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A bstract

Tests o f optimization models are often criticized when the qualitative predictions of general models cannot be falsified. However, adequate tests of specific models require both quantitative predictions and precise data. Precise modeling techniques for thermal transients permit prediction o f the egg temperature (TEexn) at which a bird should leave the nest to maximize percent time foraging (P) when constrained by incubation. Within an incubation cycle, P = (100)------- [cool— £------ 5when tcooi, theat, and tequ,i are times for *cool + *heat + *equil

egg cooling, heating, and maintenance at a constant temperature, and r is travel time. Tsexit values for 4 bird species were compared with those predicted by the model. Observed TEexn values for approximately half of all incubation cycles did not maximize P (N=243; 4 species combined). Variation in incubation patterns produced slightly different average egg temperatures for each species, possibly producing different embryo development times. However, average predicted TEexn across entire incubation periods (egg laying to hatching), were the same as observed. Thus, birds may maximize long term P by combining incubation cycles with variation in time components to compensate for non-optimal behavior. Time required for incubation reduced P from a theoretical maximum of 100% to 19.77% for Black-capped Chickadees, 28.06% for Yellow-eyed Juncos, 34.27% for Tree Swallows, and 39.4% for House Wrens. Additional optimization criteria were also considered. TEexn to maximize the rate o f net energy gain (RNEG) differed from observed for individual junco, wren, and swallow incubation cycles and entire incubation periods. TEexn to maximize RNEG for chickadee incubation cycles differed from observed, but average optimal TEexn was the same as observed for the entire incubation period. The chickadees’ ability to achieve the predicted average


optimal TEexn is linked to superior nest insulation and food caching behavior. Tsex„ to maximize foraging efficiency differed from observed for all species. Although tests for all criteria could be falsified for individual cycles, predictions for P for entire incubation periods could not be falsified. Critics o f optimization modeling would consider this a panglossian. and thus flawed, result. However, it appears valid and is likely the result o f attempts to balance reproductive effort with self-maintenance over long time periods.

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P r e d i c t i o n s a n d Q u a n t it a t iv e : T e s t s o f O p t i m a l T im e a n d T e m p e r a t u r e A l l o c a t i o n D u r i n g In t e r m i t t e n t In c u b a t i o n

By M argaret

A. Voss

BSc. State University of New York College o f Environmental Science and Forestry, 1992 MSc. State University o f New York College o f Environmental Science and Forestry, 1995

Dissertation Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Biology o f the Graduate School of Syracuse University

May 2002

Approved Dr. F. Reed Hainsworth, Professor o f Biology Date .

I ,

i-


'P r t r T z .

UMI Number: 3046866

Copyright 2002 by Voss, Margaret A. All rights reserved.

___

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Statem ent

of

C o p y r ig h t

Copyright 2002 Margaret A. Voss All Rights Reserved

R e p ro d u c e d with permission of the copyright owner. Further reproduction prohibited without perm ission.

T he G raduate School Syracuse U niversity We, the members o f the Oral Examination Committee, hereby register our concurrence that M argaret A. Voss satisfactorily defended her dissertation on Wednesday, August 15, 2001

Examiners: Scott Pitnick (Please sign)

William Shields

William Starmer

'lease sign)

J. Scott Turner

(Please sign)

Larry W olf

(Please sign)

Advisor: F. Reed Hainsworth

Oral Examination Chair:

/

James Price

V

(Please sign)


T able A

of

C ontents

b s t r a c t .......................................................................................................................................................................... i

C o m m it t e e A p p r o v a l P a

T able

of

g e

........................................................................................................................... v

C o n t e n t s ...............................................................................................................................................v i

L is t

of

T a b l e s .........................................................................................................................................................x n

L is t

of

F i g u r e s ...................................................................................................................................................... x v

G

lo ssa r y o f

N o m e n c l a t u r e ......................................................................................................................x x

A c k n o w l e d g e m e n t s ......................................................................................................................................x x iii

C h a p t e r 1 .....................................................................................................................................................................25

G e n e r a l I n t r o d u c t i o n .................................................................................................................................... 25

1.1 M o d e l i n g

t h e o p t im a l t im e a l l o c a t io n f o r c e n t r a l - p l a c e f o r a g in g .

