Initial Population

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Ecologic~ulModelling, 70 1993) 51 -61

Elsevicr Sciencc Publishcrs B.V., Amsterdam

iscounting initial population sizes for prediction

of extinction probabilities in patchy environments

Jianguo Liu

Inslilute of E cology, Unir ersity of Georgia, Athe ns , G 4 30602 U S A

Received 24 Fcbruary 1991; acceptcd 11 D ccc m be r 1992)

ABSTRACT

Liu, J., 1993. Discounting initial population sizes for prediction of extinction probabilities in

patchy environments. Ecol. Modelling, 70: 51-61.

Extinction is a major concern in conscrvation. A most urgcnt necd is to prcdict thc

rclationship of a population's initial sizc to its probability of extinction. Previous work has

led to a widcly acceptcd conclusion that thc larger an initial population size. thc lcss likcly

th e population will go cxtinct. used a spatially-explicit simulation m ode l to investigatcextinction probabilities of Bachman's Sparrow (Airnophila aeslicw1i.c.) in patchy cnviron-

ments. Contrary to the conventional wisdom, I found that the rclationship between extinc-

tion probabilities a nd initial population sizcs of thc spar row s was not correlated when initial

individuals wcre in scveral patch types. T o makc good prcdictio ns of th e sparro ws'

extinction ra tes, I have suggested discounting models which incorporated initial population

sizes and initial spatial distributions. The models discounted initial population sizcs on thc

basis of patch characteristics (patch suitability, timing of patch suitability, and duration of

patch suitability). As a result, the extinction probabilities dccreased with the logarithm of

discounted p opulatio n sizes. T h c discounting mod cls may havc implications for quantitativ epredictions of cxtinction chances of other spccies, sincc most cnvironmcnts are patchy or

spatially-subdivided. T h e discounting ap proach may bc also uscful for cvaluating im pacts of

patchy environments on population dynamics and community s tructu re .

I N T R O D U C T I O N

Initial population size is the number of individuals of a population at the

beginning of experiments or simulation studies. It is one of the most

important factors in determining ecological consequences such as competi-

tive outcomes Hutchinson, 1978) and population extinction probability.

Correspondence to (present address): J Liu, Harvard Institute for International Dcvelop-

mcnt. H aw ard Universi ty , 1 Eliot Street . Cam bridgc, M A 02138, US A.

0304-3800/93/ 06.00 993 Elscvier Scicnce Publishcrs B.V. All rights reservcd



52 L U

For prediction of population extinction probabilities, past investigators

have foun d t ha t th e high er initial popu lation sizes, the lower the extinction

probabili t ies are (Shaffer, 1981; Shaffer and Samson, 1985; Harris et al . ,

1987; H arri son et al. , 1988). This conclusion has been draw n fro m theoreti-

cal and simulation models which did not explicitly recognize environmentalheterogeneity (e.g. , Shaffer, 1978). Because most environments are patchy

o r spatially-subdivided (Levin, 1976; W iens, 1976; Kareiva, 1990; Pacala et

al., 1990; Hassell et al. , 1991), a qu estio n to t he p oint is to ask wh ethe r this

con ven tional con clusio n still hold s in a spatially-explicit con text. If n ot, how

can population extinction probabili t ies be predicted? To answer the first

question, I have developed a spatially-explicit model which could simulate

th e po pulation dynamics and extinction chances of Bachman s Sparro w in

response to changes of patchy forest s tructure in a region managed for

timber production (Liu, 1992). I will try to answer the second question by

proposing discounting models which distinguished and discounted the

contributions of various patches to population persistence.

T H E S PAT IAL L Y-E XP LIC IT S IM UL AT ION M O DE L

Bachman s Spa rrow is a possibly threa ten ed o r en dan ge red species(Pulliam et al., 1992). Its range has declined significantly since the 1930s

(Haggerty, 1986). The sparrows breed in various habitats including patchy

pine fo rests of th e sou the aste rn Unit ed States, which a re mostly mosaics of

even-aged stands. The pine forests are usually harvested at 20-60-year

cycles, depending on desired t imber products. Reproductive success of

Bach ma n s Spa rrow s differs in various stands. In m atu re 2 0 years) and

1-2-year fore st stands , a pair of adu lts pro duc e 3.0 offspring a year,

co m pa red t o 1.0 offspring in 3-5-year-old stands. No o ffspring ar e pro-duced in other age classes and clear-cuts (0-year stands following harvest-

ing) (Pulliam e t al., 1992).

In all s imulations, the hypothetical landscape for the sparrows to breed

was a pin e forest of 1000 ha (Fig. 1). I divided th e forest into a tw o-dime n-

sional array of 20*20 hexagonal cells. Each cell was 2.5 ha, which is the

averag e terri tory size for a pair of Bachman s Sparrow. I assumed tha t ea ch

forest s tand was as large as an array of adjacent 2 * 2 cells (10 ha). In th efores t , the re were two matu re s tands and the remaining s tands were

assigned with equ al prob ability as 0-19-year age classes. All sta nd s w ere

randomly dis tr ibuted. I further assumed that t he m ature s tan ds were never

harvested, bu t the remaining s tand s we re m anaged in rotat ions of 20 years

each.

