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Pesticide Resistance: Strategies and Tactics for Management. 1986. National Academy Press, Washington, D.C. Population Dynamics and the Rate of Evolution of Pesticide Resistance ROBERT M. MAY and ANDREW P. DOBSON For a wide range of organisms exposed to insecticides or the like, the number of generations taken for a significant degree of resistance to appear exhibits a relatively small range of variation, typically being around 5 to 50 generations; we indicate an explanation, and also seek to explain some of the systematic trends within these pat- terns. We review the effects of insect migration to andfrom untreated regions and of density-dependent aspects of the population dynamics of the target species. Combining population dynamics with gene flow considerations, we review ways in which the evolution of resistance may be speeded or slowed; in particular, we contrast the rate of evolution of resistance in pest species with that in their natural enemies. We conclude by emphasizing that purely biological aspects of pesticide resistance must ultimately be woven together with eco- nomic and social factors, and we show how the appearance of pes- ticide resistance can be incorporated as an economic cost (along with the more familiar costs of pest damage to crops and pesticide application I. INTRODUCTION During the 1940s, around 7 percent of the annual crop in the United States was lost to insects (Table 11. Over the past two decades, this figure has risen to hold steady at around 13 percent. Much detail and some success stories are masked by the overall numbers in Table 1, but the essential message is clear: increasing expenditure on pesticides and the increasing application of pesticides have, on average, been accompanied by increased incidence of 170

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POPULATION DYNAMICS TABLE 1 Agricultural Losses to Pests in the United States 171 Percentage of Annual Crop Lost to Year Insects Diseases Weeds Total 1942-1950 7 11 14 32 (average) 1951-1960 13 12 9 34 (average) 1974 13 12 8 33 1984 13 12 12 37 SOURCE: Modified from Pimentel (1976) and May (1977). resistance, with the net result being an increased fraction of crops lost to insects. Indeed, the fraction of all crops lost to pests in the United States today has changed little from that in medieval Europe, where it was said that of every three grains grown, one was lost to pests or in storage (leaving one for next year s seed and one to eat). Beyond these practical worries, the appearance of resistance to pesticides illustrates basic themes in evolutionary biology. The standard example of microevolution in the current generation of introductory biology texts is industrial melanism in the peppered moth. This tired tale could well be replaced by any one of a number of field or laboratory studies of the evolution of pesticide resistance that would show in detail how selective forces, genetic variability, gene flow (migration), and life history can interact to produce changes in gene frequency. We believe such intrusion of agricultural or public health practicalities into the introductory biology classroom may help to show that evolution is not some scholarly abstraction, but rather is a reality that has undermined, and will continue to undermine, any control program that fails to take account of evolutionary processes. In what follows, our focus is mainly on broad generalities. This paper complements Tabashnik s (this volume), which deals with many of the same issues in a very concrete way, giving numerical studies of models for the evolution of resistance to pesticides by orchard pests. CHARACTERISTIC TIME TO EVOLVE RESISTANCE The discussion in this paper is restricted to situations where the genetics of resistance involves only one locus with two alleles, in a diploid insect. This is the simplest assumption to begin with. It does, moreover, appear to be a realistic assumption in the majority of existing instances where detailed understanding of the mechanisms of resistance is available. The stimulating papers by Uyenoyama and Via in this volume indicate some of the important complications that may arise when two or many loci, respectively, are in

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172 POPULATION BIOLOGY OF PESTICIDE RESISTANCE valved in determining resistance. We further restrict this discussion to a closed population, in which the selective effects of a pesticide act homo- geneously in space; this assumption will be relaxed in later sections. Following customary usage, we denote the original, susceptible allele by S. and the resistance allele by R; in generation t, the gene frequencies of R and S are p' and qua, respectively (with p + q = 11. The gentoype RR is resistant, SS is susceptible, and the heterozygotes RS in general are of intermediate fitness (but see below for discussion of exceptions). In the presence of an application of pesticide of specified intensity, the fitnesses of the three genotypes are denoted WRR, WRS, WSS: we assume WRR ' WRS ' wss. The equation relating the gene frequencies of R in successive generations is then the standard expression (Crow and Kimura, 1970~: ~_ WRRpt + WRSP'q' WRRp2 + 2WRSp,q, + Wssqt In the early stages of pesticide application, the resistant allele will usually be very rare, so that p, << 1 and q, ~ 1. The initial ratio ply, will, indeed, usually be significantly smaller than the ratio wRs/wRR or wss/wRs, so that to a good approximation equation 1 reduces to ~, ~. I (1) P. + 1 /P, - WRS/WSS ~ (2) Suppose the allele R is present in the pristine population at frequency pO. By compounding equation 2, we see that the number of generations, n, that must elapse before a significant degree of resistance appears (that is, before p attains the value pf ~ 1/2, for example) is given roughly by (pf/pO) ~ (WRS/WSS) (3) We define TR to be the absolute time taken for a significant degree of resistance to appear, and Tg to be the cohort generation time (Krebs, 1978) of the insect species in question. Then n = TR/Tg, and the approximate relation of equation 3 may be rewritten as TR ~ Tg ln(pf/pO)lln(wRslwss). I, ~. ~. . . (4) 111S tO ne emphasized that equation 4 is a rough approximation. In particular, if R is perfectly recessive, we have WRS = WSS, and equation 2 is an inad- equate approximation to equation 1; even here, however, equation 4 is telling us something sensible, namely, that TR is very long when R is perfectly recessive (taken literally, equation 4 gives TR ~ °°) Equation 4 shows that TR depends directly on the organism's generation time Tg, but only logarithmically on other factors. In particular, TR depends only logarithmically on (1) the initial frequency of the resistance allele, pO;

