- Email: [email protected]
Population dynamics of a tree-dwelling aphid: individuals to populations
E[OLO61[flL mOOELLIn6 ELSEVIER
Ecological Modelling 89 (1996) 23-30
Population dynamics of a tree-dwelling aphid: individuals to populations P. K i n d l m a n n a,,, A . F . G . Dixon b a Faculty of Biological Sciences, University of South Bohemia and Czech Academy of Sciences, Brani~ot,skd 31, 37005 Ceskd Bud~jovice, Czech Republic b School of Biological Sciences, University of East Anglia, Norwich, NR4 7TJ, UK Received 9 September 1994; accepted 24 March 1995
Abstract
The model presented simulates most of the observed features of the population dynamics of tree-dwelling aphids: a sharp increase in numbers during the first 15-40 days, followed by a sharp decline to a plateau of low numbers in summer and a recovery in autumn, in some years. The larger the numbers at the beginning of the season, the larger and earlier the peak. Migration is shown to be the most important factor determining the summer decline in abundance, while changes in aphid size and food quality account for why the autumnal increase is less steep than in spring. Finally, the model suggests a possibility of a "see-saw effect" (a negative correlation between spring and autumn peak numbers) in some cases.
Keywords." Insects; Oak aphid; Population dynamics; Seasonality
1. Introduction
The population dynamics of deciduous treedwelling aphids have been studied over long periods of time and in considerable detail. The regularity of the population fluctuations from year to year, very regular 2-year cycles and chaos, as indicated by suction trap catches, has proved very attractive to modellers, who have applied time series analysis to the data (e.g., Turchin, 1990; Turchin and Taylor, 1992). The 2-year cycles can be explained by the "see-saw effect": a negative
* Corresponding author. Fax: (+ 42) 3845985.
correlation between the numbers of first generation aphids in spring (fundatrices) and the last generation in autumn (oviparae). This effect has been observed in some empirical data (Dixon, 1971), but in other species (our unpublished data) it is not present or very weak. Within a year, the dynamics of these aphids are very complicated and in looking for the mechanism of regulation this needs to be taken into consideration. For example, in the case of the Turkey-oak aphid, Myzocallis boerneri, an initial dramatic increase in population size in spring is typically followed by a steep decline in abundance during summer and sometimes a further increase in autumn. During spring and summer all the generations are parthenogenetic, short
0304-3800/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved
SSDI 0 3 0 4 - 3 8 0 0 ( 9 5 ) 0 0 1 ! 6-6
24
P. Kindlrnann, A.F.G. Dixon/Ecological Modelling 89 (1996) 23-30
lived ( 2 - 4 weeks) and fully winged. In autumn, sexuals are also produced, which mate and give rise to the overwintering eggs that hatch the following spring and give rise to fundatrices, the first parthenogenetic generation. A lot is known about the biology of the parthenogenetic generations of aphids, in particular, the o p t i m u m behaviour for maximising the instantaneous population growth rate, rm, under various environmental conditions (Kindlmann and Dixon, 1989, 1992; Kindlmann et al., 1992) and the optimal strategies for migration (Dixon et al., 1993a). Therefore, all the p a r a m e t e r s needed to build a model to explain their population dynamics on the basis of their evolutionarily stable optimal behaviour are known. Such a model will not only describe, but also explain the patterns observed. T h e parthenogenetic generations overlap in time and the environmental conditions are rapidly changing. Therefore, an individual throughout its life, as well as individuals born at different, but close instants in time, can experience quite different conditions, which results in them using different and varying reproductive strategies. Thus any model that lumps together the different age stages is inappropriate as an individual-based model is needed. In this p a p e r an individual-based model is presented which incorporates all that is known about the biology of tree-dwelling aphids. The predictions and conclusions are c o m p a r e d with an extensive data set on the population dynamics of the Turkey-oak aphid Myzocallis boerneri.
2. M a t e r i a l a n d m e t h o d s
2.1. The larval development T h e sizes of the soma and gonads of larvae and the size of the gonads and age of adults were monitored in a matrix, which contained one row for each individual. In each time step (0.01 day), different procedures were followed for larvae and adults: The changes in the size of the gonads and soma of larvae were assumed to follow the Kindl-
mann and Dixon (1989) model: ds
dt dg --
dt
-as"-Rg,
s(0) =s0 (1)
=Rg
g(0) =go
where s and g are sizes of soma and gonads at time t, respectively, R is the constraint on the gonadal growth rate, a is the assimilation rate, a = 2 / 3 is a constant (surface to volume ratio) and s o and go are birth sizes of soma and gonads. The simulations of Eq. 1 were performed in 20 000 steps, simulating a season 200 days in length with time step equal to 0.01 day, using a difference method. At birth, the size of soma was assumed to be equal to 25 units, a unit corresponding to 1 /zg for the species considered; this can be used for other species if the units are changed (i.e., if e.g. the birth size were 50 tzg, then a unit 2 /xg would have to be used). The size of gonads at birth was assumed to be that predicted by the optimisation process described in Kindlmann et al. (1992). The function go = (1 + 0.01s0)a 2
(2)
approximates the results of that optimisation fairly accurately and was used instead of repeating the lengthy optimisation. There exists, however, no analytical proof of this dependence. The constraint on the gonadal growth rate, R, was assumed to be equal to 0.5 / x g / / x g / d a y , as this is the value common in aphids (Kindlmann and Dixon, 1992). The assimilation rate, a, was not constant, as in the original model, where the predictions were verified using results of laboratory experiments done under constant conditions. In the field, the assimilation rate changes dramatically during the course of the season and follows changes in the quality of the host plant. The concentration of total free amino acids in the leaves can be used as a good indicator of host quality and Dixon et al. (1993b) indicate a parabolic dependence of this quality on time. In addition to changes in host plant quality, the (negative) influence of cumulative aphid population density on host plant quality was also consid-
P. Kindlmann, A.F.G. Dixon/Ecological Modelling 89 (1996) 23-30
ered. Therefore, the equation a(t) =
( ( t -- train)/train) 2" ( A m a x - A m i n )
behaviour of the model and therefore tmi n = 135 was used in most of the simulations. The values of A . . . . A m i n and C used in the simulations were varied over the range observed in aphids, e.g., A max = 2.5 was estimated from laboratory data ( = good food quality) and its value is unlikely to differ much from this value (Kindlmann and Dixon, 1992). When food quality is poor, a has to be low. A good criterion for testing whether Areax and Ami n are realistic, are the predicted sizes of the adults. If these were not realistic, the simulations were discontinued. The value of C was varied until the simulated population became extinct, or the model's output differed from that observed. The simulations of Eq. 1 were stopped, when
+Ami n
d/C + 1
(3) was used to estimate a, where d is the cumulative aphid population density (calculated from the model), C a scaling constant, A r e a x the maximum value of a at the beginning of the season (t = 0), A m i n the minimum value of a at time tmin (on an uninfested tree, i.e., d = 0 throughout the season). If the season begins in April, the value tmin = 135 is realistic for the time of minimum host plant quality, i.e., August. A change in the value of this p a r a m e t e r does not affect the qualitative
/
[ Gonadal growthrate~ I~
[Temperaturel + ~/J
Birth size]
[Energy partitioning model ]
I Optimisation procedure ] I Larva, deve,cpment I
j
"-,.
