Optimal switching to diapause in relation to the onset of winter

THEORETICAL

POPULATION

BIOLOGY

Optimal

18,

125-133

Switching to Diapause in Relation to the Onset of Winter FRITZ

Department

(1980)

of Biology,

University

TAYLOR

of New

Received

Mexico,

September

Albuquerque,

New

Mexico

87131

18, 1979

A model is presented that specifies optimal switching times for the induction of diapause relative to the onset of winter in a deterministic environment. Fitness is defined as an individual’s contribution to the overwintering population in diapause at the time of the fust hard frost. The fitness of a nondiapausing female is determined by the switching times of her offspring. If age-specific fecundity and survivorship do not change significantly from generation to generation, the optimal switching time precedes the onset of winter by a constant amount of time equal to the age of first reproduction plus the time to produce one offspring plus the difference in ages between the sensitive age and the diapause age. This result is independent of the time at which the original female began to reproduce. However, if fecundity or survivorship decreases toward the end of the season, the optimal switching time depends upon the time when she began to reproduce and should be more conservative by an amount of time that can be computed. Possible tests of the model are proposed.

I. Ten years of diapause

ago, Cohen induction.

grown studied

tremendously, life history

freezing

(Jungreis,

(1970) In

the

making characters

INTRODUCTION

presented intervening diapause (Saunders,

a theoretical paper on optimal timing decade the empirical literature has induction

one

of the

most

thoroughly

1976; Dingle, 1978). However, no advances have been made in the theory or in relating experimental findings to the theory. I believe the lack of coherence has resulted from the difficulty of deriving testable predictions from Cohen’s original formulation of the problem. His results were stated in terms of conditions for switching of the entire population or some proportion of it not in terms of when the switch should occur although such predictions could be derived from his formulas. This paper presents a model based on the renewal equation that provides testable hypotheses about the optimal time to switch to diapause when the season being avoided is brought on by a catastrophic event such as a hard frost. Our interest is in hibernal diapause which is a physiological condition of suspended development generally correlated with increased resistance to 1978). 125 0040-5809/80/040125-094M)2.00/0 Copyright 0 1980 by Academic Press. Inc. All rights of reproduction in any form reserved.

126

FRITZ

II. THE

TAYLOR MODEL

An absolute fitness can be assigned to a female according to her contribution to the population of diapausing individuals at the time of the first hard frost. Thus, the expected fitness of a particular female if she decides to enter diapause is her probability of entering diapauseand surviving to the time of the first hard frost. A female that does not enter diapausemust be assigneda fitness according to her contribution to the overwintering population via her offspring. The general problem that is solved in this paper is the following: At what time should the offspring of a female begin to forego reproduction during the current seasonand enter diapause given that the suitable habitat ends abruptly with the first hard frost. This is an evolutionary problem faced by many insects that have one or more generations each year in the temperate zones. As a single example, the tobacco hornworm, Manduca sextu, passes through three or four generationsper seasonin North Carolina. Pupal diapause is brought on at the end of the seasonby sensitivity to daylength (photoperiod) during earlier larval stages(Bell et al., 1975). Saunders (1976, Appendix B, Table 4) lists sensitive and/or diapausestage for well over 200 insect species with a photoperiodic induction of diapause. This section outlines the form of the model used to solve the problem stated above. The results are presented in the next section and the derivation of the results appearsin the Appendix. Consider the demographic progress of the descendentsof a female that begins reproduction at time 0. Her offspring, generation one, may in turn begin to reproduce at time OL,the age of first reproduction. Suppose age x diapausesand age x’ is the sensitive age that induces diapause. We are first interested in the number of individuals of agex through time which is given by YZ(X,t) = i

Bi(t - X) Ii(X)

i=l

consideringg generations.The term &(t) gives the number of births at time t in generation i and the term I,(X) gives survivorship to age x in generation i. Using the renewal equation which expressesBi(t) in terms of B+,(t) this can be rewritten as n(x, t) = z?,(t - x) Z,(x) + I$; Z,+,(x) s.” qt

- x - a) $(t - x - a) $&(a)da, (1)

where &tt>

= 1:

if(i-l)or t or t >

t,

- x’.

