Fluctuating environments are sufficient to drive substantial variability in species abundance across locations

Abstract

Species growing in environments that change in time and space will vary in their abundance across locations, even in the absence of persistent location preferences. Here we quantify this non-equilibrium effect by studying a minimal model of a spatially compartmentalised community with time-averaged-neutral competition but location-dependent environmental fluctuations. We analytically derive distributions of two-point inequality, defined as the log-ratio of a species' abundance across a pair of locations. We characterise how the balance of relaxation via migration and fluctuation strength determine the bulk and extreme value statistics of these distributions in the two-patch and infinite-patch cases. We demonstrate the existence of a noise-induced transition to bimodal inequality, which depends on the correlation timescale of the environmental fluctuations. Finally, we discuss the evolutionary benefit of finite migration rates in environments with temporal correlations.

Figures (11)

• Figure 1: Analytical densities for the steady-state distribution of two-point inequality in the case of a) $N=2$ given by Eq. \ref{['white noise steady state x']} and b) $N \rightarrow \infty$ given by Eq. \ref{['white noise steady state x mf']} for a range of migration rate to noise intensity ratios.
• Figure 2: a) Trajectories of Eq. \ref{['sde for log ratio']} simulated via the Euler-Maruyama method for white environmental noise (dotted line) with $M = 0.1$, $D=10$, and a coloured noise (solid line) with $M = 0.1$, $D=10$ and $\lambda = 1$. b) Histograms of trajectories obtained at $Mt = 20$ over $2 \times 10^4$ simulation runs. The analytic steady state densities are overlaid. Simulation runs are initialised as $x(0)=0, \eta(0)=0$. c) 'Phase' diagram highlighting monostable and bistable parameter regimes. These regions are separated by a supercritical pitchfork bifurcation strogatz2024nonlinear shown by the dashed black line.
• Figure 3: Tail exponent of two-point inequality in the mean-field limit obtained by numerically solving the self-consistency equation $\left<e^{y}\right> = \left<w\right>=1$. The dashed line is the boundary predicted by a saddle-point approximation (see text).
• Figure 4: Logarithmic growth rate for a species migrating between two time varying environments as a function of its migration rate. Each curve shows the numerically obtained growth rates for fluctuations with different correlation timescales $1/\lambda$. Circles show the analytic solution for temporally uncorrelated environmental fluctuations. All rates have been rescaled by the intensity of the environmental fluctuations, $D$, and $r$ set to $0$.
• Figure S1: Logarithmic growth rates for a species migrating between time varying environments in the limit of $N \rightarrow \infty$. Each curve shows the numerically obtained growth rates for environmental fluctuations with a correlation timescale of $1/\lambda$. Circles are the analytic solution found in the case of temporally uncorrelated environmental fluctuations. The dashed line is the analytically obtained growth rate in the limit $\lambda/D \rightarrow 0$, which is valid for $D < M$, beyond which the mean-field approximation breaks down. The crosses are the predictions of the growth rate from bernard2025mean which studies the case of quenched random growth rates. We make this comparison was made by taking $D \lambda = \sigma^2 / (2 \ln N)$, and keeping $2\lambda \ln N = M$ as $N \rightarrow \infty$ and $\lambda \rightarrow 0$ such that $\sigma^2 \approx MD$. The authors find that for $M > \sigma$, the long term growth rate of the system is $\zeta = 0$, while for $M < \sigma$ they find $\zeta = \sigma-M$. All rates have been rescaled by the intensity of the environmental fluctuations, $D$.