THEORETICAL
POPULATION
BIOLOGY
13, 197-213 (19%)
A Stochastic Model of a Temperature-Dependent GUY L. CURRY, RICHARD M. FELDMAN,
AND KIRBY
Population* C. SMITH
Biosystems Research Division, Department of Industrial Engineering, Texas A&I’M University, College Station, Texas 77843 Received December
23, 1976
The theoretical basis is developed for a population model which allows the use of constant temperature experimental data in predicting the size of an insect population for any variable temperature environment. The model is based on a stochastic analysis of an insect’s mortality, development, and reproduction response to temperature. The key concept in the model is the utilization of a physiological time scale. Different temperatures affect the population by increasing an individual’s physiological age by differing rates. Conditions for the temperature response properties are given which establish the validity of the model for variable temperature regimes. These conditions refer to the relationship between chronological and physiological age. Reasonable agreement between the model and field populations demonstrates the practicality of this approach.
1.
INTRODUCTION
It is a well-established fact that an insect’s rate of development is heavily dependent upon temperature. Estimates for an insect’s development rate under a specified temperature environment are obtainable by laboratory experiments; however, it is not always possible to use these laboratory estimates directly for predicting development in a field environment where actual temperatures are almost never the same as the laboratory temperatures. The most commonly used technique for predicting variable temperature development (physiological age) is the degree-day method (Hughes, 1962; Gilbert et al., 1976). The recognition of the nonlinearity of the developmental rate with respect to temperature has lead to the more general rate summation methods. Using rate summation, it has been demonstrated (Fye et al., 1969; Stinner et al., 1974) that the mean time of stadial emergence under variable temperature environments can be predicted utilizing the reciprocal of the constant temperature * The work herein reported was funded in part by a U.S. Department of Agriculture sponsored program entitled “The Expanded Southern Pine Beetle Research and Applications Program” Grant Number 89-106 (19-192) and in part by the National Science Foundation and the Environmental Protection Agency, through a Grant (NSF GB-34718) to the University of California.
197 0040-5809/78/0132-0197$02.00/0 All
Copyright Q 1978 by Academic Press, Inc. rights of reproduction in any form reserved.
198
CURRY,
FELDMAN,
AND
SMITH
emergence times. Sharpe et al. (1977) and Stinner et al. (1975) used these concepts to predict the probability distribution function of emergence times under arbitrary temperature regimes. The underlying feature which allows Sharpe et al. (1977) and also Barfield et al. (1977) to predict emergence time distributions for a variety of insects is that the probability distribution functions of the time of stadial emergence under different temperature regimes have the “same shape.” Specifically by “same shape” we mean the following: Consider several cohorts (a cohort is a group of insects of the same species and age within the same stadium) raised under different constant temperature regimes. Recording the stadial emergence times and rates (reciprocal of times) for each cohort separately, then multiplying the emergence times by the cohort mean rates, one obtains resulting probability distribution functions of “normalized” emergence times. Then if these cohort distribution functions are identical, we say they have the same shape. The same shape property has been experimentally verified for the boll weevil (Anthonomus grandis Boheman) and the cotton fleahopper (Pseudutomoscelis seriutus Reuter) by Sharpe et al. (1977) and for a boll weevil parasite Bracon mellitor Say by Barfield et al. (1977). It will be shown that for insects having this same shape characterization, the emergence time distribution under an arbitrary temperature regime can be computed using only laboratory data. In addition to stadial emergence, we shall consider reproduction and survivorship under analogous assumptions. These characteristics are utilized in the development of a general temperature-dependent single-species stochastic population model based on laboratory data. The results of the model indicate that the physiological age of an insect along with the associated normed distributions for emergence, reproduction, and survivorship is sufficient to fully describe the relevant population processes under variable temperatures. The purpose of this paper is the mathematical justification for this temperature-dependent model. It should be emphasized that the model developed in this paper is oriented toward estimating field populations based on traditional laboratory experimental data. The necessary data are obtained from constant temperature experiments and consist of: mean values for development times, percentage mortality, and total reproduction; and distribution functions for emergence times, death times, and percentage reproduction. Under reasonable assumptions, these data from just a few constant temperature experiments can be used to predict the population response for variable temperature regimes. The assumptions (stated explicitly below) are readily tested with the experimental data. Although the assumptions are apparently valid for many species, they clearly do not hold in general. Due to the practicality of this approach, research in the relaxation of the assumptions is needed. The basic notational conventions to be used throughout this paper are given in Section 2. Properties relating to exit times, emergence times, and reproduction
TEMPERATURE-DEPENDENTMODEL
199
are discussed in Sections 3,4, and 5, respectively. The integration of the cohort submodels into a population model is described and illustrated in Section 6.
2. PRELIMINARY DEFINITIONS AND NOTATION A temperature regime will be denoted by the function yGwhere #(t) is the
temperature at time t. The function &(.) denotes the constant temperature regime set at k degrees Kelvin. Consider an individual insect raised under some specified temperature regime #. The times at which the insect emerges from the egg phase, the larval phase, etc., as well as its death time and fecundity are random variables dependent on 4. Specifically for temperature regime 4, let Tl*, T,*,... be random variables denoting the lengths of time the insect is in the first (egg) stadium, second (larval) stadium, etc. The Ti’s will be called exit times and they are assumed conditionally independent given that no death occurs. If the insect dies in the ith stadium, let Ui* be a random variable denoting the time of death; thus Ui* = Ti*. To denote that death did not occur before the ith stadium, let Ut’j = co for j = 1, 2,..., i - 1. As a general rule a superscript will refer to the temperature regime and a subscript will refer to a specific stadium. Capital letters will refer to either random variables or distribution functions. For notational convenience, the subscript will be deleted whenever it is possible to do so without causing confusion. By an abuse of notation Ttk will be written as Ti”. In terms of the above notation, our purpose is to use the empirical distribution function of Tk to obtain an estimate for the probability distribution function of T*, where # is an arbitrary temperature regime. In normalizing the exit times, it is convenient to use the “instantaneous rate of development.” At temperature k, the development rate, r(k) is defined by r(k) = E[l/Tk],
(2-l)
where E[l/Tk] is the expected value of l/Tk. Constant temperature experiments are generally performed at a selected number of points over the operational temperature range. The functional form of I( .) over the temperature continuum is obtained by the standard Qr,, , Arrhenius, temperature summation rule (see Laudien, 1973, p. 359), or the more recent relationship of Sharpe and DeMichele (1977). In a manner similar to that of Butler and Watson (1974) it is assumed that the insect species under consideration can instantly adapt to temperatures within a certain interval; that is, after a temperature change the insect’s developmental rate immediately changes to that of the new temperature. Furthermore, all temperature regimes are within this interval and are assumed piecewise continuous. The additional notation specific to reproduction will be defined in Section 5.
200
CURRY,
3.