(2)

The step function &(t) identifies those femalesthat reproduce in generation i and &(a) is the net maternity function for generation i. For simplicity assume reproduction in eachgeneration beginsat ageOL and endsat age/3. The switching time, td , is defined as that time after which all individuals at the sensitive

SWITCHING TO DIAPAUSE

127

age on or after t, , i.e., born on or after time t, - x’, enter diapause,and do not reproduce. Ultimately we are interested in the number of individuals that have accumulated in the diapause age at the end of the season,time t, , as a function of the switching time. This is (3)

where 6 = x - x’. The problem then is to find

my N(b).

(4)

This formulation captures a great deal of the real problem faced by insects. They do typically diapause at a well-defined, narrow stage of development which can be represented closely as one age. The sensitive stage inducing diapause often encompassesone or more developmental stages or instars. However, if X’ is defined as the last age of the sensitive stage, the optimal proportion of individuals passing through this age that is destined to enter diapausecan be specified by the results below. The most common mechanisms for inducing diapause involve using photoperiod as an indicator of the time of year (Danilevskii, 1965; Saunders, 1976). Therefore, I have set up the theory so that the switching time is relative to the sensitive stage becausemost data involving photoperiod induction of diapauserecord the time at which individuals in the sensitive stage are switching. If data are more readily available for switching in the diapausestage itself, the results are easily restated solely in terms of the diapausestageby setting X’ = x in which caseone is asking when individuals should start entering diapauseinstead of inducing diapause.The photoperiodic response inducing diapause is usually polygenic (Hoy, 1978; Tauber and Tauber, 1979) which makes it reasonableto consider switching times that directly affect more than one generation. Finally, the appropriate time scale is the physiological time scale for the species being considered (Taylor, 1980). III.

OPTIMAL

SWITCHING

TIMES

A. The General Solution In the generalcase,optimal switching times are found by solving the following expression for tg which is derived in the Appendix.

g

= F; B&d* - x’) Zi+,(X) j-atf-(tjt8)f$i(U) du (I-1 - 1 B&d* - x’) Z&Y) = 0, i=l

(5)

128

FRITZ

TAYLOR

where g is the number of generations that needs to be considered as determined from expression (A6) in the Appendix. It says that the optimal switching time occurs when the number at the sensitive age in the current generation at time t, expected to reach the age of diapause equals the number of offspring of these individuals in next generation expected to reach the age of diapause Summed across all relevant generations. The latter point can be seen by noting that the time available to reproduce successfully for those individuals that first enter diapause would have extended from t, - x’ + (Yto tf - x, which is tr - (td + S) - cc. B. An Important Special Case Since there is generally limited overlap between the first and successive generations, an example with broad applicability is that in which the switching time only directly affects the first two generations, i.e., - x’ < 201 and - x’) = 0. Assume that &(a) = &(a) = +(a) and II(a) = Z,(a) = Z,(a) for all a. In this case Eq. (5) simplifies to t,

B&t,

t,-(t;+a) I

&4)du

= 1.

(6)

LI

Equation (6) says that if there is no reason to expkct survivorship or fecundity to change significantly from generation to generation, .the optimal switching time precedes the onset of winter by a fixed amount of time equal to the age of first reproduction plus the time to produce one female offspring plus the difference in ages between the sensitive age and the diapause age. This result can be restated as follows: The female offspring of the original female should switch to diapause at age x’ at the time when they will themselves be able to produce only one female offspring that would have enough time before the first hard frost to enter diapause at age x. The production of one net female offspring entering diapause is required because the model assumes that once in diapause a female survives to the onset of winter with probability one. If occurred earlier than some females in generation one would have entered diapause when they could have contributed more than one female offspring to the overwintering population. On the other hand, if occurred later than some females in generation one would have begun reproducing when there was not enough time for them to be replaced by their offspring. This result is particularly interesting because it predicts that one should observe the offspring of all indi&duals begin diapause at the same time relative to the end of the season no matter when these individuals began to reproduce. The photoperiodic induction of diapause appears to represent this kind of switch. It should be modified from year to year to account for changes in date of the first frost which might be accomplished by the additional use of temperature signals (Saunders, 1976; Beach, 1978). Expression (6) also gives the t,

tz,

t,

t$,

129

SWITCHING TO DIAPAUSE conditions under which none of the first generation if t, - 6 is not larger than OLby enough time to allow then tz < 0. Insects having one generation per category, if they are selected to avoid the first hard

should reproduce. That is, production of one offspring, year should fall into this frost.

