A computer simulation study of stochastic models to investigate the population dynamics of the screech owl (otus asio) under increased mortality

Ecological Modelling, 40 (1988) 233-263 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

A COMPUTER SIMULATION STUDY OF STOCHASTIC TO INVESTIGATE THE POPULATION DYNAMICS OF THE SCREECH OWL (OTUS A SIO) UNDER INCREASED M O R T A L I T Y

233

MODELS

PHILIP M. NORTH, ALEX W. BODDY and DUNCAN R. FORRESTER

Applied Statistics Research Unit, University of Kent at Canterbury, Canterbu~', Kent CT2 7NF (Great Britain) (Accepted 30 March 1987)

ABSTRACT North, P.M., Boddy, A.W. and Forrester, D.R., 1988. A computer simulation study of stochastic models to investigate the population dynamics of the Screech Owl (Otus asio) under increased mortality. Ecol Modelling, 40: 233-263. The work reported in this paper complements an earlier study in which deterministic models were used to examine the effects of additional mortality imposed on a Screech Owl, Otus asio, population. Interest lay in the compensatory mechanisms that the owls might be able to adopt in their population dynamics to counteract the overall effect of the additional mortality imposed, in particular when only a part of the population is affected. Stochastic models are used here to address the same questions, and to give more detailed information about the likely behaviour of the owl population than the information an average sizes provided by the deterministic approach.

1. INTRODUCTION I n an earlier p a p e r , N o r t h (1985) has d e s c r i b e d Leslie m a t r i x t y p e m o d e l s w h i c h w e r e u s e d to i n v e s t i g a t e the p o p u l a t i o n d y n a m i c s of the S c r e e c h O w l (Otus asio). T h e m a i n interest of t h a t s t u d y w a s h o w a S c r e e c h O w l population would behave under increased mortality (through taking prey i t e m s c o n t a m i n a t e d with a r o d e n t i c i d e a p p l i e d in o r c h a r d s ) , a n d h o w c h a n g e s in the b i r d s ' b e h a v i o u r m i g h t i n f l u e n c e the p o p u l a t i o n p a r a m e t e r s . S o m e general b a c k g r o u n d o n the Screech O w l is given in t h a t p a p e r , a n d m o r e extensive details c a n b e f o u n d in the r e p o r t o f a l o n g - t e r m s t u d y of the species b y V a n C a m p a n d H e n n y (1975). All the m o d e l l i n g d e s c r i b e d b y N o r t h (1985) w a s d e t e r m i n i s t i c in n a t u r e , i.e. the i n v e s t i g a t i o n s w e r e p u r e l y in t e r m s of w h a t w o u l d h a p p e n o n a v e r a g e 0304-3800/88/$03.50

© 1988 Elsevier Science Publishers B.V.

234 to the owl population, as predicted by the models used. However, in that paper it was suggested that it would be desirable to consider stochastic models also. Certainly, in the context of the owl population, it is particularly important to know with what probability a model predicts that the population affected by additional mortality will become extinct, as well as knowing what the predicted average behaviour is. The purpose of this present paper is to re-examine the various questions about the population dynamics within a stochastic framework, and to give the more detailed information which results, and which is important when assessing the possible development of an actual population. 2. DEVELOPMENT OF THE BASIC STOCHASTIC MODEL The stochastic models to be discussed in this paper are developed directly from the deterministic models considered by North (1985). For description of other modelling work in a similar context see Smies (1983a, b) and Grant et al. (1984). In this section we establish the basic framework for all the stochastic models that will be studied here. The deterministic models were of a Leslie matrix type, drawing on the work of Leslie (1945), and adopting an approach similar to that of Cooke and Leon (1976) in their modelling of the Oxford Great Tit (Parus major) population. We briefly recap here on the matrix model which forms the basis of those that follow. The population is assumed to develop in distinct, equal length stages (1 year each). We suppose that each stage begins just before the start of the owls' breeding season each year. Just two age classes are defined and the model describes the development of the population in terms of the numbers of individuals in these age classes. They are: (a) birds which were born in the previous breeding season, i.e. birds which are just approaching 1 year of age; (b) birds of age greater than 1 year. Just the female part of the population is considered by the model, it being assumed that males and females in the population balance. North (1985) discusses this assumption, for which justification is provided by VanCamp and Henny (1975). Clearly, in further simulation work, this assumption, and the effects of deviations from it, could be examined further. The state of the female population at time t is then represented by the column vector:

[Nj,]

N'= LNA,j where

Njt is the expected number of 'juvenile' birds (birds of type a) at time

235 t, and NAt is the expected number of 'adult' birds (birds of type b) at time t. The progress of the population is then determined by the projection matrix:

where fj is the juvenile fecundity, i.e. the expected number of daughters born in a breeding season to a female of type (a) that survive to the next breeding season; fA is the corresponding adult fecundity; Sj is the juvenile survival rate, i.e. the proportion of birds of type (a) that survive to the next breeding season; it is therefore the second-year survival rate; SA is the corresponding, constant, annual adult survival rate. (For comment on the assumption of a constant rate here, see North, 1985.) Then N t + 1 = P N t and, more generally, N~+k =PkN~. The nature of the population development (increase, decrease, stability) is then determined by the dominant eigenvalue of the matrix P (see Usher, 1972; North, 1985). The deterministic steps in the development of the population, as described by this model, are now transformed into a stochastic framework by assigning individual birds probabilities corresponding to all the steps, and by charting the progress of the individuals through the various stages. In assigning appropriate probabilities we again have to rely heavily on the parameter estimates presented by VanCamp and Henny (1975), whose study of Screech Owls is the only extensive and detailed one available. North (1985) discusses the reliability of these estimates. Despite their limitations they are the best ones available. The various steps that must be gone through to constitute the mechanism of the stochastic model are shown in the flow diagram in Fig. 1. Different choices of the probabilities involved can lead to different models based on the general framework. 3. MODEL 1: SECOND-YEAR SURVIVAL RATE EQUAL TO THE ANIMAL ADULT SURVIVAL RATE We consider briefly the behaviour of a model corresponding to North's (1985) deterministic Model 1 (section 4.4.1) with second-year survival rate equal to the annual adult survival rate. This follows from the VanCamp and Henny (1975) study: North (1985) found that the population described by this model was increasing in size. It is of interest here to check that the predictions made by the stochastic models are compatible with those of the corresponding deterministic models. From the earlier paper we have: S o = 0.305

Sa = S = 0 . 6 6 1

236

SET S3RVIVAL PAKCMETN~, SI2Z I~OBABILITIES, FIRS~ YEAR N{[~)ING I:~{OhAB]IZFf NO. (1~ TRIAI~ AND NO. OF (~NfP,ATIONS

I

s~

m R EA~

[ ~DI~mJAL

Ft~tJI~TION

ol I:~PdIATI~ P ~

,)

N

Y

Fig. 1. Flow diagram for Model 1. where So, $1 and S are the first-year, second-year and annual adult survival rates. VanCamp and Henny (1975) estimated that between 77% and 83% of first year birds attempted to breed (that is, began nesting activities) and, corresponding to North's (1985) assumption, we suppose that the probability that a first-year bird breeds is 0.8.

