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many years, population sizes of spruce budworm remain low and have little impact
on trees. When forest stands reach maturity, however, spruce budworm populations
explode, seriously affecting the forest by defoliating the trees. As a result of defoli-
ation, trees are weakened and ultimately may die. With the death of trees comes a
loss of the food source for spruce budworm and a consequent population crash.
The cycle of low spruce budworm population densities, followed by population
explosions and catastrophic collapse tends to repeat over the course of years. The
resulting damage and death of trees negatively affects the timber and paper pulp
industries of the region. Frequently, forest managers decided to spray forest stands
to control budworm populations. The dynamics inherent in the system, however,
lead it to follow its own path, making ever more extensive pest control necessary.
If those controls fail, outbreaks will be more severe and devastating than if the sys-
tem had been left to control itself, as recent experiences in the United States and
Canada show.
One of the most unused natural system controls in forestry relates to the idea
of patch size. Natural systems no doubt avoid large catastrophes because they op-
erate in patches, where the degree of maturity of adjacent patches is nearly always
different. Consequently, pests and fires find difficulty in spreading beyond a patch,
and the size of the catastrophe is kept small. Current forestry practice seems to be
disconnected from such natural system behavior.
To model the spruce budworm catastrophe
2
let us denote B as the budworm pop-
ulation size, K as the carrying capacity, S as habitat size, and GR as the budworm's
natural rate of increase. Thus,
1
dB
dt
=
B
GR
∗
B
∗
−
(14.3)
K
∗
S
would describe the population dynamics for a fixed carrying capacity and no preda-
tory influences on population growth. This is the logistic growth equation that we
have used in this topic many times before. Let us introduce the effects of predation
with a maximum predation rate C, which is assumed to be constant. At small pop-
ulation densities, predation has an insignificant effect on the budworm population
because they are well hidden in a relatively dense canopy. As population densities
increase, however, predators may increasingly feed on budworm that partially or to-
tally defoliated the trees and are then easy prey. A predation term that captures such
interactions is
B
2
C
∗
(14.4)
A
2
B
2
+
with A as a scalar that captures the effectiveness of the predators to spot and prey
on spruce budworm. In an immature forest, predation is easier than in a mature
forest with a diverse and dense canopy. Thus, A may be assumed to increase with
increased maturity of the forest, i.e. habitat size S
2
This model follows closely the model laid out in Beltrami, E. 1987.
Mathematics for Dynamic
Modeling
, Academic Press, Inc., Boston, pp. 189—-196.