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POPULATION DYNAMICS 173 (2) the choice of the threshold at which resistance is recognized, pf; and (3) the selection strength, wRs/wss, which in turn is determined by dosage levels and by the degree of dominance of R. Elsewhere in this volume, Roush suggests that pO values may range from 10-2 to 10-13; this enormous range, however, collapses to a mere factor of six separating highest from lowest when logarithms are taken. Likewise, ratios of wss/wRs ranging from 10-~ to 10-4 or less all make similar contributions to the denominator in equation 4, which involves only the logarithm of this ratio. Table 2 sets out values of TR for a variety of organisms (insects, and parasites of vertebrates), under the selective forces exerted by various in- secticides or other chemotherapeutic agents. Table 3 (see p. 188) attempts a rough summary of the general trends exhibited in Table 2: we see that for the great diversity of animal life embraced by Table 2, TR lies in the sur prisingly narrow range of around 5 to 100 generations. We argue that such relative constancy of TR, despite enormous variability in pO and wRs/wss, is because TR depends on all these factors (except Tg) only logarithmically. We will return to the systematic trends exhibited in Table 2 and crudely sum- marized in Table 3, after the discussions of migration, density dependence, and other miscellaneous factors. The approximate expression for TR in equation 4 mixes factors that are intrinsic to the genetic system underlying the resistance phenomenon (such as Tg, pO, and the degree of dominance of R) with factors that are under the direct control of the manager (such as dosage levels). Comins (1977a) sug- gests a useful partitioning of these two kinds of factors. First, define the relative fitnesses of the genotypes RR, RS, SS, to be 1: w~-~:w. Here w is the fitness of the susceptible homozygotes relative to the resistant homozy- gotes; w essentially measures the relative survivial of wild-type insects (high dosage of pesticide implies low w). The parameter ,8 measures the degree of dominance of R: if R is perfectly dominant, ,8 = 1; if R is perfectly recessive, ,8 = 0; and in general, ,8 will take some numerical value inter- mediate between O and 1. Equation 4 can now be rewritten as TR = Tolln~llw'. (5) This separates the parameter w (which measures the selection strength as determined by the dosage level) from the parameter To (which conflates intrinsic genetic factors). The quantity To is defined as To = Tg ln(pf/po)/,8 (6) Parameters such as pO or ,B usually cannot be estimated, and To should be thought of as a phenomenological constant, to be determined empirically in the laboratory or in the field (Coming, 1977a). Beyond explaining the general trends exhibited in Table 2 and other similar compilations, equations 4 or 5 (or more refined versions of them) may be

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175 * I* * * * * * * * _ o ~_ en, _ ~* * _ ~ _ v Cot * ~UP * * * * * _ . .= ~ Cat S a ~S cut a, ~ - - Cat ~_ _ ~ _ ~ ~ _ O ~ ~ ~ 9 C . '3 3 9 - 9 C - ' ' 0 8 l 7~4 8 ~ ', ~T ~2 9 ~Cot ~

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176 ._ C) C CD Cal ·C~ o 4- ·4 ~ ARC En au o m ¢ EM Cal as ·C) An * * es O ~ ^-5 =* * * * * * * * * * ED O oo ~ ~ V) ~ ~ ~ ~ ~ ~ O 3 I ~ ~ ~ A oo o o C) _ g ·C~ ~ . Ha C C ~ ~ t`O r~ V}C~ ~ ~` D ~ =:.= ~ C ~ C ~ c: ~ ~ * * * * ~* * * * * * * * * * a~ C) .= _ 3. ,y -° ~ C~ ~ C~, - - C~ ._ ~_` . . .> ~o ~ - o · - - s~ ~ o 3 ~ ~ ~ ~ ~W ~ ~ t ~ ~ Cq o ~ ~ ~ ~ ~ ° 3 " r;~ =; m x C,) =0 ~. ~ ~ ¢ ~q .~ S: 5e . ~. ·~ ~ . . St ~ ~ ~: .c :: . . . C~ .~ S: 5: x Cq ~: ~W Ct Ct

