Transient phenomena in ecology

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Science  07 Sep 2018:
Vol. 361, Issue 6406, eaat6412
DOI: 10.1126/science.aat6412


  • Two ways that long transients arise in ecology, illustrated as a ball rolling downhill.

    (A) Slow transition away from a ghost attractor: a state that is not an equilibrium, but would be under slightly different conditions. (B) Lingering near a saddle: a state that is attracting from some directions but repelling from others. Additional factors such as stochasticity, multiple time scales, and high system dimension can extend transients.

  • Fig. 1 Examples of transient dynamics.

    (A to C) Empirical examples of regime shifts occurring after long transient dynamics. (A) Population abundance of voles in northern Sweden, showing a transition from large-amplitude periodic oscillations to nearly steady-state dynamics [redrawn from (67)]. (B) Biomass of forage fishes in the eastern Scotian Shelf ecosystem; a low-density steady state changes to a dynamical regime with a much higher average density [blue line is the estimated carrying capacity; error bars are SEM; redrawn from (27)]. (C) Spruce budworm [dots; data from (68)] has a much faster generation time than its host tree, resulting in extended periods of low budworm density interrupted by outbreaks. A model (blue line) with fast budworm dynamics and slow foliage dynamics shows qualitative agreement with the data (2). (D and E) Examples of long transients on population dynamics models: (D) apparently sustainable chaotic oscillation suddenly results in species extinction (18); (E) large-amplitude periodic oscillations that persist over hundreds of generations suddenly transition to oscillations with a much smaller amplitude and a very different mean (19).

  • Fig. 2 Ghost attractors.

    Illustration of ghost attractors in a two-species competition model (A to D) and a resource-consumer-predator model (E to H). In the left column [(A), (C), (E), and (G)], there are two stable invariant sets and no ghost attractors. In the right column [(B), (D), (F), and (H)], there is a single stable invariant set, plus a ghost attractor that causes long transients. A bifurcation (tipping point) occurs for parameter values intermediate to these two cases; at this bifurcation, one stable state is lost and a ghost attractor takes its place. [(A) and (B)] Dynamics of one of the competitors depicted as a ball on a quasi-potential surface. In (A), a ball to the right of the hump at 0.07 will tend to roll toward the stable equilibrium (well) at 0.58, as in time series (C). In (B), the surface is relatively flat, rather than containing a well, to the right of ~0.1; a ball to the right will eventually roll to the stable equilibrium at 0 but will roll very slowly on the flat part of the surface, generating a long transient. There is a ghost attractor at a density around 0.3, which is visible in time series (D). [(E) to (H)] The same phenomenon with more complex invariant sets: [(E) and (G)] The system shows bistability where a chaotic three-species attractor (dark blue) coexists with a stable consumer-resource limit cycle with no predators (light blue); dark and light trajectories differ only in their initial conditions. [(F) and (H)] For parameter values on the other side of a bifurcation that turns the chaotic attractor into a chaotic saddle, any trajectory will eventually converge to the stable limit cycle, which is now the global attractor. However, convergence can be slow, as seen in (H), because the chaotic set is now a ghost. Models are as follows: [(A) to (D)] Competitor 1 is v and competes with species u: du/dt = u(1 – u) – a12unv, dv/dt = γ[v(1 – v) – a21unv] with a12 = 0.9, a21 = 1.1, γ = 10, and n = 3 [(A) and (C)] or n = 1.55 [(B) and (D)]; [(E) to (H)] from (28, 29), where the resource is R, consumer C, and predator P: dR/dt = R[1 – (R/K)] – xcycCR/(R + R0), dC/dt = xcC{[ycR/(R + R0)] – 1} – xpypPC/(C + C0), dP/dt = xpP{[ypC/(C + C0)] – 1} with xc = 0.4, yc = 2.009, xp = 0.08, yp = 2.876, R0 = 0.16129, C0 = 0.5, and K = 0.99 [(E) and (G)] or K = 1 [(F) and (H)]. Quasi-potentials in (A) and (B) were computed using (69).

  • Fig. 3 Predator-prey dynamics with and without transients.

    Predator-prey dynamics without long transients (A and B), with long transients due to crawl-bys (C and D), and with long transients due to slow-fast dynamics (E and F). In (A), (C), and (E), the intersection of the predator’s and prey’s isoclines (blue lines) produces a coexistence equilibrium. When the prey’s predator-free carrying capacity K is beyond a threshold (Hopf bifurcation), the system exhibits limit cycles around this equilibrium. [(A) and (B)] For K just beyond this threshold, relatively small limit cycles occur and there are no long transients. [(C) and (D)] With an increase in K, the cycle grows in size and closely approaches the two saddle points at (0,0) and (K,0). In (D), crawl-bys are visible at 0 and K. (E) When predator (slow) and prey (fast) dynamics occur on very different time scales, the shape of the cycle changes, and more horizontal parts of the cycle (thin arrows) proceed much more quickly than more vertical parts (thick arrows). (F) The corresponding time series for the prey shows long transients at 0 and higher prey density. The difference between (F) and (B) is entirely due to the slower predator dynamics in (F). In all panels, for prey species N and predator P, dN/dt = αN[1 – (N/K)] – γNP/(N + H), dP/dt = ε{[νγNP/(N + h)] – mP} with γ = 2.5, h = 1, ν = 0.5, m = 0.4. In (A), (B), (E), and (F), α = 1.5, K = 2.2; in (C), α = 1.5, K = 10; in (D), α = 0.8, K = 15; in (A) to (D), ε = 1; in (E) and (F), ε = 0.01.