237 We now have to account for the possibility of different clutch sizes. The probabilities are assumed to be constant, and not to vary with, for example, age of the bird laying the clutch (as might be the case in reality). In his study, and corresponding stochastic model, Smies (1983) used a normal distribution for clutch size, based on the mean clutch size observed. However, in their paper, VanCamp and H e n n y (1975) provide information on different clutch sizes in North Ohio. The probabilities assigned here are therefore based on information drawn directly from Table 10 in that paper. By this means we arrive at the following probabilities Pi, where pi = probability of clutch size i for a 'successful' nest (that is, a nest which is completed and in which eggs are laid): Pl = 0; P2 =0.011; P3 = 0.110; P4 = 0.406; P5 = 0.385; P6 = 0 . 0 8 8 ; Pi = 0 , i > 6. The mean clutch size is 4.43. Only the figures for Northern Ohio have been used, hence ignoring values for other regions, as given by VanCamp and H e n n y (1975). Those authors report on an increase in clutch size from east to west and from south to north in the United States. But, averaging the clutch size data over all regions is inappropriate, since the survival data come mainly from Ohio. If offspring of either sex are assumed to be equally likely, then the following probabilities p: can be computed from the p,, where p: is the probability that a clutch contains i females in a successful nest (successful being defined as above): !

P0 = 0.056 !

P4

=

0.106

!

Pl = 0.217 !

P5

-----

0.020

!

P2 = 0.337 !

P6 = 0.001

t

P3 = 0.263 P

p~ = 0,

i> 6

But not all nests are successful. VanCamp and H e n n y (1975) used the Mayfield exposure-day method (Mayfield, 1961) to estimate that during their study 69.2% of the nesting attempts by the Screech Owls were successful. Using this as a scaling factor, we obtain the following probabilities p* from the p~, where p* is the probability of a breeding pair producing a clutch containing i 'female' eggs: p~' = 0.347

p~' = 0.150

p~' = 0.233

p~" = 0.182

P4* = 0.073

P5* = 0.014

P6* = 0.001

p* = 0,

i> 6

But not all the eggs laid will hatch and not all the young produced will fledge. A nest survival probability, SN, therefore needs to be defined. Dividing the estimated 3.80 young fledged per successful nesting (VanCamp and Henny, 1975, table 11) by the mean clutch size for successful nestings, 4.43, we obtain: SN = 0.858 Lack of suitable data precludes us from making any assumption other than that nest survival does not depend on clutch size. Staggered laying of eggs

238 22 7

z r ~2oI u_

i

lZt

Io_

8_

6_

F - 1

17 1400

1500

1600

1700

tBO0

1900

2000

2100

2200

2300

2400

2500

2000

27D0 TOTAL

2800

2gO0

POPULATION

3000

3100

AFTER

3200 25

YEARS

Fig. 2. Distribution of population sizes after 25 years in 166 simulations of Model 1, starting with 500 females and 500 males.

may mean this is rather unlikely in practice (see, for example, Sparks and Soper, 1970). In fact, as pointed out by one of the referees, higher clutch size is likely in good food years, when more young can be successfully raised (as is observed for many bird species, including owls). The model defined by these parameters can be easily simulated on a computer, following the flow diagram in Fig. 1. The program used in the present study was written in F O R T R A N 77, and the (pseudo-)random number generator used is from the N A G subroutine G05CCF (NAG F O R T R A N Library, Mark 11, Numerical Algorithms Group Ltd., Oxford, Great Britain). In the simulations the initial population size was set at 1000 (500 females, 500 males), made up of 43% first-year birds and 57% birds of age greater than 1 (see North, 1985, for justification of these starting percentages). The dominant eigenvalue for the deterministic, matrix model was found to be 1.028, leading to a doubling of the population size in 25-26 years, and the stable age distribution contained 35.7% first-year birds. The stochastic version of the model has been simulated 166 times, over 25 years. The distribution of the eventual population size after these runs is shown in Fig. 2. We find that the minimum size of the population after 25 years was 1414

239

and the maximum 3224, with mean and median values 2218 and 2194, respectively. In every run of the model the population grew in size. The stable age distribution predicted by the deterministic model is reflected well by the stochastic model, the approach to this distribution being achieved very early in the simulated population developments. At the end of 25 years the simulated populations had between 31% and 39% first-year birds, with almost 90% of the values lying between 34% and 38%, the mean being 36.0%, compared with 35.7% for the deterministic model. The mean populations predicted by the deterministic and stochastic models are in good agreement. After 25 years the corresponding means were 1975 and 2218, respectively - a difference of 12%. The deterministic model corresponds to growth of 2.8% year-1 (i.e. a dominant eigenvalue of 1.028). The stochastic model showed an average growth of 3.3% year-1. The stochastic model is, in fact, equivalent in the mean to the deterministic model of North (1985). This is shown in the Appendix. 4. MODEL 2: SECOND-YEAR SURVIVAL RATE D I F F E R E N T F R O M TH E A N N U A L A D U L T SURVIVAL RATES

VanCamp and Henny (1975) present data (admittedly, from only a very small sample) allowing a second year-survival rate of 0.594 to be estimated, as used in North's (1985) Model 2 (section 4.4.2), with which a stochastic version will now be compared. The flow diagram for generating the stochastic model follows very easily from Fig. 1. The insertion of one additional step is required namely 'Determine new second-year population', between the steps for determining new juvenile (first-year) and adult populations, and now each of these three groups has its own survival rate. Note that while there are now three different survival rates, there are still only two age groups at the beginning of each generation of the model (birds born in the previous breeding season (juveniles), and birds of 2 years and older). A complete generation is shown schematically in Fig. 3. The equivalence of the stochastic models may be demonstrated exactly as in the Appendix. In the simulations of this model, the initial percentages of juveniles and adults were taken to be as in Model 1 : 1 0 0 simulations were carried out, each lasting for 100 generations, and with a starting population of 1000 birds (500 male, 500 female). After 100 years the population had declined only once, the minimum final population size being 996, the maximum 3616. The median size after 100 years was 1950, and the mean size 2011. This difference between the mean and the median values suggested some skewness, and this can be seen in the histogram of the distribution of final

240

Stage 1

Stage 2

Stage 3

Beginning of generation (immediately prior to breeding season)

On completion of breeding

End of generation

birds born in previous breeding season (breed with probability PB)

birds in second year (survive with probability $1)

birds in second year or older (breed with probability 1)

brids of second year of older birds older than 2 years ,z (survive with probability S)

birds born in breeding season just completed (survive with probability S0)

~

birds born in previous breeding season

Fig. 3.Schematic representation of one complete generation in a model for the Sreech Owl population.