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POPUL4TION DYNAMICS 177 used to make predictions about the way TR depends on pesticide dosage levels or on degree of pesticide persistence in specific laboratory studies. Some such work is discussed in the next section. The above ideas also apply to the back selection or regression to population- level susceptibility that may appear once a particular pesticide is no longer used. As discussed elsewhere (Coming, 1984), it is possible in principle that a pesticide may have cycles of useful life: the gene frequency of R first increases under the selection pressure exerted by use of the pesticide; even- tually R attains a frequency sufficiently high to produce a noticeable degree of resistance, and shortly thereafter the pesticide is discontinued as ineffec- tive; in the absence of the pesticide, usually wss > WRR, and selection will now cause the frequency of R to decrease. Applying equation 4, mutatis mutandis, to this back-selection process, we note that the time elapsed before the population is again effectively susceptible to the pesticide will depend on (1) the intrinsic fitness ratios wRR:wRs:wss, which measure the strength of back selection in the absence of pesticide; (2) the frequency of R when the pesticide is discontinued; and (3) how low a frequency of R is required before reuse of the pesticide becomes sensible. For factor 1 it has been shown that significant back-selection effects can indeed occur (Georghiou et al., 1983; Ferrari and Georghiou, 1981~; Roush, in this volume, estimates the rate-determining ratio wRs/wss to be in the range 0.75 to 1.0 for untreated populations. Even when demonstrably present, however, such back selection in the absence of a pesticide is typically weaker than the corresponding strengths of selection for resistance under pesticide usage, so that the denominator in equation 4 is smaller. For this reason alone, "regression times" will tend to be longer than "resistance times," TR. The influence of factor 2 is that regression will be faster if pesticide application is discontinued before the frequency of R gets too high. The possible complications discussed by Uyenoyama in this volume are more likely to arise when PR is relatively high, which gives an additional reason for prompt discontinuation of a pesticide to which resistance has appeared. For factor 3 we observe that in pristine populations the frequency of R may typically be around 10-6 to 10-8 (Roush, this volume). After use of a particular pesticide is stopped, resistance will be unobservable and effectively unmeasureable long before it attains levels as low as these pristine ones; when the frequency of R is around 10-2, the population could easily be considered to have regressed to effective susceptibility. Taking the above numbers as illustrative, we see that resistance to the recycled pesticide is likely to appear significantly more quickly than it did in the first instance (TR depends on ln(1/pO), so that TR is three or four times faster for pa = 10-2 than for pa = 10-6 or 10-~. In short, all three factors suggest that a population will usually take longer to recover susceptibility than it did to acquire resistance, and also that re . .

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178 POPULATION BIOLOGY OF PESTICIDE RESISTANCE sistance will probably reemerge significantly faster following reintroduction of the pesticide. These broad generalities need to be fleshed out by detailed studies of specific mathematical models, backed where possible by long- term laboratory studies of relevant pest-pesticide systems. MIGRATION AND GENE FLOW The above discussion assumed that pesticides would be applied uniformly to a closed population of pests. In the field, the next generation of pests will virtually always include some immigration from untreated (or more lightly treated) regions, and this flow of susceptible genes will work against the evolution of resistance. This is a particular instance of one of the central questions of evolutionary biology: under what circumstances will gene flow wash out the selective forces that are tending to adapt an organism to a particular local environment? Earlier thinking of a qualitative kind suggested that very small amounts of gene flow may be sufficient to prevent local differentiation, and that geographical isolation was usually necessary before local adaptation could lead to new races or species (Mayr, 19631. More recently, population geneticists have shown that the occurrence of local differentiation (or "clines" in gene frequency) depends on the balance be- tween the strength and the steepness of the spatial gradient of selection versus the amount and spatial scale of migration (Slatkin, 1973; Endler, 1977; Nagylaki, 19771. May et al. (1975) gives a brief review of migration theory and data. One illuminating study contrasts two examples of industrial me- lanism: BiStOn betUIaria iS relatively vagile and thus is predominantly in the melanic form over most of England's industrial midlands; individuals of GOnOJOntiS bid~entata move significantly less in each generation, leading to weaker gene flow and a patchy pattern of local adaptation with melanic forms predominating near cities and wild types predominating in the intervening countryside (Bishop and Cook, 19751. This academic literature is directly relevant to the problem of the evolution of pesticide resistance in the presence of migration. Comins (1977b) has given an analytic study of the implications for pesticide management, and Taylor and Georghiou (1979, 1982; Georghiou and Taylor, 1977) have pre- sented numerical studies of particular examples. What follows is an attempt to lay bare the essential mechanisms; the above references should be consulted for a more accurate and detailed discussion. To begin, suppose there is an infinite reservoir of untreated pests; within this untreated reservoir the gene frequency of R will therefore remain constant at the pristine value, which we denote by PR. In the treated region the next generation of larval pests will come partly from the previous generation of adults that have survived treatment (which tends to select for resistance) and have not emigrated, and partly from those among the previous generation of

