A Theoretical Model to Determine the Degree of Trapping
Required for Insect Population Contror
By D. E. WEIDHAAS AND D. G. HAILE
Insects Affecting Man and Animals Research Laboratory
Agric. Res. Serv., USDA, Gainesville, FL 32604
growth to survival and progeny production
is:
Trapping devices designed to capture insects have been
in use for many years, but they have usually been used
as survey tools to determine the abundance and distribution of types and species. Nevertheless, they have also
been considered as a potential means of control since large
numbers can be caught. Hienton (1974) recently summarized methods of trapping based on attraction to electromagnetic radiation and reported their effect on insect
control. Also, research involving attractants and chemicals that modify insect behavior has indicated that traps
may be useful in insect control.
(1)
=
mx
relating
the rate
oviposition
number of female eggs produced per live
female at age x.
= mean
Since oviposition occurs in the adult stages, this equation
can be modified to include other life history parameters
by making certain assumptions concerning rate of adult
survival and oviposition characteristics.
We assumed that
the rates of adult survival and oviposition are constant
with age and that oviposition occurs at discrete intervals
after a preoviposition time from emergence. The equation
can then be written as:
00
(2) Ro = S.m
1:
n=o
d+cn
S.
where
S,
= probability
of survival from egg to emerging
adult
s.
average daily survival of adult female
m = average number of female eggs per live
female per oviposition
d
preoviposition time in days
c = egg laying cycle in days.
=
=
This equation forms a convergent
can be written as :
(3) Ro
geometric
series and
= S,m s." .
I-s:
The immature survival term (S.) can be subdivided into
survival values for stages in the immature cycle if desired.
For example, if values for survival of eggs (S.), larvae
(S,), and pupae (S.) are used, S,
S.'SI'S"
Also, if
an avg daily survival of immature stages (5.) is assumed
=
for development time (I),
then S.
=
I
S,.
The effects of trapping adult females can be considered
as an additional survival factor in equation 3 if a constant
proportion of females are removed from the population
each day. We can describe the avg daily proportion of
females removed by a trapping system as a trapping efficiency factor, et, and convert it to an avg daily survival
factor after trapping, St = I-et. Then St can be included
in our equation as follows:
We therefore modified the basic population equation
used to define the rate of population growth over generations to include common biological parameters and a factor for trapping efficiency. Then we used this equation
to relate the degree of trapping required to obtain several
levels of control with several insect species to illustrate
both the methods and typical results.
equation
= }";lxmx
Ro net reproduction rate per generation
Ix = probability of survival to age x from birth or
In our research with mosquitoes and flies we have been
measuring and using population parameters that are associated with life history analyses in order to model and
simulate growth and control of populations (Haile and
Weidhaas 1977). This particular approach could also be
used to relate the degree of trapping efficiency required
to effect certain levels of population control. It occurred
to us that such a relationship could provide a general rule
of thumb on trapping efficiency vs. population control.
The results of such a relationship would not provide information or help on the design of trapping systems that
would capture large proportions of insect populations, but
they would provide a means of determining whether existing systems were sufficiently effective to produce control
over time.
basic
1948)
where
In spite of the continued interest in insect control by
attractants or trapping techniques, there has been little
published relating the basic biology of a given insect and
its population ecology and dynamics to the degree of trapping required for population control. Knipling (1966,
1970) discussed theoretical aspects of the potential of sex
attractants for insect control and of the possibility of an
oviposition trap for mosquitoes. His models are based on
the available biological data for given species and assumptions on the effectiveness of the attractants. His general
approach was to define the efficacy of attractants, attractive components, or systems as a ratio to what exists in
nature (or in native populations) and then to determine
the degree of control obtained when the attractive systems were applied at other ratios than those that occur
naturally. The procedure allows one to compare the relative effectiveness of control strategies and to demonstrate
theoretically how such strategies could be developed.
The
Ro
(Birch
of population
(4) Rt
The research reported in this manuscript was conducted in
part with contract funds transferred from the Medical Research
and Development Command, Office of the Surgeon General, U.S.
Army. Received for publication Oct. 17, 1977.
1
=
Slm(s.'s.)"
I-(s.'s.)<
Our interest
18
is in determining
the trapping
efficiency
Vol. 24, nO.1
1978
19
ESA BULLETIN
Table I.-Biological
parameters
used in calculations.
S, for indicated Ro
Insect species
m
c
d
s.
IX
2X
3X
A. albimanus
60
3
6
0.8
.85
0.03103
0.06205
0.09308
.80
.85
.8
.75
.80
.85
.80
.85
.98
.99
.8
.02327
.04654
.06981
.02959
.21068
.05890
.42139
.08878
.63212
5X
lOX
0.08526
0.17052
.06018
.14710
.12037
.29593
C. p. quinquefasciatus
100
M. domestica
S. calcitrans
3
7
16.5
5
M. autumnalis
4
5
8
G. morsitans
.5
C. homnivorax
150
2
4
10
17
3
6
required to reduce Ro (growth rate without trapping) to
specified levels. If values are known for each of the parameters necessary to determine Ro, then a trial and error
solution of equation 4 can be used to determine the value
of s. required to achieve a specified level of R •. We used
a programmable
calculator to perform this iterative
solution.