18_ 16_ 14_ 12_ IO_

>-. c..) z O" u2 r-~ r..

B_ 6.

4.

I 900

II06

1300 1500 1700 1900 2100

2"300 2500

2700

2900

I

I

I

3100 3300 3560 3700

TOTAL POPULATIONAFTER 1OO YEARS Fig. 4. Distribution of population sizes after 100 years in 100 simulations of model 2, starting with 500 females and 500 males.

241 population size in Fig. 4. A mean population size of 2011 after 100 years corresponds to an increase of 0.8% year -a (dominant eigenvalue 1.008), which agrees quite well with the value of 0.3% (dominant eigenvalue 1.003) given by North (1985, section 4.4.2). Perturbations of the starting percentages were examined, from which it was found that the populations invariably settled down very quickly to similar age distributions. Following the ideas of North (1985) we now examine what happens to populations described by Model 2, under varying amounts of increased mortality. 5. EFFECT OF INCREASED MORTALITY ON THE WHOLE POPULATION We now consider the introduction of an additional force of mortality affecting the whole population where, as in North (1985), the extra mortality occurs in the autumn each year. This is intended to mimic the situation where a rodenticide is applied to orchard areas at that time of year. In this way the survival rates S o (1st year), S 1 (2nd year) and S (annual adult) are reduced, but SN, the nest survival rate, is not affected. For the simulations here a starting population of 200 birds (100 male, 100 female) is used, to reflect the fact that only smaller, localised populations in orchard habitat are likely to be affected. A check first on the behaviour of the population with no increased mortality reflects what we would expect, namely greater variability than the larger population examined in the previous section. The range of values of the population size is from 6 to 1272 and, clearly, in instances such as the first, the population is on the verge of extinction. The median and mean values are 372 and 393, respectively. A number of different additional mortality factors were applied to the population, and 100 simulations carried out for each. The population was followed for 100 generations or until it became extinct. Details of the populations surviving 100 years are given in Table 1, and details of the times to extinction in Fig. 5. As well as knowing how the populations can be expected to behave in the long term under the model, it is also of interest to know what the behaviour is likely to be in the short term. This is particularly true in the practical situation under consideration here, where it is necessary to know what the potentially harmful short-term effects of the application of rodenticide might be. Figure 6 therefore summarises the simulated populations under Model 2, with additional mortality, after 10 years. In all cases 100 simulated populations contributed to these summary measures. These results tie in, of course, with the long term effects noted above. They do emphasise, however,

242 II0

100.

90.

80

70

60, >~

50,

4-

4o

O

÷

+

30,

+

4-

÷

D

+

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+

÷

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1.00

i

l

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l

l

i

i

i

1.05

I.I0

1.15

1.20

1.25

1.30

1.35

1.40

1.45

ADDITIONAL MORTALITY FACTOR

Fig. 5. Mean, median and extreme times to extinction for populations starting with 100 females and 100 males, simulated according to Model 2. *, Mean: D, median; + , extreme value. Values for additional mortality of 1.05 and 1.10 were calculated for those populations that became extinct within 100 generations (81, 99 of these for additional mortality factors of 1.05, 1.10 respectively). TABLE 1 Summary of surviving population sizes after 100 years for populations simulated according to Model 2, starting with 100 males and 100 females Additional mortality factor

Mean size

Median size

Minimum size

Maximum size

Number of populations surviving

1.00 1.05 1.10 1.15

393 10 10 0

372 10 10 0

6 2 10 0

1272 18 10 0

100 15 1 0

243 :

360

330

300

270

240

210

180

2 <

150

~

120

+

9O

60

+

+

30

+

+

+

+

+ 0

.

1.00

1.05

.

.

1.10

.

.

1.15

.

÷

~'

1.20

1.25

1.30

1.35

I

1.40

1.45

ADDITIONAL MORTALITY FACTOR

Fig. 6. Mean, median and extreme population sizes after 10 years simulated according to Model 2, starting with 100 females and 100 males, 100 times. *, Mean; t~, median; +, extreme value. the drastic reductions in population size that can occur quite quickly, even with only moderate increases in the mortality rates. As in the earlier paper (North, 1985) we n o w consider ways in which changes in the behaviour of the owl population might compensate for additional mortality encountered, and we consequently examine h o w populations develop under suitably amended versions of the stochastic model. We first of all consider the possible compensatory factors individually, and then in pairs.

244 6. EFFECT OF COMPENSATION FOR ADDITIONAL MORTALITY C H A N G I N G A SIMPLE VARIABLE

6.1. Increase in the proportion of first-year birds breeding It is possible that if an increased force of mortality is experienced by the owl population, a larger proportion of first year females may breed, possibly because a greater quantity of suitable nesting sites are left vacant. However, definite evidence of this does not appear to be available in the literature for Screech Owls. Of course, if there is no additional mortality, and if all the first-year females breed, then populations generated by Model 2, with this amendment, explode in size; 100 simulated populations, initially with 100 males and 100 females, yielded the following summary measures of the distribution of population sizes after 100 years: mean = 6058.8

median = 5747

minimum = 1176

maximum = 12 700

But even with all first-year females breeding, the populations were shown to be unable to withstand even small increases in mortality. See Table 2. In none of the simulated populations was the increase in the number of birds breeding able to overcome the effect of a 5% increase in mortality. The summary statistics of the distributions of times to extinction in simulated populations, generated under the model, with increasing additional mortality factors are given in Fig. 7. In order to view the decline in the populations in a shorter time context summary statistics for the distributions of population sizes after 10 years are given in Fig. 8. This serves to emphasise the conclusion that unless the additional mortality experienced by the owl population is very small, an increase in the proportion of first-year birds breeding is, by itself, insufficient to compensate for the extra mortality and hence to stabilise the population.