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POPULATION DYNAMICS PR 179 R high R low M migrat ion ~_ - - ~ increasing FIGURE 1 The degree of pesticide resistance that evolves in a treated region in the presence of immigration from untreated regions in each generation: PR is the gene fre- quency of R in the untreated region, and m is a measure of the amount of migration (gene flow) as a ratio to the strength of selection. This figure abstracts the more complex and more detailed results of Commons (1977b), and is discussed more fully in the text. untreated (and thus, largely susceptible) adults that have immigrated into the treated region. As discussed by Comins (1977b) and others, we assume it is the larval stage that damages the crops. As shown in detail by Comins (1977b), the rate of evolution of resistance in the treated region will, under the above circumstances, depend on (1) the gene frequency of R in the untreated reservoir, PR; (2) the degree of dom- inance of R. as measured by the parameter ,8 of equation 6 (actually, Comins uses a parameter h for arithmetically intermediate heterozygotes, rather than ,8 for geometrically intermediate heterozygotes, but this is an unimportant detail); and (3) the magnitude of migration in relation to selection, as mea- sured by a parameter m. Specifically, the migration/selection parameter m (Coming, 1977b) is defined as: m = r/~1 -r)~1 - wit. (7) Here r is the migration rate (i.e., the fraction of adults in a given area that migrate rather than "staying at home"), and w measures the strength of selection (w = wss/wRR, as in equation 51. If ,8 is low enough (R sufficiently recessive, corresponding very roughly to ~ ' 1/2), the treated region will settle to a stable state in which the gene frequency of R remains low, providing migration is sufficiently high (m sufficiently large) (Coming, 1977b). Conversely, for relatively small m-val- ues, selection overcomes gene flow and the system eventually settles to a resistant state (with PR close to unity). This situation is illustrated schemat- ically in Figure 1. In the treated region, the final steady state will be one of resistance or continued susceptibility, depending on the strength of migration relative to selection, as measured by m. There is a fairly sharp boundary between these two regions (indicated by the hatched line in Figure 11; the boundary depends weakly on the magnitude of PR, with slightly higher gene

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180 POPULATION BIOLOGY OF PESTICIDE RESISTANCE flow (higher m) being required to maintain susceptibility if PR is higher. Comins shows that there can, in fact, be two alternative stable states for m- values close to the fuzzy boundary in Figure 1, but we suppress these elegant and rather fragile details in favor of the robust generalities shown schemat- ically in Figure 1. For ,B-values approaching unity (relatively dominant R), the treated regime will eventually become resistant no matter how large the gene flow. Even here, however, TR can be very long if m is relatively large (Coming, 1977b). More generally, the untreated region will be finite. The situation is now more symmetrical, with preponderately R genes migrating out from the treated regions into the untreated ones at the same time as preponderately S genes are flowing into the treated regions. The net outcome is that the gene fre- quency of R in the untreated regions, PR, will slowly increase. As indicated in Figure 1 (by the vertical trajectory from point a to point b), for any specified value of m such increase in PR will in general eventually cause the treated region to move sharply from susceptibility (low R) to resistance (high R). Thus, in the real world, resistance is always likely to appear in the long run. Its appearance can, however, be delayed by management strategies that keep m relatively high. Such strategies include maximizing the area of un- treated regions or refugia, and keeping the dosage level as low as feasible in treated regions: both of these actions work toward higher m-values. In some situations it could pay to introduce susceptible adult males following treatment, which could enhance the gene frequency of S in the next generation without producing any additional pest larvae. These analytic and numerical insights have been corroborated by laboratory experiments on Musca domestica exposed to dieldrin at various dosage levels and with various levels of influx of susceptibles (Taylor et al., 1983~. As suggested by the mathematical models, the onset of resistance occurred sharply and at a time TR that depended in a predictable way on dosage and immi- gration levels. It would be nice to have more laboratory studies of this kind. On the other hand, one should not place too much reliance on such laboratory studies, because they unavoidably fail to include many of the density-de- pendent mortality factors that are important in nature. This leads us into the next section. DENSITY DEPENDENCE AND PEST POPULATION DYNAMICS Density-dependent effects can enter at any stage in the life cycle of a pest. Such complications can be dissected with standard techniques, such as k- factor analysis (Varley et al., 1972~. For simplicity the main density de- pendence is assumed to act on the adult population, N' in generation t. Such nonlinearity, or density dependence, in the relationship between the popu