28

1.2 A COMPLETE SYSTEM FOR POTENTIAL QUANTITATIVE TESTS: INTERMITTENT INCUBATION......................................................................................................................... 3 6 1.3 W h y p r e c i s e t e s t s f o r 1.3 A A c q u i s i t i o n 1.3B T h e W

P* a n d R N EG *

o f d e t a il e d d a t a

h a v e n o t b e e n d o n e ................................ 4 3

............................................................................................ 4 4

n e e d f o r p r e c is e m o d e l in g t e c h n iq u e s :

h y t h e r m a l t r a n s i e n t s a r e i m p o r t a n t ............................................................................. 4 5

1.4 A MODIFIED MODEL FOR P .....................................................................................................................4 8 1.5 A MODIFIED MODEL FOR

R N E G ......................................................................................................... 4 9

1.6 M o d i f i e d f o r a g i n g e f f i c i e n c y : A n a l t e r n a t i v e t o

P a n d R N E G ...................4 9

vi


C hapter 2

52

G eneral M o d e l D evelopm ent Fo r T

2 .1 A c c u r a t e l y

em perature and

T im

e

............................................5 2

p r e d i c t i n g t i m e s a n d t e m p e r a t u r e s ....................................................... 5 2

2.1 A IS A SECOND-ORDER ANALYSIS NECESSARY?...................................................................... 5 3 2 .IB W 2 .2 M

. a r e s e c o n d - o r d e r a n a l y s e s n o t c o m m o n l y u s e d ? ...............................5 6

hy

o d e l in g h e a t in g a n d c o o l in g

:

METHODS FROM .AN EQUIVALENT MECHANICAL S Y S T E M ............................................................ 5 7 2 .3 E g g

h e a t i n g a n d c o o l i n g a s s e c o n d - o r d e r p r o c e s s e s ............................................6 3

2 .4 E r r o r s w i t h f i r s t - o r d e r

a n a l y s i s ...........................................................................................6 4

C h a p t e r 3 ....................................................................................................................................................................... 6 7

S p e c if ic M e t h o d s In c u b a t e d E g

3 .1 F i n d i n g

fo r

F in d i n g T e m p e r a t u r e s

and

T im e s

for

N aturally

g s ....................................................................................................................................................... 6 7

s e c o n d - o r d e r m o d e l p a r a m e t e r s f o r n a t u r a l l y in c u b a t e d

e g g s ............................................................................................................................................................................... 6 8

3 .1 A F i n d i n g

a s y m p t o t e s .......................................................................................................................6 8

3 . IB F in d in g

r a t e c o n s t a n t s ............................................................................................................. 6 9

3 .1 C Is i t a l w a y s p o s s i b l e t o i s o l a t e t h e r e q u i r e d p a r a m e t e r s

for a

s e c o n d - o r d e r m o d e l ? ............................................................................................................................. 7 2 3 .2 E x a m p l e

analyses of

Ho use W

r e n e g g t r a n s i e n t s ................................................. 7 3

3 .2 A

F in d in g

a s y m p t o t e s ..................................................................................................................... 7 3

3 .2 B

F in d in g

r a te c o n sta n ts fo r

3 .3 E r r o r

H o use W

r e n e g g t r a n s i e n t s ........................8 1

a n a l y s i s ......................................................................................................................................... 8 6

vii

R e p ro d u c e d with perm ission of th e copyright owner. F urther reproduction prohibited without permission.

3 .3 A . E r r o r s 3 .3 A .i T

f r o m f i r s t - o r d e r m o d e l s .................................................................................86

e m p e r a t u r e p e r c e n t e r r o r s ................................................................................... 8 7

3 .3 A .H T

i m e p e r c e n t e r r o r s ........................................................................................................ 9 4

3 .3 B H o u s e W 3 .4 A n a l y s e s

r e n e g g t r a n s ie n t s : e r r o r s f r o m f ir s t -o r d e r m o d e l s

94

o f c o o l i n g a n d h e a t i n g i n o t h e r s p e c i e s .................................................96

3 .4 A E c o l o g i c a l

c o m p a r is o n s b e t w e e n in t e r m it t e n t l y in c u b a t in g

s p e c ie s a n d d e t a il s f o r c o l l e c t io n o f d a t a

3 .4 A 1 H o u s e W r e n s

{Troglodytes aedon) ..............................................................................96

3 .4 A 2 T r e e S w a l l o w s

{Tachycineta bicolor) .....................................................................9 7

3 .4 A 3 B l a c k - c a p p e d C h i c k a d e e s 3 .4 A 4 Y e l l o w - e y e d J u n c o s 3 .4 B C o m p a r i s o n o f

......................................................................96

Tbp a c r o s s

{Parus atricapillus) .............................................. 98

{Junco phaeonotus) ...........................................................99 s p e c i e s ...................................................................................100

3 .4 C C o m p a r i s o n o f c o o l i n g

k hk2,

3 .4 D C o m p a r i s o n o f h e a t i n g

k t,k2, a n d

a n d t i m e s t o c o o l a c r o s s s p e c i e s .... 100 t i m e s t o h e a t a c r o s s s p e c i e s ........106

3 .4 E N e t g a i n s i n t i m e f o r e a c h s p e c i e s f r o m v a r i a t i o n in 3 .5 S u m m a r y ':