TABLE 2 Summary of surviving population sizes after 100 years, simulated according to Model 2, but with all first-year birds breeding, starting with 100 males and 100 females, 100 times Additional mortality factor

Mean size

Median size

Minimum size

Maximum size

Number of populations surviving

1.00 1.05 1.10

6058.8 21.7 9

5747 14 9

1176 2 6

12700 96 12

100 59 2

245 II0

I00 +

90

+

80

"70

+

÷ +

50

~

4o

30

÷ 20

+

10

+

+ +

0 1.00

I

1.05

[

1.10

I

1.15

I

1.20

I

1.25

I

1.30

I

1.35

+

[

1.40

I

1.45

ADDITIONAL MORTALITY FACTOR

Fig. 7. Mean, median and extreme times to extinction for populations starting with 100 females and 100 males, simulated according to Model 2, but with all first year females breeding, 100 times. *, mean; D, median; +, extreme value. Values for additional mortality of 1.05 and 1.10 were calculated for those populations that became extinct within 100 generations (41, 98 of these for additional mortality factors of 1.05, 1.10 respectively). Simulations of a p o p u l a t i o n with a d i f f e r e n t initial ratio of juveniles to adults simply c o n f i r m this finding.

6.2. Increase in the juvenile survival rate It is n o t u n r e a s o n a b l e to c o n s i d e r that j u v e n i l e Screech Owls m a y b e able to increase their survival rate to o v e r c o m e a force o f a d d i t i o n a l m o r t a l i t y . This might arise as a result o f fewer j u v e n i l e birds b e i n g f o r c e d to disperse i n t o s u b o p t i m a l h a b i t a t (see N o r t h , 1985, for f u r t h e r discussion).

246 270

240.

210.

180

•

at

150 -

120

go c~

R

÷ 60

+

+

3O +

+

i

i

1. 10

1. 15

at

+

i

i

+

at

0 i

1.00

1.05

i

1.20

1.25

1.30

i

I . 35

I

1.40

i

1.45

ADDITIONAL MORTALITY FACTOR

Fig. 8. Mean, median and extreme population sizes after 10 years simulated according to

Model 2, starting with 100 females and 100 males, but with all first year females breeding, 100 times. *, mean; D, median; +, extreme value.

When the juvenile survival is increased from 0.305 to just 0.35 (i.e. the intrinsic survival rate, before the force of additional mortality takes effect), simulated populations (again starting with 200 birds) show that even with an additional mortality factor of 1.10, the compensation is insufficient to overcome population decline. In only 18 out of 100 cases did the population survive 100 years. The results of 100 simulations at each of a range of juvenile survival rates and additional mortality factors are given in Tables 3,

247 TABLE 3 Summary of surviving population sizes after 100 years, simulated 100 times according to model 2, but with increasing juvenile survival rate from 0.305, starting with 100 males and 100 females Juvenile survival rate, So

Additional mortality factor

Mean size

Median size

0.35

1.10

0.40 0.40 0.40 0.40

1.05 1.10 1.15 1.20

quadrupled in 25 years 403.8 340 12 9.6 8 2 . . . .

0.50 0.50 0.50

1.20 1.25 1.30

0.60 0.60 0.60

1.35 1.40 1.45

16.8

8

Minimum size 4

Maximum size

Number of populations surviving

60

18

1024 44

99 19 0

population increasing 53.3 40 2 24.0 19 12

230 50

79 6

population explodes 105.8 82 19.6 11

388 76

79 14

2 4

TABLE 4 Summary of simulated times to extinction, in years, for those populations out of 100 that became extinct within 100 years. Simulations, according to Model 2, started with 100 males and 100 females, but with increased juvenile survival rate, and with additional mortality factors applied Juvenile survival rate SO

Additional mortality factor

Mean time (years)

Median time (years)

Minimum time (years)

Maximum time (years)

0.35

1.10

67

66

31

100

82

0.40 0.40 0.40 0.40

1.05 1.10 1.15 1.20

. 79 67 42

.

. 79 32 19

79 99 98

0 1 81 100

0.50 0.50 0.50

1.20 1.25 1.30

. 79 54

.

. 30 23

99 99

0 21 94

0.60 0.60 0.60

1.35 1.40 1.45

. 77 60

.

. 58 27

100 98

0 21 86

. 79 68 39 . 82 52 . 74 58

Number of populations extinct

248 TABLE 5 Summary of population sizes after 10 years for 100 populations simulated according to Model 2, starting with 100 males and 100 females, but with increased juvenile survival rate, and additional mortality factors applied Juvenile survival rate SO

Additional mortality factor

Mean size

Median size

0.35

1.10

119.0

114

0.40 0.40 0.40 0.40

1.05 1.10 1.15 1.20

0.50 0.50 0.50 0.60 0.60 0.60

Minimum size

Maximum size

62

230

population explodes 214.8 212 125.8 124 71.0 70

100 64 30

392 238 154

1.20 1.25 1.30

population explodes 172.3 169 102.8 94

78 40

290 216

1.35 1.40 1.45

population explodes 187.6 183 129.1 130

96 44

352 228

4 and 5. Table 3 summarises the surviving populations after 100 years, Table 4 gives times to extinction, and Table 5 details of the 10 year populations. 6.3. Increase in the recruitment rate

Another way in which the owls may overcome the additional mortality is by an increase in the recruitment rate, that is the number of fledglings produced per breeding pair. In theory, this might be achieved in one of two ways - an increase in nest survival or an increase in the clutch size. To what extent this might be physically possible is not clear, though it does seem reasonable to suppose that the first method might be the less plausible. However, considering that possibility first, if the nest survival rate is 1.0, then even with additional mortality of only 5%, the 100 simulated populations declined on average, and in two cases they became extinct (in 72 and 97 years). The summary statistics (mean, median, minimum, maximum) for the population sizes at 100 years, for the 98 populations that survived that long are 168.5, 137, 6, 790. After only 10 years, the corresponding statistics are 191.4, 186, 102, 326. With only 10% additional mortality, 97 out of 100 simulated populations had become extinct within 100 years, in times ranging from 22 to 95 years (mean = 55, median = 52), while three populations survived 100 years, with sizes of only 4, 12 and 16 birds.