FELDMAN,
STADIAL
EXIT
AND
SMITH
TIMES
In this section it is demonstrated that the distribution of exit times under an arbitrary temperature regime can be obtained using the distribution function of the normalized exit times from experimental temperature regimes. Throughout this section, we shall be dealing with insect properties for a specified stadium; thus, the stadium subscript will not be used. The main assumptions regarding exit times underlying (3.1) ASSUMPTIONS. this research are as follows: (a) Exit times have the same shape characteristic. That is, the function F(.) defined by, for x > 0, F(x) = P{r(k) T” < x> is independent of k. Furthermore, F is continuous and the exit times are positive and uniformly bounded for all temperatures (that is, there are numbers u and b such that 0 < a < Tk < b < oc,for all A). (b) A small change in temperature produces no more than a small change in the exit times; that is, for any temperature regime I/ and any sequence of regimes (p) with #” -+ #, it follows that limnem T@’ = T* almost surely. (c) When an insect’s environment changes from one temperature to another, there is no residual effect due to the temperature change. In other words, after the time of the temperature change, the insect’s behavior is identical to that of an insect of the same biological age (i.e., fractional development as defined below) which was raised at the second temperature. A key concept throughout this paper is the fractional development of an insect which indicates the expected value of its age on the normalized time scale. The fractional development of an insect within the stadium at time t, denoted by d”(tO , t), is defined by d”(t, , t> = 1; r 0 W
&
(3.2)
where to is the entrance time into the stadium and # is the temperature regime. Equation (3.2) is the standard form (rate summation) used in computing physiological time (Stinner et al., 1974, Eq. (2); Allen, 1976, Eq. (1); Logan et al., 1976, Eq. (12)). Th e integrand of the equation uses the developmental rate I defined for constant temperatures composed with the temperature regime. As before, we shall write d”(t, , t) instead of dlLk(t, , t). For constant temperatures it should be noted that d”(t,, , t) = r(k)(t - t,). The main result of this section is presented in the following theorem.
(3.3)
TEMPERATURE-DEPENDENT
201
MODEL
(3.4) THEOREM. Let # be an arbitrary temperature regime. The exit time distribution for an insect enteritlg the stadium at time t, is given by P{TS < t - to} = F(d”(& , t)), where 4(t) = $(t + t,,).
Proof. Without loss of generality, the theorem will be proved for If I/J= I& for some k, then P{T* < t} = P{Tk < t} = P{r(k) Tk < r(k) thus, by Assumption (3.la) and Eq. (3.3),
to t}
= 0. and
P(T” < t} = F(dk(O, t)). Next, consider the case where the temperature changes once, say at is, define 1/,by for t < t*, W = k, for t > t*. = k,
t*;
that
(3.6)
For t < t* it is clear from the above that P{ T’J < t} = F(d@(O,t)). At time t * an organism has fractional development d%(O, t*). Select the time s* such that d%(O, t*) = d”z(0, s*).
(3.7)
2 zz :; 5 e
-------------
0
FIG.
“\/dk+ -
IF
d s*
TIME
1. The chronological times t * and s*, at constant temperature
respectively,
corresponding
to the same age d* on the fractional
regimes kl and k, , development scale.
Intuitively, s* is the time needed under temperature k, for the organism to have the same fractional development as it would have at time t* under temperature k, (this transform is illustrated in Fig. 1). For T > 0, P{T”
+P{t* t*} P{T” < t* + 7 1 T* > t”}
202
CURRY, FELDMAN, AND SMITH
and by Assumption (3.1~)
= P{T”I < t*} + P(T”1 > t*} P{Tkn < s* + r j T% > s*>. Using (3.5) and (3.7) and simplifying yields
P{ T” < t * + T} = F(dkz(O,s* + T)). Using (3.2) it is seen that d*(O, t* + T) = dlc8(0,s* + T) which implies the theorem is true for any $ having only one temperature change. Using mathematical induction, it is clear that the theorem holds for any # which is a simple function (step function with finite range). Finally, assume that I/ is an arbitrary temperature regime. Then # can be represented as the limit of an increasing sequence of simple functions, say, I,G= lim 4”. The th eorem is valid for each regime p; that is P{T”” < t> = F(d@(O, t)). Combining the uniform bounded property of { Tk) given in Assumption (3.la) and the almost sure convergence of Assumption (3.lb), one has that r(.) is a continuous function (this is convergence of the means and is given by Chung (1968, p. 93)). Ag ain, using Assumption (3.la) and the Lebesgue Convergence Theorem, limn+m d@(O,t) = d”(O, t) and thus limn+m F(d@(O, t)) = F(dti(O, t)). Since almost sure convergence implies convergence in distribution (see Chung, 1968, p. 84), it follows that
F(d”(0, t)) = &if F(d”“(0, t)) = f+xr P{T”” < t} = P{T” < t} and the proof is complete.
4. STADIAL EMERGENCE In order to develop a population model, the distribution of stadial exit times is necessary. In addition, it is important to know whether the departure was due to death or to emergence into another stadium. In this section the probabilities associated with stadial emergence are developed. Specifically consider an insect which enters the stadium at time to under temperature regime #, then the probability of emergence into the next stadium by time t is
P{Ts < t - t,, , ti > t - to}, where a&t) = #(t + t,,).
203
TEMPERATURE-DEPENDENTMODEL
(4.1) ASSUMPTIONS. It is assumed that the emergence times have the same shape characteristic up to a multiplicative temperature constant; that is, there is a distribution function G( .) such that, for x > 0, P{r(k) T” < x, r(k) U” > x} = c(k) G(x), where c(k) is the probability of ultimate emergence from the stadium under constant temperature regime k. Furthermore, it is assumed that G is continuous and there is a number b < co such that G(6) = 1 (the number b gives an upper bound for the emergence time on the fractional development scale). (4.2) PROPOSITION. The distribution function for the normalized emergence time conditioned on live emergenceis independent of temperature and is G(x), i.e., P{v(k) Tfi < x / U” = CC} = G(x).
Proof. By Assumption (4.1), we have P{V = a} = c(k). Then, since the events (r(k) Tk < x, Uk = co> and (r(k) T” < x, r(k) Uk > x} are equal, the result follows by the definition of conditional probability. As in the previous section, it will be shown that the emergence distribution under an arbitrary temperature regime is obtained using fractional development. (4.3) THEOREM. Let 4 be an arbitrary temperature regime. The emergence time distribution for an insect entering the stadium at time t, is given by P{Ts < t - t, , u$ > t - t,} = jt c 0 4(s) dG(W, to
, 4,
where the integral is a Stieltjes integral and $(t) = #(t + to). Proof. Without loss of generality, the theorem will be proved for t, = 0. If # = I+&for some k, then clearly P(T” f t, U” > t} = c(k) G(d”(0, t)).
(4.4)
Consider the case where the temperature regime $ has one step as defined by Eq. (3.6). For t < t* (4.4) holds. As before, the time s* is chosen such that the fractional development under the k, temperature regime at s* equals the fractional development under the k, regime at time t* (that is, Eq. (3.7) holds). For T > 0, P{T” < t* + ‘, u* > t* + T} = P{T* < t*, U* > t*} +P{T~>t*}P{T”t*+r/T~>t*} and, by Assumption (3.lc), = c(k,) G(d”l(0, t*)) + [l -F(d”l(O, t*))][c(k,) G(d”a(0, s* + T)) - c(kJ G(d”z(0, s*))]/[l - F(d*z(O, s*))].