R C ....................... 113

t e c h n iq u e s r e q u ir e d t o m o d e l t im e s a n d t e m p e r a t u r e s fo r

i n c u b a t e d e g g s .................................................................................................................................................127

C

hapter

4 .................................................................................................................................................................... 128

C h a p t e r IV : P r e d i c t i o n s f o r T r a d e - o f f s B e t w e e n M a x i m i z i n g

P

and

I n c u b a t i o n C y c l e A v e r a g e E g g T e m p e r a t u r e ......................................................................... 128

4 .1 I n c u b a t i o n

t i m e s a n d t e m p e r a t u r e s ....................................................................................129

4 .2 I n c u b a t i o n

p e r c e n t t im e s f o r a g in g

....................................................................................132

viii


4 .2 A S e n s i t i v i t y

a n a l y s e s f o r t h e p a r a m e t e r s r e q u ir e d t o m a k e

PREDICTIONS FOR PERCENT TIME FORAGING............................................................................ 133 4 .2 B S e n s i t i v i t y o f

TEaot o

4 .2 B 1 S e n s i t i v i t y o f 4 .2 B 2 S e n s i t i v i t y

d a t a c o l l e c t i o n t e c h n i q u e s ........................................137

TEx t o

of

7> -

TEt

To illustrate the general nature of the

18.0 °C was used. The equations used were:

2At + 18.0, based on

T e i — ( T e o — T eoo) q h

second-order:

(see below).

+

T e *^

-0.24 = (18.0)[— -----]e~02Al +18.0 ]e~2A21 +(18.0) [(2.42-0.24) (2.42-0.24)

based on equation (2 - 1 0 ) solved for Teco, or

86


Note that the first-order equation has the same rate constant as

for the second-order

equation. This ignores the transient by using only the terminal linear change in ^ 1Tei —T 1 ek versus tune. This is commonly done in first-order analyses (e.g., T T 1 E0 1 £oo Robertson and Smith, 1981; Pages et al., 1991). If, on the other hand, both transients were represented by one first-order rate constant using all the data for a transient, it would be somewhat lower than ki. In either case, errors will result in predictions of both time and temperature.

3.3A.i Temperature percent errors Temperature percent errors were found from 100 times the differences in temperatures between the second- and first-order values at the same times divided by second-order values. Percent errors varied with time, so time was varied within 0.1 min in the above equations to find when a percent error was maximum. To generalize the results for Teo = 36.0°C and Teco= 18.0°C, the maximum percent errors and the times when they occurred were found with the same first-order solutions but with larger or smaller ratios between the second-order rate constants, or ki(ki)~{ for kj>ki. This was done by keeping the second rate constant at 0.24 min-1 and varying the first rate constant from 0.2618 to 22.04 min-1. This method is similar to one described by Turner (1987a) where ratios of time constants were related to the damping ratio.

87


The damping ratio (£, dimesionless) is one o f two derived parameters that are often used to describe the second-order model represented by equation (2-4) (Turner, 1987a). W ith the natural frequency (G ^tim e1), it constitutes part o f the coefficient a in equation (2-4), as a is a>n%and b is (On. Thus, the damping ratio and natural frequency can k _|_k be expressed entirely in terms of rate constants, with -f =

1 ~ and con = k2. Note that 2-yJk\ki

these were derived from equation (18) and equation (15b) , respectively, in Turner (1987a), but equation (19), used by Turner to find con, is in error since it cannot be derived from his equation (15b) by algebra. Also note that this method for finding parameters from rate constants applies only when f(t) is a step function. When f(t) is more complex, such as a sine function, there is a different solution for equation (2-4) (Turner, 1994a; 1994b). The damping ratio and the errors from differences in ki and k 2 are o f interest because they both depend on the ratio of rate constants [ki(k2)~xwhen ^/>^ 2 ](see below and equation 18 in Turner, 1987a). Using data from an incubated House W ren egg, I will show that either ki(k2)~lor the damping ratio can be used to predict the error associated with a first-order model.

For the first- and second-order equations given in the previous section a maximum percent error o f 4.18% occurred at 1.3 minutes. Prior to and following this time the error diminished to zero at t = 0 min and as Te approached Te*, = 18.0°C (Fig. 11). The ratio o f second-order rate constants, ki(k2)~l for ki>k2, determined the timedependent maximum percent errors (Fig. 12). For Teo = 36.0 °C and Te< x>= 18.0 °C, (maximum % error )"1 = 0.02027 [kj(k2)~l ] + 0.0309 (r2=0.99996) can be used to find

88


Figure 11. C om parison of tem peratures with a first- and second-order model. Rate constants determ ined from a cooling transient fo r an incubated egg were used to find tem peratures with a first-order m odel (solid circles) and a second-order m odel (open circles). See the text fo r the equations used for each model. A m axim um percent e rro r of 4.18% in tem peratures for the first-o rd er model occurred at 1.3 m inutes (*). T his e rro r is time dependent, as the e rro r can be seen to diminish to zero at t= 0 min and as egg tem perature approached Teoo- E rro rs for tim es to reach the sam e tem perature are m axim um near zero time and dim inish with time cooling (see text for values).

