Models for the Population Dynamics of the Yellow Fever Mosquito, Aedes aegypti Author(s): Christopher Dye Source: Journal of Animal Ecology, Vol. 53, No. 1 (Feb., 1984), pp. 247-268 Published by: British Ecological Society Stable URL: http://www.jstor.org/stable/4355 . Accessed: 02/05/2014 16:57 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . British Ecological Society is collaborating with JSTOR to digitize, preserve and extend access to Journal of Animal Ecology. http://www.jstor.org This content downloaded from 130.132.123.28 on Fri, 2 May 2014 16:57:06 PM All use subject to JSTOR Terms and Conditions
Transcript
Models for the Population Dynamics of the Yellow Fever Mosquito, Aedes aegyptiAuthor(s): Christopher DyeSource: Journal of Animal Ecology, Vol. 53, No. 1 (Feb., 1984), pp. 247-268Published by: British Ecological SocietyStable URL: http://www.jstor.org/stable/4355 .
Accessed: 02/05/2014 16:57
Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp
.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].
.
British Ecological Society is collaborating with JSTOR to digitize, preserve and extend access to Journal ofAnimal Ecology.
http://www.jstor.org
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MODELS FOR THE POPULATION DYNAMICS OF THE YELLOW FEVER MOSQUITO, AEDES AEGYPTI
BY CHRISTOPHER DYE
Department of Zoology, University of Oxford
SUMMARY
(1) Analytical models are described for a field population of adult Aedes aegypti mosquitoes, and used as foundations for the development of a multi-age-class simulation model. The models bring together published and unpublished data on the larval and adult ecology of A. aegypti in Wat Samphaya, Bangkok, Thailand.
(2) The most appropriate analytical model is a generalization of a continuous time model used by Gurney, Blythe & Nisbet (1980) to describe Nicholson's blowfly populations. Despite uncertainty about egg-laying rate, local stability analysis firmly predicts that the population in Wat Samphaya is monotonically stable. Equilibrium analysis predicts that adult populations will be more sensitive to changes in death rate than to changes in either birth rate or number of larval breeding sites. Accurate prediction of equilibrium population size requires good estimates of parameters (fls) describing density- dependent mortality.
(3) Results of stability analysis with the simulation model accord with those of the analytical model: observed fluctuations in adult population size are unlikely to be driven cycles, but rather due to fluctuations in adult survivorship combined with strong density- dependent larval mortality. Equilibrium analysis reinforces the conclusion that the adult population is more sensitive to changes in adult survivorship than to changes in fecundity.
INTRODUCTION
Despite the importance of Aedes aegypti (L.) (Diptera:Culicidae) as a pantropical vector of human viral disease (Mattingly 1969), there have been few attempts to describe or explain its population dynamics mathematically. This is surprising because the widespread domestic form is found in near-insectary conditions in the field and is unusually well-suited to single species population modelling. The larvae live in water storage containers inside houses all year round, and receive an artificial food supply (e.g. maize meal accidentally dropped into the water; Subra 1983). Density-dependent predation, parasitism or inter- specific competition have not been found anywhere in the life history.
The most detailed models to date are by Gilpin, McClelland & Pearson (1976) and Gilpin & McClelland (1979). They followed Holling (1959, 1963) in carrying out an investigation of larval competition in the spirit of 'experimental components analysis'. In 10 years of laboratory experiments, they established, among other relationships, the effects of temperature on development rate, and of larval food supply on weight and age at pupation. The system components were then assembled in a computer model with fourteen parameters and seven variables. This model can qualitatively reproduce the dynamics of a laboratory population multiplying in discrete generations (Gilpin et al. 1976; Gilpin &
Correspondence: Dr C. M. Dye, Imperial College at Silwood Park, Sunninghill, Ascot, Berkshire SL5 7PY.
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McClelland 1979), but it cannot be readily extended to describe a continuously reproducing population with overlapping generations (Dye 1982a). Most importantly, when food and larvae are added to a system each day, as they typically are for domestic A. aegypti (Curtis et al. 1976), density-dependent development rate and mortality cannot maintain a stable equilibrium population size. The complexity of the model means that its defects are not easily pinpointed, and that a large experimental effort will be required before the necessary corrections can be made. The effort is unlikely to be repaid by successful application to a field population because fourteen parameters could not easily be measured, nor seven variables monitored, under field conditions. Furthermore, the model only describes events in one water container; the whole immature population resides in many.
In this paper, I take an alternative approach to modelling A. aegypti populations. Rather than starting with the physiology of the individual in the laboratory, I begin with the simplest description of the entire population in the field. The chosen population, in Wat Samphaya, Bangkok, is framed in single age-class models which have few parameters easily estimated from field data, and whose dynamic behaviour can be revealed analytically by local stability analysis (Bailey, Nicholson & Williams 1962; Maynard Smith 1968, 1974).
This combination of simple models and analytical techniques offers an extremely useful ready-reckoner in population analysis. But to answer more detailed questions about population behaviour, more detailed models are required, and one consequence of the process of elaboration is that analytical techniques become more difficult and sooner or later must be superseded by numerical simulation. Thus, in the second half of the paper, the analytical models are used as foundations for the development of a fully age-structured simulation model. The model specifies the correct order and duration of each of the life history events. Although comparatively complex, it is the simplest way of investigating, for example, the response of the adult population to changes in larval survivorship.