204
CURRY,
FELDMAN,
AND
SMITH
Using (3.7) and simplifying yields
= Wi) - WI WW t*)> + ~(4) W”(O, t* + 4 which is a Stieltjes integral for the single step function and using integration by parts leads to the proper result. Now let 1F,be a simple function. That is, for fixed times 0 = t, < t, -=za*. < tN and fixed temperatures $ , A1,..,, k, , # is defined by $(s) = Ki when ti < s < ti+l (i = O,..., N - 1) and 9(s) = kN when s > t, . Then it can be shown that
P{T* < t, + T, U” > t, + T} = c(k,) G(@(O,t, + T)) + f Wi-1) - &)I
G(@(O,ti)), (4.5)
i=l
for fixed n = 1, 2 ,..., N and t, + r < tn+l (with tN+l = co). Equation (4.5) is the Stieltjes integral for a simple function and the integral of the theorem is obtained after integration by parts. Finally, assume that $ is an arbitrary temperature regime (of bounded variation). The first term on the right-hand side of the equation of the theorem follows due to the same reasoning as in Theorem (3.4). The second term follows from the theory of the Stieltjes integral (see Kolmogorov and Fomin, 1970, p. 378).
Remarks. Assumption (4.1) implies that changing the temperature uniformly changes (raises or lowers) the probability of emergence in any physiological time interval (x1 , xe). The basis for this assumption is that the data of Sharpe et al. (1977) and Barfield et al. (1977) indicate that emergence distributions for certain insects exhibit the same shape property (as in Proposition (4.2)). For other organisms, an alternate same shape property might be appropriate. This assumption can be applied to death times instead of emergence times, yielding similar formulations. Another alternative leading to a more complicated formulation is to use the same shape property for both emergence and death times. This alternative necessitates the relaxation of the same shape exit time assumption. 5.
REPRODUCTION
The final component in a single population model is reproduction. The rate of reproduction for insects typically varies with both age and temperature. It has been observed for some species (Barfield, 1976) that the proportion of individual female’s total reproduction as a function of fractional development
TEMPERATURE-DEPENDENT
.i
20.5
MODEL
1.0
1.5
300 150 zoo
FIG. 2. Characteristic curves for reproduction. (a) Proportion of total reproduction as a function of fractional development. (b) Total reproduction as a function of temperature. is the same for all temperatures. Figure 2a displays this functional form for the boll weevil which shows, for example, that 46% of a female’s total reproduction occurs in the fractional development interval (0.5, 1.0). However, even though the proportions are independent of temperature, the actual number of eggs laid does vary with temperature (Wood and Starks 1972; Isely 1932; Cole 1970). Figure 2b displays the total expected number of eggs per adult female boll weevil which shows, for example, that at 25°C the expected number of eggs laid during the female’s lifetime is 243. Thus, the expected number of eggs laid in the fractional development interval (0.5, 1.0) at 25°C is 112. In this section, a method for determining reproduction under an arbitrary temperature regime is developed. It is known that for some species, variable temperatures can produce a more fertile insect than is possible at constant temperatures. This possibility is ignored in order to have a tractable model. The expected rate of reproduction at fractional development x and temperature k, ye(k) x), is assumed to have the form
206
CURRY,
FELDMAN,
AND
SMITH
where Jzg(x) dx = 1. The term h(K) is the total expected reproduction over the lifetime of a female adult held under the constant temperature regime K; while g(x) dx has the interpretation that it is the proportion of reproduction of each female in the fractional development interval (x, x T dx). It should bc noted that many data sets have been published of the form necessary to determine h(.); however, as pointed out by Pianka and Parker (1975, p. 457) few data sets on age specific reproduction have been published. The basic assumption for reproduction is that the (5.2) ASSUMPTION. reproduction at a given time depends only on the insect’s fractional development and the current temperature. Assumption (5.2) is consistent with the previous temperature assumptions in that historical temperatures are reflected only in fractional development. In previous sections, we have analyzed the relevant processes utilizing the probabilistic nature of the components. To describe reproduction requires more detail than was necessary for the simplified descriptions in the previous sections. To develop the stochastic model of reproduction underlying Eq. (5.1) necessitates a biological analysis at a depth considerably beyond that utilized in the previous sections. The expectation form assumed in Eq. (5.1) adequately reflects the observed response for constant temperatures. Thus, to avoid unnecessary complications, only expected values will be considered in the extension to variable temperature regimes. Under arbitrary temperature regime I/, the inverse of the function in Eq. (3.2) is needed to determine the time i in which an insect (born at ts) reaches fractional development x. This is denoted by 8f0 ; in other words
t = 6:&x)
(5.3)
if and only if x = d*(t, , t). The rate of reproduction at fractional development x is r,(~#b(x)), x) and thus the total reproduction between the development states x, and xs is
Using transformation (5.3) and Eq. (5.1), the total reproduction p on the time scale between time t, and 1, is given by
(5.4) where r(a) is the development rate, Eq. (2.1), for the adult phase.
TEMPERATURE-DEPENDENT
6. POPULATION
207
MODEL
MODEL
An insect population consists of a collection of cohorts. A cohort, as defined in this paper, is a group of insects within a stadium that started the stadium at the same time. As time varies, new cohorts are generated by two mechanisms: (1) emergence from one stadium into the next and (2) adult reproduction. For clarity of presentation, cohorts will be labeled and the index set of the cohort labels will be denoted by N. The only information necessary concerning a specific insect cohort i is the triple (#, si, t,$ where qi is the initial cohort size, si is the cohort stadium, and toi is the initiation time of the cohort. To utilize the discrete cohort concept, an implicit time increment d t exists for which all insects generated within this increment are assumed identical. The convention that d(t) = z,h(t$ to) will be continued throughout this section. To model a dynamic insect population, an initial group of cohort triples must be given. The cohort time increment At is then used to iteratively compute the population changing over time. New cohorts can be added to the population
lARVAI. COHORT LARVAI, COHORT
I.ARVAI. COHORT ADL'LT COHORT
ADULT COHORT ADVLT COHORT
EGG COHORT
.u.E TIME
FIG. 3. New cohorts are generated every dt time increment for which emergence and/or reproduction are possible. This example uses three stadia; eggs-E, larvae-L, and adults-A. The curves represent the relevant distributions for stadial emergence or reproduction.
208
CURRY,
FELDMAN,
AND
SMITH
at each time increment via emergence and reproduction as illustrated in Fig. 3. To demonstrate the iterative population update, consider a time t with a current population consisting of cohorts indexed by the set P C N. That is, for i E P the triple ($, si, tOi) when t,,j < t represents a cohort present at time t. Thus, the expected number of insects within a specified stadium M at time t is
where the sum is over all i E P such that sa’= m and the expression is computed using Theorem (3.4). For each stadium m (m > I), a new cohort I can be generated by emergence during the time interval [t, t + dt) and its triple is given by (T,“, sz, to”), where s1 = m,
(f-54
t,” = t + At/2, 7)’ =
c
qP{t - toi < T$+, < t + At - tog,U$-, > t - toi}.
