Fig 1.
A 1-dimensional grassland example.
Panel a shows grass population as a function of time starting from 0.3 kg ha−1. Panel b shows the stability criterion derived in the text, . The time point at which the initial instability transitions to subsequent stability by crossing the neutrality line (dashed in panel b) is shown by a gray vertical line, and the value at which it crosses the population curve is indicated by the horizontal line from the crossing point to t = 0.
Fig 2.
Demonstration of the two sources of amplification of initial perturbations beyond the eigenvalue based growth or decay.
Here, we use the 2 × 2 matrix A (Eq (12)) with unit norm columns (recall that unless otherwise explicitly stated, we take “norm” to mean the L2 or Euclidean norm). The vertical axes show the largest singular value of the indicated matrix at the shown θs (i.e., the curves show the most possible amplification, assuming optimal excitation).
Fig 3.
The time evolution of the solution of the grass–herbivore system.
We solve Eq (5) using 4th order Runge–Kutta time scheme with the three parameter sets (S1 Table), with a ans b showing grass and herbivore, respectively. The nontrivial fixed points (Eq (2) of S3 Appendix) are shown as horizontal lines of corresponding colors. To emphasize the solution phase dependence on parameter choices, in the top (grass) panel we connect by straight line segments the local minima of the solutions with parameter sets 1 and 3. Panel c shows the full evolution with parameter set 1, from x0 (solid circle) to year 10 (near the fixed point shown by open square).
Fig 4.
Evolution of initial perturbations about the nontrivial fixed points of the simple grass–herbivore model with parameter sets 1–3 (a–c, respectively).
Solid/squares present growth based on eℜ[λmaxℜ(J)]τ, i.e., assuming the τ → ∞ asymptotic. Dashed/triangles present the growth measure appropriate over finite times, maxi{σi[exp(Jτ)} the largest singular value of the propagator. The two curve types thus present the stability predictions made by the traditional eigenvalue-based, asymptotically-valid method and by the alternative method proposed here, respectively. The maximum growth factors and the lead times in days at which they are realized are indicated. Thin horizontal lines separate the growth region above from the decay (growth<1) region below.
Fig 5.
Evolution of initial perturbations about the near catastrophe day 800 state of the simple grass–herbivore model with parameter set 1.
As in Fig 4, solid/squares present eℜ[λmaxℜ(J)]t, the growth assuming t → ∞. Dashed/triangles present the growth measure appropriate over finite times, maxi{σi[exp(Jt)} the largest singular value of the propagator. The maximum growth factor and the lead time in days at which it is realized are indicated. Because of the near neutrality of the leading eigenvalue, the thin horizontal line separating the growth and decay regions is barely discernible.
Fig 6.
The various contributions to the overall growth by the propagator.
We use parameter set 1, t ∈ [1, 5] years, and τ = 200 days of linearized growth throughout. For all curves, v1 is the same, and is the unit norm perturbation that optimally excites the full propagator eJ200. The solid curve shows the full impact of the propagator on the magnitude of the initial perturbation, ‖EΔE−1v1‖. The contributions to this impact by the first discussed mechanism of nonnormal growth, ‖E−1v1‖, and the modifications of this by the eigenvalues, ‖ΔE−1v1‖, are given by the dash-dotted and dashed curves, respectively. The vertical solid and dashed lines show the time of grass and herbivore global minima (t = 584 and 800 d, respectively).
Fig 7.
Modal (a) and nonnormal (b) growth throughout t ∈ [1, 10] y.
Because the system is at times near neutral, panel a presents 103ℜ(λmaxℜ), with least damping for τ = 200 d indicated. Panel b shows nonnormal growth (right colorbar) as a function of both t (horizontal) and τ (vertical).