Forum
Population Theory: An Essential Ingredient in
Pest Prediction, Management, and Policy-making
Alan A. Berryman
July
n
1969, a vehicle landed softly on
the surface of the moon and a hominid
in a bright plastic suit stepped out-Hone
giant leap for mankind." But who was responsible for making this leap? The engineers
who designed the huge spaceship that bore
the humans aloft? Partly, yes. But without
the theories of planetary motion and universal gravitation, laid down more than one hundred years earlier by Kepler and Newton, the
effort would have been in vain. Putting a man
on the moon by trial and error would take
thousands, even millions, of flights and unbearable costs in money and human life. Theory was an essential prerequisite for the successful application of space technology.
In 1939, the Swiss chemist Miiller discovered that the chemical dichloro diphenyl
trichloroethane
(DDT) was extremely toxic
to insects. DDT was hailed as the miracle
insecticide, the "answer to the insect menace." The new technology was applied to
control insect populations,
but theory was
sadly ignored. Those few entomologists who
foresaw the theoretical
consequences
of
widespread DDT application were hooted
from the stage. Yet they were right: resistance, resurgence, and secondary pest outbreaks emerged as predicted by the theories
of evolution and population dynamics.
Why it is that applied entomologists pay
so little attention to the same laws of nature
I
that other technologists use so effectively? I
believe that the problem is rooted in the following:
• An education that neglects the theoretical
aspects of ecology. How many entomology departments
require that their pest
management students take courses in theoretical ecology? Compare this to the
courses in theoretical physics and control
theory required of physics and engineering
students.
• An attitude among many biologists that
there are no laws, or even reasonable theories, pertaining to biological systems, even
in the face of well-established theories of
evolution and population dynamics.
• An argument that nature is very complex
and that the theories are too simple to
explain this complexity. Yet the KeplerNewton equation of celestial dynamics and
Einstein's mass-energy equivalence equation are so simple that even high-school
students remember them. Comparatively
speaking, the equations of population dynamics are as complicated as those in the
physical sciences, whereas the systems to
which they pertain seem no more complex
than the universe.
Resistance to theory is also present (and
this is rather surprising) among some entomologists who construct population models
3600
Fig. 1. Dynamics
of Douglas-fir
tussock moth
populations
predicted by the
linear logistic
equation with
parameters
estimated from data
(~).e, Data from
Mason (1974)
and personal
communication.
138
o
3
6
9
12
15
Time
18
21
24
27
30
for use in pest management policy- and decision-making. If one scans the entomologicalliterature,
it becomes apparent that two
basic philosophies dominate the models. The
first takes the position that accurate predictions of biological events, like insect outbreaks, can be obtained only from complex,
biologically explicit simulation models. These
extremely
detailed mathematical
descriptions are usually built around the life cycle
of the insect, because entomologists usually
take the point-of-view of the insect. Detailed
high-resolution
models are fraught with
problems, however, because they are
• expensive to build and operate, requiring
huge quantities of data, large computers,
and many hours of labor and computer
time;
• so complex that it is often difficult for the
user, or even the creator, to understand
what is going on inside the model;
• often inaccurate because redundant equations are built into the model in the name
of biological realism, and small errors in
these equations can be amplified as they
pass through the many sequential operations of the simulation process.
The second philosophy revolves around
the principle of parsimony, which asserts that
modeling should start with the simplest possible structure and that complexity should be
added only when absolutely necessary (see
Berryman et al. [1990], for example). Within
this group, one can find purely empirical
modelers, who derive equations by statistical
analysis, and purely theoretical modelers, who
derive equations by logical reasoning. Empirical models can be quite useful, for they
are easy to build and often have a high degree
This paper evolved from speeches given at the
1987 symposium "Computers in Mosquito Control" of the Northwest Mosquito and Vector Control Association in Seattle, Washington; the 1989
conference "Population Dynamics of Forest Insects" of the Natural Environmental Research
Council (Institute of Terrestrial Ecology) in Edinburgh, Scotland; and the 1990 annual meeting
of the Entomological Society of America in New
Orleans, Louisiana.