C. SeasonalDeclinein Survivorship or Fecundity Without the assumptions for determining t$ is

&(t;

of constancy

- x’) [I,(x) %-“;‘“’

&(u)

across generations,

the expression

du - &c)]

+ B,(t,* - x’) [I,(x) jbt’-(fitd)c#+) du - Z&z)] = 0.

(7)

Now the optimal switching time does depend upon the time at which the original female began to reproduce. That is, the optimal switching time depends upon the age structure of her offspring relative to the time-dependent changes in survivorship and fecundity. In addition, it can be shown as follows that if fecundity or survivorship declines between generations, the switching time should become more conservative. To see this, suppose t& solves Eq. (7). Now decrease &(u) so that the integral involving this function decreases making the expression in the right-hand bracket smaller. To get the left-hand side of (7) equal to zero again, t, must be decreased to some t& . Similar arguments show that if survivorship in generations two and/or three is multiplied by a constant between 0 and 1, tz should decrease. Thus, fitness will be increased if the insect uses some signal that is correlated with decreased fecundity or survivorship in subsequent gener.ations to decrease t, by an appropriate amount which can be determined from Eq. (7).

IV.

TESTING

THE THEORY

An insect should be chosen whose activity is likely to be limited by the first hard frost because individuals not in diapause are killed by the frost or quickly starve for lack of food. The easiest test of the relationship between t, and t, is to look at the time at which most of the individuals are entering the diapause stage in the diapause condition. As a first approximation, unless clearly not true, one can assume that dl(a) = &(a) and l,(a) = &(a) = l,(a) in which case Eq. (6) should apply and t, should procede t, by a time equal to the age of first reproduction plus the time to produce one offspring. It is not specified by the current theory how tz should relate to a previous sequence of t,‘s as determined from long-term temperature records. As a first test, one might compare the observed td to the earliest t,‘s in a 20-year series, say. If t, is

130

FRITZ TAYLOR

significantly earlier than predicted by the model, then this modelshould probably be rejected, and it may be concluded that the switch to diapauseis being selected relative to an earlier event than the first hard frost. The theory should, however, be extended to include the effectsof stochasticvariation in life history parameters and t, on the switching time. If the predicted relationship does hold approximately for the conditions over a seriesof years, then it would be worthwhile to study the mechanism of induction in the sensitive stage to see if it shifts t, in a way that appearsto be adaptive in relation to yearly variations in tt , Zi(a), and &(a) as determined from Eqs. (6) and (7). APPENDIX:

DERIVATION

OF THE GENERAL SOLUTION IN EQ.(~)

For simplicity we derive the expressionfor the case where the number of generationsg = 3. Thus n(x, t) is given by the expression

n(x, t) = Il,(t - x) Z,(x) + 22(x)I” B,(t - x - a) s,p - .2”- a) 91(a) da a - x - a) S,(t - x - a) c&(a) da,

+ 4(x) j” w a

where 4(t) and i&(t) are as defined in Eq. (2). Using Leibnitz’ rule to differentiate the integral in expression(3) we obtain i3N - --Z,(x) &(t, 3g-

- x’)

(AlI

--l&

- x’)

643

Wd

+Z,(x) If:,

&

l0 B,(t

- x - a) S,(t - x - a) +&(a) da dt

d -Is(x) B&

- x’)

(A3) (A4)

Working through this expressionterm by term, term (Al) in the present model has B,(t) = A,(t + 4/k,) where M a) is the net maternity function for the female starting the birth sequencewho was age a, at time -(a - uJ. Term (A2) equals 0 -Z,(x)

1” Bl(td

OL

- x’ - a) &(a)

-Z*(x) Itd-” B,(t, (I

- x’ - u)&(u)

da da

if

td - x’ < (Y

if

td - x’ > /3

if

01< td - x’ < p

SWITCHING

TO

131

DIAPAUSE

since S,(t, - x’ - a) = Similarly,

if if

0 1

I

t, - N’ < a or a < 0 0 < a < t, - x’.

term (A4) equals

0

if

t, - s’ - iy < 01

-Z3(x) 1” B2(td - x’ - a) $,(a) da u

if

t, - x’ - 01> ,l3

-&)

if

a
/-‘-= u

Bz(t, - x’ - a)+,(a) da

since S2(& - x’ - a) = Terms (A3) and taking into account (A3) first determine a change of variable

0 I1

if if

t, - x’ - a < a or a < L, 0 < a < td - x’ - 0~.

