The control of tsetse flies in relation

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The Control of Tsetse Flies in Relation to Fly
Movement and Trapping Efficiency
Article in Journal of Applied Ecology · January 1992
DOI: 10.2307/2404359

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Journal of
Applied Ecology
1992,29,

163-179

The control of tsetse flies in relation to fly movement
and trapping efficiency
BRIAN WILLIAMS,* ROBERT DRANSFIELD and
ROBERT BRIGHTWELL
International Centre for Insect Physiology and Ecology (ICIPE), PO Box 30772, Nairobi, Kenya

Summary
1. The control of tsetse fly populations using traps or targets depends on the
movement patterns of the flies , which determines how many flies find the traps, and
on the efficiency of the traps, which determines the proportion of these flies that are
killed. In this paper we develop models to predict population loss rates under
various trapping regimes . The parameters in our models are the range of attraction
of the traps, the mortality rate imposed by the traps, the rate at which the flies
diffuse through an area, the fly population growth rate, and the distribution of the
traps or targets.
2. We derive analytical results for two limiting cases: very mobile flies and
inefficient traps; relatively immobile flies and very efficient traps. We show that if
the flies are very mobile and the traps relatively inefficient , the rate at which the fly

population is reduced is limited by the range of attraction, the trapping mortality
rate and the population growth rate; if the flies are relatively immobile and the
traps very efficient, the rate of reduction is limited by the mobility of the flies and
the population growth rate. The actual situation will lie within these limits.
Numerical simulations are used to test the validity of the analytical results. Data
from field studies in Africa are used to test the predictions of the models and to
confirm their validity.
3. We show how the efficiency of barriers constructed from lines of traps or targets
depends on the width of the barrier, the mobility of the flies and the mortality rate
within the barrier.
4. We calculate the distance beyond the range of attraction of a trap over which the
trap will reduce the fly population density significantly.
5. We investigate the relationship between trap catches and population densities
and determine the factors that effect the calibration of traps as sampling devices for
the two limiting cases.
6. We investigate the rate at which a fly front will advance into country cleared of
or previously unoccupied by flies and provide an explanation for observations
regarding the relatively slow rate at which fly fronts advance.
7. Extending our models to inhomogeneous habitats and combining them with our
knowledge of tsetse biology and information on climate and vegetation should

make it possible to predict spatial and seasonal changes in tsetse fly densities and so
to provide a sound basis for planning tsetse control operations.

Key-words: tsetse flies, Glossina, trypanosomiasis , diffusion , Fisher equation.
Journal of Applied Ecology (1992) 29, 163-179

163

Introduction

tsetse fly movement on attempts to control fly populations. In this paper we develop analytical and

The distribution of tsetse flies, the vectors of animal
and human trypanosomiasis in Africa, has changed
little in the last 50 years. This is due, at least in part,
to an inadequate understanding of the effect of

Correspondence: Dr Brian Williams, Department of
Zoology, University of Oxford, South Parks Road, Oxford,
UK.

164

Tsetse fly
movement and
control

simulation models of tsetse fly movement and trap
efficiency and combine these with a simple population model of tsetse flies. These models provide
the first quantitative results for predicting the effect
that a given distribution of traps or targets will have
on a tsetse population in relation to biologically
measurable parameters.
It is not difficult to kill large numbers of tsetse
flies in a given area . Traditional methods of control,
such as aerial spraying with insecticides to eradicate
the population, can produce dramatic results in a
short time, but small numbers of flies may survive
in sprayed areas and re-establish the population
(Turner & Brightwell 1986). And even if an area is

completely cleared of tsetse by aerial spraying, flies
may migrate in from neighbouring areas, making it
necessary to establish barriers around the cleared
area by deploying traps or targets or by ground
spraying or clearing vegetation in a suitably wide
band. More recently, it has been shown that traps or
targets , used mainly for control rather than eradication, can reduce tsetse fly densities to acceptably
low levels ( Vale et al. 1988; Dransfield et al. 1990) .
However, for such on-going control strategies to be
viable they must be cost effective to livestock producers in Africa. It is essential, therefore, to make
the most efficient use of the traps or targets .
In order to reduce tsetse fly populations, traps or
targets must kill the flies more rapidly than the flies
can reproduce or invade the control area. The life
cycle of the tsetse fly is well understood and several
models of their population dynamics have been
published (Hargrove 1988; Dransfield & Brightwell
1989; Williams, Dransfield & Brightwell 1990). The
movement of tsetse flies in field conditions is less
well understood and little work has been done

to investigate the consequences of fly movement
on control programmes . The movement of nonmigratory insects has often been represented as a
diffusion or a random-walk process. Rudd & Gandour
(1985), for example, show that the dispersion of
several different insect populations can be adequately
described using a Gaussian diffusion model with an
exponential mortality term. Bursell (1970) was the
first to recommend diffusion models to describe
tsetse dispersal. In this paper we develop a diffusion
model for tsetse fly movement.