249 With 20% additional mortality, all 100 simulated populations became extinct quite quickly, in times ranging from 13 to 52 years (mean = 23, median = 23). The summary statistics for the population sizes after 10 years are 25.3, 26, 2, 52. We see, therefore, that even with no nest mortality (a situation that could never be realised in reality) the population declines for all b u t the smallest additional mortality. Such a means of increasing the recruitment rate will therefore not be considered further in this paper and we now consider the effect of increased clutch size. Since we have no information on the way in which the distribution of the clutch size probabilities might vary as the mean size increases, we return to a partially deterministic framework for the model, to examine the effect of an increase in mean clutch size. For example, for a mean clutch size of 4, the probability of clutch size 4 is simply set equal to one, so that each breeding female lays two female eggs and two male eggs. This is not the only approach that could be adopted, of course. Some distribution, i.e. a symmetric one about the mean, could be imposed on to the clutch sizes, for instance. However, it is hoped that the approach adopted here will give some indication of the population behaviour to be expected. The clutch size probabilities given in Section 3 allows us to c o m p u t e the expected clutch size for a breeding female to be 1.53 female eggs, i.e. a mean total clutch size of 3.06. We therefore consider here the effect of increasing the mean clutch size to 4, 5, or 6. With a mean clutch size of 4 we find that simulated populations explode in size when the additional mortality is as low as 5%. However, with additional mortality of 10%, the simulated population died out within 100 years in a little more than half the simulated runs, and decreased considerably in most of the other runs. When the additional mortality was increased to 20%, the simulated populations all died out within 100 years and mostly in considerably less time. When the mean clutch size was increased to 5 the simulated populations exploded for both 5% and 10% additional mortality, but with additional mortality of 20%, the simulated populations died out within 100 years in all 100 cases, in mean time 39 years (standard error 13). With mean clutch size 6 the simulated population again exploded for additional mortalities of 5% and 10%. With increased mortality of 20%, although the simulated populations remained just above zero size in 13 out of 100 cases, the remaining 87 populations died out in mean time 67 years (standard error 18). Since the largest clutch size observed b y V a n C a m p and H e n n y (1975) in their study area was 6, it seems unlikely that the m e a n clutch size could be increased to this level in a real population. The indication is, then, that an

250 TABLE 6 Summary of the approximate changes in parameter values necessary, when parameter changes are considered one at a time, to stabilise the population. Results are given for a range of values of the additional mortality factor, and to the nearest value used in the simulation Additional mortality factor

Required proportion of first years breeding

Required juvenile survivalrate

Required mean clutch size

1.00 1.05 1.10 1.20 1.30 1.40

0.8 Not possible Not possible Not possible Not possible Not possible

0.305 Between0.305 and 0.4 0.4 0.5 0.6 Greater than than 0.6

3.06 Between3.06 and 4 Between 4 and 5 Greater than 6 (infeasible) Infeasible Infeasible

increase in mean clutch size alone could not compensate for additional mortality of 20% or more. If the additional mortality were only 10%, then the mean clutch size would need to be increased by a factor of about 1.5 to achieve population stability, while for additional mortality of only 5% the necessary mean clutch size for stability is less than 4. Of course, these results arise from simulations using a very special (simple) form of distribution for clutch size, but it is hoped that they give a reasonable indication of what might be expected from other clutch size distributions as well. The findings here are certainly in close agreement with those of N o r t h (1985). A summary of the requirements for change of population parameters, one at a time, to yield stability in the population is given in Table 6. Following N o r t h (1985) we now consider the effects of varying population parameter values in a pairwise manner. 7. EFFECT OF COMPENSATION FOR ADDITIONAL MORTALITY BY CHANGING TWO VARIABLES, TOGETHER 7.1. Increase in both the proportion of first-year birds breeding and the mean clutch size

Throughout this section the first-year survival rate, So, remains fixed at So = 0.305. We consider the compensatory effect of an increase in the proportion of first-year birds breeding from 0.8 to 1, combined with increases in the mean clutch size, for various additional mortality factors. With an additional mortality factor of only 1.05, a mean clutch size of 3.5, or even rather less, combined with the increased proportion of first-year birds breeding, is sufficient to cause the population to increase. With

251

additional mortality factor 1.10, a mean clutch size of 4 stabilised the population (mean size after 100 years = 220, standard error = 134). H o w ever, 4% of the populations simulated in this way were wiped out, or nearly so, within 100 years. A further increase in the mean clutch size caused the populations to explode. With an additional mortality factor of 1.20 a mean clutch size of 5 was insufficient to stabilise the population. In 98% of the simulations the populations had died out within 100 years, the mean time to extinction being 46 years (standard e r r o r - - 5 ) . A further increase in the additional mortality factor, to 1.30, with the mean clutch size remaining at 5, resulted in quick extinction of the simulated populations (mean time = 18 years, standard error = 4). These results indicate that even if all first-year birds were to breed, it would not be feasible for increased mean clutch size to stabilise the population, if mortality were increased by 20% or more.

7.2. Increase in both the proportion of first-year birds breeding and the juvenile survival rate Throughout this section we retain the clutch size probabilities introduced in Section 3, and we also consider throughout an increase in the proportion of first year birds breeding from 0.8 to 1. These are combined with increased juvenile survival rates to compensate for various amounts of additional mortality. With an additional mortality factor of 1.10, an increase in the juvenile survival rate from 0.305 to 0.35 is insufficient to stabilise the population. 28 out of 100 simulated populations became extinct within 100 years (mean time = 76 years, median = 79, minimum = 41, maximum = 100), and all the remaining 72 simulated populations were in decline (mean size = 53.1, median = 40, minimum = 2, maximum = 180). A further increase in the juvenile survival rate to 0.4 was enough to stabilise the population but, as we have seen from Table 6, this is true even without all first-year birds breeding. With an additional mortality factor of 1.20, but with increased juvenile survival of 0.4, 96 out of 100 simulated populations became extinct within 100 years, in mean time 57 years (median 52, minimum 27, m a x i m u m 100). The four remaining populations ranged in size from 2 to 38 birds after 100 years (mean = 16.5, median = 10). When the juvenile survival rate was increased to 0.45 the simulated populations grew in size, on average, although in about 10% of the cases, the populations became extinct within 100 years.

252 TABLE 7 Approximate values of population parameters necessary to stabilise the population under varying amounts of additional mortality. Changes summarised here are one parameter at a time, having also increased the proportion of first-year birds breeding to 1.0 Additional mortality factor

Required juvenile survival rate

Required mean clutch size

1.00 1.05 1.10 1.20 1.30 1.40

0.305 Greater than 0.305 Between 0.35 and 0.4 Between 0.4 and 0.45 Greater than 0.5 Greater than 0.55

3.06 Between 3.06 and 3.5 4 Greater than 5 (infeasible) Infeasible Infeasible

With an additional mortality factor of 1.3, even with the juvenile survival rate increased to 0.45 all 100 simulated populations became extinct in times ranging from 18 to 100 years (mean = 43 years, median = 40). Even with a juvenile survival rate of 0.5, 14 out of 100 simulated populations became extinct in times ranging from 53 to 92 years (mean = 74 years, median = 69), and most of the remaining 86 populations were in decline, the mean size after 100 years being 77.5 (median = 56, m i n i m u m = 2, m a x i m u m = 318). With additional mortality factor 1.4 and a juvenile survival rate of 0.5, the population became extinct in all 100 simulations, in times ranging from 16 to 89 years (mean = 38, median = 35). When the survival rate was increased to 0.55, 57 out of 100 simulated populations still became extinct, in times ranging from 39 to 99 years (mean = 75, median = 78). The other 43 simulated populations were also in decline, the population sizes after 100 years ranging 4 to 142 birds (mean = 36.4, median = 24). These results are well in accordance with those of North (1985). The results of the last two sections are summarised in Table 7. For comparison, simulations have also been run with the changed initial age structure in the population used earlier. Although the results are broadly similar, populations simulated in this way tended to die out rather sooner, in the cases where extinction prevailed. 7. 3. Increase in both the juvenile survival rate and the mean clutch size

With additional mortality of only 5%, increases in the juvenile survival rate from 0.305 to 0.35 and in mean clutch size from 3.06 to 3.5 resulted in population explosion. When the additional mortality is 10%, the same increases failed to prevent declining populations, and extinction, or near extinction, within 100 years in about 20% of the simulated populations.

