N$ cannot settle to a steady state until fixations occur, because of the coupled dynamics. In this case, the distributions corresponding to $x=0$ and $x=1$ are both unimodal, but peak at different values, resulting in an overall bimodal distribution of $N$.

Video 9: Histograms of the population size $N$ (left) and of the fraction of slow strains $x$ (right) for $b=2$ and $\nu=1.2$. In this case, the coupled dynamics produce different quasi-stationary distributions for $N$ (reflecting the behavior observed in Video 7). Specifically, populations fixating to $x=0$ have a unimodal equilibrium, whereas those with $x=1$ are bimodal, so the overall quasi-stationary distribution of $N$ has three peaks.

Video 10: Histograms of the population size $N$ (left) and of the fraction of slow strains $x$ (right) for b=2 and $\nu=0.2$. In this case, $N$ has a bimodal quasi-stationary distribution, regardless of $x$. However, the peak positions depend on $x$, yielding a multimodal distribution of $N$.