The Spruce Budworm Model

Module 1: Low-Dimensional Dynamics

Reading

Compulsory: Strogatz (2024), 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 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 (ludwig1978qualitative?).
$A$ 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?).

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 $Q$ (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 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.

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. ::: ## Next