Parameters of the model
The effectiveness of traps or targets will depend on
when ·the flies are active, how they move in their
active state, whether they move into the vicinity of a
trap or target, whether they are attracted to a trap or
target and, finally, whether they are trapped or
killed. Each of these factors may depend on the
climate, especially the temperature, humidity and
wind, on the structure and visibility of the trap and

on the dispensing of the odours. In this paper we

reduce all of these factors to the four key parameters
(Fig. 1): a population growth rate and a diffusion
rate for the flies , and a range of attraction and a
trapping mortality rate for the traps.
We assume that if the density of flies is very much
less than the carrying capacity of the habitat , th e
density will increase exponentially at the population
rate of increase , r . We also assume that , in the
absence of traps , the fly density converges to the
carrying capacity at the same rate. It is not, in fact ,
necessary to assume the equivalence of these two
rates and in most of what follows we will be concerned with one or the other but not both. If the
intrinsic rate of increase is high , it will be necessary
to achieve a correspondingly high trapping mortality
rate to reduce the overall population density.
The rate of diffusion of the flies will be defined by
a diffusion coefficient, lX, or equivalently by the
root-mean-square displacement in one day , A, as

defined below. If the rate of diffusion is high , the
flies will diffuse rapidly into the vicinity of the traps
and it will be correspondingly easy to reduce the
population.
We assume that the flies are attracted to traps
when they are within a circle centred on the trap, of
radius a, which we refer to as the range of attraction.
Within this circle, we assume that the flies suffer an
additional trapping mortality rate, 0., depending on
the probability that they are caught in the trap or are
killed by making contact with insecticide on the
target. The additional mortality rate imposed on the
population by the traps or targets will increase with
both the range of attraction and the mortality of the
flies within it.
In this paper we are concerned with the way in
which these four factors , r, A, a, and 0 o , determine
the rate of decline of tsetse fly populations in relation
to the distribution of traps or targets.

POPULATION

GROWTH

RATE

Female tsetse flies deposit their first larva 17 days
after they emerge from their pupae; thereafter, they
deposit one larva every 8-10 days. The larvae burrow into the soil and pupate. The pupae remain in
the ground for about 1 month before emerging. As
flies that lived for ever would produce two larvae
every 18 days, the maximum rate of increase of a
tsetse population must be less than 3·9% day- I.
The delay in the onset of reproduction because of
the pupal and pre-adult periods reduces the maximum rate of increase to about 2% day-l (Williams,
Dransfield & Brightwell 1990). To drive a population to extinction, we therefore need to impose an
additional mortality rate of more than 2% day- I.
This is easily achieved with aerial spraying (Turner
& Brightwell 1986) but can also be achieved with
traps (Dransfield et al. 1990) or targets (Vale et al.

1988).

165
B. Williams,
R. Dransfield &
R. Brightwell

o

10
Months

Fig. 1. Centre: schematic diagram of a tsetse fly trap showing the radius of attraction, a, and the trapping mortality rate, fl.
Left: schematic diagram illustrating the step length, t, and the net displacement after 16 steps, A. Strictly, A is the rootmean-square value of the net daily displacement. Right: a graph showing the increase in the number of flies, N, with time
for a population growing logistically at a rate r = 1· 5% day-I. Note that it takes about 5 months to recover from 50 to 90%
of the carrying capacity.

MOVEMENT

Field studies
In early studies of the movement of tsetse flies,
Jackson (1930, 1941) suggested that flies use one
part of their habitat for feeding and another for
breeding and that they restrict themselves to 'ambits'
from which they rarely emigrate. Bursell (1970) was
the first to suggest that flies move more or less
randomly within the boundary of their habitat and
Rogers (1977) argued that such random movement
can be used to account for Jackson's data without
invoking the concept of ambits. Hargrove (1981)
subsequently re-examined Jackson's (1946) 'large
square' experiment and was able to fit the data by
assuming that the root-mean-square displacement
in 1 day increases with age from 0 to 500m. In
a later paper, Hargrove & Lange (1989) analysed
Jackson's (1946) data further and showed that his
'spiral square' data conform well to a simple diffusion
model, while the ' large square' data can be explained
by assuming the existence of an initial escape response followed by a random diffusion. Rogers
(1977), meanwhile, showed that the observed movements of C. juscipes and C. morsitans morsitans
could be accounted for on the basis of a random
walk model, which he then applied to studies of
other subspecies of C. m . morsitans as well as
C . swynnertoni Austen and C. longipennis Corti
and obtained root-me an-square displacements of
lOO-500m in 1 day.
The assumption of random movement is supported
by studies carried out in relatively homogeneous
habitats, but tsetse fly habitats, and therefore the
ranges of movement of the flies, are restricted by
climatic factors, the nature of the vegetation and
the availability of hosts. Several studies indicate
that tsetse flies keep to cool, thick vegetation when
it is hot and dry and spread out into more open
vegetation when it is cool and wet (Nash 1937;
Hargrove & Vale 1980). At Nguruman, in southwestern Kenya, Dransfield et al. (1990) observed rapid

and synchronized ' invasions' of flies that appeared
to come from the top of the Rift Valley escarpment
into their study area 500 m below. In all of these
studies, however, fly movement could have resulted
from changes in the parameters of the diffusion
process. The results do not necessarily imply directed
movement in relation to environmental cues .
In this paper we assume that the tsetse habitat is
homogeneous and that the fly movement is diffusive.
These assumptions will provide a framework within
which to assess the evidence for directed movement
in specific cases.