253

However, when the mean clutch size was increased to 4 the populations exploded once more. With additional mortality of 20% and mean clutch size of 3.5, even a juvenile survival rate of 0.4 did not prevent a declining population, extinction occurring within 100 years in about 90% of the populations simulated. But an increase in the mean clutch size to 4.5 led to population explosion again. With additional mortality of 30% a juvenile survival rate of 0.4, combined with a mean clutch size of 5, did not prevent the population becoming extinct within 100 years in over 80% of the simulations. However, a juvenile survival rate of 0.5 and a mean clutch size of 4.5 led to population explosion. This also occurred with 40% additional mortality, when the juvenile survival rate was 0.5 and the mean clutch size was 5. 8. MODELS IN W H I C H ONLY PART OF T H E TOTAL P O P U L A T I O N IS A F F E C T E D BY A D D I T I O N A L MORTALITY

So far we have examined in some detail how a closed population might behave under the pressure of additional mortality. In this section, following North (1985) we consider the more realistic situation (in relation to the real Screech Owl population we envisage, where additional mortality is only experienced in one habitat of its range, namely orchards) where only a part of the total population is affected by the additional mortality, and where movement of birds from one part of the population to the other may occur. In the models that we adopt, movement of young birds only is allowed (see North, 1985 and VanCamp and Henny, 1975, for justification and extended discussion). Movement is also only from the unaffected part of the population to the affected part. Initially we will consider a simple immigration pattern in which a constant number of birds in each generation move from the unaffected to the affected part of the population. In the second model we consider a possibly more realistic situation though still, necessarily, a simplification in which we assume that sufficient birds enter the affected population each year to maintain a constant total of that part. This is the simplest possible form of density dependent dispersion, but it does have some practical justification. It is reasonable to suppose that the affected population occupies the optimal habitat for the species (that is, the orchard areas), and that these areas support the largest possible number of owl territories at all times. In the simulations of populations described by these models, the alternative initial age structure in the population has been used, in an attempt to ensure that we are conservative about our predictions of the owls' ability to overcome additional mortality, i.e. to try to ensure that we are not falsely optimistic about the population's development. We also consider just the

254 TABLE 8 Some examples of the effect of a constant dispersion rate on the affected and unaffected (by additional mortality) parts of the total population. (I) Affected population size initially 1% of size of unaffected population. (Initial unaffected population size 1000; 100 simulations) Additional mortality factor

Dispersion (d) females

Affected part of the population over 100 year period

Unaffected part of the population over 100 year period

1.10

0

Became extinct in all simulations in mean time 15 years

Increased to mean size 1125 (s.e. 18.7) until the affected population died out

1.10

1

Increased to mean size 18 (s.e. 0.9)

Increased to mean size 1710 (s,e. 49.7)

Both parts of the population increased together in 82% of the simulations 1.20

0

Became extinct in all simulations in mean time 9 years

Increased to mean size 1074 (s.e. 12.7) until the affected population died out

1.20

1

Decreased to mean size 8 (s.e. 0.4)

Incresed to mean size 1750 (s.e. 60.3)

Both parts of the population increased together in 22% of the simulations 1.20

2

Increased to mean size 17 (s.e. 0.6)

Increased to mean size 1502 (s.e. 53.4)

Both parts of the population increased together in 75% of the simulations 1.30

0

Became extinct in all simulations in mean time 6 years

Increased to mean size 1028 (s,e. 8.2), until the affected population died out

1.30

1

Decreased to mean size 6 (s.e. 0.3)

Increased to mean size 1774 (s.e. 56.6)

Both parts of the population increased together in 3% of the simulations 1.30

2

Increased to mean of 11.4 (s.e. 0.4)

Increased to mean size 1595 (s.e. 50.5)

Both parts of the population increased together in 41% of the simulations

255 TABLE 9 Some examples of the effect of a constant dispersion rate on the affected and unaffected (b.y additional mortality) parts of the total population. (II) Affected population size initially 10% of size of affected population. (Initial unaffected populations size 1000; 100 simulations) Additional mortality factor 1.10

Dispersion (d) females 5

Affected part of the population over 100 year period

Unaffected part of the population over 100 year period

Declined slowly, on average, to a size of mean 84 (s.e. 22). Increased in 26% of the simulations

Declined slowly, on average to a mean size of 84 (s.e. 571). Increased in 38% of the simulations. Became extinct in 4%

Both parts of the population increased together in 12% of the simulations 1.10

6

Almost constant, reaching a size of 97 (s.e. = 31). Increased in 48% of the simulations

Approximately halved, on average, to a mean size of 460 (s.e. 415). Increased in 14% of the simulations, extinct in 12%

Both parts of the population increased together in 10% of the simulations 1.10

9

Increases slightly on average until the unaffected part of the population became extinct

Became extinct in 90% of the simulations

1.20

10

Decreased by about a half, on average, until the unaffected part of the population died out

Became extinct in 88% of the simulations

1.20

13

Decreased by about a half, on average, until the unaffected part of the population died out

Became extinct in all simulations in mean time 57 years (s.e. = 9)

1.20

17

Mean size when the unaffected part of the population died out = 69

Became extinct in all simulations in mean time 44 years (s.e. = 6)

1.30

14

Mean size when the unaffected part of the population died out = 33

Became extinct in all simulations in mean time 55 years (s.e. = 10) To be continued

256

long term behaviour of the population by simulating the population's behaviour over runs of 100 years.