loove'opmenta, ,ime I
\Population~[
S
growthrate
Population] size I
+
1, 'grat'on I
25
I du', si,o I
t
Fecundity] ] Background ]
mortality
,11
Host plant] quality
Fig. 1. The organisational diagram of the aphid model.
P. Kindlmann, A.F.G. Dixon/Ecological Modelling 89 (1996) 23-30
26
d s / d t b e c a m e n e g a t i v e (at a t i m e r e f e r r e d to as D , d e v e l o p m e n t a l time), i.e., w h e n t h e a n i m a l b e c a m e adult. 2.2. Adult life and reproduction
~. io3"
F o r adults, t h e r e l a t i v e size of t h e s o m a a n d g o n a d s is i r r e l e v a n t as only t h e size o f t h e g o n a d s at m a t u r i t y , g(D), d e t e r m i n e s fecundity. T h e s h a p e o f t h e r e p r o d u c t i o n curve ( n u m b e r o f offs p r i n g b o r n p e r m o t h e r p e r day), F(t), was ass u m e d to b e positive only for 2 D > t > D a n d triangular:
F(t) = 2Rg(D)/(s o +g0)(2 - t/D),
77
6
%5
,
o i o
-5
o_ O CL
E O m~D
r
O_ O OE~ O
\,,
x
+,+
~x -" × '--
2.
"ix/
_""'- "_'_,'L*
0
Itll
,lllllllllllttllJ 50
100
/
x
1975
,[, ,i,iilIlllljtj[ 150
I 200
Time (days) Fig. 3. Empirical data on density changes of M. boerneri for 1975 (*), 1981 ( + ) and 1982 ( × ) and simulation results showing the same trends. Parameters are the same as in Fig. 2, with the following exceptions: 1975, S = 0.01; 1981, t m i n = 185;
1982,
tmi n = 100, Ama x = I.
a n d f e c u n d i t y is low. T h e p r o c e d u r e u s e d c o r r e s p o n d s with t h e e m p i r i c a l d a t a : small a d u l t a p h i d s d o n o t r e p r o d u c e every day, p r o d u c i n g a single larva at l o n g e r intervals. T h e m o r t a l i t y d u e to p r e d a t o r s , a p h i d s falling f r o m t h e leaves etc., was a s s u m e d to b e c o n s t a n t . W h e n this v a l u e was varied, it d i d n o t have any effect on t h e q u a l i t a t i v e b e h a v i o u r o f t h e m o d e l a n d t h e r e f o r e only t h e o u t p u t s using a v a l u e o f 0.15 for b a c k g r o u n d m o r t a l i t y p e r iterative s t e p are presented.
2.3. Migration
O~
° i
6. ;!",!k "~\ "~ 1981 +
(4)
w h e r e t is t h e age o f t h e i n d i v i d u a l ( K i n d l m a n n et al., 1992). A s only i n t e g e r i n c r e m e n t s in t h e n u m b e r o f i n d i v i d u a l s can b e s i m u l a t e d in t h e i n d i v i d u a l - b a s e d m o d e l , a s t e p l e n g t h 100 times l o n g e r t h a n in t h e s i m u l a t i o n of larval g r o w t h was used. F r a c t i o n s o f F(t) w e r e r o u n d e d r a n d o m l y u p o r d o w n with a p r o b a b i l i t y d e t e r m i n e d by t h e v a l u e o f t h e fraction. This is i m p o r t a n t e s p e c i a l l y w h e n t h e v a l u e s o f a a r e low, a n i m a l s a r e small
.N o3 ¢--
+
0
50
1O0
150
200
Time (doys)
T h e p r o p o r t i o n o f a p h i d s t h a t fly away f r o m a colony, m, was a s s u m e d to b e a l i n e a r f u n c t i o n o f t h e n u m b e r o f a p h i d s (larvae + adults) p r e s e n t in t h e colony, x, with positive s l o p e S a n d positive i n t e r c e p t i ( B a r l o w a n d Dixon, 1980), t i m e s a l i n e a r f u n c t i o n d e p e n d i n g o n a:
Fig. 2. The average seasonal pattern of density changes of M.
boerneri populations on two Turkey-oak trees in Norwich, England, from 1975 to 1992 (*), fitted by the model (--). Parameters: A m a x = 2.5, A r n i n = 0.01, R = 0.5, s o = 25, t m i n = 135, S = 0.005, i = 0.1, C = 2.106. Log of initial number of aphids: 1.5.
m =
(Sx+i)" (Amax- a ) A
max --
A
(5)
min
T h e s e c o n d t e r m in Eq. 5 s i m u l a t e s t h e reluct a n c e of a p h i d s to m i g r a t e , w h e n living on a g o o d
P. Kindlmann, A.F.G. Dixon/Ecological Modelling 89 (1996) 23-30
host plant, and their willingness when on a poor quality plant. The success of an aphid in finding another host plant is assumed to be in the range of 0.01 (Taylor, 1977). Therefore, the immigration from other colonies was assumed to be equal to 0.01m. For the same reason as cited for calculating fecundity, the time step used was 100 times longer than for larval development. 575-
,///j~
475-
(3_ O Ct
400-
The temporal pattern of density fluctuations of M. boerneri populations on two Turkey-oak trees
11"
q~ a
C
O
425"
375"
(b)
,,/t
500-
450-
2.4. The empirical data
12-
525-
E O L~ O
Because the output of the model was dependent on, e.g., fecundity being randomly rounded up and down, each simulation was repeated 10 times. The calculations were performed on an HP Apollo computer using F O R T R A N language.