The more detailed model is also required to corroborate the results obtained with the simpler models. Simple descriptions can easily overlook features of the population biology which affect stability and equilibrium levels, such as the order in which life history mortalities occur (Solomon 1964; Wang & Gutierrez 1980; May et al. 1981).
Drawing together the two halves of the paper in the final discussion, ome general conclusions are reached, with reservations, on the statics and dynamics of th' A. aegypti population in Wat Samphaya.
SINGLE AGE-CLASS ANALYTICAL MODELS
The dynamics of the Bangkok population of A. aegypti have already been described in a single age-class model by Hassell, Lawton & May (1976). These authors compared the dynamical behaviour of twenty-four insect populations, most of which reproduce in discrete generations, using the difference equation
N(t + T) = N(t) (1 + aN(t))-/ (1)
where N(t) and N(t + T) are the population sizes in successive generations, and Ai is the finite rate of increase (net fecundity after lifetime density independent mortalities). For A. aegypti, a is inversely proportional to the number of larval breeding sites (water containers), and fl is the maximum slope of the relationship between mortality, expressed as a k-value, and log population size.
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A FIG. 1. The stability boundaries for models (1) ( ) and (2) (----) as defined by ) and P. The solid circle is the position of A. aegypti in Bangkok determined by Hassell, Lawton & May (1976), with their 95% confidence limits. The three horizontal lines have been obtained by
re-estimating fl, with 95% confidence limits, and by redefining the range within which A lies.
All three parameters, a, ,6, and )i, determine equilibrium population size, but local and global stability of the equilibrium depends only on ft and A. When the global behaviour is a stable point, the population may approach it either by monotonic or oscillatory damping. Using data given in Southwood et al. (1972), Hassell, Lawton & May (1976) estimated the values of ,f and AL for A. aegypti in Bangkok. The estimates are shown in Fig. 1 (which is a reproduction of their Fig. 2), with 95% confidence limits. The population is expected to exhibit damped oscillations on approach to equilibrium.
Four reservations should be attached to this result. First, A. aegypti is exceptional amongst the twenty-four populations in reproducing continuously. Generations overlap and the population is more appropriately described with a time-delayed differential equation than with a difference equation. The distinction may not be very important because, as noted by May (1973) and May et al. (1974), time-delayed models can produce a similar range of dynamical behaviour irrespective of whether the delays are included implicitly (difference) or explicitly (differential). Nonetheless, both of the new mosquito models discussed later in this section are written in differential equations.
A more important problem, partly a consequence of the fact that A. aegypti reproduces continuously, is that the rate of increase of the population, Ai, cannot be calculated from the data presented in Southwood et al. (1972). Southwood and colleagues summarized the immature life budget in monthly time-specific life tables. There is no direct relationship between the pupae counted in one month and the eggs counted in the next because the monthly life tables do not represent separate generations, because samples were not necessarily taken in consecutive months, and because the egg and pupal populations were sampled in different ways.
The third question concerns the choice of density-dependent function in simple population models. The function used in model (1) is originally due to Hassell (1975). It is one of a range of density-dependent functions (see Bellows (1981) for a brochure) that have been used in discrete generation population models. A short preamble here on its relationship with two other functions, well-known in discrete generation models, will set the scene for the development of a general, continuous time model in the next section.
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The first is a one-parameter exponential function used, for example, by May (1974), MacDonald (1976) and O'Neill, Gardner & Weller (1982) in the model
N(t + T) = AN(t) exp (-aN(t)) (2)
to describe populations exhibiting cyclic or chaotic behaviour. It is a special case of model (1), arising when cr - 0 and f/ -- oo, and aft = constant = a (Hassell, Lawton & May 1976). The stability of a population modelled by eqn (2) depends only on A, as shown by the vertical lines in Fig. 1. This one-parameter exponential function is mentioned here because it has been suggested for general use in a continuous time analog of eqn (2), which is described later.
In general, two-parameter density-dependent functions are much more flexible (they can describe a wider variety of data sets), yet still simple enough to allow analytical treatment of the population model in which they are used. One of the most versatile has been used by Thomas, Pomerantz & Gilpin (1980) and by Bellows (1981). It is a straightforward elaboration of the exponential form in eqn (2) producing
N(t + T) = AN(t) exp (-aN(t)A) (3)
This function also has generic appeal as the parent from which most of the other commonly used forms can be derived (Bellows 1981). As with model (1), the regions of
dynamical behaviour are defined by AL and /3; they are shown graphically in the paper by Thomas et al. (1980). There is a stable equilibrium if f/ In A/ < 2, which is approached monotonically if /3 In /A < 1. It will become clear in the following sections that the choice between one- and two-parameter density-dependent functions can make an important difference to the predicted behaviour of the A. aegypti population in Wat Samphaya.
The fourth reservation is that the / value of 1-9 used by Hassell, Lawton & May (1976) is too high. They considered only k1 measured by Southwood et al. (1972) (Fig. 2a) to be density-dependent. Closer inspection of the data, however, reveals that mortality during the second and third instars (k2) is also likely to be density-dependent (Fig. 2b), with a
slope much less than 1. So the true value of /f should represent the net effect of an over- compensating mortality followed by an undercompensating one, and will be nearer to 1 than 2 (see below). As well as reducing the value of /f, the assertion that k2 is density- dependent will increase the value of Ai, which includes all density-independent life history mortalities.