&W+l
The final expression
in (6.2) is computed
using Theorem
(4.3) and the equality
P{t < TB < t + At, I? > t + At} =P{Tst+At}-P{T4,(t,U6>t}. For the first stadium, a new cohort j can be generated from the reproduction of the adult cohorts (let si = a for the reproduction stadium) during [t, t + At) and its triple is given by (vi, sj, toi), where
$ = c qip*(t, t + At), 8=a
(6.3)
01is the expected proportion of females, sj = 1, and t,,j = t + At/2. The expression in (6.3) is computed using Eq. (5.4). Using Eqs. (6.1)-(6.3) the insect population can be iteratively generated over time given an initial cohort structure. External influences, such as migration, insecticides, etc., can be incorporated into the model if their effect on the cohort structure is known. The mechanics for updating the population are analogous to the Leslie matrix method (Leslie, 1945). A continuous analog is given by Von Foerster (1959); however, the numerical solution of the continuous
TEMPERATURE-DEPENDENT
MODEL
209
model usually approximates time by discrete increments, which again reduces the method to the Leslie matrix approach (Wang et al., 1977). To demonstrate the applicability of these concepts for modeling an insect population, model results for the boll weevil (Anthonomus grandis Boheman) are compared in Fig. 4 with field counts of Walker and Niles (1971). The data utilized in the two-stadium model (immatures and adults) were obtained from published literature. Immature developmental rates and survival percentages are from Isely (1932, Figs. 2 and 6). The distribution function for immature emergence is from Sharpe et al. (1977, Fig. 6). Since the distribution F(.) of immature exit times has not been reported, it was assumed to be the same as the conditional emergence distribution G(.). Adult developmental rates and the physiological age dependent reproduction profile (Fig. 2a) are from the data of Cole (1970). Cole’s data also indicate that the ratio of males to females in the adult population is approximately l/l. The total expected reproduction of a female adult as a function of temperature (Fig. 2b) is from Isely (1932, Table 10). Adult mortality based on physiological age is from the data of Sterling and Adkisson (1970, Fig. 3). Figure 4 displays four years of model versus field adult weevil counts. The population peaks and troughs in the adult counts are due to the lag time (about 2-3 weeks) between egg laying and adult emergence. To obtain an accurate assessment of the model, the predicted expected population sizes should be shown together with the predicted standard deviation of the population size with respect to time. For constant temperature regimes, such a comparison would be possible using the theory from age-dependent branching processes (Jagers, 1975). However, under variable temperature regimes, equivalent results are not yet available. Temperatures for the model were generated from actual daily weather station min-max temperatures using a sinusoidal curve fit to obtain hourly values. Included in the field data of Walker and Niles (1970, Fig. 1) was the date at which reproductive sites first became available. This was included in the model so that reproduction could not be initiated until after this date. During the second in-field generation, density-dependent factors usually become limiting so that the population growth no longer is solely temperature dependent. Thus, the projections of the temperature-dependent model are not valid for this nonn’Ialthusian growth period. During the two-month period in which the model assumptions are valid, Fig. 4 shows a reasonable agreement between the predicted and actual adult population sizes.
653/13/z-3
210
3 a
CURRY,
FELDMAN,
AND
SMITH
TEMPERATURE-DEPENDENT
MODEL
211
212
CURRY, FELDMAN,
AND SMITH
ACKNOWLEDGMENTS We gratefully acknowledge the invaluable contributions of Drs. Don W. DeMichele and Peter J. H. Sharpe to the concepts underlying this paper. The suggestion and comments of the referees are also appreciated.
REFERENCES ALLEN, J. C. 1976. A modified sine wave method for calculating degree days, Environ. Etomol. 5, 388-396. BARFIELD, C. S. 1976. “A Model for Interactions between Bracon mellitor Say and Boll Weevil, Anthonomous grandis Boheman,” Ph.D. Dissertation, Department of Entomology, Texas A&M University. BARFIELD, C. S., SHARPE, P. J. H., AND BOTTRELL, D. G. 1977. A temperature driven developmental model for the parasite Bracon mellitor Say, Canad. Entomol. 109, 15031514. BUTLER, G. D., JR., AND WATSON, F. L. 1974. A technique for determining the rate of development of Lygus hesperus in fluctuating temperatures, Florida Entomol. 57, 225248. CHUNG, K. L. 1968. “A Course in Probability Theory,” Harcourt, Brace & World, New York. COLE, C. L. 1970. “Influence of Certain Seasonal Changes on the Life History and Diapause of the Boll Weevil Anthonomus grandis Boheman,” Ph.D. Dissertation, Department of Entomology, Texas A&M University. FYE, R. E., PATANA, R., AND MCADA, W. C. 1969. Developmental periods for Boll Weevils reared at several constant and fluctuating temperatures, J. Econ. Entomol. 62, 14021405. GILBERT, N., GUTIERREZ, A. P., FRAZER, B. D., AND JONES, R. E. 1976. “Ecological Relationships,” Freeman, Reading, Mass./San Francisco. HUGHES, R. D. 1962. A method for estimating the effects of mortality on aphid populations, J. Anim. Ecol. 31, 389-396. ISELY, D. 1932. Abundance of the Boll Weevil in Relation to Summer Weather and to Food, Arkansas Agricultural Experiment Station Bulletin No. 271. JAGERS,P. 1975. “Branching Processes with Biological Applications,” Wiley, London. KOLMOGOROV, A. N., AND FOMIN, S. N. 1970. “Introductory Real Analysis,” PrenticeHall, Englewood Cliffs, N. J. and Life” (H. Precht, LAUDIEN, H. 1973. Changing reaction systems, in “Temperature J. Christophersen, H. Hensel, and W. Larcher, Eds.), pp. 355-398, Springer-Verlag, New York/Berlin. LESLIE, P. H. (1945). On the use of matrices in certain population mathematics, Biometrika 33, 183-212. LOGAN, J. A., WOLLKIND, D. T., HOYT, J. C., AND TAMGOSHI, L. K. 1976. An analytic model for description of temperature’ dependent rate phenomena in arthropods, Environ. Entomol. 5, 113-l 140. PIANKA, E. R., AND PARKER, W. S. 1975. Age-specific reproductive tactics, Amer. Natur. 109, 453-464. SHARPE, P. J. H., CURRY, G. L., DEMICHELE, D. W., AND COLE, C. L. 1977. Distribution model of organism development times, /. Theor. Biol. 66, 21-38. SHARPE, P. J. H., AND DEMICHELE, D. W. 1977. Reaction kinetics of poikolotherm development, J. Theor. Biol. 64, 649-670.
TEMPERATURE-DEPENDENT
MODEL.
213
STERLING, W. L., AND ADKWGON, P. L. 1970. Seasonal rates of increase for a population of the Boll Weevil, Anthonomw grandis, in the high and rolling plains of Texas, .4nn. Entomol. Sot. Amer. 63, 1696-1700. STINNER, R. E., GUTIERREZ, A. P., AND BI:TI.EH, G. D., JH. 1974. An algorithm for temperature-dependent growth rate simulation, Canad. Entomol. 106, 519-524. QTINNW, R. E., BUTLER, G. D., BACHELEH, J. S., AND TCITLE, C. 1975. Simulation of temperature-dependent development in population dynamics mod&, Cannd. f?n~omo/. 107, 1167-I 174. VON FOERSTER, H. (1959), Some remarks on changing populations, itt “The Kinetics Ed.), pp. 382-407, Grunt k Stratton, of Cellular Proliferation” (F. Stohlman, Sew York. Dynamics of the Boll W’ecvil \vAI.KER, J. K., JR., AND NILIS, G. A. 1971. “Population and Modified Cotton Types: Implications for Pest Management, ” Texas Agricultural Experiment Station Bulletin Xo. 1109. WANG, Y., GUTIERREZ, A. P., OSTER, G., AND DAXI., R. 1977. A general model for plant growth and development, Conad. Entomol. 109, 1359-1374. WOOD, E. A., JR., AND STARKS, K. J. 1972. Effect of temperature and host plant interaction on the biology of three biotypes of the greenbug, En&on. Enzomol. 1, 230-234.