(a)
550-
03 N
27
10"
9"
O CL O (3_
.,,'; , / I /
8"
7-
350" 325300275
,
10
r
~
,
, ,
, , I
i
,
10
10 2
J
i
i
,
i
J I
10 2
InitiGI number of aphids
Initial number of ephids
451 (c)
43
40N 09
38-
E 0 O 00 El-
,35-
"\.
. . . .
33 ~
-" - " > - - _ ' 7 \
3O ~ 28 25 10
Initial number of aphids
10 ~
Fig. 4. M o d e l predictions of the p e a k densities (a), the s u m m e r m i n i m u m densities (b) and o f the final densities (c) plotted against the initial densities for h m i n = 0.5 (solid line), 0 . 4 , . . . ,0.1 (the smaller Zmin, the shorter the dashes). P a r a m e t e r s used: Ama x = 2.5, R = 0.5, $0 = 25, t m i n = 150, S = 0.005, i = 0.1, C = 5 - 104. Log o f initial n u m b e r of aphids: 2.
28
P. Kindlmann, A.F.G. Dixon/Ecological Modelling 89 (1996) 23-30
in Norwich, England, from 1975 to 1992, were used. The populations were sampled at weekly intervals from the beginning of May to the end of N o v e m b e r each year. Each week, 10 leaves were selected at random at each of 8 fixed sampling points around the circumference of the trees and the numbers of the various life stages recorded.
3. Results
The organisational diagram of the model described in the previous section is shown in Fig. 1. Not all the simulation results are depicted. The c o m m o n feature of almost all the simulations was the sharp increase in total numbers at first, followed by a decline and then another increase in abundance. The p e a k n u m b e r was reached very early each year in all cases: during the first 15-40 days. The more or less broad trough in population numbers never preceded the instant when a was lowest and occurred some 15-30 days later. The possibility of simplifying the model by deleting some of the p a r a m e t e r s was explored by varying those that were not measured directly. All such simplifications resulted in the model deviating from reality.
14-
For example, deletion of the second term in Eq. 5, i.e. omission of the assumption that the tendency of aphids to migrate depends not only on their own density, but also on host plant quality, did not change the " u p - d o w n - u p " population profile, but resulted in the peaks being of nearly the same size for a wide range of initial conditions, irrespective of the values of the other parameters. As the empirical data show a significant positive correlation of the peak value with the initial density, the second term in Eq. 5 cannot be omitted. The average seasonal pattern of density changes of the Turkey-oak aphid, from May to November, for the period 1975-1992, was fitted by the model in Fig. 2. From the same period, 1975-1992, three years with very different patterns of density changes were chosen and data plotted in Fig. 3 : 1 9 8 1 - a high spring peak followed by a monotonous decline, second peak not present; 1982 - both spring and autumn peaks relatively low and of the same size; 1975 the spring p e a k followed by a decline and a long plateau in autumn. To show the model's ability to mimic different types of aphid behaviour, parameters were chosen to simulate the dynamics in these years, but not fitted to the data, in Fig. 3.
(a)
451 (b) 43
,
13
*
40
12
• o 38
._Nll 03 E O10
* *
k
1
35~
, *
*
,
,
~v
•
*
28-1
•
*
25~
* , ,
O C~. 9 O (2_ 8
{
{
w
w
w
w
i
[
{
w
i
10
Initiol number of ophids
{
{
{
w
w
[ {
10 ~
25
* w
1
{
i
{
{
{
w i
{
{
w
10
i
i
{
{
{
{ {
10 2
Initiol number of ophids
Fig. 5. Predictions of the final and minimum densities against the initial densities obtained from 10 runs of the model using different random numbers. Z m i n = 0.5. Other parameters are the same as in Fig. 4.
P. Kindlmann, A.F.G. Dixon/Ecological Modelling 89 (1996) 23-30
A global view of the model's behaviour is given in Fig. 4, where the average peak densities, summer minimum densities and final densities obtained from 10 runs of the model are plotted against the initial densities for several values of Amin. The peak densities are clearly dependent on the initial numbers but weakly so on the value of Ami n. Both the summer minimum and the final autumnal densities are dependent on Amin; the lower Amin, the lower the corresponding density. They also decline when initial numbers and therefore also spring peak numbers (Fig. 4a) are increased (the "see-saw effect"), this being particularly so in the case of the final densities and negligible so for low Ami n and the minimum densities. Plotting the final and minimum densities against the initial densities obtained from 10 runs of the model reveals that the "see-saw effect" may be overshadowed by other phenomena (Fig. 5). Although the appropriate trend is there, variation due to random factors is very large. As the empirical data indicates, the numbers of aphids present in summer are extremely low. This suggests that the actual values of C and A min are close to those used in the simulations illustrated in Figs. 2 and 3.