AMERICAN ENTOMOLOGIST
of accuracy when applied to conditions similar to those under which they were built.
However, they may be completely wrong
when applied to new conditions, and they
rarely provide useful insights into the biological processes that give rise to the observed dynamics. (I once heard of an empirical model of forest dynamics that grew fir
trees 183 meters tall.)
Theoretical models, on the other hand, are
built upon fundamental principles and logical argument and, because of this, can provide powerful insights into the underlying
causes of population fluctuations. In addition, the models are usually quite simple and
can be fit to data with standard statistical
procedures. Models constructed in this way
are
• cheap to build and operate, because they
can be constructed with little labor (but
sometimes much thought) from data that
already exist, and they can be run on desktop computers or even pocket calculators;
• transparent,
because their simplicity allows the user to see how they work and
what causes their outcomes and responses
to management actions;
4
Population
Dynamics
a
Theory
There are two elementary principles
(or laws) of population dynamics:
•
(1.) The first principle (or Malthus'
Law) states that populations
of
living organisms grow exponentially, or logarithmically,
when
unhindered by shortages of food
or space, or by their natural enemies. This principle can be stat·
ed mathematically:
N, = N,_, exp{R},
(1)
where Nt is the density of the population at time t, and R is its per-capita
instantaneous rate of growth over the
time interval t - 1 to t.
(2.) The second principle (or Verhulst's Law) states that the realized per-capita rate of growth, R
of equation (1), must be negatively affected by high population
densities. This principle, which
includes the ecological concept
of density dependence, the control theory notion of negative
feedback, and the economics of
supply and demand, can be stated
mathematically:
-4
Population
90
294
Densi ty
392
(t-
490
2)
b
72
54
36
18
0
where A is the maximum per-capita
rate of growth, L is the time lag in the
density-dependent
(negative-feedback) response, and Q is the coefficient of nonlinearity in the densitydependent process(es).
The combination of these theoretical principles gives rise to a generalization of the logistic equation:
196
98
3
6
9
12
15
18
21
24
27
30
Time
380
C
304
228
152
This equation can be made more complicated and general by including multiple-equilibria
and underpopulation
(or Allee) effects. However, we will
not use these refinements in the current discussion.
76
0
5
10
15
20
25
Time
Fall 1991
30
35
40
45
50
Fig. 2. (a) Fit of
the generalized
logistic R function
(equation [2]) to
data from sampling
of small Douglas-fir
tussock moth larvae
in the California
Sierra Nevada
(Mason & Wickman
1988). (b) Steadystate analysis of the
Sierra Nevada
model (no random
variability). (c)
Behavior of the
Sierra Nevada
model when a small
amount of white
noise (mean, 0;
standard deviation,
0.3) is added to the
calculated rate of
increase, R.
139
Table 1. Parameter values of the nonlinear logistic R function (equation [2J) estimated from data obtained by sampling several Douglas-fir tussock
moth populations and an eastern blackheaded budworm population
Population
Douglas-fir tussock moth in Central Arizona'
Douglas-fir tussock moth in Sierra Nevada'
Douglas-fir tussock moth in Blue Mountains'
Douglas-fir tussock moth in Cascades'
Eastern black headed budwormd
Estimated parameter values'
No. yr
sampled
A
K
Q"
L
r
10
10
17
12
12
1.5
2.0
1.0
2.0
1.8
16.0
17.5
7.5
3.0
96.0
0.31
0.33
0.33
0.20
0.54
2
2
2
2
2
0.90
0.91
0.78
0.60
0.91
Correlation,
• Parameters estimated by nonlinear regression (Berryman & Millstein 1988).
A value of Q < 1 means that the R function has a concave shape, as in figure 2a.
, Data from Mason (1974) and personal communication, Mason & Wickman (1988) converted to 10 square meters of host tree foliage.
d Data from Morris (1959) converted to 100 square feet of host foliage.
b
• surprisingly accurate, because they require
minimal input of data and, being based on
theory, can predict beyond the realm of
the original data. In addition, theoretical
models usually contain density-dependent
feedback, which means that current population densities are dependent to some
extent on past densities. This intrinsic
"memory"
provides valuable predictive
power.