POPULATION
BIOLOGY
13, 197-213 (19%)
A Stochastic Model of a Temperature-Dependent GUY L. CURRY, RICHARD M. FELDMAN,
AND KIRBY
Population* C. SMITH
Biosystems Research Division, Department of Industrial Engineering, Texas A&I’M University, College Station, Texas 77843 Received December
23, 1976
The theoretical basis is developed for a population model which allows the use of constant temperature experimental data in predicting the size of an insect population for any variable temperature environment. The model is based on a stochastic analysis of an insect’s mortality, development, and reproduction response to temperature. The key concept in the model is the utilization of a physiological time scale. Different temperatures affect the population by increasing an individual’s physiological age by differing rates. Conditions for the temperature response properties are given which establish the validity of the model for variable temperature regimes. These conditions refer to the relationship between chronological and physiological age. Reasonable agreement between the model and field populations demonstrates the practicality of this approach.
1.
INTRODUCTION
It is a well-established fact that an insect’s rate of development is heavily dependent upon temperature. Estimates for an insect’s development rate under a specified temperature environment are obtainable by laboratory experiments; however, it is not always possible to use these laboratory estimates directly for predicting development in a field environment where actual temperatures are almost never the same as the laboratory temperatures. The most commonly used technique for predicting variable temperature development (physiological age) is the degree-day method (Hughes, 1962; Gilbert et al., 1976). The recognition of the nonlinearity of the developmental rate with respect to temperature has lead to the more general rate summation methods. Using rate summation, it has been demonstrated (Fye et al., 1969; Stinner et al., 1974) that the mean time of stadial emergence under variable temperature environments can be predicted utilizing the reciprocal of the constant temperature * The work herein reported was funded in part by a U.S. Department of Agriculture sponsored program entitled “The Expanded Southern Pine Beetle Research and Applications Program” Grant Number 89-106 (19-192) and in part by the National Science Foundation and the Environmental Protection Agency, through a Grant (NSF GB-34718) to the University of California.
197 0040-5809/78/0132-0197$02.00/0 All
Copyright Q 1978 by Academic Press, Inc. rights of reproduction in any form reserved.
198
CURRY,
FELDMAN,
AND
SMITH
emergence times. Sharpe et al. (1977) and Stinner et al. (1975) used these concepts to predict the probability distribution function of emergence times under arbitrary temperature regimes. The underlying feature which allows Sharpe et al. (1977) and also Barfield et al. (1977) to predict emergence time distributions for a variety of insects is that the probability distribution functions of the time of stadial emergence under different temperature regimes have the “same shape.” Specifically by “same shape” we mean the following: Consider several cohorts (a cohort is a group of insects of the same species and age within the same stadium) raised under different constant temperature regimes. Recording the stadial emergence times and rates (reciprocal of times) for each cohort separately, then multiplying the emergence times by the cohort mean rates, one obtains resulting probability distribution functions of “normalized” emergence times. Then if these cohort distribution functions are identical, we say they have the same shape. The same shape property has been experimentally verified for the boll weevil (Anthonomus grandis Boheman) and the cotton fleahopper (Pseudutomoscelis seriutus Reuter) by Sharpe et al. (1977) and for a boll weevil parasite Bracon mellitor Say by Barfield et al. (1977). It will be shown that for insects having this same shape characterization, the emergence time distribution under an arbitrary temperature regime can be computed using only laboratory data. In addition to stadial emergence, we shall consider reproduction and survivorship under analogous assumptions. These characteristics are utilized in the development of a general temperature-dependent single-species stochastic population model based on laboratory data. The results of the model indicate that the physiological age of an insect along with the associated normed distributions for emergence, reproduction, and survivorship is sufficient to fully describe the relevant population processes under variable temperatures. The purpose of this paper is the mathematical justification for this temperature-dependent model. It should be emphasized that the model developed in this paper is oriented toward estimating field populations based on traditional laboratory experimental data. The necessary data are obtained from constant temperature experiments and consist of: mean values for development times, percentage mortality, and total reproduction; and distribution functions for emergence times, death times, and percentage reproduction. Under reasonable assumptions, these data from just a few constant temperature experiments can be used to predict the population response for variable temperature regimes. The assumptions (stated explicitly below) are readily tested with the experimental data. Although the assumptions are apparently valid for many species, they clearly do not hold in general. Due to the practicality of this approach, research in the relaxation of the assumptions is needed. The basic notational conventions to be used throughout this paper are given in Section 2. Properties relating to exit times, emergence times, and reproduction
TEMPERATURE-DEPENDENTMODEL
199
are discussed in Sections 3,4, and 5, respectively. The integration of the cohort submodels into a population model is described and illustrated in Section 6.
2. PRELIMINARY DEFINITIONS AND NOTATION A temperature regime will be denoted by the function yGwhere #(t) is the
temperature at time t. The function &(.) denotes the constant temperature regime set at k degrees Kelvin. Consider an individual insect raised under some specified temperature regime #. The times at which the insect emerges from the egg phase, the larval phase, etc., as well as its death time and fecundity are random variables dependent on 4. Specifically for temperature regime 4, let Tl*, T,*,... be random variables denoting the lengths of time the insect is in the first (egg) stadium, second (larval) stadium, etc. The Ti’s will be called exit times and they are assumed conditionally independent given that no death occurs. If the insect dies in the ith stadium, let Ui* be a random variable denoting the time of death; thus Ui* = Ti*. To denote that death did not occur before the ith stadium, let Ut’j = co for j = 1, 2,..., i - 1. As a general rule a superscript will refer to the temperature regime and a subscript will refer to a specific stadium. Capital letters will refer to either random variables or distribution functions. For notational convenience, the subscript will be deleted whenever it is possible to do so without causing confusion. By an abuse of notation Ttk will be written as Ti”. In terms of the above notation, our purpose is to use the empirical distribution function of Tk to obtain an estimate for the probability distribution function of T*, where # is an arbitrary temperature regime. In normalizing the exit times, it is convenient to use the “instantaneous rate of development.” At temperature k, the development rate, r(k) is defined by r(k) = E[l/Tk],
(2-l)
where E[l/Tk] is the expected value of l/Tk. Constant temperature experiments are generally performed at a selected number of points over the operational temperature range. The functional form of I( .) over the temperature continuum is obtained by the standard Qr,, , Arrhenius, temperature summation rule (see Laudien, 1973, p. 359), or the more recent relationship of Sharpe and DeMichele (1977). In a manner similar to that of Butler and Watson (1974) it is assumed that the insect species under consideration can instantly adapt to temperatures within a certain interval; that is, after a temperature change the insect’s developmental rate immediately changes to that of the new temperature. Furthermore, all temperature regimes are within this interval and are assumed piecewise continuous. The additional notation specific to reproduction will be defined in Section 5.
200
CURRY,
3.