4. Discussion
The model satisfactorily describes the withinyear population dynamics of the Turkey-oak aphid: a sharp increase in numbers in spring, which is usually followed by a sharp decline in summer and sometimes by a subsequent increase in autumn. The autumnal increase follows the improvement in food quality with a delay and is never so steep as in spring. The delay has a simple biological explanation: small summer individuals feeding on food of low, although rapidly improving quality, are not able to produce large offspring and their fecundity is also low. Therefore the effect of the autumnal improvement in food quality is reflected in the population dynamics but with a delay and the subsequent increase in numbers is less steep than for the large aphids living in the lush conditions prevailing in spring. The model further reveals the regulatory
29
mechanism that is responsible for the summer decline in numbers: migration. This increases linearly with density and declines with improving food quality. The model outputs indicate that migration is important in shaping the population dynamics. If the decline were mainly due to low fecundity and high mortality, then the average realised number of surviving offspring per aphid would have to be - because of parthenogenesis smaller than one! This seems unlikely, as even in the worst conditions the average reproductive output per aphid is about 1 offspring per 1-2 days and the reproductive life span is 10-20 days. Empirical data confirm that the tendency to migrate depends on density and food quality in the above mentioned way. In the model, the extent of the summer decline in aphid abundance is determined by the degree to which the food quality declines in summer, which influences aphid size, fecundity, density and subsequently their tendency to migrate. Therefore, changes in food quality determine in this rather complicated way the changes in summer numbers. This prediction is rather difficult to test empirically and is a task for the future. In contrast to most other groups, the empirical data do not lend support to a marked effect of natural enemies, or a severe scramble for resources by these aphids. Therefore, these factors do not seem to play a substantial role in regulating the abundance of these aphids. Food quality determines the rate of development and size of an individual more than intraspecific competition, and food quality is not affected by aphid numbers substantially (Dixon, 1971). Finally, a comparison of data presented here and by Dixon (1971) further supports the hypothesis that migration is responsible for the summer decline in numbers: while in nature there is no second increase in population numbers in the lime aphid, in glasshouse experiments there is (Dixon, 1971), which is possibly because the experiment was done in an enclosed space in which the probability of finding another host plant was greater than in the field. The "see-saw effect", a negative correlation between the numbers of first generation aphids in spring (fundatrices) and the last generation in
30
P. Kindlmann, A.F.G. Dixon/Ecological Modelling 89 (1996) 23-30
a u t u m n ( o v i p a r a e ) , has b e e n o b s e r v e d in s o m e e m p i r i c a l d a t a (Dixon, 1971), b u t in o t h e r s p e c i e s ( o u r u n p u b l i s h e d d a t a ) it is not p r e s e n t o r very w e a k . T h e m o d e l p r e s e n t e d h e r e reveals, why this is t h e case: t h e " s e e - s a w e f f e c t " is a statistical p h e n o m e n o n , r a t h e r t h a n a d e t e r m i n i s t i c one. It is m o r e likely to o c c u r w h e n t h e initial n u m b e r s a n d / o r t h e s u m m e r p o p u l a t i o n a r e very high a n d w h e n t h e a p h i d has some, b u t n o t a l a r g e effect o n t h e host plant. I f no c u m u l a t i v e d e n s i t y effect on a p h i d a b u n d a n c e is a s s u m e d in t h e m o d e l ( C = ~ in Eq. 3), it d o e s n o t p r e d i c t any " s e e - s a w e f f e c t " (simul a t e d , b u t n o t d e p i c t e d ) . This is quite s t r a i g h t f o r w a r d , as t h e r e is a lag o f several g e n e r a t i o n s between the summer and autumn peaks. Theref o r e t h e r e m u s t be s o m e " m e m o r y " in t h e system, which t r a n s f e r s i n f o r m a t i o n on a b u n d a n c e f r o m s p r i n g to a u t u m n . It is u n c l e a r w h a t b i o l o g i c a l f o r m this " m e m o r y " takes. A lag c a u s e d by p r e d a t o r s is unlikely, as e x p l a i n e d above. T h e c u m u l a t i v e d e n s i t y effect a s s u m e d in t h e m o d e l m a y b e a c o n s e q u e n c e of e i t h e r a p e r s i s t i n g a n d therefore accumulating influence of the aphids on t h e p l a n t , o r o f s o m e c a r r y - o v e r effect bet w e e n a p h i d g e n e r a t i o n s . T h e r e is n e i t h e r e m p i r i cal e v i d e n c e for t h e f o r m e r , n o r t h e o r e t i c a l supp o r t for t h e latter. P o p u l a t i o n census d a t a are usually subject to a d e s c r i p t i v e statistical analysis using a t i m e series analysis a p p r o a c h , with t h e T u r k e y - o a k a p h i d d a t a u s e d h e r e b e i n g n o e x c e p t i o n (they w e r e analysed, e.g., in T u r c h i n , 1990; T u r c h i n a n d Taylor, 1992). This p a p e r analyses t h e m f r o m a d i f f e r e n t p o i n t o f view: t h a t b a s e d on a d e t a i l e d k n o w l e d g e o f t h e b i o l o g y o f t h e g r o u p involved, aphids. T h i s approach reveals the underlying mechanisms, r a t h e r t h a n just d e s c r i b i n g t h e p o p u l a t i o n t r e n d s . This m o d e l p r o v i d e s a g o o d basis for a p r e d i c t i v e
m o d e l . F o r this, however, m o r e e m p i r i c a l d a t a on t h e s e a s o n a l c h a n g e s in f o o d quality similar to t h o s e for s y c a m o r e ( D i x o n et al., 1993b) a r e n e e d e d along with a b e t t e r u n d e r s t a n d i n g o f t h e m i g r a t o r y b e h a v i o u r of aphids.
Acknowledgements This w o r k was s u p p o r t e d by t h e N E R C g r a n t GR3/8026A and NATO research grant CRG920553.
References Barlow, N.D. and Dixon, A.F.G., 1980. Simulation of Lime Aphid Population Dynamics. Pudoc, Wageningen, 165 pp. Dixon, A.F.G., 1971. The role of intra-specific mechanisms and predation in regulating the numbers of the lime aphid Eucallipterus tiliae L. Oecologia (Berl.), 8: 179-193. Dixon, A.F.G., Horth, S. and Kindlmann, P., 1993a. Migration in insects: costs and strategies. J. Anim. Ecol., 62: 182-190. Dixon, A.F.G., Wellings, P.W., Carter, C. and Nichols, J.F.A., 1993b. The role of food quality and competition in shaping the seasonal cycle in the reproductive activity of the sycamore aphid. Oecologia (Berl.), 95: 89-92. Kindlmann, P. and Dixon, A.F.G., 1989. Developmental constraints in the evolution of reproductive strategies: telescoping of generations in parthenogenetic aphids. Funct. Ecol., 3: 531-537. Kindlmann, P. and Dixon, A.F.G., 1992. Optimum body size: effects of food quality and temperature, when reproductive growth rate is restricted. J. Evol. Biol., 5: 677-690. Kindlmann, P., Dixon, A.F.G. and Gross, L.J., 1992. The relationship between individual and population growth rates in multicellular organisms. J. Theor. Biol., 157: 535542. Taylor, L.R., 1977. Migration and the spatial dynamics of an aphid, Myzus persicae. J. Anim. Ecol., 46: 411-423. Turchin, P., 1990. Rarity of density dependence or population regulation with lags? Nature, 344: 660-663. Turchin, P. and Taylor, A.D., 1992. Complex dynamics in ecological time series. Ecology, 73: 289-305.