In the remainder of this paper, I will illustrate
how simple theoretical models can be employed in the prediction and management of
forest insect populations.
Predicting
Outbreaks
In the early seventies, outbreaks of Douglas-fir tussock moth (Orgyia pseudotsugata
(McDunnough))
flared up in many parts of
western North America. However, since the
last outbreak in the sixties, Rachel Carson's
Silent Spring (1962) had been published and
the newly formed Environmental Protection
Agency (EPA) had banned DDT, the only
pesticide known to kill tussock moth larvae
effectively. Application was made to EPA for
special permission to use DDT and, as part
of the special-use package, the Forest Service
was instructed to mount a large system study
to develop alternative control methods.
Natural
logarithm of the
average number of
male Douglas-fir
tussock moths
captured in
pheromone traps in
northern Idaho.
Numbers of male
moths captured per
trap in year t
roughly equal
numbers of larvae
hatching per 10
square meters of
host foliage, Nt.
., Predictions from
the Sierra Nevada
nonlinear logistic
5
Fig. 3.
equation (table 1).
140
A major challenge to the scientists of the
tussock moth project was to develop a model
of tussock moth population dynamics. Program managers (all entomologists) took the
view that a highly detailed, biologically explicit simulation model should be built, and
this eventually was achieved at considerable
cost (Colbert et al. 1979). This large model
of more than fifty parameters faithfully mimics the 1972-74 outbreak but cannot predict
the occurrence or intensity of future tussock
moth outbreaks. Nor has it yielded, as far as
I know, any valuable insights into the forces
that regulate tussock moth population dynamics or that cause the periodic nine-year
outbreak cycles (see Vezina & Peterman
[1985], for example).
During the early years of the tussock moth
project, I began some independent research
on simple theoretical models of tussock moth
dynamics. I was amazed to find that a timedelayed modification of the elementary logistic model, with parameters estimated from
published data, predicted the observed nineyear tussock moth outbreak cycles (fig. 1). In
addition, the model predicted that the amplitude of the cycles, or the probability of
attaining
outbreak
densities,
would be
strongly related to the favorability of the
habitat for tussock moth reproduction
and
survival. Thus, the model anticipated outbreaks in warm, dry locations or when warm,
•
fJ)
~
c..
~
~
4
,
3
fJ)
.t::
.•...
2
~
~
C)
....•
1
0
1979 1980 1981 1982 1983 1984 1985 1986 1987
dry weather preceded the population peak
(favorable habitat conditions). This prediction was later supported by the work of
Stoszek et al. (1981). The model also predicted that cycles would be driven by delayed
density-dependent
processes (such as parasitoids or predators), synchronized over large
areas by widespread climatic events like extreme cold or hot weather, and desynchronized by local disasters (as is discussed later).
Finally, the model predicted that suppression
tactics would not influence the timing of population cycles, and may even increase their
amplitude, but that cultural practices could
eliminate high amplitude outbreak oscillations completely. These are rich insights from
such a simple model.
Since the publication of this simple model
(Berryman 1978), additional data have been
collected from tussock moth populations in
different areas (Mason & Wickman 1988).
These new data clearly show that the relationship between tussock moth per-capita
rates of change (R of equation [2]) and the
correctly lagged population density is not linear, as the original model assumed (fig. 2a,
table 1). Unlike the linear model, these curvilinear equations generate damped-stable
dynamics in a constant environment and require external environmental shocks to sustain their cyclical behavior (fig. 2 b and c).
Thus, they predict that non-outbreak tussock
moth populations will be found in relatively
stable (consistent) environments, while out~
break populations will be associated with less
stable (variable) environments. This prediction also seems to hold because outbreaks
have never been observed in the equable climates west of the Cascade Mountains, even
though tussock moths do exist there.
Simple theoretical models, with parameters estimated from real data, generate useful
insights into the behavior of pest populations, but can they predict future population
densities? A crucial test came in the early
1980s, when pheromone traps indicated tussock moth populations rising again toward
outbreak levels (fig. 3). Both the logistic model and empirical evidence led to the prediction that the cycle would
Year
reach
its
peak
in
1982 or 1983, nine years after the last peak.