FELDMAN,
STADIAL
EXIT
AND
SMITH
TIMES
In this section it is demonstrated that the distribution of exit times under an arbitrary temperature regime can be obtained using the distribution function of the normalized exit times from experimental temperature regimes. Throughout this section, we shall be dealing with insect properties for a specified stadium; thus, the stadium subscript will not be used. The main assumptions regarding exit times underlying (3.1) ASSUMPTIONS. this research are as follows: (a) Exit times have the same shape characteristic. That is, the function F(.) defined by, for x > 0, F(x) = P{r(k) T” < x> is independent of k. Furthermore, F is continuous and the exit times are positive and uniformly bounded for all temperatures (that is, there are numbers u and b such that 0 < a < Tk < b < oc,for all A). (b) A small change in temperature produces no more than a small change in the exit times; that is, for any temperature regime I/ and any sequence of regimes (p) with #” -+ #, it follows that limnem T@’ = T* almost surely. (c) When an insect’s environment changes from one temperature to another, there is no residual effect due to the temperature change. In other words, after the time of the temperature change, the insect’s behavior is identical to that of an insect of the same biological age (i.e., fractional development as defined below) which was raised at the second temperature. A key concept throughout this paper is the fractional development of an insect which indicates the expected value of its age on the normalized time scale. The fractional development of an insect within the stadium at time t, denoted by d”(tO , t), is defined by d”(t, , t> = 1; r 0 W
&
(3.2)
where to is the entrance time into the stadium and # is the temperature regime. Equation (3.2) is the standard form (rate summation) used in computing physiological time (Stinner et al., 1974, Eq. (2); Allen, 1976, Eq. (1); Logan et al., 1976, Eq. (12)). Th e integrand of the equation uses the developmental rate I defined for constant temperatures composed with the temperature regime. As before, we shall write d”(t, , t) instead of dlLk(t, , t). For constant temperatures it should be noted that d”(t,, , t) = r(k)(t - t,). The main result of this section is presented in the following theorem.
(3.3)
TEMPERATURE-DEPENDENT
201
MODEL
(3.4) THEOREM. Let # be an arbitrary temperature regime. The exit time distribution for an insect enteritlg the stadium at time t, is given by P{TS < t - to} = F(d”(& , t)), where 4(t) = $(t + t,,).
Proof. Without loss of generality, the theorem will be proved for If I/J= I& for some k, then P{T* < t} = P{Tk < t} = P{r(k) Tk < r(k) thus, by Assumption (3.la) and Eq. (3.3),
to t}
= 0. and
P(T” < t} = F(dk(O, t)). Next, consider the case where the temperature changes once, say at is, define 1/,by for t < t*, W = k, for t > t*. = k,
t*;
that
(3.6)
For t < t* it is clear from the above that P{ T’J < t} = F(d@(O,t)). At time t * an organism has fractional development d%(O, t*). Select the time s* such that d%(O, t*) = d”z(0, s*).
(3.7)
2 zz :; 5 e
-------------
0
FIG.
“\/dk+ -
IF
d s*
TIME
1. The chronological times t * and s*, at constant temperature
respectively,
corresponding
to the same age d* on the fractional
regimes kl and k, , development scale.
Intuitively, s* is the time needed under temperature k, for the organism to have the same fractional development as it would have at time t* under temperature k, (this transform is illustrated in Fig. 1). For T > 0, P{T”
+P{t* t*} P{T” < t* + 7 1 T* > t”}
202
CURRY, FELDMAN, AND SMITH
and by Assumption (3.1~)
= P{T”I < t*} + P(T”1 > t*} P{Tkn < s* + r j T% > s*>. Using (3.5) and (3.7) and simplifying yields
P{ T” < t * + T} = F(dkz(O,s* + T)). Using (3.2) it is seen that d*(O, t* + T) = dlc8(0,s* + T) which implies the theorem is true for any $ having only one temperature change. Using mathematical induction, it is clear that the theorem holds for any # which is a simple function (step function with finite range). Finally, assume that I/ is an arbitrary temperature regime. Then # can be represented as the limit of an increasing sequence of simple functions, say, I,G= lim 4”. The th eorem is valid for each regime p; that is P{T”” < t> = F(d@(O, t)). Combining the uniform bounded property of { Tk) given in Assumption (3.la) and the almost sure convergence of Assumption (3.lb), one has that r(.) is a continuous function (this is convergence of the means and is given by Chung (1968, p. 93)). Ag ain, using Assumption (3.la) and the Lebesgue Convergence Theorem, limn+m d@(O,t) = d”(O, t) and thus limn+m F(d@(O, t)) = F(dti(O, t)). Since almost sure convergence implies convergence in distribution (see Chung, 1968, p. 84), it follows that
F(d”(0, t)) = &if F(d”“(0, t)) = f+xr P{T”” < t} = P{T” < t} and the proof is complete.
4. STADIAL EMERGENCE In order to develop a population model, the distribution of stadial exit times is necessary. In addition, it is important to know whether the departure was due to death or to emergence into another stadium. In this section the probabilities associated with stadial emergence are developed. Specifically consider an insect which enters the stadium at time to under temperature regime #, then the probability of emergence into the next stadium by time t is
P{Ts < t - t,, , ti > t - to}, where a&t) = #(t + t,,).
203
TEMPERATURE-DEPENDENTMODEL
(4.1) ASSUMPTIONS. It is assumed that the emergence times have the same shape characteristic up to a multiplicative temperature constant; that is, there is a distribution function G( .) such that, for x > 0, P{r(k) T” < x, r(k) U” > x} = c(k) G(x), where c(k) is the probability of ultimate emergence from the stadium under constant temperature regime k. Furthermore, it is assumed that G is continuous and there is a number b < co such that G(6) = 1 (the number b gives an upper bound for the emergence time on the fractional development scale). (4.2) PROPOSITION. The distribution function for the normalized emergence time conditioned on live emergenceis independent of temperature and is G(x), i.e., P{v(k) Tfi < x / U” = CC} = G(x).
Proof. By Assumption (4.1), we have P{V = a} = c(k). Then, since the events (r(k) Tk < x, Uk = co> and (r(k) T” < x, r(k) Uk > x} are equal, the result follows by the definition of conditional probability. As in the previous section, it will be shown that the emergence distribution under an arbitrary temperature regime is obtained using fractional development. (4.3) THEOREM. Let 4 be an arbitrary temperature regime. The emergence time distribution for an insect entering the stadium at time t, is given by P{Ts < t - t, , u$ > t - t,} = jt c 0 4(s) dG(W, to
, 4,
where the integral is a Stieltjes integral and $(t) = #(t + to). Proof. Without loss of generality, the theorem will be proved for t, = 0. If # = I+&for some k, then clearly P(T” f t, U” > t} = c(k) G(d”(0, t)).
(4.4)
Consider the case where the temperature regime $ has one step as defined by Eq. (3.6). For t < t* (4.4) holds. As before, the time s* is chosen such that the fractional development under the k, temperature regime at s* equals the fractional development under the k, regime at time t* (that is, Eq. (3.7) holds). For T > 0, P{T” < t* + ‘, u* > t* + T} = P{T* < t*, U* > t*} +P{T~>t*}P{T”t*+r/T~>t*} and, by Assumption (3.lc), = c(k,) G(d”l(0, t*)) + [l -F(d”l(O, t*))][c(k,) G(d”a(0, s* + T)) - c(kJ G(d”z(0, s*))]/[l - F(d*z(O, s*))].