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POP UL4TlON DYNAMICS 183 in the population dynamics of the pest. These ideas are developed more fully and more rigorously by Comins (1977b). Another way of setting out the ideas encapsulated in Figure 4 is to observe that, other things being equal, resistance will appear more quickly in pop- ulations with overcompensating density dependence and more slowly in pop- ulations with undercompensating density dependence than in populations with perfect density dependence; that is, TR increases as the density-dependence parameter b of equation 8 decreases. Several studies have attempted to assess lo-values of insect populations in the field and in the laboratory (Hassell et al., 1976; Stubbs, 1977; Bellows, 19811. (These studies all use more complex models than equation 8, but the distinction between overcompensating and undercompensating density de- pendence remains clear and valid). Most, although not all, populations that have been studied in the field show undercompensating density dependence. Among these studies the field population exhibiting the most pronounced degree of overcompensation is the Colorado potato beetle, which elsewhere in this volume (see Georghiou) is singled out as notorious for the speed with which it has developed resistance to a wide range of pesticides. In contrast to field populations, most laboratory populations in the above surveys show marked overcompensation. This difference between field and laboratory pop- ulations probably derives from the many natural mortality factors that com- monly are not present in the laboratory; whatever the reason, this difference underlines the need for caution in extrapolating laboratory studies of the evolution of resistance into a field setting. Comins (1977b) gives an interesting discussion of the detailed dependence of TR on b and m. For b = 1, we simply have the results summarized in the preceding section. These amount to the rough estimate that, in the pres- ence of a high level of migration, TR(m; b = 1) = TR (0; b = 1) Migration/1 -wit. (9) Here TR(O; b = 1) is the time for resistance to appear in a closed population, and TR(m; b = 1) is the time for it to appear in the presence of migration; w is the selection strength, as defined earlier (equation 5~; and the factor labeled migration is a complicated term, involving m and other parameters, that measures the effects of migration. We see that TR(m; b = 1) will increase as selection becomes weaker (w larger), but that the dependence on w is more pronounced at low dosage (TR ~ °° as w ~ 1) than at high dosage (TR is roughly independent of w for w << 11. For b < 1, the expression for TR(m; by is more complicated than given in equation 9. Because undercompensating density dependence makes mi- gration relatively more important, TR(m; b < 1) is always greater than TR(m; b = 1) for given values of m and w. At low levels of selection (w ~ 1) the differences created by subsequent density-dependent effects are relatively

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184 POP UNCTION BIOLOGY OF PESTICIDE RESISTANCE 4 3 2 In 41) 1 ._ V, o o 4 b3 2 1 PESTS _ I 1 1 1 NATURAL ENEMIES J l - O 10 20 30 40 50 60 70 number of generations FIGURE 5 The number of generations taken for pesticide resistance to appear in species of orchard pests is contrasted with the corresponding patterns among their natural enemies (data from Tabashnik and Croft, 1985~. unimportant, but at high levels of selection (w << 1), density-dependent effects cause migration to assume increasing importance when b ~ 1. The result is that, for b < 1, TR is longest at low and high selection levels, and shortest at intermediate values of w. These theoretical insights of Comins (1977b) are concordant with the numerical simulations and laboratory experiments of Taylor et al. (1983) on flies with undercompensating density dependence. These authors found that (for a given level of immigration) resistance evolved fastest at intermediate dosage levels. POPULATION DYNAMICS OF PESTS AND THEIR NATURAL ENEMIES The propensity for pest species to evolve resistance more quickly than their natural enemies do has often been remarked (Tabashnik, this volume; Roush, this volume). Table 3 summarizes the trends for some groups of pests and their natural enemies, and Figure 5 presents detailed evidence for orchard

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POPUC4TION DYNAMICS 185 crop pests and their predators. Clearly, such systematic differences in the rate of evolution of pesticide resistance can cause problems. One reason for these differences might be that the convolution between plants and phytophagous insects has preadapted the latter to the evolution of detoxifying mechanisms, whereas this is much less the case for the natural enemies of such insects. Laboratory studies show that there are in fact no simple, general patterns of this kind, and that under controlled conditions the rate of evolution of resistance in prey and in predator populations depends on the detailed molecular mechanisms underlying detoxification (Croft and Brown, 1975; Mullin et al., 19821. This in turn has prompted a search for pesticides that may be less lethal for natural enemies than for pests (Plapp and Vinson, 1977; Rock, 1979; Rajakulendran and Plapp, 1982; Roush and Plapp, 1982), or even the release of natural enemies that have been artificially selected for resistance to specific pesticides (Roush and Hoy, 19811. An alternative explanation for the typically swifter evolution of resistance by pests than by their natural enemies lies in the population dynamics of prey-predator associations (Morse and Croft, 1981; Tabashnik and Croft, 1982; Tabashnik, this volume). Suppose a pesticide kills a large fraction of all prey and all predators in the treated region. For the surviving prey life is now relatively good (relatively free from predators), and the population is likely to increase rapidly. Conversely, for the surviving predators life is relatively bad (food is harder to find), and their population will tend to recover slowly. This argument can be supported by a standard phase plane analysis for Lotka-Volterra or other, more refined, prey-predator models. Such anal- ysis shows that, in the aftermath of application of a pesticide that affects both prey and predator, prey populations will tend to exhibit overcompen- sating density-dependent effects (essentially with b ~ 1), while predator populations will tend to manifest undercompensation (b < 11. Returning to the arguments developed in the preceding section and illustrated schematically in Figure 4, we can now deduce that, for a given level of migration and pesticide application, pest species (which effectively have overcompensating density dependence) will tend to develop resistance faster than will their natural enemies (which effectively have undercompensating density depend- ence). The detailed numerical studies of Tabashnik and Croft (1982) and Ta- bashnik (this volume) also make the above point, but in more detailed and specific settings. We think it is useful to buttress these concrete studies with the very general observation that pesticide resistance is likely to appear faster among pests than among their natural enemies, by virtue of the interplay between population dynamics and migration; in this sense, the phenomenon illustrates the general arguments made in the previous section. Other work in this area includes the numerical studies by Gutierrez and collaborators on management of the alfalfa weevil, taking account of pest

