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 $(s,K_+, K_-,
x_0)=(0.02,450,50,0.5)$.

**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.

**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 $N$.