AMERICAN ENTOMOLOGIST
Table 2. Predictions of small larval densities of Douglas-fir tussock moth on host foliage in 1986 using a variety of methods
Model
Trend index
Logarithmic regression
Nonlinear logistic, Central Arizona data
Nonlinear logistic, Sierra Nevada data
Nonlinear logistic, Blue Mountains dara
Nonlinear logistic, Cascades data
Predicted density,
no. larvae/m'
Predicted
density minus
observed density
Damage probability
16.4
13.7
6.4
9.2
13.0
3.5
13.5
10.8
3.5
6.3
10.1
0.6
0.55
0.46
0.21 (0.00)
0.31 (0.04)
0.43 (0.09)
0.12 (0.01)
Predicted year of
next outbreak
Cannot predict
Cannot predict
1992
1993
1992
1992
Estimated observed density in 1984, 1985, and 1986 was 1.25,4.53, and 2.92 small larvae per square meter. Damage probability is defined as predicted
density + 30, the economic damage level. In parentheses is the damage probability predicted by the logistic model when iterated one hundred times with
standard normal deviation N(O,S),where S is the standard deviation obtained from filting the model 10 data.
This expectation held true in western Montana and southern Idaho, but the peak was
delayed by two to three years in northern
Idaho (fig. 3). This desynchronization of the
northern Idaho population seems to have
been caused by the eruption of Mount St.
Helens, which laid down a blanket of abrasive ash over the forests of northern Idaho,
killing tussock moth larvae that had JUSt
hatched from their overwintering eggs (Mason et al. 1984). Pheromone trap captures
were reduced to zero in 1980, the year during
which the eruption occurred (fig. 3).
Although tussock moth populations in
western Montana and central Idaho declined
in 1983 before reaching economic damage
levels, trap captures in northern Idaho continued to increase and, in 1985, exceeded the
critical level of twenty-five moths per trap.
(This trapping threshold was being used to
indicate a high probability of economic damage for the following year.) In the winter of
1985, the decision was made to spray certain
areas of northern Idaho with a virus formulation.
At this point, it is useful to compare the
prediction methods available to forest pest
managers. The most commonly used method
is the empirical trend index, I = N,fN'_Il
which predicts expected population levels in
the following year by extrapolation-that
is,
N,+, = IN •. Another simple method involves
extrapolating from the linear regression of
the log-transformed population counts against
time; for example, fit a linear regression to
the pheromone trap counts (fig. 3) over the
period 1980-85, and then estimate the 1986
count by extrapolation. Both of these methods predicted that tussock moth populations
would increase four- or five-fold in 1986,
reaching close to the economic damage level
(table 2). But, in fact, the population declined
to such an extent that the virus spray operation was canceled in midAight (fig. 3).
The failure of the traditional methods of
trend index and linear extrapolation to predict the downturn of the tussock moth population is not surprising because neither can
reverse itself. This is because they make predictions entirely by extrapolation and, therefore, always show the same trend as the data
on which the predictions were based. The
Fall 1991
logistic equation, on the other hand, is an
equilibrium model and can therefore predict
trend reversals. All the nonlinear logistic
models predicted that tussock moth densities
in northern Idaho would remain below damaging levels in the mid-eighties (table 2). Although some of these predictions were quantitatively imprecise in that they did not predict
the downturn, they were qualitatively accurate in the sense that they predicted little or
no economic damage to the forest. These
models also predict that tussock moth populations will reach peak densities in northern
Idaho earlier than expected in the ninetiesthat is, 1992 or 1993 rather than 1995 (table
2). This is because, theoretically, cycles with
lower amplitudes will have higher frequencies (Berryman 1978). This prediction will be
tested in the near future.
model. Complexity can be added in an ad
hoc manner (complexity for complexity's
sake) or in a planned series of steps (complexity because it will improve understanding
or prediction). For example, increased complexity may be required if questions are to
be asked about another variable, like parasitoids, or if there is reason to believe that
another variable has predictive or explanatory power. I will explore the problem of
adding dimension by considering the population dynamics of the eastern blackheaded
budworm (Acleris variana (Fernald)) and its
guild of insect parasitoids.