204
CURRY,
FELDMAN,
AND
SMITH
Using (3.7) and simplifying yields
= Wi) - WI WW t*)> + ~(4) W”(O, t* + 4 which is a Stieltjes integral for the single step function and using integration by parts leads to the proper result. Now let 1F,be a simple function. That is, for fixed times 0 = t, < t, -=za*. < tN and fixed temperatures $ , A1,..,, k, , # is defined by $(s) = Ki when ti < s < ti+l (i = O,..., N - 1) and 9(s) = kN when s > t, . Then it can be shown that
P{T* < t, + T, U” > t, + T} = c(k,) G(@(O,t, + T)) + f Wi-1) - &)I
G(@(O,ti)), (4.5)
i=l
for fixed n = 1, 2 ,..., N and t, + r < tn+l (with tN+l = co). Equation (4.5) is the Stieltjes integral for a simple function and the integral of the theorem is obtained after integration by parts. Finally, assume that $ is an arbitrary temperature regime (of bounded variation). The first term on the right-hand side of the equation of the theorem follows due to the same reasoning as in Theorem (3.4). The second term follows from the theory of the Stieltjes integral (see Kolmogorov and Fomin, 1970, p. 378).
Remarks. Assumption (4.1) implies that changing the temperature uniformly changes (raises or lowers) the probability of emergence in any physiological time interval (x1 , xe). The basis for this assumption is that the data of Sharpe et al. (1977) and Barfield et al. (1977) indicate that emergence distributions for certain insects exhibit the same shape property (as in Proposition (4.2)). For other organisms, an alternate same shape property might be appropriate. This assumption can be applied to death times instead of emergence times, yielding similar formulations. Another alternative leading to a more complicated formulation is to use the same shape property for both emergence and death times. This alternative necessitates the relaxation of the same shape exit time assumption. 5.
REPRODUCTION
The final component in a single population model is reproduction. The rate of reproduction for insects typically varies with both age and temperature. It has been observed for some species (Barfield, 1976) that the proportion of individual female’s total reproduction as a function of fractional development
TEMPERATURE-DEPENDENT
.i
20.5
MODEL
1.0
1.5
300 150 zoo
FIG. 2. Characteristic curves for reproduction. (a) Proportion of total reproduction as a function of fractional development. (b) Total reproduction as a function of temperature. is the same for all temperatures. Figure 2a displays this functional form for the boll weevil which shows, for example, that 46% of a female’s total reproduction occurs in the fractional development interval (0.5, 1.0). However, even though the proportions are independent of temperature, the actual number of eggs laid does vary with temperature (Wood and Starks 1972; Isely 1932; Cole 1970). Figure 2b displays the total expected number of eggs per adult female boll weevil which shows, for example, that at 25°C the expected number of eggs laid during the female’s lifetime is 243. Thus, the expected number of eggs laid in the fractional development interval (0.5, 1.0) at 25°C is 112. In this section, a method for determining reproduction under an arbitrary temperature regime is developed. It is known that for some species, variable temperatures can produce a more fertile insect than is possible at constant temperatures. This possibility is ignored in order to have a tractable model. The expected rate of reproduction at fractional development x and temperature k, ye(k) x), is assumed to have the form
206
CURRY,
FELDMAN,
AND
SMITH
where Jzg(x) dx = 1. The term h(K) is the total expected reproduction over the lifetime of a female adult held under the constant temperature regime K; while g(x) dx has the interpretation that it is the proportion of reproduction of each female in the fractional development interval (x, x T dx). It should bc noted that many data sets have been published of the form necessary to determine h(.); however, as pointed out by Pianka and Parker (1975, p. 457) few data sets on age specific reproduction have been published. The basic assumption for reproduction is that the (5.2) ASSUMPTION. reproduction at a given time depends only on the insect’s fractional development and the current temperature. Assumption (5.2) is consistent with the previous temperature assumptions in that historical temperatures are reflected only in fractional development. In previous sections, we have analyzed the relevant processes utilizing the probabilistic nature of the components. To describe reproduction requires more detail than was necessary for the simplified descriptions in the previous sections. To develop the stochastic model of reproduction underlying Eq. (5.1) necessitates a biological analysis at a depth considerably beyond that utilized in the previous sections. The expectation form assumed in Eq. (5.1) adequately reflects the observed response for constant temperatures. Thus, to avoid unnecessary complications, only expected values will be considered in the extension to variable temperature regimes. Under arbitrary temperature regime I/, the inverse of the function in Eq. (3.2) is needed to determine the time i in which an insect (born at ts) reaches fractional development x. This is denoted by 8f0 ; in other words
t = 6:&x)
(5.3)
if and only if x = d*(t, , t). The rate of reproduction at fractional development x is r,(~#b(x)), x) and thus the total reproduction between the development states x, and xs is
Using transformation (5.3) and Eq. (5.1), the total reproduction p on the time scale between time t, and 1, is given by
(5.4) where r(a) is the development rate, Eq. (2.1), for the adult phase.
TEMPERATURE-DEPENDENT
6. POPULATION
207
MODEL
MODEL
An insect population consists of a collection of cohorts. A cohort, as defined in this paper, is a group of insects within a stadium that started the stadium at the same time. As time varies, new cohorts are generated by two mechanisms: (1) emergence from one stadium into the next and (2) adult reproduction. For clarity of presentation, cohorts will be labeled and the index set of the cohort labels will be denoted by N. The only information necessary concerning a specific insect cohort i is the triple (#, si, t,$ where qi is the initial cohort size, si is the cohort stadium, and toi is the initiation time of the cohort. To utilize the discrete cohort concept, an implicit time increment d t exists for which all insects generated within this increment are assumed identical. The convention that d(t) = z,h(t$ to) will be continued throughout this section. To model a dynamic insect population, an initial group of cohort triples must be given. The cohort time increment At is then used to iteratively compute the population changing over time. New cohorts can be added to the population
lARVAI. COHORT LARVAI, COHORT
I.ARVAI. COHORT ADL'LT COHORT
ADULT COHORT ADVLT COHORT
EGG COHORT
.u.E TIME
FIG. 3. New cohorts are generated every dt time increment for which emergence and/or reproduction are possible. This example uses three stadia; eggs-E, larvae-L, and adults-A. The curves represent the relevant distributions for stadial emergence or reproduction.
208
CURRY,
FELDMAN,
AND
SMITH
at each time increment via emergence and reproduction as illustrated in Fig. 3. To demonstrate the iterative population update, consider a time t with a current population consisting of cohorts indexed by the set P C N. That is, for i E P the triple ($, si, tOi) when t,,j < t represents a cohort present at time t. Thus, the expected number of insects within a specified stadium M at time t is
where the sum is over all i E P such that sa’= m and the expression is computed using Theorem (3.4). For each stadium m (m > I), a new cohort I can be generated by emergence during the time interval [t, t + dt) and its triple is given by (T,“, sz, to”), where s1 = m,
(f-54
t,” = t + At/2, 7)’ =
c
qP{t - toi < T$+, < t + At - tog,U$-, > t - toi}.
