Abstract:
Environmental changes greatly influence the evolution of populations.
Here, we study the dynamics of a population of two strains, one growing slightly
faster than the other, competing for resources in a time-varying binary environment
modeled by a carrying capacity switching either randomly or periodically between
states of abundance and scarcity. The population dynamics is characterized by
demographic noise (birth and death events) coupled to a varying environment.
We elucidate the similarities and differences of the evolution subject to a
stochastically- and periodically-varying environment. Importantly, the population
size distribution is generally found to be broader under intermediate and fast
random switching than under periodic variations, which results in markedly different
asymptotic behaviors between the fixation probability of random and periodic switching.
We also determine the detailed conditions under which the fixation
probability of the slow strain is maximal.
This document contains videos to support the main text and supplementary material. The first section
shows how the dependence of the fixation probability of the slow-growing species, $\phi$,
on the switching rate, $\nu$, changes as the public good parameter $b$ is increased,
for random and periodic switching. The second section shows how the quasi-stationary population size
distribution (PSD) changes as the switching rate, $\nu$, is increased for random
and periodic switching.
Throughout this document, equations, figures and sections in the main text and supplementary material (S) are refered to by
their relevant label.
These videos show the different behaviours in the public good scenario as the parameter $b$ is raised, see
Section S7 and Fig. S4.
The first part of each video shows the public good counterpart of Figs. 3(e) and S2(a), showing the $(\gamma,\delta)$-
parameter space for the dependence of $\phi_r$ (random switching) on $\nu$ for different values of $b$. Increasing $b$
effectively rescales $s$, squashing the triangular region to the right. The dot corresponds to $(\gamma,\delta)$ (fixed),
which resultantly occupies different regions of the parameter space as $b$ increases. Similar behaviour is oberserved
for $\phi_p$ (periodic switching), but here the effective theory does not result in a rescaling of $s$ (see Section S7).
The second part of each video shows $\phi_r(\nu)$ and $\phi_p(\nu)$ as $b$ increases from $0$ to $1$, symbols show simulation
results (taken over $10^4 - 10^5$ runs), solid lines show theoretical predictions from (S42) and (S43).
For all videos in this section, $(K_0,x_0) = (250, 0.6)$ and other parameters are in the captions.
Video 1: Dependence of fixation probability on $\nu$ as $b$ increases: $\phi_r$ is non-monotonic
when $b=0$, becoming monotonically decreasing for large enough $b$. Similar behaviour is observed for periodic switching.
Other parameters are: $(\gamma,\delta,s)=(0.9,0.7,0.05)$.
Video 2: Dependence of fixation probability on $\nu$ as $b$ increases: $\phi_r$ is monotonically increasing for
$b=0$, becoming non-monotonic for large enough $b$ and monotonically decreasing if $b$ is further increased.
Similar behaviour is observed for periodic switching.
Other parameters are: $(\gamma,\delta,s)=(0.9,0.6,0.04)$.
These videos show how the PSDs for random and periodic switching change as the switching rate $\nu$ is increased
(see Sections S2 and S3). Results are shown for the six characteristic regions of the $(\gamma,\delta)$-parameter space.
Quasi-stationary population size distributions from simulations are shown in blue/red for random/periodic switching
$(P_{\nu}^{(r)}(N)/P_{\nu}^{(p)}(N))$ (calculated from $\sim 10^6 / 10^9$ samples for random/periodic noise, where results from an
initial short transient are discarded). The predictions from the piecewise deterministic Markov process $(P_{\nu}^{\rm PDMP }(N)$- (S22))
and piecewise periodic process $(P_{\nu}^{\rm PPP }(N)$- (S19)) are shown in light blue/purple. The vertical light blue lines show
$N=N_{\rm min/max}$ (i.e. the support of $P^{\rm PPP}$ - see Section S2.3 in SM). The vertical purple lines show $N=K_{\pm}$
in the slow switching regime, and $N=K_{+/-}$ for $\delta >/<0$ in the intermediate switching regime.
For large $\nu$, (S19) is replaced by the prediction from the Kapitsa method $(P_{\nu}^{\rm Kap}(N)$- (S15)). Where applicable,
$N^*$, the smaller solution
to (S23), is shown with a light blue dot.
For each $(\gamma,\delta)$ parameter set, there are various different scenarios divided into three general regimes: slow,
intermediate and fast
switching. The regime for $P^{\rm PDMP}_{\nu}(N)$
is indicated on the right hand side of the video, with more details in the captions and Section S3.1. Additional details
for $P^{\rm PPP}_{\nu}(N)$ and $P^{\rm Kap}_{\nu}(N)$ can be found in Sections S2.3 and S2.2 respectively.
In all cases, $P^{(r),(p),{\rm PDMP}, {\rm PPP}}(N)$ are peaked around $K_{\pm}$ (bimodal) when $\nu \ll 1$ and
$P^{(r),(p),{\rm PDMP}, {\rm Kap}}(N)$ are peaked around $\mathcal{K}=K_0\frac{(1-\gamma^2)}{1-\delta\gamma}$ (unimodal) for $\nu \gg 1$.
Another motif is that the variance of $P^{(p)}(N)$ is larger than $P^{(r)}(N)$ for $0.1 \lesssim \nu \lesssim 100$, explaining why we
observe faster transitions for periodic switching in the fixation probability (Figs. 3(a)-(e), S2(c)-(e), S4(b)-(e)),
mean fixation time (Fig. S3(a)) and total population size (Fig S3(b)).
For all videos in this section, $(K_0,s,x_0,b) = (250,0.05, 0.6,0)$ and other parameters are in the captions.
Video 3: PSD and PDMP/PPP for random/periodic show in blue/red and light blue/purple for $(\gamma,\delta)=(0.8,0)$.