Ecological Modelling 89 (1996) 23-30
Population dynamics of a tree-dwelling aphid: individuals to populations P. K i n d l m a n n a,,, A . F . G . Dixon b a Faculty of Biological Sciences, University of South Bohemia and Czech Academy of Sciences, Brani~ot,skd 31, 37005 Ceskd Bud~jovice, Czech Republic b School of Biological Sciences, University of East Anglia, Norwich, NR4 7TJ, UK Received 9 September 1994; accepted 24 March 1995
Abstract
The model presented simulates most of the observed features of the population dynamics of tree-dwelling aphids: a sharp increase in numbers during the first 15-40 days, followed by a sharp decline to a plateau of low numbers in summer and a recovery in autumn, in some years. The larger the numbers at the beginning of the season, the larger and earlier the peak. Migration is shown to be the most important factor determining the summer decline in abundance, while changes in aphid size and food quality account for why the autumnal increase is less steep than in spring. Finally, the model suggests a possibility of a "see-saw effect" (a negative correlation between spring and autumn peak numbers) in some cases.
Keywords." Insects; Oak aphid; Population dynamics; Seasonality
1. Introduction
The population dynamics of deciduous treedwelling aphids have been studied over long periods of time and in considerable detail. The regularity of the population fluctuations from year to year, very regular 2-year cycles and chaos, as indicated by suction trap catches, has proved very attractive to modellers, who have applied time series analysis to the data (e.g., Turchin, 1990; Turchin and Taylor, 1992). The 2-year cycles can be explained by the "see-saw effect": a negative
* Corresponding author. Fax: (+ 42) 3845985.
correlation between the numbers of first generation aphids in spring (fundatrices) and the last generation in autumn (oviparae). This effect has been observed in some empirical data (Dixon, 1971), but in other species (our unpublished data) it is not present or very weak. Within a year, the dynamics of these aphids are very complicated and in looking for the mechanism of regulation this needs to be taken into consideration. For example, in the case of the Turkey-oak aphid, Myzocallis boerneri, an initial dramatic increase in population size in spring is typically followed by a steep decline in abundance during summer and sometimes a further increase in autumn. During spring and summer all the generations are parthenogenetic, short
0304-3800/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved
SSDI 0 3 0 4 - 3 8 0 0 ( 9 5 ) 0 0 1 ! 6-6
24
P. Kindlrnann, A.F.G. Dixon/Ecological Modelling 89 (1996) 23-30
lived ( 2 - 4 weeks) and fully winged. In autumn, sexuals are also produced, which mate and give rise to the overwintering eggs that hatch the following spring and give rise to fundatrices, the first parthenogenetic generation. A lot is known about the biology of the parthenogenetic generations of aphids, in particular, the o p t i m u m behaviour for maximising the instantaneous population growth rate, rm, under various environmental conditions (Kindlmann and Dixon, 1989, 1992; Kindlmann et al., 1992) and the optimal strategies for migration (Dixon et al., 1993a). Therefore, all the p a r a m e t e r s needed to build a model to explain their population dynamics on the basis of their evolutionarily stable optimal behaviour are known. Such a model will not only describe, but also explain the patterns observed. T h e parthenogenetic generations overlap in time and the environmental conditions are rapidly changing. Therefore, an individual throughout its life, as well as individuals born at different, but close instants in time, can experience quite different conditions, which results in them using different and varying reproductive strategies. Thus any model that lumps together the different age stages is inappropriate as an individual-based model is needed. In this p a p e r an individual-based model is presented which incorporates all that is known about the biology of tree-dwelling aphids. The predictions and conclusions are c o m p a r e d with an extensive data set on the population dynamics of the Turkey-oak aphid Myzocallis boerneri.
2. M a t e r i a l a n d m e t h o d s
2.1. The larval development T h e sizes of the soma and gonads of larvae and the size of the gonads and age of adults were monitored in a matrix, which contained one row for each individual. In each time step (0.01 day), different procedures were followed for larvae and adults: The changes in the size of the gonads and soma of larvae were assumed to follow the Kindl-
mann and Dixon (1989) model: ds
dt dg --
dt
-as"-Rg,
s(0) =s0 (1)
=Rg
g(0) =go
where s and g are sizes of soma and gonads at time t, respectively, R is the constraint on the gonadal growth rate, a is the assimilation rate, a = 2 / 3 is a constant (surface to volume ratio) and s o and go are birth sizes of soma and gonads. The simulations of Eq. 1 were performed in 20 000 steps, simulating a season 200 days in length with time step equal to 0.01 day, using a difference method. At birth, the size of soma was assumed to be equal to 25 units, a unit corresponding to 1 /zg for the species considered; this can be used for other species if the units are changed (i.e., if e.g. the birth size were 50 tzg, then a unit 2 /xg would have to be used). The size of gonads at birth was assumed to be that predicted by the optimisation process described in Kindlmann et al. (1992). The function go = (1 + 0.01s0)a 2
(2)
approximates the results of that optimisation fairly accurately and was used instead of repeating the lengthy optimisation. There exists, however, no analytical proof of this dependence. The constraint on the gonadal growth rate, R, was assumed to be equal to 0.5 / x g / / x g / d a y , as this is the value common in aphids (Kindlmann and Dixon, 1992). The assimilation rate, a, was not constant, as in the original model, where the predictions were verified using results of laboratory experiments done under constant conditions. In the field, the assimilation rate changes dramatically during the course of the season and follows changes in the quality of the host plant. The concentration of total free amino acids in the leaves can be used as a good indicator of host quality and Dixon et al. (1993b) indicate a parabolic dependence of this quality on time. In addition to changes in host plant quality, the (negative) influence of cumulative aphid population density on host plant quality was also consid-
P. Kindlmann, A.F.G. Dixon/Ecological Modelling 89 (1996) 23-30
ered. Therefore, the equation a(t) =
( ( t -- train)/train) 2" ( A m a x - A m i n )
behaviour of the model and therefore tmi n = 135 was used in most of the simulations. The values of A . . . . A m i n and C used in the simulations were varied over the range observed in aphids, e.g., A max = 2.5 was estimated from laboratory data ( = good food quality) and its value is unlikely to differ much from this value (Kindlmann and Dixon, 1992). When food quality is poor, a has to be low. A good criterion for testing whether Areax and Ami n are realistic, are the predicted sizes of the adults. If these were not realistic, the simulations were discontinued. The value of C was varied until the simulated population became extinct, or the model's output differed from that observed. The simulations of Eq. 1 were stopped, when
+Ami n
d/C + 1
(3) was used to estimate a, where d is the cumulative aphid population density (calculated from the model), C a scaling constant, A r e a x the maximum value of a at the beginning of the season (t = 0), A m i n the minimum value of a at time tmin (on an uninfested tree, i.e., d = 0 throughout the season). If the season begins in April, the value tmin = 135 is realistic for the time of minimum host plant quality, i.e., August. A change in the value of this p a r a m e t e r does not affect the qualitative
/
[ Gonadal growthrate~ I~
[Temperaturel + ~/J
Birth size]
[Energy partitioning model ]
I Optimisation procedure ] I Larva, deve,cpment I
j
"-,.