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186 POPULATION BIOLOGY OF PESTICIDE RESISTANCE population dynamics, natural enemies, and the evolution of resistance (Gu- tierrez et al., 1976; Gutierrez et al., 1979), and Hassell's (in press) inves- tigation of the dynamical behavior of pest species under the combined effects of pesticides and parasitoids. There is much scope for further work, both in the laboratory and with analytic or computer models. MISCELLANEOUS TOPICS This section comprises brief notes on a variety of factors that complicate the analyses presented above. Life History Details Throughout we have considered pests with deliberately oversimplified life cycles, in which pesticide application and density dependence acted only on one stage. Comins (1977a,b; 1979) indicates how the analysis can be ex- tended, rather straightforwardly, to a life cycle with n distinct stages (pupae, several stages of larvae, adults). The numerical models of Tabashnik and of Gutierrez and collaborators also include such realistic complications. High Dosage to Make R Electively Recessive As we noted earlier, if R is perfectly recessive, resistance will evolve much more slowly than is otherwise the case (Crow and Kimura, 19701. It has been argued that dosage levels high enough to kill essentially all het- erozygotes may thus slow the evolution of resistance by making R. in effect, perfectly recessive. This strategy, however, will work only if pesticide dosage can be closely controlled in a closed population (Coming, 19841. This is roughly the case for acaricide dipping of cattle against ticks, for example (Sutherst and Comins, 19791. In general, lack of close control and/or the immigration of pests from untreated regions is likely to render such a strate~v infeasible. Heterozygote Superiority cat There appear to be some instances among insects where the RS genotypes are more resistant to an insecticide than either RR or SS (Wood, 19811. The spotted root-maggot Euxesta notada may exhibit such heterozygous advan- tage in the presence of DDT or dieldrin (Hooper and Brown, 19651. Although familiar for rat resistance to warfarin, such heterozygous superiority raises questions that do not seem to have been discussed for pesticides directed at insects.

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POPULATION DYNAMICS 187 Pesticide Resistance Compared with Drug Resistance Resistance to antibiotics and antihelminths poses growing problems in the control of infections among humans and other animals. Reviewing recent work, Peters (in press) concludes that both high dosage rates and the use of drug mixtures may tend to retard the evolution of resistance. Drug admin- istration to humans and other animals often does permit close control in a closed population, such that these strategies have a chance to work (rather than be washed out by gene flow; see Life History Details, above). Pesticide Resistance Compared with Herbicide Resistance Herbicide resistance has usually been slower to evolve than pesticide re- sistance, even when the longer generation time of most weeds is taken into account (Gressel and Segel, 1978; Gressel, this volume). Gressel suggests that this is due to the presence of seed banks in the soil (corresponding, in effect, to gene flow over time instead of space) and to the lower reproductive fitness of resistant genotypes. Gressel and Segel's analysis (1978) leads to an expression tantamount to equation 4 for TR, but with the denominator replaced by: lnEwRs/wss] ~ lnE1 + (WRs/Wss)({Rs~lfss)(llTsoi~] (10) Here fRs/fss is the ratio of the reproductive success of the two genotypes, which may be 0.5 or less; Toil represents the number of years that a typical seed spends in the seed bank, which can be 2 to 10 years. These two factors can diminish the RR1SS selective advantage by an order of magnitude, leading to significantly longer TR The array of complications discussed above helps to explain several of the general trends set out in Table 3. ECONOMIC COST OF PESTICIDE RESISTANCE The foregoing discussion has dealt exclusively with biological aspects of the evolution of pesticide resistance. Such a discussion, however, only makes sense if embedded in a larger economic context. Some broad insight into the economic costs of pesticide resistance can be obtained by the following modification of a more detailed analysis by Comins (1979~. Agricultural costs associated with pests are of at least three kinds: the damage done to crops, the cost of pesticide application, and the more subtle costs arising from the need to develop new pesticides as the appearance of resistance retires old ones. To a crude approximation we may think of the parameter w (which measured the strength of selection in our previous analysis) as determining the fraction of the pest population surviving pesticide