(A5) require differentiation of the inside integral while the effect of the switch in functions S, and S, . For term when t, enters into evaluation of the inner integral. By to u = t - x - a one gets

t4z+cd t--(x+E) W4 M - x - 4 du I =o

if

t<

x + 01or

t, -

r-b+4 l-(s+s) W) Mt - x - 4 du =I

if

t<

x + /3 and

s’ <

td -

t-

x’ >

(x + /I)

t-

(x + LX)

=sy;-,, W)Mt- fi- 4lfu

if t>x+fiandt,-x’
t-(r+a)

B,(u)+,(t-x-u)du

id-X’ =s =s

if t<.t+fandt,-x’>t-(x+or)

0

0

B,(u)c$,(t

- x - u) du

if

t

< .x + /3 and

td t - x’ <

- (x + a).

So if tf < t, + 6 + ~1,then the integral does not depend on td and its partial with respect to t, is zero. If t, > t, + 6 + OL,then the integrand of term (A3) is

a at

y

’

J-‘:r:,,, B,(u)+,(t

- x - u) du = &(t,

- X’)Mt

- (td + 8))

132

FRITZ

TAYLOR

and term (A3) becomes Z,(x) Bl(td - x’) jff-(td+a) q$(u) du cl after a change of variable to u = t - (td + 8). In similar fashion term (A5) is found to be

Finally from the pattern that has developed one gets the following for the required partial derivative in the general case:

expression

where 0

if

(i-l)or>td-X’>$---

t,-z’-k-2h

Bi-l(td -

i s01 Bi(td -

X’)

=

if

(i - l)a <

X’ td

- a) fjiel(a) da - X’ < (i - 2)a + /3

’ Bi-l(td - X’ - a) dieI da sa i if td - x’ > (j - 2)a + /I,

(-47)

i = 2, 3,.., .

B,(t) is given above. That equation (A6) specifies a maximum when set equal to zero can be seen by decreasing t, from l$ by some small amount. This increases the upper limits for the integrals involving the $i(u) making the expression positive.

ACKNOWLEDGMENTS I am especially indebted to Henry Harpending who helped in the early formulation of the problem and David Sanchez who provided guidance along the way. Cliff Crawford, Evelyn Ewing, Henry Harpending, and David Sanchez all provided helpful comments on an earlier draft of this manuscript. This research was supported in part by NIH Grant GM0 7661-02 to the Department of Mathematics at the University of New Mexico.

SWITCHING

TO

133

DIAPAUSE

REFERENCES

day number and timely induction of diapause in geographic LTdes atropalpus, J. Insect Phys. 24, 449-455. BELL, R. A,, RASUL, C. G. AND JOACHIN, F. G. 1975. Photoperiodic induction of the pupal diapause in the tobacco hornworm, Menduca sexta, /. Insect Phys. 2 1, 147 l-l 480. Co~nu, D. 1970. Theoretical model for the optimal timing of diapause, dmer. Nutw. 104, 389-400. DANILEVSKII, A. S. 1965. “Photoperiodism and Seasonal Development of Insects,” 1st English ed. Oliver & Boyd, Edinburgh. DINGLE, H. (Ed.) 1978. “Evolution of Insect Migration and Diapause,” Springer-Verlag, New York. HOY, M. A. 1978. Variability in diapause attributes of insects and mites: Some evolutionary and practical implications, in “Evolution of Insect Migration and Diapause” (H. Dingle, Ed.), pp. 101-126, Springer-Verlag, New York. JZINGREIS, A. M. 1978. Insect dormancy, in “Dormancy and Developmental Arrest” (M. E. Clutter, Ed.), pp. 49-112, Academic Press, New York. SAUNDERS, D. S. 1976. “Insect Clocks,” Pergamon, New York. TAUBFX, M. J. AND TAUBER, C. A. 1979. Inheritance of photoperiodic responses controlling diapause, Bull. Entomol. Sot. Amer. 25, 125-128. TAYLOR, F. Ecology and evolution of physiological time in insects, Amer. N&w., in press. BEACH,

strains

R. 1978. Required of the mosquito,

Printed

in Belgium