Budworm and Forest: Fast and Slow

Module 1: Low-Dimensional Dynamics

With the forest fixed, an outbreak never ends. Ludwig, Jones and Holling let the forest change on its own, slower time scale and obtained the outbreak and collapse cycle from three differential equations (ludwig1978qualitative?). This page summarizes their model. All equations and parameter values come from that paper.

Time scales

Ludwig et al. estimate that the budworm density can change several hundredfold in a few years, so its characteristic time is of the order of months. Trees need 7 to 10 years to replace their foliage and live 100 to 150 years (ludwig1978qualitative?). The budworm is the fast variable; the forest is slow.

The three-variable model

The state is the budworm density $B$ (larvae per acre), the total branch surface area $S$ (branches per acre) and an “energy reserve” $E$ that measures the condition of the trees, with maximum $K_E = 1$:

\[ \begin{aligned} \frac{dB}{dt} &= r_B B\left(1 - \frac{B}{K' S}\right) - \beta\,\frac{B^2}{(\alpha' S)^2 + B^2},\\ \frac{dS}{dt} &= r_S S\left(1 - \frac{S}{K_S}\,\frac{K_E}{E}\right),\\ \frac{dE}{dt} &= r_E E\left(1 - \frac{E}{K_E}\right) - P\,\frac{B}{S}. \end{aligned} \tag{1}\]

This is eqns (3), (10) and (11) of (ludwig1978qualitative?) with the carrying capacity $K_B = K'S$ and the half-saturation density $\alpha = \alpha' S$ both proportional to branch area (eqns 13 and 14). The budworm equation is the model of The Spruce Budworm Model; its dimensionless groups become

\[ R = \frac{\alpha' r_B}{\beta}\,S, \qquad Q = \frac{K'}{\alpha'} , \]

so $Q$ is fixed and $R$ grows with the forest (ludwig1978qualitative?, eqn 15).

The outbreak cycle

Ludwig et al. describe the cycle qualitatively (ludwig1978qualitative?):

A young forest has small $S$, so $R$ is small and the budworm stays at the refuge level, controlled by predators.
As the forest grows, $R$ increases until it crosses the upper fold of the $(Q, R)$ diagram. The refuge disappears and the budworm jumps to the outbreak level.
During the outbreak, consumption $PB/S$ exceeds the recovery of $E$ when $M = PK'/(r_E K_E) > 1/4$, so $E$ and then $S$ decline.
The smaller forest lowers $R$ below the lower fold and the outbreak collapses.

Figure 1 integrates Equation 1 with the Level III parameter values of the paper ($r_B = 1.52$, $K' = 335$, $\beta = 43\,200$, $\alpha' = 1.11$, $r_S = 0.095$, $K_S = 25\,440$, $r_E = 0.92$, $P = 0.00195$), which give $Q \approx 302$ and $M \approx 0.71$ (ludwig1978qualitative?, Table 1 and p. 328).

Figure 1: One outbreak in the Ludwig, Jones and Holling model with Level III parameters, from a young forest (S = 0.1 K_S). B/S is larvae per branch; S is relative branch area. The integration stops when E falls to 0.05, where the model stops being valid.

The budworm stays low for about three decades while the forest grows, then rises by more than two orders of magnitude within a few years, and the energy reserve collapses. Ludwig et al. state that with realistic parameter values the outbreak leads to a collapse of the forest, and show one outbreak cycle in their Fig. 5 (ludwig1978qualitative? and 328). The exact timing in Figure 1 depends on the initial state chosen here.

Limitations

With these parameters $E$ becomes negative after the outbreak. Ludwig et al. note that the model then describes one cohort of trees and must be restarted with a young forest, or modified for small $E$ (their eqns 20 to 22) (ludwig1978qualitative?).
The model is continuous in time, while the insect has one generation per year.
Birds, parasites and weather are summarized in two predation parameters.

Management

Ludwig et al. point out that if budworm densities are partially controlled, for example by spraying, the forest may stay at an equilibrium with high budworm, foliage and branch area. They link this to the “perpetual outbreak” observed in New Brunswick after insecticide spraying (ludwig1978qualitative?). Keeping the budworm below the outbreak level also keeps the forest in the state where the next outbreak is possible.

Exercise

Using the Level III values, compute $R$ for a mature forest ($S = K_S$). Is it above the value $1/2$ needed for outbreaks when $Q$ is large?

Solution

$R = \alpha' r_B K_S / \beta = 1.11 \times 1.52 \times 25\,440 / 43\,200 \approx 0.99 > 1/2$. A mature forest is past the fold, so it cannot hold the budworm at the refuge level. The outbreak starts once $S$ exceeds roughly half of $K_S$.

Session 5 turns to epidemics.