loove'opmenta, ,ime I
\Population~[
S
growthrate
Population] size I
+
1, 'grat'on I
25
I du', si,o I
t
Fecundity] ] Background ]
mortality
,11
Host plant] quality
Fig. 1. The organisational diagram of the aphid model.
P. Kindlmann, A.F.G. Dixon/Ecological Modelling 89 (1996) 23-30
26
d s / d t b e c a m e n e g a t i v e (at a t i m e r e f e r r e d to as D , d e v e l o p m e n t a l time), i.e., w h e n t h e a n i m a l b e c a m e adult. 2.2. Adult life and reproduction
~. io3"
F o r adults, t h e r e l a t i v e size of t h e s o m a a n d g o n a d s is i r r e l e v a n t as only t h e size o f t h e g o n a d s at m a t u r i t y , g(D), d e t e r m i n e s fecundity. T h e s h a p e o f t h e r e p r o d u c t i o n curve ( n u m b e r o f offs p r i n g b o r n p e r m o t h e r p e r day), F(t), was ass u m e d to b e positive only for 2 D > t > D a n d triangular:
F(t) = 2Rg(D)/(s o +g0)(2 - t/D),
77
6
%5
,
o i o
-5
o_ O CL
E O m~D
r
O_ O OE~ O
\,,
x
+,+
~x -" × '--
2.
"ix/
_""'- "_'_,'L*
0
Itll
,lllllllllllttllJ 50
100
/
x
1975
,[, ,i,iilIlllljtj[ 150
I 200
Time (days) Fig. 3. Empirical data on density changes of M. boerneri for 1975 (*), 1981 ( + ) and 1982 ( × ) and simulation results showing the same trends. Parameters are the same as in Fig. 2, with the following exceptions: 1975, S = 0.01; 1981, t m i n = 185;
1982,
tmi n = 100, Ama x = I.
a n d f e c u n d i t y is low. T h e p r o c e d u r e u s e d c o r r e s p o n d s with t h e e m p i r i c a l d a t a : small a d u l t a p h i d s d o n o t r e p r o d u c e every day, p r o d u c i n g a single larva at l o n g e r intervals. T h e m o r t a l i t y d u e to p r e d a t o r s , a p h i d s falling f r o m t h e leaves etc., was a s s u m e d to b e c o n s t a n t . W h e n this v a l u e was varied, it d i d n o t have any effect on t h e q u a l i t a t i v e b e h a v i o u r o f t h e m o d e l a n d t h e r e f o r e only t h e o u t p u t s using a v a l u e o f 0.15 for b a c k g r o u n d m o r t a l i t y p e r iterative s t e p are presented.
2.3. Migration
O~
° i
6. ;!",!k "~\ "~ 1981 +
(4)
w h e r e t is t h e age o f t h e i n d i v i d u a l ( K i n d l m a n n et al., 1992). A s only i n t e g e r i n c r e m e n t s in t h e n u m b e r o f i n d i v i d u a l s can b e s i m u l a t e d in t h e i n d i v i d u a l - b a s e d m o d e l , a s t e p l e n g t h 100 times l o n g e r t h a n in t h e s i m u l a t i o n of larval g r o w t h was used. F r a c t i o n s o f F(t) w e r e r o u n d e d r a n d o m l y u p o r d o w n with a p r o b a b i l i t y d e t e r m i n e d by t h e v a l u e o f t h e fraction. This is i m p o r t a n t e s p e c i a l l y w h e n t h e v a l u e s o f a a r e low, a n i m a l s a r e small
.N o3 ¢--
+
0
50
1O0
150
200
Time (doys)
T h e p r o p o r t i o n o f a p h i d s t h a t fly away f r o m a colony, m, was a s s u m e d to b e a l i n e a r f u n c t i o n o f t h e n u m b e r o f a p h i d s (larvae + adults) p r e s e n t in t h e colony, x, with positive s l o p e S a n d positive i n t e r c e p t i ( B a r l o w a n d Dixon, 1980), t i m e s a l i n e a r f u n c t i o n d e p e n d i n g o n a:
Fig. 2. The average seasonal pattern of density changes of M.
boerneri populations on two Turkey-oak trees in Norwich, England, from 1975 to 1992 (*), fitted by the model (--). Parameters: A m a x = 2.5, A r n i n = 0.01, R = 0.5, s o = 25, t m i n = 135, S = 0.005, i = 0.1, C = 2.106. Log of initial number of aphids: 1.5.
m =
(Sx+i)" (Amax- a ) A
max --
A
(5)
min
T h e s e c o n d t e r m in Eq. 5 s i m u l a t e s t h e reluct a n c e of a p h i d s to m i g r a t e , w h e n living on a g o o d
P. Kindlmann, A.F.G. Dixon/Ecological Modelling 89 (1996) 23-30
host plant, and their willingness when on a poor quality plant. The success of an aphid in finding another host plant is assumed to be in the range of 0.01 (Taylor, 1977). Therefore, the immigration from other colonies was assumed to be equal to 0.01m. For the same reason as cited for calculating fecundity, the time step used was 100 times longer than for larval development. 575-
,///j~
475-
(3_ O Ct
400-
The temporal pattern of density fluctuations of M. boerneri populations on two Turkey-oak trees
11"
q~ a
C
O
425"
375"
(b)
,,/t
500-
450-
2.4. The empirical data
12-
525-
E O L~ O
Because the output of the model was dependent on, e.g., fecundity being randomly rounded up and down, each simulation was repeated 10 times. The calculations were performed on an HP Apollo computer using F O R T R A N language.