Each of these points is dealt with in the following sections. I first describe single age- class models written in continuous rather than discrete time, which are new to the world of mosquitoes. I then examine the suitability of different density-dependent functions for describing the Bangkok data and estimate parameter values in the models. This leads to a
comparison of the performance of the different models, and an assessment of the size and stability of the A. aegypti population in Wat Samphaya.
Continuous time, single age-class models
Gurney, Blythe & Nisbet (1980) distinguished between time-lagged density-dependent recruitment to the adult population and instantaneous adult death rate in describing the
FIG. 2. The four mortalities measured by Southwood et al. (1972) plotted as functions of logl0 population size: k, and k2 are assumed to be density-dependent, and the solid lines are the least squares regressions on the two-parameter exponential function (see Table 1). k3 and k4 are
density-independent with means 0.35 and 0.369.
Here P is the maximum per capita daily egg production rate (corrected for egg to adult survival), 1/a is the size at which the population reproduces at its maximum rate, and 6 is the per capita daily adult death rate. The foremost assumption made by this model (see Gurney, Blythe & Nisbet (1980) for others), as with model (2), is that a one-parameter exponential function is adequate to describe density-dependent mortality. This may be true for Nicholson's cycling blowfly population, but as noted above, a two-parameter exponential function (Thomas, Pomerantz & Gilpin 1980; Bellows 1981) will be more flexible. Thus,
dN(t) = - PN(t - T) exp (-aN(t - T)/) - -N(t) dt
(5)
is a more versatile form of Gurney, Blythe & Nisbet's (1980) model. When T is the generation time, the stability properties of model (5) are similar, though not identical, to those of model (3) with A/ = P1S. The stability analysis of model (5) is outlined in the Appendix. When ft = 1, model (4) is recovered and stability depends only on P/6 and T (c.f. model (2)).
1.6- (a)
1.4-
1-2-
K1
30
251
0
I 0
O-C
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All the above models require a minor alteration before they can be used to describe A. aegypti because, for this insect, density-dependent mortality acts on the eggs, not the adults. The adjustment affects the calculation of equilibrium population size, though not stability. Thus, converting model (5), the dynamics of the adult population, A, are described by
where E is the egg production rate of adults, not corrected for density-independent survival between the egg stage and adulthood. The equilibrium adult population size is then
A* [In (P5)/c]a E
Alternatively, if density dependent mortality is described by Hassell's (1975) function in eqn (6),
A* (Pb3)" - 1 (8) aE
Describing density-dependent mortality and estimating parameter values for Aedes aegypti in Bangkok
The estimates of parameter values given in this section have been obtained from data published in Sheppard et al. (1969) and Southwood et al. (1972), and from unpublished reports of the World Health Organization's Aedes Research Unit (ARU) in Bangkok.
In the foregoing discussion, I have mentioned three functions which could be used to define the density-dependent mortalities suffered by larval A. aegypti in the Wat; they are the one- and two-parameter exponential, and the two-parameter form of Hassell (1975). In fitting these' functions, I have assumed that k1 and k2 (Fig. 2) have no significant density-independent components. This is probably not true for k, at least because a proportion of eggs laid will never be submerged in water, and of those which are submerged, some will be infertile. However, if the density-independent components are extracted from k1 and k2, following Bellows' (1981) suggestion, no density-dependent component remains! These data do not pass any of the statistical tests for density- dependence (e.g. Varley, Gradwell & Hassell 1973); the only evidence for density- dependent mortality comes from visual inspection of the data (Figs 2 & 3). I have also assumed that there are no interactions between larval stages. Though interactions between age cohorts, either negative (competition) or positive (dead larvae recycled as food), seem likely from laboratory experiments described elsewhere (Dye 1982a, b), there is no evidence, from plots of k1-k4 against the numbers of larvae in different instars, that they occurred during the Bangkok study.
Table 1 gives the results of fitting all three functions to the data shown in Fig. 2(a) and (b), and to the sum of k1 and k2 as a function of logl0 eggs (Fig. 3), using a least squares technique available in BMDP (1979). The goodness-of-fit of the models to the data is determined by r2, the per cent of variance explained. The first column in Table 1 shows the variable performance given by the one-parameter exponential. In one case, r2 < 0, i.e., the residual sums of squares about the regression line is greater than the sums of squares about the mean.
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FIG. 3. The net density-dependent mortality larval life for A. aegypti in Bangkok, from data in Southwood et al. (1972) (see Fig. 2). The solid line is the least squares regression on the two-
parameter exponential function, with c12 and f12 as in Table 1.
TABLE 1. Estimates of parameter values, their standard deviations (n = 9), and the goodness-of-fit of three density-dependent functions to data from Southwood et al.
Hassell's function gave the best description of the data for k1 and k, + k2, but a and ft could not be estimated for k2. Using BMDP, the value of ft for kl is somewhat lower than the estimate of 1.9 given by Hassell, Lawton & May (1976). Of greater importance though is the fact that ft for k, + k2 is very much lower with a value near one. The inference is that overall density-dependent mortality is almost exactly compensating rather than over- compensating. As anticipated, the two-parameter exponential was most flexible: estimates of a and ft could be obtained for all three data sets. For k, and k2, these values can be inserted directly into the simulation model described in the second half of the paper. However, for k1 + k2, since mortality is acting on the egg population, N must be multiplied by the egg production per adult before density-dependent mortality can be applied in the single age-class models.