&W+l
The final expression
in (6.2) is computed
using Theorem
(4.3) and the equality
P{t < TB < t + At, I? > t + At} =P{Tst+At}-P{T4,(t,U6>t}. For the first stadium, a new cohort j can be generated from the reproduction of the adult cohorts (let si = a for the reproduction stadium) during [t, t + At) and its triple is given by (vi, sj, toi), where
$ = c qip*(t, t + At), 8=a
(6.3)
01is the expected proportion of females, sj = 1, and t,,j = t + At/2. The expression in (6.3) is computed using Eq. (5.4). Using Eqs. (6.1)-(6.3) the insect population can be iteratively generated over time given an initial cohort structure. External influences, such as migration, insecticides, etc., can be incorporated into the model if their effect on the cohort structure is known. The mechanics for updating the population are analogous to the Leslie matrix method (Leslie, 1945). A continuous analog is given by Von Foerster (1959); however, the numerical solution of the continuous
TEMPERATURE-DEPENDENT
MODEL
209
model usually approximates time by discrete increments, which again reduces the method to the Leslie matrix approach (Wang et al., 1977). To demonstrate the applicability of these concepts for modeling an insect population, model results for the boll weevil (Anthonomus grandis Boheman) are compared in Fig. 4 with field counts of Walker and Niles (1971). The data utilized in the two-stadium model (immatures and adults) were obtained from published literature. Immature developmental rates and survival percentages are from Isely (1932, Figs. 2 and 6). The distribution function for immature emergence is from Sharpe et al. (1977, Fig. 6). Since the distribution F(.) of immature exit times has not been reported, it was assumed to be the same as the conditional emergence distribution G(.). Adult developmental rates and the physiological age dependent reproduction profile (Fig. 2a) are from the data of Cole (1970). Cole’s data also indicate that the ratio of males to females in the adult population is approximately l/l. The total expected reproduction of a female adult as a function of temperature (Fig. 2b) is from Isely (1932, Table 10). Adult mortality based on physiological age is from the data of Sterling and Adkisson (1970, Fig. 3). Figure 4 displays four years of model versus field adult weevil counts. The population peaks and troughs in the adult counts are due to the lag time (about 2-3 weeks) between egg laying and adult emergence. To obtain an accurate assessment of the model, the predicted expected population sizes should be shown together with the predicted standard deviation of the population size with respect to time. For constant temperature regimes, such a comparison would be possible using the theory from age-dependent branching processes (Jagers, 1975). However, under variable temperature regimes, equivalent results are not yet available. Temperatures for the model were generated from actual daily weather station min-max temperatures using a sinusoidal curve fit to obtain hourly values. Included in the field data of Walker and Niles (1970, Fig. 1) was the date at which reproductive sites first became available. This was included in the model so that reproduction could not be initiated until after this date. During the second in-field generation, density-dependent factors usually become limiting so that the population growth no longer is solely temperature dependent. Thus, the projections of the temperature-dependent model are not valid for this nonn’Ialthusian growth period. During the two-month period in which the model assumptions are valid, Fig. 4 shows a reasonable agreement between the predicted and actual adult population sizes.
653/13/z-3
210
3 a
CURRY,
FELDMAN,
AND
SMITH
TEMPERATURE-DEPENDENT
MODEL
211
212
CURRY, FELDMAN,
AND SMITH
ACKNOWLEDGMENTS We gratefully acknowledge the invaluable contributions of Drs. Don W. DeMichele and Peter J. H. Sharpe to the concepts underlying this paper. The suggestion and comments of the referees are also appreciated.
REFERENCES ALLEN, J. C. 1976. A modified sine wave method for calculating degree days, Environ. Etomol. 5, 388-396. BARFIELD, C. S. 1976. “A Model for Interactions between Bracon mellitor Say and Boll Weevil, Anthonomous grandis Boheman,” Ph.D. Dissertation, Department of Entomology, Texas A&M University. BARFIELD, C. S., SHARPE, P. J. H., AND BOTTRELL, D. G. 1977. A temperature driven developmental model for the parasite Bracon mellitor Say, Canad. Entomol. 109, 15031514. BUTLER, G. D., JR., AND WATSON, F. L. 1974. A technique for determining the rate of development of Lygus hesperus in fluctuating temperatures, Florida Entomol. 57, 225248. CHUNG, K. L. 1968. “A Course in Probability Theory,” Harcourt, Brace & World, New York. COLE, C. L. 1970. “Influence of Certain Seasonal Changes on the Life History and Diapause of the Boll Weevil Anthonomus grandis Boheman,” Ph.D. Dissertation, Department of Entomology, Texas A&M University. FYE, R. E., PATANA, R., AND MCADA, W. C. 1969. Developmental periods for Boll Weevils reared at several constant and fluctuating temperatures, J. Econ. Entomol. 62, 14021405. GILBERT, N., GUTIERREZ, A. P., FRAZER, B. D., AND JONES, R. E. 1976. “Ecological Relationships,” Freeman, Reading, Mass./San Francisco. HUGHES, R. D. 1962. A method for estimating the effects of mortality on aphid populations, J. Anim. Ecol. 31, 389-396. ISELY, D. 1932. Abundance of the Boll Weevil in Relation to Summer Weather and to Food, Arkansas Agricultural Experiment Station Bulletin No. 271. JAGERS,P. 1975. “Branching Processes with Biological Applications,” Wiley, London. KOLMOGOROV, A. N., AND FOMIN, S. N. 1970. “Introductory Real Analysis,” PrenticeHall, Englewood Cliffs, N. J. and Life” (H. Precht, LAUDIEN, H. 1973. Changing reaction systems, in “Temperature J. Christophersen, H. Hensel, and W. Larcher, Eds.), pp. 355-398, Springer-Verlag, New York/Berlin. LESLIE, P. H. (1945). On the use of matrices in certain population mathematics, Biometrika 33, 183-212. LOGAN, J. A., WOLLKIND, D. T., HOYT, J. C., AND TAMGOSHI, L. K. 1976. An analytic model for description of temperature’ dependent rate phenomena in arthropods, Environ. Entomol. 5, 113-l 140. PIANKA, E. R., AND PARKER, W. S. 1975. Age-specific reproductive tactics, Amer. Natur. 109, 453-464. SHARPE, P. J. H., CURRY, G. L., DEMICHELE, D. W., AND COLE, C. L. 1977. Distribution model of organism development times, /. Theor. Biol. 66, 21-38. SHARPE, P. J. H., AND DEMICHELE, D. W. 1977. Reaction kinetics of poikolotherm development, J. Theor. Biol. 64, 649-670.
TEMPERATURE-DEPENDENT
MODEL.
213
STERLING, W. L., AND ADKWGON, P. L. 1970. Seasonal rates of increase for a population of the Boll Weevil, Anthonomw grandis, in the high and rolling plains of Texas, .4nn. Entomol. Sot. Amer. 63, 1696-1700. STINNER, R. E., GUTIERREZ, A. P., AND BI:TI.EH, G. D., JH. 1974. An algorithm for temperature-dependent growth rate simulation, Canad. Entomol. 106, 519-524. QTINNW, R. E., BUTLER, G. D., BACHELEH, J. S., AND TCITLE, C. 1975. Simulation of temperature-dependent development in population dynamics mod&, Cannd. f?n~omo/. 107, 1167-I 174. VON FOERSTER, H. (1959), Some remarks on changing populations, itt “The Kinetics Ed.), pp. 382-407, Grunt k Stratton, of Cellular Proliferation” (F. Stohlman, Sew York. Dynamics of the Boll W’ecvil \vAI.KER, J. K., JR., AND NILIS, G. A. 1971. “Population and Modified Cotton Types: Implications for Pest Management, ” Texas Agricultural Experiment Station Bulletin Xo. 1109. WANG, Y., GUTIERREZ, A. P., OSTER, G., AND DAXI., R. 1977. A general model for plant growth and development, Conad. Entomol. 109, 1359-1374. WOOD, E. A., JR., AND STARKS, K. J. 1972. Effect of temperature and host plant interaction on the biology of three biotypes of the greenbug, En&on. Enzomol. 1, 230-234.