The dimension of a model is defined by
the number of components or variables contained explicitly within the model structure.
For instance, we could increase dimension
by considering eggs, larvae, pupae, and adults
as separate variables, or by adding parasitoids, predators, pathogens, or food plants as
variables in the model. The question is how
to increase dimension in a planned manner
that improves model behavior and predictions but does not add unnecessary detail. In
the case of the eastern blackheaded bud-
Adding Dimension and Complexity
Although simple theoretical models that
have been fit to real population data can provide valuable insights and predictions, additional detail can sometimes improve the
Table 3. Regression of eastern blackheaded budworm (R,,) and parasitoid (Rp) rates of increase on
densities of the two organisms, Band P, and the parasitoid / prey ratio (PI B)
Dependent
variable
Independent
variable
Intercept"
Slopeb
Correlation
coefficient
-0.004·
-0.013···
-4.395···
-0.011·'
-0.002
-5.623'"
0.451
0.721
0.870
0.537
0.169
0.955
Simple regressions
0.838
0.975
1.842
1.039
0.671
2.441
B
P
PIB
B
P
PIB
Multiple regressions'
B
PIB
2.152
P
2.442
-0.003"
-4.131'"
-0.001
-5.535'"
PIB
0.924
0.955
The observed rate of increase is obtained from the first principle (equation [1]) (i.e., R = log,N, log,N 1l where Nt is the density of either population at time t.
" Simple regression intercept, A; multiple regression intercept, Ai'
b Simple regtession slope, S; multiple regression slope, S;., or Si)' ., •• , ••• , Significantat the 90%, 95%,
and 99% levels, respectively.
, The multiple regression equations, or the R function for each species, are
t_
Rb = Ab
+ SblB + Sb'P
IB
(4.1)
and
Rp
= Ap
+ Sp,P + Sp,PIB
(4.2)
(see Berryman [1990]).
141
worm, for example, analysis of sample data
(table 1) indicated that the observed cycles
of abundance probably were caused by a delayed density-dependent
process, much like
the Douglas-fir tussock moth. This diagnostic
clue, together with empirical evidence, suggested that budworm population cycles could
be generated by interaction with larval parasitoids (Morris 1959, Berryman 1984). If this
hypothesis is true, then the following should
hold:
• budworm densities should have a significant positive correlation with the per-capita rate of increase of parasitoids;
• parasitoid densities should have a significant negative correlation with the per-capita rate of change of budworms; and
• the ratio of parasitoids to their prey should
be negatively correlated with the per-capita rates of change of both populations.
The hypothesis was tested by regression
analysis (table 3). We see that all but one of
the hypothesized
interactions
were significant, and all significant interactions
were
classified correctly. Because these results generally support the hypothesis that eastern
blackheaded budworm population dynamics
are driven by interactions with parasitoids, it
seemed reasonable to increase the dimension
of the original model by adding an equation
for the parasitoid population. According to
table 3, the most significant independent vari-
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142
able was the parasitoidjbudworm
ratio. The
data were therefore fit to ratio-dependent
predator-prey
equations
by least-squares
multiple regression (table 3). Although the
fitted equations explained a high percentage
of the variation in the data (85% and 91%,
respectively), they only increased predictive
power by 2% (from 83% to 85%, tables 1
and 3)-hardly a significant improvement over
the single-species model.
If our objective was to predict eastern
black headed budworm densities, then adding parasites as a second variable hardly seems
justifiable. Before rejecting the two-species
model, however, some other factors should
be considered. First, in order to predict budworm densities from the single-species model, one must have sampled the budworm
population in the two preceding years, N'_I
and N'_2' To predict budworm densities from
the two-species model, on the Olher hand,
one needs to measure only budworm larval
density and the number of larvae parasitized
in the current year. This may be an important
consideration in a management situation.
Another significant feature of the two-species model is the emergence of a critical parasitoidjprey
ratio. For instance, if we set Rb
in equation (4.1) (table 3) to zero we find the
critical ratio
(PjB)*
=
=
-AjSb2
- SblBjSb2
0.521 - 0.0007B,
where budworm populations
remain constant. A larger ratio means that the budworm
population will probably decline next year;
a smaller ratio means that it will probably
increase. Thus, the parasitoidjbudworm
ratio provides quick field assessment of the direction of budworm population change.