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188 POPULATION BIOLOGY OF PESTICIDE RESISTANCE TABLE 3 Some Possible Trends in the Way TR/Ts (the number of genera- tions that elapse before resistance is noticed, as cataloged in Table 2) Depends on the Biological and Environmental Setting Generations to Resistance Organism Variation in Life History Parameter or Efficiency of Treatment 2 s 10 20 50 00 Gut Coccidia Gut Nematodes Mosquitoes House Flies Rats Cattle Ticks Phytophagous Insects Weeds Entomophagous Insects Tsetse Fly Mediterranean Fruit Fly Large proportion of population treated High population densities and strong density-dependent effects Asexual reproduction and high mutation rates Increasing mobility into and out of treated area Increasing proportion of lifetime fecundity prior to treatment Seed Banks Low population density and reduced contact rate between organisms and control agent NOTE: The first column sets a scale (measured logarithmically in generations); the second column places some organisms along this scale in a very approximate way; and the third column comments on some rough correlations between the time scale and life histories or treatment efficiencies. application; the cost of insect damage to the crop may then be estimated as Aw. Comins (1979) argues that application costs are likely to be related logarithmically to the fraction killed, whence these costs may be estimated as B ln(1/w). A and B are proportionality constants that can be empirically determined. Finally we need to estimate the amount of money that must be set aside each year such that after TR years, when resistance necessitates the introduction of a new pesticide, its development costs (C') will be met. If the set-aside money compounds at an annual interest rate 8, a standard calculation gives the average "cost of resistance" as C' [exp(~) - 11/Lexp(8TR) - 11. (This is a more realistic estimate of the cost than that used by Comins, 1979.) The total annual cost that pests pose to the farmer is thus Total cost = Aw + B in (1/w) + (STO)C/[exp(6TO/ln(1/w))- 11. (1 1)

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POP UL4TION DYNAMICS Lo _ tote I cost ~ ~" / ~ _ "/ ° 0.5 ~/'-_ o - dose I ever 0 _ - O ~ ~ I I -2 -1 O ~2 log(8TO ) 189 2 o In o O FIGURE 6 The solid curve shows the pesticide dosage (measured by ln(1/w)) that min- ~mizes the total economic costs associated with pests (crop damage, cost of pesticide application, cost of developing new pesticides as resistance renders old ones ineffective). The dashed line correspondingly shows the minimized total costs. The curves are based on equation 11, with the parameters A, B. C here having the representative values 1.0, 0.2, 0.2, respectively (in some arbitrary monetary units); the basic features of Figure 6 are not qualitatively dependent on these parameter values. Both dosage levels and total costs are shown as a function of the parameter combination bTo, which is essentially the ratio between the intrinsic time scale associated with the evolution of resistance and the doubling time of invested money (at interest rate b: for more precise definitions, see the text). Here equation 5 has been used to express TR in terms of the intrinsic time scale for resistance, To' and the selection strength, 1/w. The cost constant C is defined as C = C' [exp(~) - 11/~8To); in the limit ~ ~ O. C is essentially the insecticide development cost per year, C = C'/To. In accord with common sense, equation 11 says that as dosage levels increase (that is, as w decreases), the cost associated with pest damage to the crop decreases, but the cost of pesticide application increases, as does the cost associated with developing new pesticides (because this task becomes more frequent). For any specific set of values of A, B. C, and bTo, some intermediate level of w (between O and 1) will minimize the total cost. Figure 6 shows this optimal dosage level (solid line) and the associated total cost (dosage + application t pesticide development; dashed line) as a function of BTo for representative values of A, B. and C. For a combination of low interest rates and/or intrinsically short times to evolve resistance (8To << 1), the optimum strategy suggests relatively low dosage rates (and the lowest possible total cost is necessarily relatively high). Conversely, if BTo