253
0
0 0
0 1 1 1 1 1 1
14-
1-2-
1.0
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The data in Fig. 2(a) and (b) and Fig. 3 could, of course, have been described with linear rather than curvilinear functions for density-dependence. Indeed, linear regression gives a better fit to the data: r2 values for kl, k2 and kI + k2 are 65.0%, 53-0% and 49.0% respectively (c.f. Table 1). However, because curvilinear functions are more appropriate for use in population models (May et al. 1974; Hassell 1975), I proceed with the best of the three described here.
In order to assess the stability and equilibrium levels of the Bangkok population using the models described above, estimates of ,A, T, P, and 6 are also required. For each of Ai, and P, since it is not possible to obtain a good estimate of the mean with confidence limits, an overestimate and an underestimate are given. These are useful in defining limits within which the true mean values probably lie.
First, F is calculated, the expected mean number of eggs produced by an adult during its lifetime, knowing the average time between the emergence of a female and her first blood meal, the length of the gonotrophic cycle, female fecundity, daily adult survivorship, and the sex ratio.
Most newly-emerged females will have taken a blood meal by 36 h in the field (unpublished ARU report, May 1970). For convenience, I assume that the time to the first blood meal is 37.7 h or 1.57 days. This is the time step used in the simulation model described in the second half of the paper; the approximation will make little difference to the calculated reproductive rate, and has the advantage that direct comparisons can be made between analytical and numerical results. Sheppard et al. (1969) were unable to estimate the length of the gonotrophic cycle from their mark-recapture data but guessed it to be about 3 days at an average temperature of 28.5 ?C. This is consistent with the findings of a later study carried out elsewhere in Bangkok (unpublished ARU report, May 1970), with the 4 days observed by McClelland & Conway (1971) at 24 ?C in Dar-es-Salaam, and with 2.2-3.8 days recorded by Hervy (1977) during the hot season in Upper Volta. I take the duration of the gonotrophic cycle to be 3.14 days, or two time steps in the simulation model.
Figure 4 shows that adult daily survival rate is variable (ranging from 0.63 to 0.88) but not density-dependent. Between July 1966 and July 1967 there are approximately four clockwise cycles; changes in survival rate drive changes in population size and not the reverse. Sheppard et al. (1969) could not explain the fluctuations in daily sutvival rate; for example, they were not correlated with temperature or rainfall. The mean rate, corrected for movement and the permanent loss of some marked individuals, was 0.88. Thus 6 - 0.12. The mean survivorship over 1.57 days, s = 0.818, is used in calculating F.
The final component of F is R, the average number of eggs laid by a female during each gbnotrophic cycle. In the March 1968 Progress Report of the ARU, five bimonthly averages are given for the number of eggs which could be laid (including eggs retained, counted by dissection) by wild caught females in Wat Samphaya. The 10 month average (? S.E.) is 57.8 + 3.35. The number of gonotrophic cycles a female is likely to complete in her lifetime is now
s s(s + S3 + S +...)=
and the total lifetime fecundity per adult is Rs2
F = (9) 2(1 - s2) (9)
where the sex ratio is 1: 1, as observed by Sheppard et al. (1969). The mean value of F is 58-4.
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0'62 1 I I I I I I 1 2-7 28 2'9 3.0 3.1 3-2 3-3 3-4
log1o population size FIG. 4. The relationship between daily adult survival rate and logl0 adult population size. The
data are taken from Tables 13 and 14 of Sheppard et al. (1969).
To obtain A, F must be corrected for density-independent mortalities during immature life. Of the four mortalities measured by Southwood et al. (1972), k1 and k2 are probably density-dependent, as noted above, and k3 and k4 are variable and density-independent (Fig. 2(c) & (d)). On average, only 19 1% of larvae survive from mid third instar to mid pupa. If the density-independent components of k, and k2 are ignored, as well as mortality during the latter half of pupal life, an overestimated mean A of 11.2 is obtained. This is somewhat higher than the value of 10.6 given by Hassell et al. (1976), though well within the bounds of their 95% confidence limits.
A second estimate of A can be obtained from the recorded changes in adult population size in the Wat (Sheppard et al. 1969). As the population increases from its lowest levels, it will be subject to least density-dependent mortality, though regulatory mortalities will not be entirely absent. Assuming the adults of one generation do not overlap with the adults of the following generation, the maximum observed increase in population size over the generation time, 27 days (egg to adult 18.6 days (Southwood et al. 1972), adult 8.5 days (Sheppard et al. 1969)), will give an underestimate of A. Figure 5 shows five periods of population increase during 12 months. The highest rate of increase occurred during the last period, beginning in April 1967, when the adult population multiplied 3.1 times in 27 days. So the true value of A could lie anywhere in the range 3.1-11 2.
The range of P can now be obtained from the range of AL; A is the number of offspring produced by each adult which survive to adulthood in the absence of density-dependent mortality. In the Bangkok population, these offspring are produced over 8.5 days on average, so the daily rate, P, ranges from 0 367 to 1 31.
