Memory reshapes stability landscapes: resilience-resistance tradeoffs and critical transitions

TL;DR

This work studies how memory reshapes bistable stability landscapes and regime shifts using a minimal bistable model with a fractional derivative that controls memory strength, and connects landscape geometry to classical notions of resilience and resistance.

Abstract

Regime shifts in biology, ecology, and other complex systems are often interpreted through stability landscapes and early warning signals that implicitly assume dynamics without memory effects. Yet many real systems exhibit these effects, thus present dynamics depend on past states and past forcing. Here, we study how memory reshapes bistable stability landscapes and regime shifts using a minimal bistable model with a fractional derivative that controls memory strength. We connect landscape geometry to classical notions of resilience and resistance by quantifying basin curvature and the perturbation magnitude required to cross the unstable threshold, and we track how these quantities evolve in time after perturbations. Memory typically flattens basin floors, slowing recovery, while often increasing the perturbation threshold for stability transitions, revealing a tradeoff between resilience and resistance. Because the landscape becomes history-dependent and time-evolving, memory generates qualitative behaviors that do not appear in memory-free models, including delayed collapse or recovery after stress ends, rebound after apparently successful transition, and broadened hysteresis under gradual parameter change. Finally, we show that fitting a memory-free model to memory-driven data can reproduce trajectories while systematically shifting equilibrium branches and tipping locations, risking incorrect diagnosis and management of regime shifts. These results motivate a moving landscape view and provide practical guidance for interpreting observed anomalies and distinguishing memory-driven effects from noise.

Figures (41)

• Figure 1: (a) The top panel shows a bistable time series of a univariate system. The middle panel illustrates its stability landscapes at different times as conditions change. The bottom panel projects the stability landscape onto a bifurcation diagram. In the stability landscape, valleys represent stable equilibria and peaks represent unstable states. The bifurcation diagram highlights equilibria between two tipping points, with filled circles indicating stable states and unfilled circles indicating unstable ones. Changes in system conditions can drive transitions between stable states. Here, the conditions are altered by endogenous perturbations that are tuned parameter values. The purple region denotes the parameter range in which bistability occurs. (b) This panel represents resilience, interpreted as the system’s rate of recovery (Fig. \ref{['fig: RL']}). When the system is in a stable state (the ball is at the bottom of a basin of attraction), a mild or brief perturbation may deform the stability landscape and momentarily shift the system, but the state returns to its original equilibrium after the disturbance ends. (c) This panel illustrates resistance, defined as the minimum strength (magnitude) of perturbation required to push the system into an alternative stable state. Suppose the system is initially in a stable state and experiences a perturbation of sufficient amplitude, regardless of its duration. In that case, the landscape is deformed enough that the system transitions to a different basin of attraction and does not return to the original state after the perturbation ends. The critical perturbation amplitude at which this transition occurs quantifies the system’s resistance. (d) Metrics of the basin of attraction. The two main properties are the flatness (inverse of curvature at the stable point) and the depth (distance from the valley bottom to the nearest hilltop, or unstable point). Flatter, shallower valleys correspond to slow recovery (low resilience) and lower resistance to shifts, while steeper, deeper valleys are associated with faster recovery (high resilience) and greater resistance to state shifts.
• Figure 2: (a) Summary of the main findings, the impact of memory on stability metrics, resilience, and resistance of 1000 randomly parameterized polynomial models: memory can either increase or decrease potential depth, tends to flatten the bottom of the basin of attraction, slows recovery (decreasing resilience), and raises the endogenous perturbation threshold for switching to an alternative stable state (increasing resistance). (b) Conceptual comparison of stability landscapes in one-dimensional bistable systems, with and without memory. Three phases are shown: before, during, and after a perturbation. $X_{S1}$ (set to 0 in this study) and $X_{S2}$ (positive; initial state) are stable states, while $X_{U}$ is an unstable state. $t_0$ marks the initial time or endogenous perturbation onset (the derivative is zero at this point and does correspond to the systematic flattening of the basin floor under memory, whereas the size of the resistance gain depends on the turns to the original stable state or transitions to an alternative one. In memory-free systems, the landscape remains unchanged before and after perturbation. In contrast, systems with memory exhibit evolving landscapes during and after the perturbation, which alter the system’s response.
• Figure 3: Schematic of committed transition detection for a noisy bistable system. Time increases vertically, and the horizontal axis is the state. Left: memory-free case with five committed transitions. Right: memory-driven case with one committed transition. Diamond mark detected onsets of alternating committed transitions.
• Figure 4: Memory broadens hysteresis in the quorum sensing model \ref{['eq: quorum1']} (see Models and methods). It shows system trajectories on the bifurcation diagram with stable equilibrium branches ($X_{S1}$ and $X_{S2}$) shown in gray dots and the unstable branch ($X_U$) shown as a dashed curve. The system experiences both exogenous perturbations and gradual shifts of the endogenous control parameter $\rho$: it decreases from 0.38 to 0.29 to induce collapse, then returns to 0.38 for recovery. Red traces correspond to a system with memory equal to 0.25, and blue traces to a memory-free system. Circle markers indicate the collapse leg (as $\rho$ moves from 0.38 to 0.29), and triangle markers indicate the recovery leg (as $\rho$ returns from 0.29 to 0.38). With memory, the collapse and recovery paths separate more, so returning to the upper stable branch $X_{S2}$ requires stronger or longer favorable conditions.
• Figure 5: Misplaced bifurcation structure when neglecting memory. Red curves (baseline, memory-driven system that generated the data) show stable (solid) and unstable (dashed) equilibria versus the control parameter $B$. Blue curves depict the bifurcation diagram of the best memory-free fit to those time series. Although the fit reproduces trajectories well, it shifts the equilibrium branches along $B$, moving the tipping points to lower $B$ and distorting (shrinking) the bistable interval. Thus, thresholds inferred from the fitted model are incorrect.