(a)
550-
03 N
27
10"
9"
O CL O (3_
.,,'; , / I /
8"
7-
350" 325300275
,
10
r
~
,
, ,
, , I
i
,
10
10 2
J
i
i
,
i
J I
10 2
InitiGI number of aphids
Initial number of ephids
451 (c)
43
40N 09
38-
E 0 O 00 El-
,35-
"\.
. . . .
33 ~
-" - " > - - _ ' 7 \
3O ~ 28 25 10
Initial number of aphids
10 ~
Fig. 4. M o d e l predictions of the p e a k densities (a), the s u m m e r m i n i m u m densities (b) and o f the final densities (c) plotted against the initial densities for h m i n = 0.5 (solid line), 0 . 4 , . . . ,0.1 (the smaller Zmin, the shorter the dashes). P a r a m e t e r s used: Ama x = 2.5, R = 0.5, $0 = 25, t m i n = 150, S = 0.005, i = 0.1, C = 5 - 104. Log o f initial n u m b e r of aphids: 2.
28
P. Kindlmann, A.F.G. Dixon/Ecological Modelling 89 (1996) 23-30
in Norwich, England, from 1975 to 1992, were used. The populations were sampled at weekly intervals from the beginning of May to the end of N o v e m b e r each year. Each week, 10 leaves were selected at random at each of 8 fixed sampling points around the circumference of the trees and the numbers of the various life stages recorded.
3. Results
The organisational diagram of the model described in the previous section is shown in Fig. 1. Not all the simulation results are depicted. The c o m m o n feature of almost all the simulations was the sharp increase in total numbers at first, followed by a decline and then another increase in abundance. The p e a k n u m b e r was reached very early each year in all cases: during the first 15-40 days. The more or less broad trough in population numbers never preceded the instant when a was lowest and occurred some 15-30 days later. The possibility of simplifying the model by deleting some of the p a r a m e t e r s was explored by varying those that were not measured directly. All such simplifications resulted in the model deviating from reality.
14-
For example, deletion of the second term in Eq. 5, i.e. omission of the assumption that the tendency of aphids to migrate depends not only on their own density, but also on host plant quality, did not change the " u p - d o w n - u p " population profile, but resulted in the peaks being of nearly the same size for a wide range of initial conditions, irrespective of the values of the other parameters. As the empirical data show a significant positive correlation of the peak value with the initial density, the second term in Eq. 5 cannot be omitted. The average seasonal pattern of density changes of the Turkey-oak aphid, from May to November, for the period 1975-1992, was fitted by the model in Fig. 2. From the same period, 1975-1992, three years with very different patterns of density changes were chosen and data plotted in Fig. 3 : 1 9 8 1 - a high spring peak followed by a monotonous decline, second peak not present; 1982 - both spring and autumn peaks relatively low and of the same size; 1975 the spring p e a k followed by a decline and a long plateau in autumn. To show the model's ability to mimic different types of aphid behaviour, parameters were chosen to simulate the dynamics in these years, but not fitted to the data, in Fig. 3.
(a)
451 (b) 43
,
13
*
40
12
• o 38
._Nll 03 E O10
* *
k
1
35~
, *
*
,
,
~v
•
*
28-1
•
*
25~
* , ,
O C~. 9 O (2_ 8
{
{
w
w
w
w
i
[
{
w
i
10
Initiol number of ophids
{
{
{
w
w
[ {
10 ~
25
* w
1
{
i
{
{
{
w i
{
{
w
10
i
i
{
{
{
{ {
10 2
Initiol number of ophids
Fig. 5. Predictions of the final and minimum densities against the initial densities obtained from 10 runs of the model using different random numbers. Z m i n = 0.5. Other parameters are the same as in Fig. 4.
P. Kindlmann, A.F.G. Dixon/Ecological Modelling 89 (1996) 23-30
A global view of the model's behaviour is given in Fig. 4, where the average peak densities, summer minimum densities and final densities obtained from 10 runs of the model are plotted against the initial densities for several values of Amin. The peak densities are clearly dependent on the initial numbers but weakly so on the value of Ami n. Both the summer minimum and the final autumnal densities are dependent on Amin; the lower Amin, the lower the corresponding density. They also decline when initial numbers and therefore also spring peak numbers (Fig. 4a) are increased (the "see-saw effect"), this being particularly so in the case of the final densities and negligible so for low Ami n and the minimum densities. Plotting the final and minimum densities against the initial densities obtained from 10 runs of the model reveals that the "see-saw effect" may be overshadowed by other phenomena (Fig. 5). Although the appropriate trend is there, variation due to random factors is very large. As the empirical data indicates, the numbers of aphids present in summer are extremely low. This suggests that the actual values of C and A min are close to those used in the simulations illustrated in Figs. 2 and 3.