The size and stability of the Aedes aegypti population in Bangkok
The number of adult mosquitoes in the Wat Usually, the aim in mosquito control is to reduce A*, the equilibrium adult population
size. The relationship between A* and the net reproductive rate, P/l, is shown in Fig. 6(a),
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Months FIG. 5. The harmonic mean estimates of daily population size of adult A. aegypti in Wat
Samphaya. From Table 14 of Sheppard et al. (1969).
400- (
~0 C)
.0
0
-0
.0
-a
300-
200-
100 -
a)
Range within which P/8 lies in the Wat
2 4 6 8 10 12
0 2 4 6 8 10 12
P/8 FIG. 6. The equilibrium population size of adult A. aegypti in Wat Samphaya as a function of the reproductive rate, P/b. P/b is varied by changing (a) P (and hence E) with 6 = 0- 12 or (b) 6. Lines 1, 3 and 5 with Hassell's (1975) function; lines 2, 4 and 6 with the two-parameter exponential function for density-dependent mortality. Lines 3, 4: P = 0-367; lines 5, 6: P = 1-31.
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for models with both of the two-parameter functions describing density-dependent mortality. Here, A* is calculated by holding the death rate (6) constant, and changing the birth rate (P and E). Although the two functions fit the data equally well (Table 1), they differ markedly in their predictions of the population's response to changes in the birth rate, with Hassell's offering the more pessimistic view for mosquito control.
They agree, however, that the number of adult A. aegypti in the Wat is unlikely to exceed 400. This is much lower than the average population size estimated by Sheppard et al. (1969), 1120. But considering the confidence intervals associated with a12 and /12, and the way in which A* depends on ,f in eqns (7) and (8), the discrepancy is not remark- able. For example, from eqn (7), with a12 as estimated and /12 lower by one standard deviation, the equilibrium population size of adults is expected to be around 100 000 when P1/ = 10. Clearly, accurate assessment of A* requires, above all, good estimates of fl.
In Fig. 6(b), A* is again plotted as a function of P/1, but now the death rate is adjusted, with the birth rate constant. A comparison of Fig. 6(a) and (b) shows that, over the range of possible P1i/ in the Wat, the adult population is more effectively suppressed by increasing the death rate than by decreasing the birth rate, especially if P is small and if density-dependent mortality is described by Hassell's function. This is so because whenever density-dependent mortality occurs somewhere in the life cycle of an insect pest, an attack on the offending stage must be more effective than a similar attack elsewhere in the life cycle. For A. aegypti, density-dependent mortality occurs after the egg stage and before adulthood.
Finally, equilibrium adult population size should be directly proportional to the number of larval breeding sites because all density-dependent mortality occurs early in larval life. Thus, a in eqn (8) may be inversely proportional to the number of breeding sites (water containers), though a in eqn (7) is not a simple measure of the number of these sites. According to eqn (8), between minimum and maximum P in the Wat, fractional reduction in the number of larval breeding sites is more effective in suppressing A* than fractional reductions in the birth rate.
Stability The complete horizontal lines in Fig. 1 represent the range of reproductive rates within
which the true value is likely to lie. For A. aegypti, model (2) describes a population which can exhibit damped oscillations or limit cycles depending on the reproductive rate. Models (1) and (3), however, which have an extra parameter defining density-dependence, are much more stable, and only monotonic damping is possible following a small perturbation. For model (1) this is clear from the contours in Fig. 1; for model (3) the maximum value of ft In A is 0 73; quasi-cyclic behaviour begins at ,3 In A = 1. Similarly, for the continuous time models the addition of an extra parameter to the density-dependent function has a strongly stabilizing effect. Local stability analysis of model (5) shows (see Appendix) again that only monotonic damping is possible, and given the similar behaviour of models (1) and (3), it is clear that model (5) with density-dependent mortality described instead by Hassell's function would also allow only monotonic damping.
These analytical results were confirmed by numerical simulation of model (6), either setting /3 = 1, or using the best estimate given in Table 1. Figure 7(a) shows, that, when ft = 1, the population undergoes a large change in dynamical behaviour between the lower and upper limits to reproductive rate, ranging from a small overshoot of equilibrium to limit cycles. In contrast, with ft = 0.302, only monotonic damping is possible, as Fig. 7(b) shows.
C. DYE 257
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FIG. 7. Simulations with model (6). (a) / = 1, ct= 5.73 x 10-4. (b) = 0.302, a= 0.229. Each run starts with ten egg-laying females.
Thus, model (6) so far gives the most reliable assessment of A. aegypti population behaviour, but the very different dynamical characteristics of models with one- and two- parameter density-dependent functions prompts the question of whether the addition of more life history detail to model (6) will add important qualifications to the results it has already produced. This is one reason for developing the simulation model described in the next section.
A SIMULATION MODEL FOR AEDES AEGYPTI IN BANGKOK
I now proceed with a more detailed analysis of population behaviour using computer simulation. Model (6) is elaborated by dissecting out components of P, , ac and f,, by specifying the order and duration of life history events, and by monitoring the immature as well as the adult population. The functions of the model are (i) to substantiate the results obtained in the first half of the paper, (ii) to extend them by investigating the sensitivity of a multi-age-class population to single rather than amalgamated demographic parameters, and (iii) to erect a flexible framework to which more information can be added as it becomes available.