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190 POPULATION BIOLOGY OF PESTICIDE RESISTANCE 1, optimum dosage rates are relatively high (and total costs are relatively low). In other words the right-hand side of Figure 6 corresponds to characteristic resistance times being longer than the time it takes for invested money to double (which is proportional to 1/~; resistance is effectively far off, and optimal dosage can thus be high. The left-hand side of Figure 6 corresponds to characteristic resistance times being short compared with the doubling time of invested money; resistance looms, and therefore useful pesticide life should be extended by lower dosages. An essential point, which is given little attention elsewhere in this volume, is that not all actors in this drama discount the future at the same rate. Pesticide manufacturers may often tend to inhabit the right-hand side of Figure 6, seeing money as fungible, and taking ~ to be relatively high. Many farmers, however, may tend instead to inhabit the left-hand side of Figure 6, with assets tied up in their land, the future of which they would wish to discount slowly. In short even with goodwill and a clear biological understanding of how best to manage pesticide resistance, different groups can come to different decisions. This is a particular case of a more general phenomenon, discussed lucidly by Clark (1976) for fishing, whaling, and logging. CONCLUSION Our aim has been to combine population biology with population genetics, to show how migration and density-dependent dynamics can affect the rate of evolution of resistance to pesticides. To advance this enterprise we need a better understanding of the detailed genetic mechanisms underlying resis- tance and more information about the population biology of pests and natural enemies in the laboratory and in the field. Insofar as the dynamical behavior of pest populations influences the rate of evolution of resistance, we must be wary of extrapolating the laboratory studies into field situations; it would be nice to see more control programs being designed with a view to acquiring a basic understanding at the same time as they serve practical ends. If dosage levels, migration, refugia, natural enemies, and other factors are to be managed to slow down the evolution of pesticide resistance, efforts must be coordinated over large regiOns. Some crops lend themselves to this, and some do not. Often the best interests of individuals will differ from those of groups, leading to problems that are social and political rather than purely biological. Beyond this, even with good biological understanding and coherent plan- ning of group activities, it can be that different sectors pesticide manufac- turers, farmers, planners responsible for feeding people-have different aims stemming from different rates of discounting the future and the absence of

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POPUL4TION DYNAMICS 191 a truly common coinage. Population biology can clarify these tensions, but it cannot resolve them. ACKNOWLEDGMENTS This work was supported in part by the National Science Foundation, under grant BSR83-03772 (RMM), and by the North Atlantic Treaty Or- ganization Postdoctoral Fellowship Program (APD). REFERENCES Bellows, T. S., Jr. 1981. The descriptive properties of some models for density dependence. J. Anim. Ecol. 50:139-156. Bishop, J. A., and L. M. Cook. 1975. Moths, melanism and clean air. Sci. Am. 232(1):9~99. Brazzel, J. R., and O. E. Shipp. 1962. The status of boll weevil resistance to chlorinated hydrocarbon insecticides in Texas. J. Econ. Entomol. 55:941-944. Brown, A. W. A., and R. Pal. 1971. Insecticide Resistance in Arthropods. Geneva: World Health Organization. Chapman, H. D. 1984. Drug resistance in avian coccidia (a review). Vet. Parasitol. 15:11-27. Clark, C. W. 1976. Mathematical Bioeconomics. New York: John Wiley and Sons. Comins, H. N. 1977a. The management of pesticide resistance. J. Theor. Biol. 65:399-420. Comins, H. N. 1977b. The development of insecticide resistance in the presence of migration. J. Theor. Biol. 64:177-197. Comins, H. N. 1979. Analytic methods for the management of pesticide resistance. J. Theor. Biol. 77:171-188. Comins, H. N. 1984. The mathematical evaluation of options for managing pesticide resistance. Pp. 454-469 in Pest and Pathogen Control: Strategic, Tactical and Policy Models, G. R. Conway, ed. New York: John Wiley and Sons. Croft, B. A., and A. W. A. Brown. 1975. Responses of arthropod natural enemies to insecticides. Annul Rev. Entomol. 20:285-335. Crow, J. F., and M. Kimura. 1970. An Introduction to the Theory of Population Genetics. New York: Harper and Row. Endler, J. A. 1977. Geographic Variation, Speciation and Clines. Princeton, N.J.: Princeton Uni- versity Press. Ferrari, J. A., and G. P. Georghiou. 1981. Effects of insecticidal selection and treatment on reproductive potential of resistant, susceptible, and heterozygous strains of the southern house mosquito. J. Econ. Entomol. 74:323-327. Georghiou, G. P., and C. E. Taylor. 1977. Genetic and biological influences in the evolution of insecticide resistance. J. Econ. Entomol. 70:319-323. Georghiou, G. P., A. Lagunes, and J. D. Baker. 1983. Effect of insecticide rotations on evolution of resistance. Pp. 183-189 in Pesticide Chemistry: Human Welfare and the Environment, J. Miyamoto, ed. New York: Pergamon. Graves, J. B., and J. S. Roussel. 1962. Status of boll weevil resistance to insecticides in Louisiana during 1961. J. Econ. Entomol. 55:938-940. Gressel, J., and L. A Segel. 1978. The paucity of plants evolving genetic resistance to herbicides: Possible reasons and implications. J. Theor. Biol. 75:349-371. Gutierrez, A. P., U. Regev, and C. G. Summers. 1976. Computer model aids in weevil control. Calif. Agric. April:8-18.

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