Laboratory studies
Tsetse flies have been observed to fly at speeds of up
to 4 m S- 1 on flight-mills in the laboratory (Hargrove
1975) . Mean speeds of about 6ms- 1 and maximum
speeds of lOms- 1 have been recorded in the field
with video-cameras (Gibson & Brady 1988). This
high flight speed is balanced by the relatively short
flight duration: flights in the laboratory generally
last for 1-2 min (BurseIl1978; Brady 1988). Warnes
(1989) has monitored the flight activity of tsetse flies
in concentrations of carbon dioxide ranging from
0·04 to 10%. Because the flies were constrairled by a
1 m x 1 m x lOcm cage, individual flights between
landings lasted on average only 3·3 s. However, the
flies executed a series of such flights, each series
lasting for about 22s. These bursts of activity were
interspersed with periods of inactivity lasting on
average 86 s. Brady (1988), in a study of the activity
patterns of tsetse flies, shows that bursts of activity
last for 30-50s irrespective of the hunger state
of the flies, but that the interval between bursts
decreases as the flies become hungrier.
From the above evidence, it is reasonable to suppose that the duration of each flight in the field is
between 1 and 2 min. With an average flight speed
of 5 m S-1, each flight will cover between 300 and
600 m. A diffusive model in which flies move randomly in steps of 50-200 m, when not in the presence
of host stimuli, should provide a reasonable starting

166
Tsetse fly
movement and
control

point for our calculations. Bursell & Taylor (1980)
also argue that the flying time of tsetse flies is further
limited to about 15 - 30 min day- I. With a flight speed
of 5 m S- I, the total distance flown per day would be
4·5-9 km. As we shall see below, in a random walk
model, a step length of 50 m and a total flight
distance of 4·5 km give a root-mean-square displacement in 1 day of 167 m, while a step length of 200 m
and a total flight distance of 9 km give a root-meansquare displacement in 1 day of 1·3 km. These displacements are consistent with the displacements
deduced from movement observed in the field
experiments discussed above.

RANGE

OF

ATTRACTION

Unbaited traps and targets
Dransfield (1984) carried out a series of experiments
in which serveral unbaited biconical traps were
arranged in a row in open grassland and the catches
monitored as the traps were brought progressively
closer together. When the ranges of attraction overlapped, the catch in the centre trap fell. The results of these experiments gave ranges of attraction
of IS - 20m for G. paUidipes and 1O- 15m for
G. brevipaipis.
Gibson & Young (1991) measured the angular
receptive fields of ommatidia in the eyes of male
G. m . morsitans and G. paUidipes and obtained
values of 1- 1·6°. Hardie, Vogt & Rudolph (1989)
measured the angular sensitivity of one central
rhabdomere and one peripheral rhabdomere in the
eyes of G. morsitans and G . paipaiis , obtaining
values of 1·7° and 2·7°, respectively. Taking a value
of 2°, a tsetse fly would not be able to discriminate a
1 m x 1 m object from the background at a distance
of more than about 30m, which is consistent with
Dransfield's estimate of the range of attraction. In
this paper, we shall assume that the range of attraction of unbaited stationary traps is 10- 20m .
Baited traps and targets
Traps and targets are commonly baited with cow
urine (Dransfield et ai. 1986) or a mixture of phenols
found in cow urine (Bursell et ai. 1988; Vale, Hall &
Gough 1988). Acetone (Vale 1980a) and octenol
(Hall et at. 1984), both components of cow breath,
are also used. Carbon dioxide attracts many haematophagous insects (Sutcliffe 1987) , including tsetse
(Frezil & Carnevale 1976), but is too expensive to
use in control programmes.
It is still not clear to what extent odour baits
increase catches by increasing the range of attraction
or by changing the behaviour of the flies in the
immediate vicinity of the trap. Torr (1988), for
example, has argued that the concentration of carbon
dioxide breathed out by cows will fall to the back-

ground level in the environment over a few tens of
metres and that carbon dioxide thus provides a
directional cue for host-finding only over short distances . More recently , however, Torr (1990) has
shown that in the presence of acetone the number of
flies caught on an electric screen in front of a target
increases steadily as the release rate of carbon dioxide increases up to 2001 min- I. He attributes this
largely to increases in the effective length of the
odour plume.
In early experiments on G. swynnertoni, Bax
(1973) found that flies kept in a cage reacted to the
unseen passage of men and cattle at 50 m but not at
100 m upwind or more. More recent field experiments have shown that tsetse flies respond to odours
from a cow at distances of up to 90m downwind of
the odour source (Vale 1977) . These distances probably define the limit of attractiveness for odourbaited traps. Other insects are able to locate odour
sources only over similar distances. Less than 20%
of moths reach a pheromone source 80 m upwind
(Elkinton et ai. 1987) and blowflies reach a 'struck'
sheep up to only 20m upwind (Eisemann 1988) .
Once flies enter an odour plume, they tend to fly
upwind (Vale 1974; Gibson & Brady 1988). In wind
tunnel experiments in the laboratory, flies take off
upwind in the presence of suitable odours (Bursell
1984, 1987 ; Torr 1988). However, the role of wind
direction as a navigational aid is by no means clear.
In typical tsetse habitats, wind speeds are usually
less than about 2ms- 1 (Brady, Gibson & Packer
1989), and the wind flow below the canopy of a
woodland is usually turbulent and unpredictable
(Elkinton et ai. 1987; Brady, Packer & Gibson
1990). In a typical mopane woodland habitat in
Zimbabwe, it has been shown that at a distance of
15 m from an odour source placed in the mean wind
direction, the wind comes from within 10° of the
direction of the source only 30% of the time (Brady,
Gibson & Packer 1989); the rest of the time , the
wind comes randomly from all other directions. In
this paper we assume that the radius of attraction of
baited traps is between 50 and 100 m.

