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\begin{document}
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\paragraph*{S6 Appendix.}
\label{Appendix_S6}
{\bf Linear vs. Nonlinear Flow Throughout the State Space.}
In the main text, we address the possibility that the optimal growth
timescale $\tau=196$ d we find on day 800 (the time of global minimum
herbivore population) exceeds the expected utility of the linearized
dynamics. We offer two tests for this possibility, with both
cautiously suggesting that the linearization is in fact valid over the
196 d timespan. 

We test the validity of this objection using two methods, both
suggesting the linearization is reasonably valid over the $\approx$200
day timespan. First, we note that $\|{\bf J}_{800}-{\bf
  J}_{996}\|^{\rm Frob}/ (0.5\|{\bf J}_{800}+{\bf J}_{996}\|^{\rm
  Frob})\approx 0.16$, the Frobenius norm of the difference between
the $t=800$ and $t=996$ d Jacobians (the numerator) is only 16\% that
of the mean of the two Jacobians (the denominator). In the second, we
compare $e^{{\bf J}_{800}}$ and $e^{{\bf J}_{996}}$, the propagators
evaluated at $t=800$ and 996 d. For these propagators,
$\sigma_1\approx 0.2$ at both times, and ${\bf v}_1\approx
(\;0.1\;1\;)^T$ and $(\;0.2\;1\;)^T$ respectively.  The reasonable
similarity of these characteristics of the propagators at the two
bracketing times mostly vindicates the validity of the linearization
over this timespan, indicating that the linearized growth time does
not significantly exceed the validity of the linearization.

Next, we embed this result in the broader context of the flow
throughout the domain of interest under both the full nonlinear
dynamics and the approximate linearized dynamics (top and bottom
panels of S2 Figure, respectively). The similar structure and
magnitude of the two flow fields indicates that the good validity of
the linearized approximation we find locally for $t=800$ is the rule
rather than the exception, with the local linearization closely
tracking the full nonlinear dynamics throughout the domain of
interest.

\nolinenumbers

\clearpage

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\end{thebibliography}

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