Bistability

Module 1: Low-Dimensional Dynamics

For some values of $r$ and $k$ the budworm model has two stable fixed points separated by an unstable one. The system is then bistable: the long-run state depends on where you start (Strogatz 2024, sec. 3.7).

Two stable states

With $r = 0.5$ and $k = 10$ the positive fixed points are $x_a \approx 0.683$, $x_b = 2$ and $x_c \approx 7.317$ (see Fixed Points and Stability).

$x_a$ is stable. Strogatz calls it the refuge level: predation keeps the budworm scarce.
$x_c$ is stable. It is the outbreak level: the budworm is limited by food, not by predators.
$x_b$ is unstable. It is the threshold between the two.

Ludwig et al. describe the same two regimes as limitation by predators and limitation by food (ludwig1978qualitative?).

Basins of attraction

The basin of attraction of a stable fixed point is the set of initial conditions whose trajectories approach it. In one dimension trajectories cannot cross fixed points, so the basins here are intervals:

$0 < x_0 < 2$ converges to the refuge $x_a$;
$x_0 > 2$ converges to the outbreak $x_c$.

Figure 1: Trajectories of the budworm model for r = 0.5, k = 10 from several initial conditions. The dotted line is the unstable threshold x = 2.

Two features of Figure 1 deserve attention.

Trajectories that start near $x = 2$ take a long time to leave it, because $|f'(2)| = 0.14$ is small.
Trajectories never overshoot. In a one-dimensional autonomous ODE, solutions are monotonic in time, so oscillations are impossible (Strogatz 2024, sec. 2.6).

Ecological interpretation

Bistability means that the same forest conditions ($r$ and $k$ fixed) can support two very different budworm densities. Which one you observe depends on history: a large enough immigration or a good year for the insect can push the population across the threshold, and it will then grow to the outbreak level without any change in the parameters.

The model also shows what it cannot do. With $r$ and $k$ constant, an outbreak never ends. The observed cycles of outbreak and collapse require the forest to change, which is the subject of Budworm and Forest: Fast and Slow.

Bistability with a threshold is common in ecological models of sudden regime shifts in lakes, forests and other systems (scheffer2001catastrophic?).

Exercise

For $r = 0.5$ and $k = 10$, the population sits at the refuge $x_a \approx 0.68$. What is the smallest instantaneous increase $\Delta x$ that triggers an outbreak?

Solution

The population must cross the unstable fixed point $x_b = 2$, so $\Delta x > 2 - 0.683 \approx 1.32$ in dimensionless units, that is about $1.32A$ in larvae per acre.

Session 3 turns this analysis into code.

References

Strogatz, Steven H. 2024. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Chapman; Hall/CRC.

--- title: "Bistability" subtitle: "Module 1: Low-Dimensional Dynamics" format: html --- For some values of $r$ and $k$ the budworm model has two stable fixed points separated by an unstable one. The system is then **bistable**: the long-run state depends on where you start [@strogatz2024nonlinear, Section 3.7]. ## Two stable states With $r = 0.5$ and $k = 10$ the positive fixed points are $x_a \approx 0.683$, $x_b = 2$ and $x_c \approx 7.317$ (see [Fixed Points and Stability](fixed-points-stability.qmd)). - $x_a$ is stable. Strogatz calls it the **refuge** level: predation keeps the budworm scarce. - $x_c$ is stable. It is the **outbreak** level: the budworm is limited by food, not by predators. - $x_b$ is unstable. It is the threshold between the two. Ludwig et al. describe the same two regimes as limitation by predators and limitation by food [@ludwig1978qualitative, p. 320]. ## Basins of attraction The basin of attraction of a stable fixed point is the set of initial conditions whose trajectories approach it. In one dimension trajectories cannot cross fixed points, so the basins here are intervals: - $0 < x_0 < 2$ converges to the refuge $x_a$; - $x_0 > 2$ converges to the outbreak $x_c$. ```{python} #| echo: false #| label: fig-bistable-trajectories #| fig-cap: "Trajectories of the budworm model for r = 0.5, k = 10 from several initial conditions. The dotted line is the unstable threshold x = 2." #| fig-width: 7 #| fig-height: 3.5 import matplotlib.pyplot as plt import numpy as np from scipy.integrate import solve_ivp from amlab.odes_1d.spruce_budworm import spruce_budworm t = np.linspace(0, 60, 600) fig, ax = plt.subplots(figsize=(7, 3.5)) for x0 in [0.2, 1.0, 1.9, 2.1, 4.0, 9.5]: sol = solve_ivp(spruce_budworm, (0, 60), [x0], args=(0.5, 10.0), t_eval=t) ax.plot(sol.t, sol.y[0], color="tab:blue" if x0 < 2 else "tab:red") ax.axhline(2.0, color="k", ls=":", lw=1) ax.set_xlabel("Dimensionless time τ") ax.set_ylabel("x") ax.grid(alpha=0.3) plt.tight_layout() plt.show() ``` Two features of @fig-bistable-trajectories deserve attention. - Trajectories that start near $x = 2$ take a long time to leave it, because $|f'(2)| = 0.14$ is small. - Trajectories never overshoot. In a one-dimensional autonomous ODE, solutions are monotonic in time, so oscillations are impossible [@strogatz2024nonlinear, Section 2.6]. ## Ecological interpretation Bistability means that the same forest conditions ($r$ and $k$ fixed) can support two very different budworm densities. Which one you observe depends on history: a large enough immigration or a good year for the insect can push the population across the threshold, and it will then grow to the outbreak level without any change in the parameters. The model also shows what it cannot do. With $r$ and $k$ constant, an outbreak never ends. The observed cycles of outbreak and collapse require the forest to change, which is the subject of [Budworm and Forest: Fast and Slow](budworm-forest-fast-slow.qmd). Bistability with a threshold is common in ecological models of sudden regime shifts in lakes, forests and other systems [@scheffer2001catastrophic]. ::: {.callout-note} ## Exercise For $r = 0.5$ and $k = 10$, the population sits at the refuge $x_a \approx 0.68$. What is the smallest instantaneous increase $\Delta x$ that triggers an outbreak? ::: ::: {.callout-tip collapse="true"} ## Solution The population must cross the unstable fixed point $x_b = 2$, so $\Delta x > 2 - 0.683 \approx 1.32$ in dimensionless units, that is about $1.32A$ in larvae per acre. ::: ## Next Session 3 turns this analysis into code.