8.1. Constant immigration / emigration We now consider an unaffected population initially of 1000 birds (500 female, 500 male) and, in addition, consider affected populations of various sizes relative to the unaffected part. The initial percentage sizes of the affected populations to be considered, relative to the unaffected part, are 1%, 10% and 20%. Each generation a constant number, d, of first-year females move from the unaffected to the affected population. The total number of birds moving is therefore 2d (d males, d females) each year. If the unaffected population contains less than the required number of first-year birds, then all of the first-year birds are transferred to the affected population. At the lowest percentage size considered for the affected part of the population, as we would expect, very small numbers of birds migrating each year are sufficient to stabilise the population, for additional mortality factors up to 1.3. Further details are given in Table 8. When the initial affected population size was 10% of the unaffected population size, and was affected by additional mortality of 10%, it was found that six females per year moving between populations were enough to keep the affected population at an average level of just slightly less than the initial population. However, the effect on the unaffected population was to reduce it to less than half the initial size, on average, in the 100 year period, and in 12% of the simulations it became extinct. In contrast, there was a 1 in 10 chance, on the evidence of the simulation, that both populations would increase. When only five females per year moved both populations declined

T A B L E 9 (continued) Additional mortality factor

Dispersion (d) females

Affected part of the p o p u lation over 100 year period

U n a f f e c t e d p a r t of the p o p u l a t i o n over 100 year period

1.30

18

M e a n size w h e n the unaffected part of the population died out = 44

Became extinct in simulations in m e a n time 44 years (s.e. = 6)

1.30

27

M e a n size w h e n the unaffected part of the population died out = 65

Became extinct in all simulations in m e a n time 33 years (s.e. = 4)

257 T A B L E 10 Some example of the effect of a c o n s t a n t dispersion rate o n the affected a n d unaffected (by additional mortality) parts of the total population. (III) Affected p o p u l a t i o n size initially 20% of size of the unaffected population. (Initial unaffected p o p u l a t i o n size = 1000; 100 simulations) Additional mortality factor

Dispersion (d) (females)

Affected part of the population over a 100 year period

U n a f f e c t e d part of the p o p u l a t i o n over a 100 year period

1.10

9

Halved, on average, to a m e a n size of 111 (s.c. = 38). W h e n the unaffected part of the p o p u l a t i o n became extinct

Became extinct in 80% of the simulations

1.10

12

Decreased, on average, to a m e a n of 130, until the size of the unaffected part of the p o p u l a t i o n died out

Became extinct in all simulations in m e a n time 63 years

1.10

18

Decreased slightly o n average to a m e a n size of 185 until the unaffected part of the p o p u l a t i o n died out

Became extinct in all simulations in m e a n time 44 years

1.20

19

Decreased o n average to a m e a n size of 79 until the unaffected part of the p o p u l a t i o n died out

Became extinct in all simulations in m e a n time 41 years

1.20

25

Halved, o n average, to a m e a n size of 101 until the unaffected part of the p o p u l a t i o n died out

Became extinct in all simulations in m e a n time 33 years

1.20

35

Decreased slowly, on average, to a m e a n size of 154 until the unaffected part of the p o p u l a t i o n died out

Became extinct in all simulations in m e a n time 24 years

1.30

27

M e a n size w h e n the unaffected part of the population died out = 72

Became extinct in all simulations in m e a n time 32 years

1.30

35

M e a n size w h e n the unaffected part of the population died out = 91

Became extinct in all simulations in m e a n time 27 years

1.30

35

M e a n size w h e n the unaffected part of the population died out = 136

Became extinct in all simulations in m e a n time 22 years

258 very slowly on average, but there was only a 4% chance of extinction, of the unaffected population within 100 years, and both populations increased in 12% of the simulations. Higher rates of dispersion wiped out the unaffected population in most of the simulations. In reality extinction of the unaffected population would seem unlikely since a reduced population would result in less pressure for juveniles to move away to join the affected population. As suggested by one of the referees, if, for example, suboptimal habitat is good enough to supply emigrants, then it is probably good enough to keep and attract birds. In this case it is unlikely that it would be 'sucked dry' of birds. Thus in Section 8.2 we will consider density-dependent population movements. With the additional mortality factors of 1.20 and 1.30 it became impossible to maintain the affected population without rapidly wiping out the unaffected population. A more detailed summary of results is given in Table 9. With the largest size of affected population considered here, additional mortality rates of 1.10 and higher caused almost certain extinction of the unaffected population when the rate of dispersion was high enough to maintain the affected population. Results are summarised in Table 10.

8.2. Density-dependent immigration/ emigration Again we consider an initial unaffected (by the additional mortality) population of 1000 birds (500 male, 500 female) with initial affected popula-

TABLE 11 Some examples of the effect of density dependent dispersion on the unaffected part of the population. (I) Affected population size initially 1% of unaffected population size. (Initial unaffected population size = 1000; 100 simulations) Additional mortafity factor

Unaffected part of the population over a 100 year period

1.10

Increased to mean size of 1976.29 (s.e. = 70.91). Decreased in 3% of the simulations to a minimum value of 680. (The affected population became extinct in 9% of the simulations after a mean time of 50 years)

1.20

Increased to mean size of 1797.11 (s.e. = 58.74). Decreased in 4% of the simulations to a minimum value of 878. (The affected population became extinct in 14% of the simulations after a mean time of 50 years)

1.30

Increased to mean size of 1653.79 (s.e. = 74.2). Decreased in 9% of the simulations to a minimum value of 308. (The affected population became extinct in 22% of the simulations after a mean time of 51 years)

259 tions of sizes equal to 1%, 10% and 20% of this. As far as it is possible, the affected part of the population is maintained at its initial level by an influx of first-year birds from the unaffected part of the population. If there are too few first-year birds in the unaffected part of the population to maintain the affected part then all of the first-year birds are transferred to the affected part. With the initial affected population size only 1% of the unaffected population size, and with this dispersion process operating, the mean size of the unaffected population increased for all additional mortality factors up to 1.3. However, at this highest level of mortality, 9% of the simulations of the unaffected population decreased over the 100 year period. Even with an additional mortality factor of only 1.1, 9% of the simulations resulted in the affected part of the population becoming extinct within 100 years (in mean time 50 years), and at the highest level of additional mortality considered (factor = 1.3), this happened in 22% of the simulations (in mean time 51 years). For further details, see Table 11. When the initial population size is 10% of the unaffected population size the unaffected part of the population grew in nearly three quarters of the simulations, over a period of 100 years, and with an additional mortality factor of 1.05. However, with an additional mortality factor of 1.10 the population increased in only about one quarter of the simulations, and the mean levels of the unaffected part of the population after 100 generations was only about two-thirds of its initial level. Additional mortality rates of 1.20 and 1.30 caused extinction of the unaffected part of the population in most of the 100 year simulations. See Table 12 for further details.