TRAPPING

MORTALITY

RATE

The trapping mortality rate within the range of
attraction is the most difficult parameter to estimate.
Once flies enter a circle of attraction, we assume
that they actively seek the trap . We know little,
however, about their subsequent behaviour. If they
find the trap, they may then circle it without landing
or land on it without entering. We do not know how
long flies will remain in the vicinity of a trap if they
are not caught. In our model, we assume that the
flies move in the same way inside and outside the
circle of attraction but that while they are within the
circle of attraction , they experience an additional
trapping mortality rate. Situations where the trapp-

167
B. Williams,
R. Dransfield &
R . Brightwell

ing mortality rate is high but the flies remain in the
vicinity of the trap only for a short time will appear
similar to situations in which the trapping mortality
rate is low but the flies remain in the vicinity of the
trap for a long time .
To obtain an admittedly crude first estimate of the
mortality rate inside the circle of attraction, we can
argue as follows . We know that at anyone time ,
some flies are inactive: those that have just fed will
be resting while they digest their blood meal, heavily
pregnant females will be seeking larvae-position
sites rather than hosts, and flies that have recently
flown will be building up their reserves of proline.
We will assume that (i) the proportion of flies that
are potentially active at anyone time is between 0·6
and 0·75; (ii) of the flies that enter the circle of
attraction, the proportion that approach the trap is
0·8 (Vale 1980b); (iii) of the flies that approach the
trap , the proportion that are trapped lies between
0·3 and 0·6 (R. Dransfield & R. Brightwell, unpub!') ;
and (iv) the time spent in the vicinity of the trap is
5 min. It then follows that the proportion of the flies
within the circle of attraction that are killed in 5 min
is between 0·14 and 0·36. If each fly spends 30min
per day flying, the equivalent instantaneous mortality
rate is 0·91 and 2·67 day - l (see Appendix). In this
paper, we shall consider mortality rates within the
circle of attraction ranging from 0·30 to 3·0 day - I.
We shall provide further arguments below to support
a value of about 1·0 day - I.

sive steps, and spatial scales that are much greater
than the length of successive steps. It then follows
from the central limit theorem that the probability
distribution of the position of a fly released from the
origin at time zero will be Gaussian with mean zero
and variance in the x-direction equal to V(x,,) , with a
similar expression for the variance in the y-direction.
Furthermore, the central limit theorem enables us to
relax the restriction that the step length should be
constant and we demand only that I corresponds to
the root-mean-square step length . If a fly is at the
origin of the co-ordinate system at time zero, the
probability density function of its position at time t is
therefore
eqn 3
where the variance, a(t)2, is
eqn 4
The normalization factor in equation 3 ensures that
the probability that the fly is somewhere is 1. We
now define )".1 by
eqn 5
so that for a fly starting from the origin at time zero,
the mean-square displacement per unit time in the xor y-direction is )".112, while the mean-square displacement measured in any direction from the origin
is )...2.

THE

Diffusion models of movement
To avoid confusion and to establish our notation ,
we outline below some of the key results relating
random walk processes to diffusion (Murray 1989;
Pielou 1977).
RANDOM

=

leosO i

EQUATION

An alternative approach to the problem of random
movement starts from the equation used in the
analysis of gaseous diffusion (Pielou 1977). If the
density of tsetse flies at the position r at time tis per,
t), then the time rate of change of the density is
given by

WALK

eqn 6

For a random walk in two dimensions, we assume
that fly movement takes place in a series of steps,
each of length I, and that the direction of each
successive step is randomly and uniformly distributed
around a circle. The x-component of the ith step in
the direction Oi is
Xi

DIFFUSION

eqn 1

and the x-component of the displacement after n
steps is
eqn 2
From equation 2, the expected displacement, E(x,,),
and the variance of the displacement, Vex,,), after n
steps are 0 and n1 2 /2, respectively. If each step takes
place in a time 1: , the number of steps taken in time t
is th.
We will consider only temporal scales that are
much greater than the time interval between succes-

where
eqn 7
In this paper we assume that the diffusion co-efficient
0: does not vary in space or time. On a regular, onedimensional grid of size d,
eqn 8
so that the rate at which flies enter a particular grid
cell is proportional to the number of flies in each of
the two adjacent cells and the rate at which they
leave a particular grid cell is proportional to the
number in that cell. The difference gives the rate of
change in the number of flies in each cell due to the
diffusion process.
By substituting p(r, t) in equation 3 for p(r, t) in
equation 6 and evaluating the derivative, it follows
that the descriptions based on the random walk and
the diffusion equation are equivalent provided

168
Tsetse fly
movement and
control

eqn 9
In a heterogeneous habitat, the analysis is more
complicated; in particular, the diffusion co-efficient
will be a function of position.

