Multiseasonal Management of an Agricultural Pest. * H Development of the Theory

Mangel, M. and Plant, R.E., 1983. Multiseasonal management of an agricultural pest. I: Development of the theory. Ecol. Modelling, 20: 1-19. A framework for analyzing the trade-off between economic yield from a crop and buildup of resistance to pesticide caused by repeated applications of pesticide is developed. The analysis begins with the case of age-independent pest dynamics, in which pests infest a field by arriving from an external pool. Initially, it is assumed that the pest genetics of interest are single locus, two allele, with resistance to pesticide dominant and susceptible pests more fit in the absence of spraying. The pesticide is applied only once during the season, with timing and intensity of the application as control variables. Interseasonal pest and crop dynamics are studied by solving appropriate ordinary differential equations. Intraseasonal pest dynamics are assumed to follow the Hardy-Weinberg formula. It is shown that the three class diploid model can be replaced by a two class haploid model with essentially no change in the results. A model based on partial differential equations is developed, for the case in which pest dynamics depend upon age, and it is shown that the partial differential equation model can be replaced by a pair of coupled ordinary differential equations. The main operational conclusion in this paper is that the timing of the application of pesticide can be used to control buildup of resistance and that the intensity of the application can be used to control the crop yield. INTRODUCTION: THE AGRICULTURAL DECISION PROBLEM It is common in many agricultural enterprises to use pesticides to control pests. It is also commonly observed that as pesticides are applied, resistance to the pesticide builds up as susceptible pests are removed from the population. Thus, an agricultural manager is faced with a decision problem of the following kind: By spraying his field this year, to increase the present year's economic yield, he increases resistance to the pesticide, which, in principle, * Giannini Foundation Paper. 0304-3800/83/$03.00 © 1983 Elsevier Science Publishers B.V. reduces future yields. It is this decision problem that is studied here and in an accompanying paper by Plant et al. (1983). The problem is of sufficient complexity that the two papers use only deterministic approaches. A third paper will include stochastic effects in our models. The goals in these papers are to develop a framework in which questions about the trade-off between resistance and yield can be posed and then to answer some of the more important questions. The models are formulated in a way that is sufficiently realistic to provide useful results, but, at the same time, is sufficiently simple to understand the behavior of the model. Many of the ideas used here have previously appeared in the literature in some modified form, see for example the papers by Hall and Norgaard (1973); Hueth and Regev (1974); Gutierrez et al. (1975); Regev, Gutierrez and Feder (1976); Georghiou and Taylor (1976; 1977a, b); Georghiou (1980); Headley (1981); and Shoemaker (1982). The approach in Shoemaker's paper is conceptually similar to this one, but the operational formulation and analytical tools are quite different. One agricultural system that examplifies the kind of problem of interest is the cotton-spider mite (Aracina: Tetranychidae) system in the San Joaquin valley in California. Some of the pest dynamics were examined by Carey and Bradley (1981). This system has the following features. The cotton is an annual crop grown every other year, frequently in rotation with wheat, a crop that does not support the mites. Although some mites over-winter in the field, mites also immigrate to the field at some rate throughout the entire season, coming from external sources such as fruit orchards and weed patches. These external sources of the mites appear to be much more important than the over-wintering mites already in the field. It is also known that in greenhouses, where there is no alternation of crops and no external source of non-resistant mites, resistance to the commonly used pesticides builds up quickly. The modelling approach is based on a submodel for the pest dynamics and a submodel for the crop dynamics. The pest submodel has the following goal: given the fractions of resistant and susceptible pests in the population at year n and the spraying strategy in year n, how does one find the respective fractions in year n + 1? In order to answer this question, one must include population dynamics and relatively simple genetics in the pest submodel. The submodel for the crop dynamics has the following goal: given the pest populations at the start of year n, and a spraying strategy in year n, what is the yield of crop in year n? Using this submodel one can also determine the optimal spraying strategy within a single season. The models do not include the effect of biological controls such as predators. In the case of the spider mite-cotton system in the San Joaquin valley, there is some question as to whether natural predators play an important role in controlling outbreaks, but in any case the inclusion of the effects of predators in the model at this time would not provide useful information. The three primary parameters in the application of a pesticide in a given season are the number of applications, the timing of the applications, and the intensity of the applications. In the next paper (Plant et al. 1983) the first parameter is considered; for the present it is assumed that a single pesticide application occurs each year. This allows one to study the effects of timing and intensity. In Sections 2 and 3, a simple age-independent model of the pest-crop interaction is given and the effects of varying the timing and intensity of pesticide application are examined. Since the accumulation of pesticide resistance in the population is important, the effect of assumptions about the genetics on the behavior of the model is studied. Previous simulation studies (e.g. Regev et al., 1976), have shown that the age structure of the pest population plays an important role in determining pest control strategies. Dependence on age structure occurs in two aspects of the model: susceptibility to the pesticide, and consumption of the crop. The introduction of age structure greatly complicates any model. This complication is minimized by assuming that pests inhabit colonies founded by a