TABLE 12 Some examples of the effect of density-dependent dispersion on the unaffected part of the population. (II) Affected population size initially 10% of unaffected population size (Initial unaffected population size = 1000; 100 simulations) Additional mortality factor

Unaffected part of the population over a 100 year period

1.05

Increased on average to a mean size of 1284 (s.e. = 489). Decreased in 26% of the simulations to a minimum value of 308

1.10

Decreased on average to a mean size of 591 (s.e. = 431). Increased in 26% and became extinct in 2% of the simulations

1.20

Decreased in all simulations and became extinct in 75% of the simulations

1.30

Became extinct in all simulations in mean time 57 years (s.e. = 10)

260 TABLE 13 Some examples of the effect of density-dependent dispersion on the unaffected part of the population. (III) Affected population size initially 20% of unaffected population size (Initial unaffected population size = 1000; 100 simulations) Additional mortality factor

Unaffected part of the population over a 100 year period

1.05

Decreased slightly on average to a mean size of 654 (s.e. = 524). Became extinct in 8% of the simulations

1.10

Increased in 36% of the simulations. Became extinct (or nearly so) in 94% of the simulations in a mean time of 70 years (s.e. = 14)

1.20

Became extinct in all simulations in mean time of 41 years (s.e. = 4)

1.30

Became extinct in all simulations in mean time of 32 years (s.e. = 4)

With the largest size of affected population considered here, even an extra mortality of 1.05 caused the unaffected part of the population to decrease on average (although it did increase in about one third of the simulations), and additional mortality factors of 1.10 and higher caused almost certain extinction. See Table 13 for further details. 9. CONCLUSION This paper complements the earlier one by N o r t h (1985), in which a deterministic modelling view of the Screech Owl population dynamics was presented. Here we have seen the corresponding stochastic modelling view of the population development. This is important, especially in the real life situation where the aim is to assess the hazards to the species imposed by externally imposed environmental stresses. In this way we are able to see that even with populations whose structure dictates that, on average, they will grow, there may be a non-negligible probability of extinction within a moderate time-span. The value of the stochastic models presented here has been in enabling a detailed analysis of the population to be presented, in which the population parameters are varied and aspects such as time to extinction considered. The models allow the earlier deterministic results to be viewed in a much fuller context, indicating more clearly the full range of possible outcomes for the population. These are important, if the possibilities for development of an actual population are being considered. The results presented here also indicate that a simple census of a population after a few years may not reveal that it is on the way to irreversible decline. Detailed study of the age structure and population parameters would be

261

necessary to give such information with any confidence. This conforms with what is already known from deterministic life history studies in small or moderate sized populations. If the results of this study appear to indicate a lack of resilience of the modelled Screech Owl population to increased mortality, it should be remembered that only the simplest forms of density dependence have been incorporated in the models studied here. It is quite possible that although direct evidence of density dependent regulation was not found in the available data, this may occur in a real Screech Owl population, leading to possibly different indications of the birds' resilience to additional mortality. ACKNOWLEDGEMENTS

Computational assistance from Simon Ashberry is gratefully acknowledged. We are grateful to three referees for their comments on an earlier version of this paper. Part of this work was supported financially by ICI plc, and we are grateful to Richard Brown for his encouragement to pursue this study, which originated in earlier discussions with ICI plc. APPENDIX

Equivalence of the deterministic and stochastic models of the owl population The stochastic model discussed here is equivalent in the mean to the deterministic~model of North (1985). This is shown as follows. Each generation in the stochastic model commences immediately prior to breeding, so let MAt and Mjt be random variables representing the adult (2nd year and older) and juvenile (1st year) populations at generation t. We also define the following random variables, the index i ranging over individuals in the population: Pi =J with probabilityp7 RNi =

fl

R°i---{10

with probability S N otherwise withpr°babilityS°otherwise

Rli = { 10 with otherwise pr°bability S1 R~ ={10

otherwiseWithpr°babilityS

Bi = ( ;

withprobabilitypBotherwise

262 The stochastic model is then given by: MA, Mj, MAt+I = E R i + E R x i i=1 i=1 that is (Part of old adult (Part of old juvenile population which + population which survives to become survives) adult)

(New adult population) and:

MA~( Mjt+l = E i=1 that is

j•_IROjRNj

(New juvenile population)

+ Y~ Bi i=1

RojRNj

(Juveniles produced by (Juveniles produced part of old juvenile + population which by adult population) breeds)

Taking expectations and using the result that if ( X k } is a sequence of mutually independent binomial random variables, and N is a random variable taking finitely many integer values, then the expectation of E,=lXk u is E(N) E(Xk). (See Feller, 1986, p. 286.) Then UAt+l = NAtS +NjtS 1

(3a)

Njt+l -~-NAtaps N g(ei) --~NjtPBapS N E( Pi)

= NAt& + Njtfj

(3b) Equations (3a) and (3b) are identical to those given by North (1985) in the deterministic model, and thus to demonstrate equivalence we need only show that the parameters of the stochastic model take the same values. S, S 1, PB have been assigned the same values as in the deterministic model, and

fA = SoSN~P? j = 0.305 × 0.858 × 1.53 = 0.400 J fJ =PBfA in the stochastic model. These values are again as given in the deterministic model (North, 1985, p. 116, gives fA = 0.395, the difference in the third decimal place being due to rounding, error in the PT') Hence the stochastic model is shown to be equivalent, in the mean, to the deterministic model.

263 REFERENCES Cooke, D. and Leon, J.A., 1976. Stability in population growth determined by 2 x 2 Leslie matrix with density-dependent elements. Biometrics, 32: 435-442. Feller, W., 1986. An Introduction to Probability Theory and its Applications (3rd Edition), Vol. 1. Wiley, New York, 509 pp. Grant, W.E., Fraser, S.O. and Isakson, K.G., 1984. Effect of vertebrate pesticides on non-target wildlife populations: evaluation through modelling. Ecol. Modelling, 21: 85-108. Leslie, P.H., 1945. On the use of matrices in certain population mathematics. Biometrika, 33: 183-212. Mayfield, H., 1961. Nesting success calculated from exposure. Wilson Bull., 73: 255-261. North, P.M., 1985. A computer modelling study of the population dynamics of the Screech Owl (Otus asio). Ecol. Modelling, 30: 105-143. Smies, M., 1983a. Simulation of small bird populations. I. Development of a stochastic model. Ecol. Modelling, 20: 259-277. Smies, M., 1983b. Simulation of small bird populations. II. Reduced breeding in two British raptor species. Ecol. Modelling, 20: 279-296. Sparks, J. and Soper, T., 1970. Owls. Their Natural and Unnatural History. Seventh impression 1978. David and Charles, Newton Abbot, 206 pp. Usher, M.B., 1972. Developments in the Leslie matrix model. In: J.N.R. Jeffers (Editor), Mathematical Models in Ecology. Blackwell, London, pp. 29-60. VanCamp, L.F. and Henny, C.J., 1975. The Screech Owl, its life history and population ecology in Northern Ohio. Rep. No. 71, North American Fauna, United States Department of the Interior Fish and Wildlife Service, 65 pp.