Population dynamics
The equations given above describe the diffusive
movement of a fixed population of flies, but we are
interested in knowing how this movement interacts
with changes in the fly population due to births,
deaths and the additional mortality rate imposed by
the traps. If we let f(r, t) describe the time rate of
change of the density at r in the absence of movement, then
op(r, t)/ot = f(r , t).

eqn 10

Our description of the population dynamics of the
flies in the absence of diffusion, the functionf(r , t),
can be made as detailed as we choose (Hargrove
1988; Dransfield & Brightwell 1989; Williams ,
Dransfield & Brightwell 1990), but for simplicity we
shall consider only a logistic form, so that

f(r, t) = rp(r , t)[l- p(r, t)/K),

eqn 11

where r is the population growth rate and K is the
carrying capacity.
To include the effect of trapping, we assume that
the additional population mortality rate, 6, imposed
by the traps is independent of the density of the flies,
so that

f(r, t) = rp(r, t)[l- p(r, t)1 K] - 6p(r , t).

eqn 12

This equation can be reduced to the same form as
equation 11 if we replace r by an effective growth
rate r* and K by an effective carrying capacity K*,
where
r*

= r(l- 61r) and K* = K(I- 6Ir).

eqn 13

The effect of the trapping is -to reduce both the
growth rate and the carrying capacity and if the
mortality rate imposed by the traps is sufficiently
high, both are negative. Integrating the logistic
equation gives
p(r, t)

= K*p(r, O)er'II[K* - p(r, 0) +
p(r, O)e r• / ],

eqn 14

where p(r, 0) is the initial density of the flies .
A study in Lambwe Valley , in western Kenya ,
carried out by Turner & Brightwell (1986) provides
an estimate of the growth rate parameter in equation
(14). An aerial spraying campaign in the valley
effectively reduced the density of flies in the thickets
of the valley to about 0·1 % of its original value over
4 months. One year later the population had fully
recovered. Turner & Brightwell were able to fit the
recovery to a logistic curve and obtained a population growth rate of 1·5% day - I. As there may

have been some reinvasion from neighbouring areas ,
this figure gives an upper limit to the growth rate.
To eradicate a population using traps or targets,
equation (13) shows that we need to impose a
sustained mortality of more than 1·5% day- I.
Combining equations 6, 10, 12 and 13, the equation
that governs the growth and spread of the fly populations is
op(r, t)/ot = exV 2 p(r, t) + r*p(r, t)[l p(r, t)1 K* ],

eqn 15

which is known as the Fisher equation (Murray
1989).

Analytical approximations
In general, we can solve equation 15 only numerically. However, we can obtain a number of useful
results for particular situations. We begin by considering a situation in which there is no diffusion so
that we can use the results of fairly sophisticated
models of the population dynamics to relate the
trapping mortality rate imposed on the flies to the
overall population loss rate , which we can then use
in our logistic model. We will next consider two
limiting cases. In the ｦｩｲｳｾ＠
case we assume that the
flies are extremely mobile so that the mortality rate
imposed on the population is limited only by the
trap efficiency; in the second case we assume that
the traps are very efficient so that the mortality rate
imposed on the population is limited only by the
rate at which the flies reach the traps. The actual
situation must always lie between these two extremes.

NO

DIFFUSION:

OVERLAPPING TRAPS

Suppose that the traps are placed so that their radii
of attraction overlap. We can then ignore diffusion,
as all flies in the area are subject to the same
trapping pressure. Such a deployment of traps would
not be economically viable over a large area , but
might be used within a barrier line of traps.
In trapping, target or spraying operations we kill
only flies , not pupae, so we need to consider the
relationship between the trapping mortality rate and
the rate of decline of the population. When a population is at equilibrium , the pupal period and the
mean adult lifetime are both about 1 month , so that
a mortality rate of 6 applied to the flies is equivalent
to a mortality rate of about 6/2 applied to the
population as a whole. More detailed models of the
population dynamics provide more reliable estimates
of the effective mortality rate in terms of the applied
trapping mortality rate. Williams, Dransfield &
Brightwell (1990) , for example, have calculated the
loss rate of a population of G. pallidipes at Nguruman
as a function of pupal and adult mortality rate,
assuming that each is independent of age. For pupal

169
B . Williams,
R. Dransfield &
R . Brightwell

mortalities less than 5% per day and adult mortalities
less than 10% per day, the equation
s = - 0·021

+ 0·60b. + 0·41b p

eqn 16

can be used to calculate the population loss rate, s,
from the adult mortality rate , b., and the pupal
mortality rate, b p (Williams , Dransfield & Brightwell
1990) . Using equation 16, a pupal mortality rate of
1% day- l and an adult mortality rate of 2·8% day- l
give a stable population and we take these as estimates of the natural mortalities . Assuming that the
population is regulated by density-dependent mortalities (rather than density-dependent emigration
or fecundity , for ' example), either or both of these
natural mortalities must decrease as the population
density falls. If the density-dependent factors act on
the adults and not on the pupae, we can assume that
at low densities the natural adult mortality rate is
effectively zero while b p remains constant at 1%
day- I. The population growth rate is then 1·7%
day- I, close to the value of 1·5% day- I obtained by
Turner & Brightwell (1986), and at low densities the
b. in equation 16 is due solely to the traps. In other
words, this model indicates that if we use a logistic
equation we should set the intrinsic growth rate to
1·7% day- l and multiply the trapping mortality rate
applied to the flies by 0·6 to get the effective trapping
mortality rate, bt . If the effective trapping mortality
rate exceeds the population growth rate, the population will converge exponentially to zero at a rate s
equal to b - r. To achieve a 99·9% reduction in 1
year s must be 1·9% day- I and if r is 1·7% day- I b t
must be 3·6% day-t and ba , the trapping mortality
applied to the adult flies only must be 6% day- I.