Finally, the parasitoid-budworm
model
enables us to predict the effect of various
management strategies-such
as the spraying
of selective or nonselective pesticides, inoculative or mass augmentation of the parasitaid population,
sterile male release, alternative foods for parasitoids, pheromone traps,
and other strategies-on
the stability and amplitude of budworm population fluctuations.
Getting Practical
Simple theoretical models that have been
fit to field data seem to have greater practical
utility in applied entomology than large detailed models or simple empirical models,
because they are cheap to build, easy to operate, and usually more accurate over greater
ranges of conditions than either of the other
types. In addition, they can provide valuable
diagnostic clues to the forces regulating population abundance and can be used to test
pest management alternatives. This does not
mean that other kinds of models are not useful. Detailed mechanistic models, for ex-
ample, can be useful learning and hypothesisgenerating devices, while purely empirical
models may give good predictions under certain conditions. I would argue, however, that
both detailed and simple models will have
much greater credibility and utility if they
rest upon a firm theoretical foundation.
0
References
Cited
Berryman, A. A. 1978. Population cycles of the
Douglas-fir tussock moth: the time delay hypothesis. Can. Entomol. 100: 513-518.
1984. On the dynamics of blackheaded budworm populations. Can. Entomol. 118: 775779.
1990. Population analysis system: POPSYSseries
2, two-species analysis user manual. Ecological
Systems Analysis, Pullman, Wash.
Berryman, A. A. & J. A. Millstein. 1988. Population analysis system: POPSYS series 1, singlespecies analysis user manual. Ecological Systems
Analysis, Pullman, Wash.
Berryman, A. A., J. A. Millstein & R. R. Mason.
1990. Modelling Douglas-fir tussock moth
population dynamics: the case for simple theoretical models, pp. 369-380. In A. D. Wan, S.
R. Leather, M. D. Hunter & N.A.C. Kidd [eds.],
Population dynamics of forest insects. Intercept,
Andover, U.K.
Carson, R. 1962. Silent spring. Houghron Mifflin, Boston.
Colbert, J. J., W. S. Overton & C. White. 1979.
Documentation of the Douglas-fir tussock moth
outbreak population model. U.S. Dep. Agric.
For. Servo Gen. Tech. Rep. PNW-89, Pacific
Southwest Forest Range Experiment Station,
Ponland, Oreg.
Mason, R. R. 1974. Population change in an
outbreak of the Douglas-fir tussock moth, Orgyia pseudotsugata (Lepidoptera: Lymantriidae), in central Arizona. Can. Entomol. 106:
1171-1174.
Mason, R. R. & B. E. Wickman. 1988. The
Douglas-fir tussock moth in the interior Pacific
Nonhwest, pp. 179-209. In A. A. Berryman
[ed.], Dynamics of forest insect populations. Plenum, New York.
Mason, R. R., B. E. Wickman & H. G. Paul. 1984.
Effect of ash from Mount St. Helens on survival
of neonate larvae of the Douglas-fir tussock moth
(Lepidoptera: Lymantriidae). Can. Enromol. 116:
1145-1147.
Morris, R. F. 1959. Single-facror analysis in population dynamics. Ecology 40: 580-588.
Stoszek, K. J., P. G. Mika, J. A. Moore & H. L.
Osborne. 1981. Relationships of Douglas-fir
tussock moth defoliation ro site and stand characteristics in nonhern Idaho. For. Sci. 27: 431442.
Vezina, A. & R. M. Peterman. 1985. Tests of
the role of a nuclear polyhedrovirus in the population dynamics of its host, Douglas-fir tussock
moth, Orgyia pseudotsugata (Lepidoptera: Lymantriidae). Oecologia (Berl.) 67: 260-266.
Alan A. Berryman is professor of entomology and natural resource sciences at
Washington State University, Pullman,
Washington 99164-6432.
AMERICAN ENTOMOLOGIST