A diffusion approximation for systems with frequent weak resetting

Abstract

We develop a diffusion approximation for systems subject to fast random resetting by small amplitudes. Equivalently, this describes systems with frequent but small catastrophes. We demonstrate the validity of the approximation by computing the stationary distribution and mean first-passage times of simple one-dimensional systems. The approximation captures dynamically induced correlations in multi-particle systems, and it can be used to generalise the conditionally independent and identically distributed structure recently found in systems with full resetting. Finally, we show that resetting can induce cycles and patterns, which can be characterised using the diffusion approximation.

Figures (5)

• Figure 1: Resetting process and stochastic differential equation for the logistic equation $\dot x=x(1-x)$, with $\lambda=0.1$, $D=0$ and proportional resets $g(x)=x$. Panel (a) $s=0.5$ ($50\%$ removal), panel (b) $s=0.02$. Black curves show a realisation of the system with resetting, green arrow in (a) illustrates one resetting event. Red curves realisations of the SDE (\ref{['eq:effective_eq']}). Blue curves are the deterministic limit ($s=0$).
• Figure 2: (a) Stationary distribution for the model with $f(x)=\alpha+\gamma x$, parameters $s=0.1$, $\alpha=1$, $\lambda=1$, $\gamma=0.5$, $D=0.1$. Orange histogram is from simulations of Eq. (\ref{['eq:1dmodel']}) with resetting. Full SDE means Eq. (\ref{['eq:full_main']}). LNA refers to the linear-noise appproximation, and 'limit $s=0$' to an approximation neglecting the randomness of the resets altogether (for details see the SM). (b) Mean time for a random walk with (partial) resetting to $x_0$ to find a target at the origin ($x_0=1$, $\alpha=\gamma=0$, $D=1$). Results for $s=0.1$ in red, for $s=1$ in black. Markers are from simulations, dashed lines from the diffusion approximation. Solid black line is the exact solution for full resetting from evans2011diffusion.
• Figure 3: Individual-based population dynamics [Eq. (\ref{['eq:ibm']})] with catastrophes. Panel (a) shows a segment of one realisation ($\lambda=0.1, s=0.1, \Omega=1000$). The green arrow indicates one reset, on average $10\%$ of the popoulation die in such an event. Panel (b) shows the quasi-stationary distribution. The black histogram is from simulations of the individual-based model, the red histogram from simulations of the diffusion approximation [Eq. (\ref{['eq:ibm_sde']})]. The smooth black line is from the theory, based on the linear-noise approximation.
• Figure 4: Resetting induced cycles in a predator-prey system [Eq. (\ref{['eq:LV']}), $k=1, a=0.1, b=1, c=2.2$]. Panel (a) shows an example of a trajectory (predator time series) for $\lambda=0.1, s=0.1$. Panel (b): Power spectra of the predator time series for different sizes $s$ of the catastrophes. Wiggly lines are from simulations (average over 500 realisations), solid lines from the theory [Eq. ( \ref{['eq:spectrum']})].
• Figure 5: Main panel: Resetting induced pattern (plankton species $\psi$) in the Levin--Segel model in two dimensions. Inset: Spectrum of flucutations $\left\langle{|\delta \tilde{\psi}(k,\omega=0)|^2}\right\rangle$ for the model in one dimension as a function of the wavenumber $k$. Wiggly line from simulations (average over 300 realisations), solid red line from the diffusion and linear-noise approximations. Parameters: $b=d=e=1/2$, $p=1$, $D_\psi=0.1$, $D_\phi=2.5$, $\lambda=0.1$, $s=0.05$. Model operates on discrete patches ($\Delta x=1$), lateral extent in the main panel is $256\times 256$.