INFINITE

DIFFUSION:

TRAP

LIMIT E D

Consider now the limiting case of infinite diffusion .
As the flies are infinitely mobile, the probability that
a fly is within the radius of attraction at any time is
Jta 2 n, the proportion of the 'a rea that is covered by
the traps, where n is the density of the traps . The
effective mortality rate is then 0·6b aJta 2 n and the
density of traps needed to reduce the population at a
rate s is

n = (r + s)/(0·6Jta2 b. ).

eqn 17

Field data

In a campaign using about 4 traps km - 2 to control
G. pallidipes and G. morsitans in the Rifa triangle in
Zimbabwe (Vale et al. 1988), the populations declined at rates of up to 6% day-I . Assuming that
the range of attraction for the odour-baited targets
was 100 m, and the intrinsic growth rate was 1·7%
day - I, the infinite diffusion limit implies an instantaneous trapping mortality rate for the adult flies
within the circle of attraction of 1·0 day-I .
In the suppression campaign carried out at

Nguruman, it is estimated that 5-7% of the flies
present were trapped on each day (Dransfield et al.
1990). With an effective density of about 2 traps
km - 2 and assuming that the radius of attraction was
100 m, the infinite diffusion approximation implies
that the mortality rate within the radius of attraction
must have been 0·8-1,1 day- I. These estimates are
consistent with estimates made above , which ranged
from 0·93 to 2·67 day- I.
To illustrate the application of equation 17, we
see that if the mortality rate within the radius of
attraction is 1·0 day- I, the population growth rate is
1·7% day- I, and the desired rate of reduction is
99·9% year- l so that s is 1·9% day- t, the trap
density should be greater than 0.019/a2 , or the average distance between the traps should be about 7·2a .
Assuming that the radius of attraction for baited
traps is 100m, 1·9 traps km -2 , or a mean trap
spacing of about 720 m , would be required to achieve
a 99·9% reduction in 1 year. Assuming that the
radius of attraction for unbaited traps is 10 m, 190
traps km -2, or a mean trap spacing of about 72 m ,
would be required to eradicate the population.
Under these conditions unbaited traps would not
offer an economically viable method of control over
a large area no matter how mobile the fly.
Lines of traps

We are also interested in what happens if one uses
lines of closely spaced traps , in which case th e
problem is essentially one-dimensional. Again , we
assume that the probability that a fly is within the
radius of attraction at any time is given by the
proportion of the area that is covered by the traps ,
now 2a/d where a is the range of attraction and dis
the distance betwen the lines of traps. The effective
mortality rate over the whole area is then 0·6b a x
2a/d. To achieve a 99·9% reduction of the population
in 1 year, the distance between the lines of traps
must be
d

= 1·2b aa/(r + s) .

eqn 18

If the mortality rate imposed on the flies is 1·0 day- I
and the intrinsic growth rate is 1·7% day- I, the
distance between the lines of traps must be less than
33 times the radius of attraction of the traps to bring
about a 99·9% reduction in 1 year. The spacing
between the lines of traps should therefore be less
than about 3·3 km if the traps are baited , and less
than about 330 m if the traps are unbaited .

PERFECT TRAPS:

DIFFUSION

LIMITED

Let us consider another limit in which we imagine
that we have perfect traps so that all flies that enter
the circle of attraction are killed immediately. The
rate at which flies are killed is then limited by the
rate at which they diffuse into the circle of attraction.

170

Tsetse fly
movement and
control

With a relatively immobile fly such as G. Juscipes
Juscipes on Rusinga Island in Lake Victoria (M. 1.
Mwangelwa, pers. comm .), this might be close to
the actual situation .
Figure 2 shows a regular Cartesian grid of traps,
with each trap a distance d from its four nearest
neighbours. If the flies diffuse into the circle of
attraction of the traps faster than the population is
able to grow in the regions between the traps , we
will eliminate the population between them. If the
flies diffuse into the circle of attraction more slowly
th an the population grows between the traps, the
population will persist. Now let the density at the
centre of the square of four traps be p(O) and
assume that at the boundary of each circle of attraction the fly density is zero. We want to know how
close together the traps must be for p(O) to be zero.
Using the Fisher equation, we see that to achieve a
loss rate of s when p « K ,
- sp(O) = aV2p(r )lr=o + rp(O)

eqn 19

and if we approximate V2 over five points, the centre
of the region and one point at the boundary of each
of the four nearest traps, V2p(r)lr=o is equal to
- 8p(0)/(d - Y2a)2. Substituting in eq uation 19 and
rearranging gives
8a/(d - y2a)2 = r + s

eqn 20

or
d = y2a + Y2"f../(r + S)1I2.