89


5

4 t. 3 fcd 2

• •_

38 0

36 •

o • O

• 0

3

6

12

9

15

T im e, m in

34 32

'I* o

30

•°

u

o

•°

28

H

• °• O

26

• o

24 22

8 # 8 * » 98 8

20

8 8

8 8 ee

18 0

4

6

Time, min

90


10

Figure 12. F o r Teo = 36°C an d Te*, = 18.0°C, the ratio of the second ord er-rate constants, kifkrf1 for ki>k 2 , determ ines the tim e-dependent tem perature m axim um percent errors. This is tru e for any kf>k 2 , not just for the exam ple given in the text used to construct this figure. The lowest m axim um percent errors will occur w hen the first rate constant is large relative to the second. For this exam ple a ratio > 48.0, which corresponds to a dam ping ratio of 3.54, produced a m axim um percent e rro r < 1.0%. T he solid circles show the m ean a n d 95% confidence intervals for the sam pling distribution of

fo r cooling House W ren eggs (see table 2).

L a rg e r eggs w ould be located to the left [see text fo r errors based on analyses of cooling eggs by T u rn e r (1987)]. The 95% confidence interval fo r m axim um percent errors d u rin g cooling for H ouse wren eggs from Teo = 36°C tow ard

T e* , =

18°C, ranged from 1.11% to 1.92%, but the average

e rr o r is skewed tow ard higher values from the shape of this function.

91


Dam ping Ratio =

2->ykxk2 5.0

2.5

o .o

Temperature Maximum

% Error

20.0

15.0

10.0

5.0

0.0 0 .0

60.0

30.0 ki

for kx > k2

92


90.0

temperature maximum % errors for kj>k2. The larger the first rate constant was relative to the second, the shorter was the initial transient and the lower was the maximum percent error. For these cooling conditions a ratio > 48.0, which corresponds to a damping ratio o f 3.54, would produce a maximum percent error 1(Teoo)0.135(/«.^//)+0.28(/coo/)+0.04 (Nest); r2=0.34;r=0.58; n=80]. In this case, the time eggs were allowed to cool (p=0.0033; r= 0.33), and the time eggs were held at equilibrium after heating (p=0.0489; r= -0.22) were important indicators of time required to heat. Shorter cooling times and longer equilibrium times were associated with decreased time to heat. Equilibrium egg temperature ( Te< x> ) was also borderline significant, suggesting

111


environmental factors contributed to the variation as well (p=0.06; r= -0.22). This result suggests House Wrens may be very dependent on environmental factors that influence egg cooling and heating, and that they could anticipate the time they will need to remain on the nest, possibly to adjust the average egg temperature to compensate for time off the nest. Some o f the variation was also explained by the nest in which eggs were heated (p=0.05; r=0.22), suggesting microclimate and nest construction are also important. Brood patch temperature did not seem to be as important in this analysis, but this is probably due to only a subset of TbPvalues being estimated directly from heating curves. A proportion of the data were analyzed using average values for TbP (see section 3.2B). In spite o f this, there was a borderline significance for a positive correlation with ki (p=0.06; r=0.33), suggesting high brood patch temperatures increase the rate of egg heating, as would be expected.

For Tree Swallows, the second rate constant also accounted for most of the explained variation in heating times [theat= -8.81 (&2)-0.887(7&p) +0.082(£/)+0.039(7£-o,)0.003(^0,/)+0.24(fcoo/)+0.143 (Nest); r2=0.34; r=0.55; n=49]. As with the Junco heating times, k 2 (p=0.0136; r= -0.28) and TbP (p=0.0001; r= -0.50) accounted for most of the variation the regression model could explain. Higher values of both had the effect of decreasing the time required to heat. In this model, Tex> (p=0.005; r= -0.32), tcoo/ (p=0.001; r=0.50), and the individual nest (p=0.0001; r=0.62) were positively correlated with time required to heat. So, for Tree Swallows, decreased ambient temperature and brief times away from the nest prior to heating both reduced egg heating time. Individual nest construction and location are probably somewhat important in decreasing heating time as well. 112


The multiple regression analysis for Black-capped Chickadees resulted in theat= 7.9 (k2)-0.63(Tbp) +0.115(Ar/)-0.038(re)-0.0242(re