Fig 1.
Schematic illustration of a three-species community in the presence of memory and perturbations.
(A) Mutual interaction model introduced by Gonze et al. [25] to illustrate the emergence of alternative stable states in human gut microbial communities. The model describes the dynamics of species abundances Xi as functions of growth rates bi, death rates ki, and inhibition functions fi, where Kij and n denote interaction constants and Hill coefficients, respectively. (B) Standard perturbations include pulse, periodic, and stochastic variation in species immigration, death, or growth rates. Such perturbations may trigger shifts between alternative states. (C) Memory (bolded circles) can be incorporated into dynamical models by substituting the integer-order derivatives with fractional derivatives of order μi (see [37] and Methods). As decreasing μi values correspond to increasing memory, memory is measured as 1 − μi. When all community members have the same memory (μi = μ for all i), the system is said to have commensurate memory, otherwise incommensurate. Increasing memory changes community dynamics, in particular by introducing inertia and modifying the stability landscape around stable states. (D) Ecological memory can change the system dynamics under perturbations.
Fig 2.
Impact of commensurate memory on community resistance and resilience.
(A) A pulse perturbation is applied to the community (left panel): the growth rate of the blue species is lowered while that of the green species is simultaneously raised. The perturbation temporarily moves the community away from its initial stable state, characterized by blue species dominance (middle panel). Introducing commensurate memory (right panel) increases resistance to perturbation since the community is not displaced as far from its initial state compared to the memoryless case (shown in superimposition). The effect on resilience depends on the time scale considered: while memory initially hastens the recovery after the perturbation, it slows down the later stages of the recovery (starting around the time step 150). (B) A slightly stronger pulse perturbation is applied (left panel), triggering a shift toward an alternative stable state dominated by the green species (middle panel). Memory can prevent the state shift (right panel), thus increasing both resistance and resilience to perturbation.
Fig 3.
Multi-pulse and periodic perturbations: Commensurate memory impact on hysteresis and transient oscillations.
(A) Two opposite pulse perturbations are applied successively: the blue species growth rate is first briefly lowered, and then raised for a longer time. (B) The top panel shows the hysteresis in the system: the state shift towards the dominance of the green species occurs faster after the first perturbation than the shift back to the initial stable state after the second perturbation. Introducing commensurate memory (middle and bottom panels) delays the first state shift, thus increasing resistance, and hastens the second state shift, thus mitigating the hysteresis effect and increasing long-term resilience. (C) Rapidly alternating opposite perturbations are applied to the blue species growth rate with a regular frequency. (D) Without memory (top), the hysteresis effect leads to a permanent shift towards the green-dominated alternative stable state after a few oscillations. Adding commensurate memory mitigates the hysteresis, thus extending the transitory period (middle), which may generate longstanding oscillations in community composition before the community converges to a stable state (bottom).
Fig 4.
Stochastic perturbations with commensurate memory effects.
(A) Species growth rates bi vary stochastically through time according to an Ornstein-Uhlenbeck process (see Table in S1 Table). (B) Dynamical behavior of the system in response to the stochastic perturbation for equal initial species abundances and varying memory level: in addition to slowing down community dynamics, increasing memory limits the overall variation in species abundances, thus enhancing the overall resistance of the system and promoting species coexistence. (C) For some memory strengths, the final state of the system can be sensitive to slight variations in memory, with drastic consequences on community composition.
Fig 5.
Impact of incommensurate memory on the community stability landscape: Regime shifts without perturbation.
(A) A 15-species version of Gonze’s mutual interaction model (see Gonze et al. [25]). The 15 species form three groups, blue, red, and green, and between-group species interactions are stronger than within-group interactions. The resulting system exhibits three stable states, each dominated by a different group. (B) Starting from random initial conditions, the blue group of species dominates the community in the stable state when no memory is present (top). Starting from the same initial conditions, imposing memory on the blue species leads to a temporary rise in abundance, but ultimately another (red) group of species dominates instead (bottom). (C) Ternary plots represent the stable state distributions of 50 simulations with random initial conditions and noise in model parameters. Each dot shows, for one simulation at convergence time, the identity of the dominant group (color) and the average relative abundances of the three groups (position in the triangle; see S1 Appendix for details). In the memoryless system (top), the three groups roughly have the same chance of dominating the stable state, whereas imposing memory effects on the blue set of species (bottom) favors stable states where those species are not dominant. (D) Setting incommensurate memory in the blue species around the threshold separating two stable states (here, 0.14816) leads to an abrupt regime shift after a long period of subtle, gradual inclines, without changing any model parameters or adding noise.
Fig 6.
Memory induces long transient dynamics in a region of the parameter space adjacent to the multistable region.
Bifurcation diagrams for the 3-species Gonze model showing the relative abundances at time 1,000 of the blue, red, and green species as functions of the blue species’ growth rate, for three different initial conditions (leading to three distinct curves per plot), and in the absence (left column) or presence (right column) of memory. The light and dark yellow regions exhibit several alternative states for the same parameter values. However, it can be shown analytically that the system only admits a single stable state in the dark yellow region, whereas it admits multiple stable states in the light yellow region. Therefore, alternative states at time 1,000 in the dark yellow region correspond to instances of long transient dynamics, where the system remains stuck near ghost attractor states. See S4 Fig for illustrations of the dynamics in the dark yellow region.
Fig 7.
Impact of memory on the dynamics of an empirically parameterized bistable community.
(A-B) Dynamics of a bistable community composed of Bacteroides uniformis (BU) and Bacteroides thetaiotaomicron (BT), as simulated from a gLV model fitted to a set of empirical time series by Venturelli et al. [39] in which we have introduced memory effects (Eq (2) in the Methods). (A) Dynamics for the same initial conditions as one of the empirical time series used to fit the model. The original empirical data are indicated by squares and (truncated) error bars centered on the squares, representing the mean and standard deviation across biological replicates, respectively. (B) Dynamics for initial conditions leading to the alternative stable state dominated by BU. (A) and (B) illustrate that increasing memory lengthens the convergence time to the stable state, although it may hasten it in initial stages. (C-D) For each matrix entry in (E-F), a pulse perturbation is applied to the growth rate of BU (C), which temporarily displaces the community away from its original stable state dominated by BT (D). Convergence is achieved once species abundances reach an arbitrarily set convergence interval (in purple). (E-F) Recovery time to stable state after perturbation as a function of memory strength in BU and BT (the upper-left cell corresponds to the memoryless case). Both color and matrix entries indicate the recovery time after a pulse perturbation starting from the BT-dominated stable state, as illustrated in (C-D). (E) Recovery times are measured using a loose convergence interval (Bray-Curtis dissimilarity of 0.02 or lower between the community’s current state and its initial stable state), thus capturing the early stages of the recovery. Recovery time decreases with increasing memory, i.e., memory effects increase resilience. (F) Recovery times are now measured using a much tighter convergence interval (Bray-Curtis dissimilarity of 7e − 4 or lower), thus capturing the later stages of the recovery. Recovery time now decreases with increasing memory, i.e., memory effects decrease resilience. Hence, as for the two-species Gonze model in S8 Fig, the effect of memory on resilience depends on the time scale considered: memory hastens the recovery at first (E) but slows it down further in time (F).
Fig 8.
An intuitive interpretation of the memory introduced by fractional derivatives and of its effect on convergence time for the logistic curve.
(A) The F function in Eq (4) for the standard logistic equation, and a sketch of the memory effects introduced by fractional derivatives on a time series: the weight of past states on the present decreases as a power-law of time. (B) Influence of a range of memory strengths on the classic logistic growth curve.