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In general structure, the model is similar to several others which have been used to describe overlapping generations in mosquito populations (e.g. Miller, Weidhaas & Hall 1973; Dietz 1974). In detail, it is closely tailored to the population of A. aegypti in Wat Samphaya. I have first considered what data are available (Sheppard et al. 1969; Southwood et al. 1972; unpublished World Health Organization reports) and then structured the model accordingly.
First, a suitable time step, t, is chosen by examining the transition times observed between stages in the field. Unfortunately, average stage durations have been accurately measured only for immature A. aegypti in Bangkok (Southwood et al. 1972). The time step has been selected using this information alone, although it will be clear later that the choice of t is not incompatible with the rough estimates available for adult transition times. The intervals required, first of all, are those between the midpoint of one stage and the midpoint of the following stage; these are the intervals over which Southwood et al. (1972) measured the immature mortalities shown in Fig. 2. From their data, we can calculate that the four mortalities, k,-k4, occurred on average over 3.33, 3.21, 4.86 and 4.36 days. The length of t is the approximate common denominator of these figures which economizes on computer time, but which specifies the length of immature life with the required degree of precision. On balance, I have chosen to set (3-33,3.21) 2~ t and (4-86,4.36) - 3t. The best estimate of t is the value which minimizes
i.e. t = 1.57 (the sum of squared residuals is 0.187). To complete immature life, 2t are added for the first half of the egg stage and the second half of the pupal stage. Thus, the model assumes that each individual spends 12t = 18-84 days between egg-laying and adult emergence, compared with 18-57 days measured by Southwood et al. (1972).
The immature mortality rates per unit time are determined from the data presented in Fig. 2. The two-parameter exponential function is used to define density-dependent mortalities kl and k2, with estimates of and ,f given in Table 1. For the density-independent mortalities k3 and k4, mean values are used, that is, 0.35 and 0.369, or, as survivorship per unit model time, 0.765 and 0.754.
For the adult population, there is no information on male maturation rates and mating frequencies. I therefore assume that the number of matings is never a factor limiting egg production, and consider only females explicitly in the model. Their maturation rate, length of gonotrophic cycle, survivorship per unit model time and maximum average fecundity are all given above. The minimum average fecundity is calculated from the underestimated value of /, from mean k3 and k4, and then from eqn (9). It is 16.2 eggs per gonotrophic cycle. The complete model is summarized in a flow diagram in Fig. 8.
Simulations
First runs with the model began with ten egg-laying females and monitored the ascent to equilibrium. Figure 9(a) shows the population trajectories at minimum and maximum average fecundity, R, and confirms the result obtained earlier with the single age-class model (6) (Fig. 7b): whatever the true value of R, equilibrium is approached monotonically. Figure 9(b) shows that the larval population behaves in the same way.
Figure 9(a) also shows that equilibrium adult population size, A*, is less sensitive to changes in R, between minimum and maximum R, than expected from analysis of model (6) (Fig. 6), and that A* is lower than anticipated by model (6) over all R. The explanation
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FIG. 8. Flow chart for the simulation model. dim = density independent mortality, ddm = density dependent mortality. Arrows indicate direction of program flow which is opposite to the
direction of mosquito development.
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/ , I I / I I I I, 0 100 200 300 400 0 100 200 300 400
Model time units (x 1 57 days)
FIG. 9. Trajectories of adult and larval populations, starting with ten egg-laying females: (a) adults, R = 16.2 (lower line) or 58.7 (upper line); (b) larvae with R as in (a); (c) and (d)
adults, R = 16.2 with O1% uniformity distributed random noise on /,i and s5 respectively.
is that /,8 and /,2 acting sequentially in the simulation model impose more severe density-
dependent mortality than f,2 used in model (6). In contrast to Fig. 9(a), Fig. 9(b) indicates that L* is strikingly sensitive to changes in R.
A more detailed comparison of relations between L*, A* and R is made in Fig. 10(a). Note that although the rate of change of A* with R is smaller than the rate of change of
L* with R between minimum and maximum R, the reverse will be true if R is reduced
below its minimum value. An indication of the sensitivity of larval and adult populations to 10% changes in other
parameters, in comparison with R, is given in Table 2. The entries in the Table are relative
changes in the equilibria pictured in Fig. 9(a) and (b). Because the loss of individuals is more conveniently expressed as a survival rate (a large number) than as a death rate (a small number) in the simulation model, the importance of this loss over the birth rate, already apparent from eqns (7) and (8), is magnified. Both A* and L* are most sensitive to
changes in density-independent adult survivorship, s^, particularly adults when fecundity is low. Figure 10(b) shows in detail how A* depends on s5 at high and low R. In addition, Fig. 9(c) and (d) show that adult population size is comparatively sensitive to random fluctuations about the mean value of s5 when R is low. Taking these observations together with the pattern seen in Fig. 4, it is not surprising that if the anual mean value of s5 is
replaced in the model by the monthly averages given in Sheppard et al. (1969), a faithful
reproduction (Fig. II11) of the observed pattern of population change (Fig. 5) is obtained: five distinct peaks in population size between August 1966 and July 1967, with fewer adults from November to January. The similarity between Figs 5 and 11 arises because
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FIG. 10. Equilibrium analysis with the simulation model.
larval mortality is strongly density-dependent (see Varley, Gradwell & Hassell 1973, p. 91, for another example of this). In a similar analysis, Miller, Weidhaas & Hall (1973) also identified s5 as a key parameter.