pregnant female. Spider mites behave this way (Carey and Bradley, 1981). The colonies have an age structure that depends on the age of the colony; for example, younger colonies have more eggs and larvae, and older colonies have more adults. Carey and Bradley (1981) have studied the demography of these mite colonies. The model is formulated in terms of age structure of colonies; it is assumed that from this the age structure of the population may be inferred. This greatly simplifies the formulation. It does, however, have one serious consequence. If the pest population has an age-specific susceptibility to the pesticide, then each application of the pesticide will change the age structure of the already established pest colonies. This is of no consequence if crop consumption is not age specific and if only one application of the pesticide is made. For the present, the change in age structure due to pesticide application is neglected. In the study of the age structured model, a simplified formulation that adequately represents the age-dependent effects is presented. This model does not require the assumption of a single pesticide application per season. The unit of time in the model is the physiological time, degree-days. It is assumed that the crop and pests are affected by physical temperature in the same way. This assumption could be dropped in a simulation. The utility of the models, when compared to simulations, is that because of the simplifying assumptions it is much easier to understand the crop, pest, and resistance dynamics than it is in simulations. THE AGE-INDEPENDENT MODEL To start, consider a model for the case in which pesticide susceptibility and crop consumption are independent of age. Let X(t) be the total number of pests in the field at time t, and let C(t) be a measure of the value of the crop. The most commonly used such measure is the leaf mass (Gutierrez et al., 1975). Assume that the crop is harvested before density dependent effects become important. The equation for the crop dynamics is therefore dC IrcC-vX : C ( t ) > 0 a t ~0 " C ( t ) = 0 (1) C(O)= Co, O < t < T where C O is a positive constant. The parameter ro is the intrinsic growth rate of the crop, v is a measure of the pest's unit rate of consumption of the crop, and T is the length of the season. According to eq. 1, the crop decreases as soon as rcC < vX. This is the simplest possible assumption about the crop pest interactions. Other assumptions, such as replacing vX by vXC give similar qualitative results. T is fixed and finite; for the parameters used here, C (T) is always bounded away from zero. Consider now the model for the pest dynamics, starting with the simplest model and then increasing in complexity. All models are based on the following operational picture. At the start of season n the agricultural field has negligibly few pests in it. There is an external source of pests, which we call a pool, and pests leave the pool and arrive at the field at a rate I ( t ) , where t denotes time within a season. The length of the agricultural season is T; at the end of the season the remaining pests in the external pool and pests in the agricultural field are mixed together for the overwintering process. To simplify the derivation of the pest dynamics, begin by considering a model with only one class; thus, effectively, assume that all pests are susceptible to the pesticide. When it is important to distinguish the pest populat ion in year n, it will be denoted by X(t; n). When there is no possibility for confusion, X(t) will be used. Within a season, the intensity of spraying in year n is denoted by s(t; n). In most cases, assume that the pesticide is applied only once and that s(t; n) takes the form If! s( t ;n)= n): t s (n)<t<ts(n)+8(n ) (2) Q ( n ) + 8 ( n ) < t < T Here ~/(n) is the intensity of the spraying in year n and 8(n) is the length of time that the pesticide is active in year n. As before, the argument n is dropped when no confusion results. As with the crop component, it is assumed that the pest population does not reach its carrying capacity before the end of the season. The model is therefore linear. A consequence of this assumption is that the growth rate of the pest population is independent of the value of C, the crop. Therefore, the differential equation for the pest component is d X dt r( t; n ) X + I( t; n) (3) X ( 0 ; n ) = 0 , 0 < t < T Here, I(t; n) is the rate of immigration from the pool at time t in year n and r(t; n) is the intrinsic rate of increase of the population at time t in year n. It depends upon n and t through the spraying function; i.e., when pesticide is applied, X(t) does not instantaneously decrease, but decreases over some finite period of time. This decrease can be modeled by making r(t; n) negative. In particular, the following form is chosen r ( t ; n ) = r o 1 (4) e+s(t;n) In the formula, r 0 is the intrinsic growth rate in the absence of spraying, and w and e measure the effects of the spraying. The parameter o~ measures the maximal pesticide effect in the sense that as s( t ) approaches infinity, r( t ; n) approaches r0(1 w). The parameter e is related to LDs0 (the intensity required to kill 50% of the pests) *. If e is small (as would be the case with susceptible pests) then the LDs0 is reached at low levels of s( t) ; if e is large (as would be the case with resistant pests) then the required value of s(t) is large. The next step is to incorporate genetics into the models. The philosophy here is that in dealing with the genetics of the system, one should use the simplest assumptions possible, even at the cost of sacrificing some accuracy in the model. Hence, no effect is made to take into account the particular genetic characteristics of a pest species (such as the haplo-diploidy of mites, or the ratio of inbreeding to outcrossing). Rather, it is assumed that whatever these characteristics are, their effects may be accounted for by simple, empirical parameters. The simplest point of departure is a single locus, two allele, genetic model, * I f ~/5o is the in tens i ty c o r r e s p o n d i n g to LDs0, and the pest icide is active for 8 uni ts of time, then e = ~ / [ 1 l / S r o In(0.5)] -1 ~/. with resistance to pesticide dominant. Let )(1 (t; n) = number of homozygous resistant (RR) pests in the field at time t in season n X2(t; n) = number of heterozygous resistant (RS) pests in the field at time t in season n X 3 (t; n ) = number of homozygous susceptible (SS) pests in the field at time t in season n (5)