eq n 21

If the intrinsic growth rate, r, is 1·7% day- l and the

root-mean-square displacement in 1 day,"f.., is 100m ,
equation 20 indicates that using perfect traps a mean

spacing between traps of 889 m or a density of a little
more than 1 trap km - 2 would be sufficient to
achieve a 99·9% reduction in 1 year. If the rootmean-square displacement in 1 day is 500 m, a mean
spacing of 3·9 km between the traps, or a density of
1 trap 15 km -2 , would suffice to achieve this reduction. Repeating the calculation carried out above
for the one-dimensional case, we obtain
eqn 22
so that the spacing between the lines of traps in the
one-dimensional case is slightly greater than the size
of the grid in the two-dimensional case. In the onedimensional case, the Fisher eq uation can be solved
analytically when p« K (Skellam 1951). Instead of
the factor y2 that appears before the "f.. in equ ation
22, we get yrt/2, so that in one dimension the exact
solution differs from the approximate solution only
by 11 % , giving us confidence in the two-dimensional
result given by equation 21.

ASYMPTOTIC

B E HAVIOUR

Our traps are not perfect and the flies are not
infinitely mobile. For finite trapping mortalities
and finite step lengths, we cannot solve the Fisher
equation analytically but we can derive useful results
concerning the behaviour of the fly density far from
the traps . At points far away from a trap , the fly
density must approach the carrying capacity asymptotically , so that
aV2 p*(r) = r*p*(r) ,

where p*(r) is per) - K* , and r* is the rate of
convergence to K*. The solution to equation 23
(Skellam 1951) is
p*(r) = (k/yr)e- V(r"cx)"

d*

Fig. 2. Schematic diagram of an array of traps to indicate

the distances involved in calculating the loss rate of a fly
population as a function of inter-trap spacing for perfect
traps and relatively immobile flies. We assume that the
density of Hies is zero witll!n each of the circles of attraction
and expand the diffusion equation over the five large solid
dots. d* = dy2 - a.

eqn 23

eqn 24

where k is a constant. The density therefore converges exponentially to the carrying capacity over a
distance of about y(a/ r*) or "f../2Yr*, which we shall
call the convergence length. For example, if the
root-mean-square displacement in 1 day is 100 m
and the rate of convergence to equilibrium is 1·7%
day- I, there will be a 'depletion zone' extending a
distance of the order of the convergence length,
which is 383 m around the trap, within which the
fly density will be less than about one-half of the
carrying capacity. If the root-mean-square displacement in 1 day is 500m , the depletion zone will
extend for about 1·9 km around the trap .
For points far away from a line of traps, the
eq uilibrium solution in the one-dimensional case is
an exponential
p*(x)

= K e - V(r"a)x,

eqn25

so that the convergence length is the same as in the
two-dimensional case.

171
B. Williams,
R. Dransfield &
R . Brightwell

BARRIERS

Burnett (1970) has reviewed the role of physical
barriers in tsetse control and argues that barriers
are usually inadequate when conditions favour fly
advances. Wooff (1967) reckoned that a barrier 3-4
miles wide was needed to restrain G. morsitans in
Uganda, and in Rhodesia, now Zimbabwe, Cockbill
(1964) regarded a width of 3 miles as dangerously
narrow. Nash (1948) found that G. tachinoides would
cross a clearing of 1· 75 miles and doubted if 2 miles
would be adequate in the wet season . A barrier
nearly 3 miles long failed to exclude G. tachinoides
from the Komady Gana scheme in Nigeria (MacLennan & Kirkby 1958). Nash & Steiner (1957)
showed that G. palpalis was excluded by a clearing
of 1·5 miles but not by a clearing of only 1 mile.
Glasgow & Duffy (1951) found immigration of G.
[uscipes across clearings of 2·5 and 3 miles along a
river. In Kenya , cleared barriers each 10 miles wide
were originally used to exclude G. [uscipes in the
Kuja-Migori scheme, but these were later replaced
by barriers of sprayed vegetation each 3 miles
in width. We need to understand the factors that
determine the effici ency of barriers as a function of
barrier width.
If the barri er is made by clearing vegetation, the
probability of a fly crossing the barrier will depend
on the behavioural response of the flies to the
change in the vegetation cover. If the barrier is
made with traps, targets, or by ground spraying,
however, we can apply equation 25 to the asymptotic
convergence within the barrier , provided the population dynamics predict an exponential convergence
to zero. For example, if the mortality rate within a
barrier is 0·5 day - I, and the root-mean-square displacement in 1 day is 100m, the fly density will
decrease exponentially to zero with a decay constant
equal to 70 m, and if we regard a reduction in
density of 104 tim es across the width of the barrier as
being adequate, the barrier should be 70 x ln10 4 , or
about 645 m wide. If the root-mean-square displacement in 1 day is 500 m, the barrier would have to be
3·2km wide.

boundary. In our simulations we consider a line of
traps in the y-direction running through the origin of
the x co-ordinate system and calculate the fly densities between -d/2 and +d/2 in the x-di rection so
that the program simulates an infinite array of trap
lines, each one a distance d from the next.