  • Fig. 4 Examples of additional mechanisms leading to long transients.

    (A) Spatial structure in a simple population model leads to very long transients when the local population growth rate is high [from (7); local dynamics are governed by Nt+1 = Nt exp[r(1 – Nt)] with r = 3.5; the total population density summed across all localities is plotted here]. (B) For these parameter values (α = 1.5, K = 1.5, γ = 2.5, h = 1, ν = 0.5, m = 0.4, ε = 1), the deterministic predator-prey model from Fig. 3 exhibits short transient cycles, then converges to a stable equilibrium point (blue curve). However, when stochasticity is added, the same model will exhibit sustained cycles with approximately the same period (red line; here, stochasticity was incorporated by representing the prey’s intrinsic growth rate, α, as a random variable with mean 1.5).


  • Table 1 Key concepts used in this paper.

    See (70, 71) for further elaboration of ideas from dynamical systems.

    Asymptotic dynamicsThe behavior that a system will eventually exhibit and then retain indefinitely if unperturbed (i.e., dynamics that are not
    transient). Examples would include equilibria or limit cycles of predator-prey systems.
    BifurcationA qualitative change in a system’s asymptotic dynamics as a parameter is varied, caused by gain, loss, or change in
    stability of an invariant set. Examples are crises, Hopf bifurcations, and saddle-node bifurcations.
    Regime shiftA qualitative change in a system’s dynamics after a long period of apparent stasis. Can occur at tipping points where a bifurcation
    is crossed, or at a transition from transient dynamics to asymptotic dynamics (or from one transient to another).
    Tipping pointThe conditions (or value of a changing parameter) at which a bifurcation occurs, producing qualitatively different asymptotic
    TransientNonasymptotic dynamics.
    Long transientNonasymptotic dynamics that persist over ecologically relevant time scales of roughly dozens of generations (or longer).
  • Table 2 Empirical evidence for long ecological transients.
    Population(s)Observed patternDuration
    Laboratory population
    of beetles
    (Tribolium spp.) (25)
    Switch from a regime with an almost constant density
    to large-amplitude oscillations
    (70 weeks)
    Growth of macrophytes in
    shallow eutrophic lakes
    in the Netherlands (46)
    Switch from a macrophyte-dominated state to a turbid
    water state
    1 to 51 to 5
    Population of
    benthic fishes
    on the Scotian Shelf
    of Canada’s
    east coast (27)
    Switch from a forage fish (and macroinvertebrate)–dominated
    state to a benthic fish–dominated state
    5 to 820
    Coral and microalgae in
    the Caribbean (47, 48)
    Shifts from coral to macroalgal dominance on coral reefs20 to 25 (corals);
    50 to 100 (macroalgae)
    Voles, grouse in
    Europe (59)
    Switch between cyclic and noncyclic regimes, or between
    cyclic regimes with different periodicity
    60 (voles); 20 to 30
    (lemmings); 5 (grouse)
    Dungeness crab
    (Cancer magister) (53)
    Large-amplitude transient oscillations with further
    relaxation to equilibrium
    10 to 1545
    interactions in
    temperate lakes in
    Germany (26)
    Variation of amplitude and period of predator-prey
    oscillations across the season
    80 to 100 (algae);
    5 to 8 (zooplankton)
    Planktonic species in
    chemostat and
    lakes (72)
    Long-term variation of species densities, with extinction
    of some species
    40 to 100~0.05 to 0.15
    (3 to 8 weeks)
    Laboratory microbial
    communities (56)
    Slow switch between alternative stable states20 to 400.11 to 0.21
    (6 to 12 weeks)
    Grass community
    in abandoned
    agricultural fields
    in the Netherlands (57)
    Long-term existence of a large number of alternative
    transient states
    Extinction debt
    phenomena as
    a consequence
    of habitat loss
    [plants, birds, fish,
    lichens, and others (60)]
    Long-term extinction of populations, occurring
    either steadily or via oscillations
    20 to 100 (or more)1 to 100
    Fish and invertebrates
    in watersheds in western
    North Carolina,
    USA (49)
    Influence of past habitat structure on present
    biodiversity patterns after restoration
    10 to 20 (fish);
    40 (invertebrates)
    Modeled spruce budworm
    outbreaks in balsam
    fir forests (2)
    Budworm outbreaks driven by slow
    changes in condition of fir foliage
    5 (refoliation);
    50+ (budworm)
  • Table 3 Overview of long transient (LT) classification and mechanisms.
    Type of LTRelationship to
    invariant set
    Relationship to
    Dynamics mimicked
    by LT
    Possibility of
    recurrent LTs?
    Ghost (Fig. 2)No invariant setOccurs past a
    where stable
    is lost
    Equilibrium, cycles,
    NoForage fish (27)
    (Fig. 3B)
    (Fig. 3, C and D)
    Caused by
    invariant set
    None necessaryEquilibrium, cycles,
    models (26)
    Slow-fast systems
    (Fig. 3, E and F)
    None necessaryMultiple time scalesPeriodic or
    aperiodic cycles
    Yes, if invariant
    set(s) present
    Univoltine insects (2)
    (Fig. 3C)
    High dimension
    (e.g., time delays,
    space) (Fig. 4A)
    None necessaryNone necessaryEquilibrium, cycles,
    YesChemostat microbial
    communities (57)
    (Fig. 4B)
    If invariant set present:None necessaryAperiodic cycles, chaosYesCancer crabs (53)
    If invariant set absent:Past a bifurcation
    where cycles/chaos
    are lost
    Quasi-periodic cycles

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