In reality, the winter reduction in the number of adults in the Wat may also be due to reduction in late larval survivorship at that time of the year. Southwood et al. (1972) found that, in the following winter between November 1967 and February 1968, k4 (Fig. 2) was the key larval mortality. It was low in October and February, but high in November and December (it was not measured in January).
Finally, amongst the parameters describing density-dependent mortality, it is clear from Table 2, as it was from eqns (7) and (8), that accurate estimation of ft is more important than accurate estimation of a.
In summary, the simulation results are broadly in agreement with those produced by the earlier analytical models.
DISCUSSION
The population dynamics and vectorial capacity of Aedes aegypti
The most appropriate analytical model for A. aegypti is the continuous time formulation (6). It permits overlapping generations, and includes re-estimates of parameters in a flexible
262
*0
0~ a
1000 300 -
200 -
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1966 1967 FIG. 11. Changes in adult population size in the Wat during 1966-67 anticipated by the simulation model. R = 578, s5 varying as in Table 17 of Sheppard et al. (1969) (means of two estimates for each 15-day period), density-dependent mortalities k1 and k2 described by the two-
parameter exponential function (Table 1), other parameters as given in the text.
function describing density-dependent mortality. Although an improvement on model (1), the conclusions reached with model (6) differ from those of Hassell, Lawton & May (1976), not because of the structural differences between the two models, but because of the different parameter estimates.
Both model (6) and the simulation model suggest that the Bangkok population is monotonically rather than oscillatorily stable. This correction of the result of Hassell, Lawton & May (1976), however, only reinforces their conclusion that, although simple models can display exotic dynamical behaviour, the parameter values at which they do so are not characteristic of animal populations. Indeed, Thomas, Pomerantz & Gilpin (1980) have argued that group selection might lead to combinations of growth rate and density- dependence which place populations well within the bounds of locally stable parameter space. This gives them a safety margin with which to weather other destabilizing forces, such as seasonal changes in adult survivorship (Sheppard et al. 1969), which can induce marked fluctuations in population size (Fig. 11).
If the rationale behind the analysis of mosquito populations is disease control, then we need to see how changes in the parameters of the population model affect vectorial capacity, which is proportional to A*P"/-In P (Garrett-Jones 1964; Clements & Paterson 1981). Here p is the probability of the mosquito surviving through one day and n is the length (days) of the extrinsic cycle of the virus in the mosquito. Clearly, vectorial capacity is outstandingly sensitive to changes in adult survivorship, both via the term P"/-ln P, and because of the pronounced concomitant effect of adult survivorship on adult population size (Table 2). Changes in any of the other parameters can affect only A*, and none are
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The entries are relative changes ((new - old)/old) in the equilibria shown in Figs 9(a) and (b); upper line + 10%, lower line -10% for each parameter. s5: adult survivorship, 4: late larval survivorship, as and ,Bs describe density-dependent mortality, and R is eggs laid per female in each gonotrophic cycle.
as significant as adult survivorship. In decreasing order of importance they are: S4, late larval survivorship; a a function of the number of larval breeding sites; and R, fecundity.
The fact that adult population size is insensitive to changes in fecundity infers that the release of sterile males in Wat Samphaya would be ineffective in suppressing the population (Figs 6 & lOa), unless R is somewhat near its minimum estimated value, or unless large changes in R could be achieved. This is exactly the result obtained by McDonald, Hausermann & Lorimer (1977) with a population of A. aegypti at the Kenya coast. They found that adult males released with heterozygote translocations could, by mating competitively with wild females, impose up to 64% sterility on the egg population. But this had no effect on the rate at which pupae were produced, presumably because of compensating larval mortality.
When using sensitivity analysis as a tool in mosquito control, it should be remembered that not all population parameters will be equally amenable to change. Here adult population size is also sensitive to changes in late larval survivorship (Table 2), and it may in practice be easier to treat the larval than the adult population. Thus, Bang & Pant (1972) carried out a successful field trial with Abate larvicide in Bangkok. Sensitivity analysis must therefore be complemented by some assessment of the cost-effectiveness of manipulating each, or a combination, of the parameters.
Shortcomings of the analysis The errors associated with the estimates of adult survival rate from mark-recapture
data (Sheppard et al. 1969), and of late larval survivorship (Southwood et al. 1972), are unknown. The confidence limits associated with estimates of a and ,6, however, are known to be large (Table 1).
In addition, the precise relationship between A* and R depends on the choice of function describing density-dependent mortality (Fig. 6). Although the two-parameter exponential function was used in model (6), if the number of eggs per unit larval food are dispersed amongst water pots according to the negative binomial distribution, then Hassell's (1975)
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function may be more appropriate. De Jong (1979) demonstrated that, if mortality (expressed as a k-value) from egg to adult within identical patches could be described simply by k = aN, and eggs were distributed amongst patches in negative binomial fashion, then the mortality over all patches could be summarized by k = fl log (1 + aN). For A. aegypti, there is little information on the distribution of container qualities (food available per larva in water jars), on the distribution of eggs amongst containers, and on the relationship between density and mortality within (and hence over all) containers.