Here because $\nu_{\pm}={\nu}$ the dependence of $P^{\rm PDMP}(N)$ splits into two regimes for $\nu$:
Slow switching ($\nu <1$) with peaks at $K_{\pm}$ and fast switching ($\nu >1$) with a single peak at $N^*$.
Video 4: PSD and PDMP/PPP for random/periodic show in blue/red and light blue/purple for $(\gamma,\delta)=(0.8,0.7)$.
Here, in Region I of Fig. S1 the dependence of $P^{\rm PDMP}(N)$ splits into three regimes for $\nu$:
Slow switching $(\nu <(1+\delta)^{-1})$ and fast switching $(\nu >(1-\delta)^{-1})$ where the peaks are defined as for Video 3, and a new intermediate
switching regime, $((1+\delta)^{-1}<\nu <(1-\delta)^{-1})$, where $P^{\rm PDMP}(N)$ is bimodal with peaks at
$N^*$ and $K_+$.
Video 5: PSD and PDMP/PPP for random/periodic show in blue/red and light blue/purple for $(\gamma,\delta)=(0.8,0.85)$.
Here, in Region II of Fig. S1 the dependence of $P^{\rm PDMP}(N)$ splits into three regimes for $\nu$:
Slow switching $(\nu <(1+\delta)^{-1})$ and fast switching $(\nu >(1-\delta)^{-1})$ where the peaks are defined as for Video 3.
The intermediate switching regime $((1+\delta)^{-1}<\nu <(1-\delta)^{-1})$ splits into three sub-regimes:
(A) $(1+\delta)^{-1}<\nu<\nu_1$ where $P^{\rm PDMP}(N)$ is bimodal with peaks at $N^*$ and $K_+$,
(B) $\nu_1<\nu<\nu_2$ where $P^{\rm PDMP}(N)$ is unimodal with peak at $K_+$,
and
(C) $\nu_2<\nu<(1-\delta)^{-1}$ where $P^{\rm PDMP}(N)$ is bimodal with peaks at $N^*$ and $K_+$.
$\nu_{1}/\nu_2$ is the smaller/larger solution to (S24).
Video 6: PSD and PDMP/PPP for random/periodic show in blue/red and light blue/purple for $(\gamma,\delta)=(0.65,0.85)$.
Here, in Region III of Fig. S1 the dependence of $P^{\rm PDMP}(N)$ splits into three regimes for $\nu$:
Slow switching $(\nu <(1+\delta)^{-1})$ and fast switching $(\nu >(1-\delta)^{-1})$ where the peaks are defined as for Video 3.
The intermediate switching regime $((1+\delta)^{-1}<\nu <(1-\delta)^{-1})$ splits into two sub-regimes:
(A) $(1+\delta)^{-1}<\nu<\nu_1$ where $P^{\rm PDMP}(N)$ is bimodal with peaks at $N^*$ and $K_+$,
(B) $\nu_1<\nu<(1-\delta)^{-1}$ where $P^{\rm PDMP}(N)$ is unimodal with peaks at $K_+$.
Video 7: PSD and PDMP/PPP for random/periodic show in blue/red and light blue/purple for $(\gamma,\delta)=(0.25,0.85)$.
Here, in Region IV of Fig. S1 the dependence of $P^{\rm PDMP}(N)$ splits into three regimes for $\nu$:
Slow switching $(\nu <(1+\delta)^{-1})$ and fast switching $(\nu >(1-\delta)^{-1})$ where the peaks are defined as for Video 3, and
the intermediate switching regime, $((1+\delta)^{-1}<\nu <(1-\delta)^{-1})$, where $P^{\rm PDMP}(N)$ is
unimodal with a peak at $K_+$.
Video 8: PSD and PDMP/PPP for random/periodic show in blue/red and light blue/purple for $(\gamma,\delta)=(0.8,-0.5)$.
Here, when $\delta <0$, the dependence of $P^{\rm PDMP}(N)$ splits into three regimes for $\nu$:
Slow switching $(\nu <(1-\delta)^{-1})$ and fast switching $(\nu >(1+\delta)^{-1})$ where the peaks are defined as for Video 3, and
the intermediate switching regime, $((1-\delta)^{-1}<\nu <(1+\delta)^{-1})$, where $P^{\rm PDMP}(N)$ is
unimodal with a peak at $K_-$.
This video shows how the fixation probabilities for random and periodic switching ($\phi_r$ and $\phi_p$ respectively) change
as the switching rate $\nu$ is increased, particularly how the rate of convergence is faster in the periodic case when $s > s_c$
(see Figures 3(a,b,c)). All three plots use a log-log scale.
In all plots, red/blue lines and symbols refer to periodic/random switching. The coloured lines show the predictions from
the saddle point approximation for the fixation probability (see Section 4.2 of SM, with results given by (4) in the main text and
(S35) in SM), while symbols (red squares - periodic switching, blue circles - random switching) show results from simulations (taken over $10^6$ - $10^7$ runs). In the main figure, black horizontal dashed
lines show the predictions for the slow and fast switching limits, $\phi^{(0)}$ and $\phi^{(\infty)}$ respectively,
and solid black lines are the predictions for the fixation probability from (S38) and (S39) (for random and periodic switching respectively).
The gray dashed lines
in the smaller figures are eyeguides $\propto (s/\nu)^2$ in the bottom figure and $(s/\nu)$ in the top figure. Parameters are shown in the
caption.
Video 9: Difference in the dependence of the fixation probability on $\nu$ for random and periodic switching
in the case $s>s_c$. Parameters are $(K_0,\gamma,s,x_0,\delta,b) = (800,0.7,0.025,0.5,0.2,0)$.