4. Discussion
The model satisfactorily describes the withinyear population dynamics of the Turkey-oak aphid: a sharp increase in numbers in spring, which is usually followed by a sharp decline in summer and sometimes by a subsequent increase in autumn. The autumnal increase follows the improvement in food quality with a delay and is never so steep as in spring. The delay has a simple biological explanation: small summer individuals feeding on food of low, although rapidly improving quality, are not able to produce large offspring and their fecundity is also low. Therefore the effect of the autumnal improvement in food quality is reflected in the population dynamics but with a delay and the subsequent increase in numbers is less steep than for the large aphids living in the lush conditions prevailing in spring. The model further reveals the regulatory
29
mechanism that is responsible for the summer decline in numbers: migration. This increases linearly with density and declines with improving food quality. The model outputs indicate that migration is important in shaping the population dynamics. If the decline were mainly due to low fecundity and high mortality, then the average realised number of surviving offspring per aphid would have to be - because of parthenogenesis smaller than one! This seems unlikely, as even in the worst conditions the average reproductive output per aphid is about 1 offspring per 1-2 days and the reproductive life span is 10-20 days. Empirical data confirm that the tendency to migrate depends on density and food quality in the above mentioned way. In the model, the extent of the summer decline in aphid abundance is determined by the degree to which the food quality declines in summer, which influences aphid size, fecundity, density and subsequently their tendency to migrate. Therefore, changes in food quality determine in this rather complicated way the changes in summer numbers. This prediction is rather difficult to test empirically and is a task for the future. In contrast to most other groups, the empirical data do not lend support to a marked effect of natural enemies, or a severe scramble for resources by these aphids. Therefore, these factors do not seem to play a substantial role in regulating the abundance of these aphids. Food quality determines the rate of development and size of an individual more than intraspecific competition, and food quality is not affected by aphid numbers substantially (Dixon, 1971). Finally, a comparison of data presented here and by Dixon (1971) further supports the hypothesis that migration is responsible for the summer decline in numbers: while in nature there is no second increase in population numbers in the lime aphid, in glasshouse experiments there is (Dixon, 1971), which is possibly because the experiment was done in an enclosed space in which the probability of finding another host plant was greater than in the field. The "see-saw effect", a negative correlation between the numbers of first generation aphids in spring (fundatrices) and the last generation in
30
P. Kindlmann, A.F.G. Dixon/Ecological Modelling 89 (1996) 23-30
a u t u m n ( o v i p a r a e ) , has b e e n o b s e r v e d in s o m e e m p i r i c a l d a t a (Dixon, 1971), b u t in o t h e r s p e c i e s ( o u r u n p u b l i s h e d d a t a ) it is not p r e s e n t o r very w e a k . T h e m o d e l p r e s e n t e d h e r e reveals, why this is t h e case: t h e " s e e - s a w e f f e c t " is a statistical p h e n o m e n o n , r a t h e r t h a n a d e t e r m i n i s t i c one. It is m o r e likely to o c c u r w h e n t h e initial n u m b e r s a n d / o r t h e s u m m e r p o p u l a t i o n a r e very high a n d w h e n t h e a p h i d has some, b u t n o t a l a r g e effect o n t h e host plant. I f no c u m u l a t i v e d e n s i t y effect on a p h i d a b u n d a n c e is a s s u m e d in t h e m o d e l ( C = ~ in Eq. 3), it d o e s n o t p r e d i c t any " s e e - s a w e f f e c t " (simul a t e d , b u t n o t d e p i c t e d ) . This is quite s t r a i g h t f o r w a r d , as t h e r e is a lag o f several g e n e r a t i o n s between the summer and autumn peaks. Theref o r e t h e r e m u s t be s o m e " m e m o r y " in t h e system, which t r a n s f e r s i n f o r m a t i o n on a b u n d a n c e f r o m s p r i n g to a u t u m n . It is u n c l e a r w h a t b i o l o g i c a l f o r m this " m e m o r y " takes. A lag c a u s e d by p r e d a t o r s is unlikely, as e x p l a i n e d above. T h e c u m u l a t i v e d e n s i t y effect a s s u m e d in t h e m o d e l m a y b e a c o n s e q u e n c e of e i t h e r a p e r s i s t i n g a n d therefore accumulating influence of the aphids on t h e p l a n t , o r o f s o m e c a r r y - o v e r effect bet w e e n a p h i d g e n e r a t i o n s . T h e r e is n e i t h e r e m p i r i cal e v i d e n c e for t h e f o r m e r , n o r t h e o r e t i c a l supp o r t for t h e latter. P o p u l a t i o n census d a t a are usually subject to a d e s c r i p t i v e statistical analysis using a t i m e series analysis a p p r o a c h , with t h e T u r k e y - o a k a p h i d d a t a u s e d h e r e b e i n g n o e x c e p t i o n (they w e r e analysed, e.g., in T u r c h i n , 1990; T u r c h i n a n d Taylor, 1992). This p a p e r analyses t h e m f r o m a d i f f e r e n t p o i n t o f view: t h a t b a s e d on a d e t a i l e d k n o w l e d g e o f t h e b i o l o g y o f t h e g r o u p involved, aphids. T h i s approach reveals the underlying mechanisms, r a t h e r t h a n just d e s c r i b i n g t h e p o p u l a t i o n t r e n d s . This m o d e l p r o v i d e s a g o o d basis for a p r e d i c t i v e
m o d e l . F o r this, however, m o r e e m p i r i c a l d a t a on t h e s e a s o n a l c h a n g e s in f o o d quality similar to t h o s e for s y c a m o r e ( D i x o n et al., 1993b) a r e n e e d e d along with a b e t t e r u n d e r s t a n d i n g o f t h e m i g r a t o r y b e h a v i o u r of aphids.
Acknowledgements This w o r k was s u p p o r t e d by t h e N E R C g r a n t GR3/8026A and NATO research grant CRG920553.
References Barlow, N.D. and Dixon, A.F.G., 1980. Simulation of Lime Aphid Population Dynamics. Pudoc, Wageningen, 165 pp. Dixon, A.F.G., 1971. The role of intra-specific mechanisms and predation in regulating the numbers of the lime aphid Eucallipterus tiliae L. Oecologia (Berl.), 8: 179-193. Dixon, A.F.G., Horth, S. and Kindlmann, P., 1993a. Migration in insects: costs and strategies. J. Anim. Ecol., 62: 182-190. Dixon, A.F.G., Wellings, P.W., Carter, C. and Nichols, J.F.A., 1993b. The role of food quality and competition in shaping the seasonal cycle in the reproductive activity of the sycamore aphid. Oecologia (Berl.), 95: 89-92. Kindlmann, P. and Dixon, A.F.G., 1989. Developmental constraints in the evolution of reproductive strategies: telescoping of generations in parthenogenetic aphids. Funct. Ecol., 3: 531-537. Kindlmann, P. and Dixon, A.F.G., 1992. Optimum body size: effects of food quality and temperature, when reproductive growth rate is restricted. J. Evol. Biol., 5: 677-690. Kindlmann, P., Dixon, A.F.G. and Gross, L.J., 1992. The relationship between individual and population growth rates in multicellular organisms. J. Theor. Biol., 157: 535542. Taylor, L.R., 1977. Migration and the spatial dynamics of an aphid, Myzus persicae. J. Anim. Ecol., 46: 411-423. Turchin, P., 1990. Rarity of density dependence or population regulation with lags? Nature, 344: 660-663. Turchin, P. and Taylor, A.D., 1992. Complex dynamics in ecological time series. Ecology, 73: 289-305.