At least three further aspects of A. aegypti ecology merit investigation: (i) Notably absent from Table 2 are parameters which allow changes in the durations of
the life-history stages. They have not been included in the simulation model because, in the analysis by Miller, Weidhaas & Hall (1973), changes in the durations of either larval or adult stages had trivial effects on equilibrium levels. Thus, changes in the generation time due, for example, to seasonal variation in temperature are unlikely to be significant for population statics. Also, from the Appendix, because /f In (P/b) < 1, whatever the true value of P, the length of the generation time (included in r) cannot make any difference to the underlying population dynamics.
Nonetheless, improved versions of the models used here may have to allow for varying stage duration because generation time is likely to be density-dependent as well as temperature-dependent. A recurrent theme of earlier laboratory experiments (Dye 1982a, b), backed by observations of Southwood et al. (1972) in the field, is that larval life can be greatly extended when food per larva is in short supply. This may be important for two reasons. First, it is qualitatively clear that density-dependent development rate will contribute to population stability, and Nisbet & Gurney (1983) have demonstrated this formally. Secondly, dynamically varying stage duration could lead to systematic errors in the assessment of larval mortality. Southwood et al. (1972) calculated mortality by comparing numbers of larvae counted in consecutive instars. This comparison can only be made after the numbers have been corrected by dividing by the different instar durations. For illustration, if the mean stage duration is used throughout, then the numbers of larvae counted in each instar will be over- or underestimated depending on whether there are greater or fewer than the mean number of larvae present. The relationship between development rate and larval density is unknown for A. aegypti in Wat Samphaya, so the extent of such errors in the mortalities calculated by Southwood et al. (1972) cannot be assessed.
A less important assumption made in the simulation model is that all individuals in an age cohort take the same time to complete each life-history stage. Birley (1979) has described the distribution around mean development times for the immature stages of A. aegypti in Bangkok. Unfortunately, the information cannot be included in this model because the distribution around mean stage durations is not the same as the distribution around the mean time between midpoints of consecutive stages used here. If the distribution of development times in an age cohort is density-independent, then allowing for differences in development time between individuals will provide cosmetic treatment of model output, smoothing changes in population size, but is unlikely to make any qualitative difference to the underlying population dynamics.
(ii) Earlier work (Dye 1982a, b) has pointed to the importance of interactions between rather than within larval instars. Gilpin & McClelland's (1979) model allows interactions between age cohorts, but the models described here do not. If such interactions prove important in the field, models will have to include them explicitly, in the manner of, for example, May et al. (1974), Bellows (1982) and Tschumy (1982).
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(iii) From the data in Fig. 4, I have concluded that population regulatory processes do not occur during adult life. However, the consequences of larval competition for adult survivorship and fecundity, already revealed in laboratory studies (Bar-Zeev 1957; Moore & Fisher 1969; Saul, Novak & Ross 1980), have not been explored in a field population; neither has the impact of predators, such as retaliating hosts from which blood meals are taken (Klowden & Lea 1978; Waage & Nondo 1982), been thoroughly investigated.
Finally, it is worth emphasizing that the Bangkok study by the WHO Aedes Research Unit is the only attempt to investigate the life budget of A. aegypti in the field. This mosquito is a disease vector throughout the tropics; a comparative study of the bionomics of another population would obviously be of value.
ACKNOWLEDGMENTS
I wish to thank Dr David Rogers and Professor Michael Hassell for percipient comments on the manuscript. Professor T. R. E. Southwood kindly showed me the unpublished reports of the World Health Organization's Aedes Research Unit in Bangkok. The work was funded by a Natural Environment Research Council Studentship.
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To investigate local stability, we study the dynamics of a small perturbation, x(t), from equilibrium, N*. An approximate differential equation describing these dynamics is obtained by Taylor-series expansion of f[N] about the equilibrium point, i.e.
dx(t) [df 1* dt =-x(t) + x(t-T)[dN] (A2) dt LdN
ignoring terms of x2 and higher. Now,
f[N*] = PN* exp (-aN*A)
and
aN* 3 = In (P/b)
so
d] - 611 --b ln (P1/)] (A3)
and
dx(t) =- 6x(t) + 3x(t - T) [1 - / In (P/1)] (A4)
dt
x(t) and x(t - T) have solutions x(O) eAt and x(O) eAt- Trespectively, so by substitution,
A - -6 + 6e-AT[ 1 - / In (P/1)] (A5)
Letting A' = A/1, r = 6T and b - [ 1- / In (P/1)],
' = -1 + be-A' (A6)
The system is locally stable if A' has negative real parts, i.e. when
27r- cos-'(1/b) (A7)
(b2- 1)
If A' is complex the population trajectory is an oscillation. Exponential damping occurs when condition (A7) holds and when A' is real, i.e. when b > -exp (-r - 1) (Maynard Smith 1974).
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![Aedes aegypti pres[1] control preventiv de la enfermedadd de Dengueo](https://static.documents.pub/doc/80x56/568c3b901a28ab0235aaa206/aedes-aegypti-pres1-control-preventiv-de-la-enfermedadd-de-dengueo.jpg)