For many insects, it is relatively easy to find data concerning factors such as c, d, and m and to develop estimates of avg values of these parameters typical of populations in nature. Estimates of s. and S, typical of natural populations are more difficult to find. However, if one
knows the general values of R. and s. that can be expected with such populations, equation 3 can be used to
determine appropriate estimates of St. Then one can USe
equation 4 to develop estimates of the trapping efficiency
necessary to maintain Ro at 1 or reduce it to a level less
than 1 (population reduction).
We used this approach to make calculations for 2 mosquito species (Anopheles albimanus Wiedemann and CuTable 2.-Percent
tial growth rates.
.61032
.67617
.10987
.21974
.32960
.33226
.66449
.06206
.12409
.51578
.45730
.02482
.01241
.03723
lex pipiens quinquefasciatus Say) ; the house fly (Musca
domestica L.) ; the stable fly (Stomoxys calcitrans (L.) ;
the face fly (Musca autumnalis DeGeer);
a tsetse fly
(Glossina morsitans (Westwood»;
and the screwworm
(Cochliomyia homnivorex (Coquere1».
The biological
parameters used for various growth rates (Ro) are presented in Table 1 and represent averages of data based
on the experience of workers. The calculated values for
the trapping efficiency required to reduce the various Ro
values to growth rates of 1, 0.5, and 0.1 (representing
population reductions of 0, 50, and 90% per generation
from the initial level) are presented in Table 2.
To illustrate the density trends achieved with trapping,
we used the dynamic life history model developed by
Haile and Weidhaas (1977) that was based on the biological parameters for A. albimanus presented above. For
an uncontrolled growth rate of 3X and trapping rates of
13, 21, and 380/0 (from Table 2), the simulation model
produced the expected density trends (Fig. 1) resulting
in growth rates of 1, 0.5 and O.1X per generation (18.6
of population to be trapped out each day to produce indicated rate of growth at different poten-
Percent of population to be trapped out each day with indicated species
Potential
growth
rate (no
trapping)
IX
2X
3X
5X
lOX
• A
Growth
rate
with
trapping'
A.
albimallus
0.5X
0.1X
IX
0.5X
O.1X
1X
0.5X
O.1X
IX
0.5X
O.1X
IX
0.5X
O.1X
growth rate with trapping of
C. p.
quinquefasciatus
7
23
7
14
30
11
18
34
15
22
36
22
28
42
8
26
8
16
34
13
21
38
17
24
40
24
32
47
IX,
0.5X
and
O.IX
S.
M.
domestica
is equivalent to
calcitrans
M.
autumnalis
9
29
9
18
36
14
23
41
19
27
44
25
32
48
10
33
10
20
43
16
26
48
21
31
51
31
40
58
9
31
9
19
40
15
24
45
22
31
51
31
40
57
0, SO,
and
90%
G.
moreitalls
1
7
1
2
7
reduction per generation, respectively.
C.
homnivorax
8
26
8
16
34
13
21
38
18
26
42
26
34
48
20
ESA
Vol. 24, no. 1 1978
BULLETIN
With 13%/day Trapping R
=
1
FIG. 1.- Trends in population density for A. albimanus
efficiencies of 0, 13, 21, and 38%/day.
(potential
day generation time). The oscillations in Fig. I resulted
from the instability caused by the introduction of trapping
in a normally stable population. These curves point out
that equivalent results are obtained with a simulation
model and the use of a mathematical model (equation 4)
when both are based on the same biological parameters.
The simulation model approach can be used to determine
density trends resulting from the combined use of trapping and other methods of control, such as sterile males
and insecticides.
This analysis of trapping requirements is not concerned
with making trapping systems work in total population
contra\. However, it does provide guidelines and methods
of evaluations. With specific insects and trapping systems,
mark-release-recapture
experiments could be used to determine whether the theoretical trapping requirements can
be obtained.
The calculations in Table 2 for flies and mosquitoes
(with the exception of the tsetse fly) show some variation
in the degree of trapping required to cause population reduction, but the differences are not large. Tsetse flies
with their low reproductive capacity and long-lived adults
show the lowest trapping requirements per day. Population growth rates influence the required degree of control
as would be expected. For example, a 500/0 reduction per
generation at a potential IX growth rate requires trapping out from 1-10% of the different species. To bring
about the same reduction per generation with a potential
growth rate of lOX requires trapping out 28-40% each
day. The difficulty (relatively high proportion of insects
that must be trapped) of using trapping systems for total
population control is thus plain.
growth
rate of 3X/generation)
REFERENCES
Birch, L. C.
with trapping
CITED
1948. The intrinsic rate of natural increase of an insect population. J. Anim. Eco\. 17:
15-26.
Haile, D. G., and D. E. Weidhaas.
1977. Computer
simulation of mosquito populations (AnoPheles albimanus) for comparing the effectiveness of control
technologies. J. Med. Entomol. 13; 553-67.
Hienton, T. E. 1974. Summary of investigations of
electric insect traps. USDA Tech. Bull. No. 1498.
136 pp.
Knipling, E. F. 1966. Population models to test theoretical effects of sex attractants used for insect control.
USDA Agric. Inform. Bull. 20 pp.
1970. A theoretical appraisal of the value of ovitraps
for the suppression of container-breeding mosquitoes.
WHO/VBC70.186.
14 pp.