The Spruce Budworm Model

Module 1: Low-Dimensional Dynamics

Reading

Compulsory: Strogatz (2024), Section 3.7, “Insect outbreak”.
Recommended: Ludwig, Jones, and Holling (1978).

The spruce budworm is a forest insect whose outbreaks defoliate balsam fir forests. Ludwig, Jones and Holling modelled the budworm and forest system of New Brunswick, Canada, with the budworm as a fast variable and the forest as a slow one (Ludwig, Jones, and Holling 1978). On this page you keep the forest fixed and study the budworm alone, as in Strogatz (2024), Section 3.7.

The dimensional model

Let $N(t)$ be the budworm density. With the forest held fixed,

\[ \frac{dN}{dt} = RN\left(1 - \frac{N}{K}\right) - \frac{BN^2}{A^2 + N^2}. \tag{1}\]

$RN(1 - N/K)$ is logistic growth with rate $R$ and carrying capacity $K$. The carrying capacity depends on the amount of foliage.
$p(N) = BN^2/(A^2 + N^2)$ is predation, mainly by birds. It saturates at the maximum rate $B$ when budworms are abundant. It vanishes like $N^2$ at low density, because birds switch to other food when budworms are rare. This is a Holling type III functional response (Ludwig, Jones, and Holling 1978, 317).
$A$ sets the density at which predation starts to saturate: $p(A) = B/2$.

Ludwig et al. chose these forms because they need few parameters, and they state that the conclusions depend only on the qualitative shape of the terms (Ludwig, Jones, and Holling 1978, 317).

Figure 1: Type III predation term p(N)/B as a function of N/A. It is small at low density and saturates at 1.

Removing the units

Equation 1 has four parameters $R, K, A, B$. Rescale the density by $A$ and time by $A/B$:

\[ x = \frac{N}{A}, \qquad \tau = \frac{B}{A}t . \]

Since $dN/dt = B\, dx/d\tau$, dividing Equation 1 by $B$ gives

\[ \frac{dx}{d\tau} = r x\left(1 - \frac{x}{k}\right) - \frac{x^2}{1 + x^2}, \qquad r = \frac{RA}{B}, \quad k = \frac{K}{A}. \tag{2}\]

Only two dimensionless groups remain (Strogatz 2024, sec. 3.7). Ludwig et al. obtain the same groups under the names $R$ and $Q$ (Ludwig, Jones, and Holling 1978, eqn 8).

$r$ compares the budworm growth rate with the maximum predation rate.
$k$ is the carrying capacity measured in units of the predation scale $A$.

The rest of the module uses Equation 2. Parameter values quoted as $r$ and $k$ are always the dimensionless ones.

Exercise

Check that $r$ and $k$ are dimensionless. Assume $N$ is measured in larvae per acre and $t$ in years.

Solution

$R$ has units 1/year. $A$ has units larvae/acre. $B$ is a rate of larvae eaten, so it has units larvae/acre/year. Then $RA/B$ has units (1/year)(larvae/acre)/(larvae/acre/year), which cancel. $K/A$ is a ratio of two densities.

References

Ludwig, D., D. D. Jones, and C. S. Holling. 1978. “Qualitative Analysis of Insect Outbreak Systems: The Spruce Budworm and Forest.” Journal of Animal Ecology 47 (1): 315–32. https://doi.org/10.2307/3939.

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

--- title: "The Spruce Budworm Model" subtitle: "Module 1: Low-Dimensional Dynamics" format: html --- ::: {.callout-note} ## Reading - **Compulsory:** @strogatz2024nonlinear, Section 3.7, "Insect outbreak". - **Recommended:** @ludwig1978qualitative. ::: The spruce budworm is a forest insect whose outbreaks defoliate balsam fir forests. Ludwig, Jones and Holling modelled the budworm and forest system of New Brunswick, Canada, with the budworm as a fast variable and the forest as a slow one [@ludwig1978qualitative]. On this page you keep the forest fixed and study the budworm alone, as in @strogatz2024nonlinear, Section 3.7. ## The dimensional model Let $N(t)$ be the budworm density. With the forest held fixed, $$ \frac{dN}{dt} = RN\left(1 - \frac{N}{K}\right) - \frac{BN^2}{A^2 + N^2}. $$ {#eq-budworm-dimensional} - $RN(1 - N/K)$ is logistic growth with rate $R$ and carrying capacity $K$. The carrying capacity depends on the amount of foliage. - $p(N) = BN^2/(A^2 + N^2)$ is predation, mainly by birds. It saturates at the maximum rate $B$ when budworms are abundant. It vanishes like $N^2$ at low density, because birds switch to other food when budworms are rare. This is a Holling type III functional response [@ludwig1978qualitative, p. 317]. - $A$ sets the density at which predation starts to saturate: $p(A) = B/2$. Ludwig et al. chose these forms because they need few parameters, and they state that the conclusions depend only on the qualitative shape of the terms [@ludwig1978qualitative, p. 317]. ```{python} #| echo: false #| label: fig-predation #| fig-cap: "Type III predation term p(N)/B as a function of N/A. It is small at low density and saturates at 1." #| fig-width: 6 #| fig-height: 3 import matplotlib.pyplot as plt import numpy as np x = np.linspace(0, 6, 400) fig, ax = plt.subplots(figsize=(6, 3)) ax.plot(x, x**2 / (1 + x**2)) ax.axhline(1, color="gray", ls="--", lw=1) ax.axvline(1, color="gray", ls=":", lw=1) ax.set_xlabel("N / A") ax.set_ylabel("p(N) / B") ax.grid(alpha=0.3) plt.tight_layout() plt.show() ``` ## Removing the units @eq-budworm-dimensional has four parameters $R, K, A, B$. Rescale the density by $A$ and time by $A/B$: $$ x = \frac{N}{A}, \qquad \tau = \frac{B}{A}t . $$ Since $dN/dt = B\, dx/d\tau$, dividing @eq-budworm-dimensional by $B$ gives $$ \frac{dx}{d\tau} = r x\left(1 - \frac{x}{k}\right) - \frac{x^2}{1 + x^2}, \qquad r = \frac{RA}{B}, \quad k = \frac{K}{A}. $$ {#eq-budworm} Only two dimensionless groups remain [@strogatz2024nonlinear, Section 3.7]. Ludwig et al. obtain the same groups under the names $R$ and $Q$ [@ludwig1978qualitative, eqn 8]. - $r$ compares the budworm growth rate with the maximum predation rate. - $k$ is the carrying capacity measured in units of the predation scale $A$. The rest of the module uses @eq-budworm. Parameter values quoted as $r$ and $k$ are always the dimensionless ones. ::: {.callout-note} ## Exercise Check that $r$ and $k$ are dimensionless. Assume $N$ is measured in larvae per acre and $t$ in years. ::: ::: {.callout-tip collapse="true"} ## Solution $R$ has units 1/year. $A$ has units larvae/acre. $B$ is a rate of larvae eaten, so it has units larvae/acre/year. Then $RA/B$ has units (1/year)(larvae/acre)/(larvae/acre/year), which cancel. $K/A$ is a ratio of two densities. :::