CO NV ERGE N CE

TO

THE

CA R RY IN G

CAPAC ITY

To confirm the results for the convergence to the
carrying capacity inside and outside a barrier , we
ran a simulation with the inter-trap distance set to
10 km and the radius of attraction set to 5 km . The
intrinsic growth rate was set to 2% day- l and the
trapping mortality rate to 8% day - l in order to
reflect the mortality rate imposed by a set of traps .
The simulation was run until the density reached
equilibrium , at which tim e the predicted densities
were as shown by the line in Fig. 3. The fly densities
converge to zero within the barrier and to the carrying capacity outside the barrier. The fly densities
predicted using equation 25 (dots in Fig. 3) agree
well with the results of the simulation model (line in
Fig. 3) except in the region close to the edge of the
suppression zone (at 5 km in Fig. 3). The agreement
between the analytical and simulation mod els confirms the analytical results derived above and also
provides a check on the simulation model.
Field data

The control programme at Ngurum an has been in
operation since 1987. During this time a line of 10
traps about 1·2 km apart was used to monitor the

100

'"
ｾ＠

'0,., 50
Ｎｾ＠

.,

a

01 - - - - -

Numerical simulations
To speed up the simulations, we consider only lines
of closely spaced traps so that we are concerned with
the variation in the fly density at right angles to the
trap lines and th e problem can be treated in one
dimension .
As numerical simulations extend over a finite
region of space, we need to consider the conditions
at the boundary of the region we are simulating.
With cyclic boundary conditions, the solution at one
boundary matches the solution at the other and one
can imagine that a fly leaving one boundary is
immediately replaced by a fly entering at the opposite

Di stance (km)

Fig. 3. Numerical simulation to illustrate the distribution
of fly densities at the boundary of a suppression zone .
Lines of traps were spaced 25 km apart and the range of
attraction was set to 5 km, the intrinsic growth rate to 2%
day- I and the mortality rate within the range of attraction
to 8% day- I to simulate conditions within and without
a suppression zone. The solid line is the result of the
simulation after it has been run to a steady state. The dots
indicate the convergence to zero inside and the convergence
to the carrying capacity outside the suppression zone calculated separately from the analytical approximation given
by equation 25. Except in the region close to the barrier ,
the agreement is excellent.

172
Tsetse fly
movement and
control

tsetse fly density along a north-south line running
from the centre of the suppression zone to several
kilometres outside the suppression zone (Dransfield
et al. 1990). The study area at Nguruman is subjected
to intermittent invasions of tsetse flies both from the
north and from the escarpment to the west, so that
any simple interpretation of the variation of apparent
fly density along this line of traps must be treated
with caution. However, the tsetse densities do show
an approximately exponential decline from north to
south, as illustrated in Fig. 4, for October 1989.
To determine the diffusion coefficient from the
data in Fig. 4, we also need to know the trapping
mortality rate. Dransfield et al. (1990) estimated
the trapping mortalities for male and female G.
pallidipes at Nguruman in 1987 with the results
shown in Table 1. Taking an average population loss
rate of 3·5% day - I the slopes of the graphs of the
logarithm of the catches against distance for each
month in 1989 give the root-mean-square displacements in 1 day shown in Fig. 5. The highest values
are found at the beginning of the rains in March,
when A is 1·1 km day - 1I2 for females and 900m
day - 112 for males. The lowest values are found in the
dry season in September, when Ais 360m day - 112 for
females and 320 m day - 112 for males. If we use the
wet and dry season population loss rates for 1989
given in Table 1 instead of the average loss rate of
3·5% day-I, the root-mean-square displacements in
March are reduced to 670m day - 112 for females and
549 m day - 1I2 for males, whereas the values for
September are increased to 370 m day-112 for females and 401 m day-112 for males. These seasonal
changes in the diffusion of the flies may explain in
part the observations made by Dransfield et al.
(1990) at Nguruman that invasions of flies from the
escarpment into the study area occur in the wet

6
P---e/

"

,,

4

,l

"

2

,,
o

__ .0-

-6),

)/

\,

',,/

ＭＲｌ

,"

ｾ＠

__ｾ＠
2

''

__ｾＭｌ＠
3

4

__ｾＭｌ＠

5

6

__ｾ

7

8

Ｍｌ

9

Ｎ＠

10

Trap number

Fig. 4. Trap catches of males (0 ) and females (.) along a
transect running north from the centre of the suppression
zone (trap 1) at Nguruman in October 1989. Trap 8 is just
outside the suppression zone and the traps are approximately 1·2 km apart. In traps 9 and 10 the catches converge
to the carrying capacity outside the suppression zone. 95%
confidence limits for the data points are from +0·68 to
-3·37 when the In(catch day-I) is -2 and ± 0·03 when the
In (catch day-I ) is 5.

Table 1. Estimated mortality rates for female and male
G. pallidipes at Nguruman in the dry and wet seasons in
1987 (Dransfield et al. 1990). The population loss rate is
calculated using equation 16 assuming a pupal mortality
rate of 1% day - I. All figures expressed as a percentage
per day
Dry season

Wet season

Female Male

Female

Male

3
2
5
1·3

3
2
5
1·3

Trapping mortality rate 5
Natural mortality rate
4
Net adult mortality rate 9
Population loss rate
3·7

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8
4

12
5·5

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