--- title: "Budworm and Forest: Fast and Slow" subtitle: "Module 1: Low-Dimensional Dynamics" format: html --- With the forest fixed, an outbreak never ends. Ludwig, Jones and Holling let the forest change on its own, slower time scale and obtained the outbreak and collapse cycle from three differential equations [@ludwig1978qualitative]. This page summarizes their model. All equations and parameter values come from that paper. ## Time scales Ludwig et al. estimate that the budworm density can change several hundredfold in a few years, so its characteristic time is of the order of months. Trees need 7 to 10 years to replace their foliage and live 100 to 150 years [@ludwig1978qualitative, p. 316]. The budworm is the fast variable; the forest is slow. ## The three-variable model The state is the budworm density $B$ (larvae per acre), the total branch surface area $S$ (branches per acre) and an "energy reserve" $E$ that measures the condition of the trees, with maximum $K_E = 1$: $$ \begin{aligned} \frac{dB}{dt} &= r_B B\left(1 - \frac{B}{K' S}\right) - \beta\,\frac{B^2}{(\alpha' S)^2 + B^2},\\ \frac{dS}{dt} &= r_S S\left(1 - \frac{S}{K_S}\,\frac{K_E}{E}\right),\\ \frac{dE}{dt} &= r_E E\left(1 - \frac{E}{K_E}\right) - P\,\frac{B}{S}. \end{aligned} $$ {#eq-ludwig} This is eqns (3), (10) and (11) of @ludwig1978qualitative with the carrying capacity $K_B = K'S$ and the half-saturation density $\alpha = \alpha' S$ both proportional to branch area (eqns 13 and 14). The budworm equation is the model of [The Spruce Budworm Model](budworm-model.qmd); its dimensionless groups become $$ R = \frac{\alpha' r_B}{\beta}\,S, \qquad Q = \frac{K'}{\alpha'} , $$ so $Q$ is fixed and $R$ grows with the forest [@ludwig1978qualitative, eqn 15]. ## The outbreak cycle Ludwig et al. describe the cycle qualitatively [@ludwig1978qualitative, pp. 322-323]: - A young forest has small $S$, so $R$ is small and the budworm stays at the refuge level, controlled by predators. - As the forest grows, $R$ increases until it crosses the upper fold of the $(Q, R)$ diagram. The refuge disappears and the budworm jumps to the outbreak level. - During the outbreak, consumption $PB/S$ exceeds the recovery of $E$ when $M = PK'/(r_E K_E) > 1/4$, so $E$ and then $S$ decline. - The smaller forest lowers $R$ below the lower fold and the outbreak collapses. @fig-ludwig integrates @eq-ludwig with the Level III parameter values of the paper ($r_B = 1.52$, $K' = 335$, $\beta = 43\,200$, $\alpha' = 1.11$, $r_S = 0.095$, $K_S = 25\,440$, $r_E = 0.92$, $P = 0.00195$), which give $Q \approx 302$ and $M \approx 0.71$ [@ludwig1978qualitative, Table 1 and p. 328]. ```{python} #| echo: false #| label: fig-ludwig #| fig-cap: "One outbreak in the Ludwig, Jones and Holling model with Level III parameters, from a young forest (S = 0.1 K_S). B/S is larvae per branch; S is relative branch area. The integration stops when E falls to 0.05, where the model stops being valid." #| fig-width: 8 #| fig-height: 4 import matplotlib.pyplot as plt import numpy as np from scipy.integrate import solve_ivp from amlab.odes_1d.budworm_analysis import LUDWIG_LEVEL_III as p from amlab.odes_1d.budworm_analysis import ludwig_forest_model def low_energy(t, y): return y[2] - 0.05 low_energy.terminal = True s0 = 0.1 * p["k_s"] sol = solve_ivp(ludwig_forest_model, (0, 60), [0.2 * s0, s0, 1.0], method="LSODA", rtol=1e-8, atol=1e-6, events=low_energy, dense_output=True) t = np.linspace(0, sol.t[-1], 800) b, s, e = sol.sol(t) fig, ax1 = plt.subplots(figsize=(8, 4)) ax1.plot(t, b / s, "tab:red", label="B/S (larvae per branch)") ax1.set_xlabel("Time (years)") ax1.set_ylabel("Larvae per branch") ax2 = ax1.twinx() ax2.plot(t, s / p["k_s"], "tab:green", label="S / K_S") ax2.plot(t, e, "tab:blue", label="E") ax2.set_ylim(0, 1.05) ax2.set_ylabel("S / K_S and E") fig.legend(loc="upper left", bbox_to_anchor=(0.1, 0.9)) plt.tight_layout() plt.show() ``` The budworm stays low for about three decades while the forest grows, then rises by more than two orders of magnitude within a few years, and the energy reserve collapses. Ludwig et al. state that with realistic parameter values the outbreak leads to a collapse of the forest, and show one outbreak cycle in their Fig. 5 [@ludwig1978qualitative, pp. 324 and 328]. The exact timing in @fig-ludwig depends on the initial state chosen here. ## Limitations - With these parameters $E$ becomes negative after the outbreak. Ludwig et al. note that the model then describes one cohort of trees and must be restarted with a young forest, or modified for small $E$ (their eqns 20 to 22) [@ludwig1978qualitative, p. 324]. - The model is continuous in time, while the insect has one generation per year. - Birds, parasites and weather are summarized in two predation parameters. ## Management Ludwig et al. point out that if budworm densities are partially controlled, for example by spraying, the forest may stay at an equilibrium with high budworm, foliage and branch area. They link this to the "perpetual outbreak" observed in New Brunswick after insecticide spraying [@ludwig1978qualitative, pp. 322-323]. Keeping the budworm below the outbreak level also keeps the forest in the state where the next outbreak is possible. ::: {.callout-note} ## Exercise Using the Level III values, compute $R$ for a mature forest ($S = K_S$). Is it above the value $1/2$ needed for outbreaks when $Q$ is large? ::: ::: {.callout-tip collapse="true"} ## Solution $R = \alpha' r_B K_S / \beta = 1.11 \times 1.52 \times 25\,440 / 43\,200 \approx 0.99 > 1/2$. A mature forest is past the fold, so it cannot hold the budworm at the refuge level. The outbreak starts once $S$ exceeds roughly half of $K_S$. ::: ## Next Session 5 turns to epidemics.