Sparrow population sizes at the beginning of simulations were propor-

tional to th e nu m be r of 1-5-year patches and ma tur e patches. T h e spatial



PKEDIC I IONS OF EXI INC I ION P KOR AR I I . I S I ES 5

Fig. 1.A sam ple landscap e from th e spatially-explicit simulation mo del. E ach hexagonal cell

represents 2.5 ha of pine woodlands which equals a sparrow territory s izc . Nu mb er in each

cell is the age of pines.

distributions of initial individuals within each patch type were random.

With data from field studies and a literature survey, adult and juvenile

survivorships have been estimated as 0.60 and 0.40 respectively in all

patches (Pulliam et al., 1992). For simplicity, I just considered a single sex

(female) in the simulation model because the sex ratio is 1 :1 (Pulliam et

al., 1992). If the parents died, one juvenile stayed in its natal patch, but

other juveniles had to search for new territories. If there were suitableneighboring patches (mature or l-5-year stands), a juvenile settled in one

of them. Otherwise a juvenile moved into any of the adjacent patches until

it found an unoccupied suitable patch, or until it died. The dispersal

survivorship was assumed to be 0.90. The forest boundary was constructed

to be reflective. When a juvenile reached a forest edge, it moved back to

the forest. If a breeding patch became unsuitable, an adult might stay there

unless there was one or more suitable neighbors.



The simulat ion model was coded in Borland C 2.0 (an object-ori-

ented programming language) and implemented in a Zeni th 386 computer

with a math coprocessor . Populat ion dynamics were s imulated for 100

years. If no individuals existed before or in the last simulation year, the

popu lat ion was con sidered ext inct. Ea ch s imulation run had 100 replicates .The extinction probabil i ty was calculated as the rat io of extinction fre-

quency over total replicates (Harris et al . , 1987).

SIMULATION RESULTS: EFFECTS O F HABITAT HETEROGENEITY ON THE

RELATIONSHIP BETWEEN POPULATION SIZE AND EXTINCTION PROBABIL

ITY

Tradit ionally, p op ulation size is counte d as th e total of individuals in al lpatches. In my simulation model, individuals in mature patches and 1-2-

year patches produced more of fspr ing than those in o ther patches . The

tradit ional concept of population size ignores suitabil i ty differences among

patches , so i t may be ca lled nominal popula t ion size. T o und ers tand the

consequences of disregarding the variance in patch suitabil i ty, I have done

simulations with init ial individuals in one patch type and in several patch

types. T h e simu lation results have show n th at if all individuals w ere initially

in one patch type, the relat ionship of nominal ini t ial population sizes toextinction rates was significantly correlated. For example, when init ial

individuals were only in mature patches, the Spearman rank correlat ion

coefficient r -0.94 ( n 11, P 0.01). This is consistent with the con-

ventional theory that higher init ial population sizes result in lower chances

of extinction. If individuals were initially in several types of patches,

however , the re was n o correlat ion betwee n nominal init ial populat ion s izes

and extinction probabil i t ies (see Fig. 2, Spearman rank correlat ion coeffi-cient r -0.1182, n 11 P >> 0.05).

The above resul ts are due to the differences in spat ial dis t r ibut ions of

init ial populations. In the case that individuals init ial ly were in one patch

type, the init ial population was made up of individuals in patches with the

sa m e rep rodu ctive success. Wh en individuals initial ly w ere in several types

of patches, however, the init ial population was composed of individuals

with different reproductive success, and individuals in poor patches (with

low reproduct ive success) were t reated the same as those in good patches(with high reproductive success). The discounting models below were

attempts to differentiate individuals in various patches.

THE DISCOUNTING MODELS

Population persistence may be influenced by three patch characterist ics:

patch suitabil i ty (how suitable is a patch?) (Pull iam, 1988; Pull iam and



I R E D I CI I O N S OF t N T I N C I I O N P R O B AB I I. I I l E S

ominal initial population size

Fig. 2. T h e relationship bctw een cstinction probabilities and nom inal initial populatio n sizesof B ach m an s Spa rrow in patchy cnvironrncn ts w a s not correla tcd a t a l l . Notc that the

fitting curve was almost parallel to X-axis.

D an iels on , 1991; Pulliam e t al. , 1992), timing of pa tch suitability (w hen is a

patch suitable?) (Liu, 1992), an d du ration of patc h suitabili ty (how long is a

patch suitable?) (Liu, 1992; Pull iam et al . , 1992). In my simulation model,

patch sui tabi l i ty was measured in terms of reproduct ive success because I

assum ed tha t al l ot he r condi tions were t h e sam e for individuals in different

patches except that reprodu ct ive success varied.

propo sed a discount ing ap pro ach to d ifferent iate individuals in var ious

patches. That is , individuals in poorer patches were discounted as equiva-

lents to a cer tain number of individuals in the best patches so that al l

individuals after discounting were essential ly the same. In the following, I

wi l l demonst ra te the discount ing models and procedures . S teps I a n d 2

below will integrate temporal characterist ics of patches (t iming of patchsuitability and duration of patch suitability), step 3 will incorporate spatial

difference in patch suitabil i ty, step 4 will calculate discounted population

s izes in o ne patch, an d s tep 5 will co m pu te discoun ted p opulat ion s izes in

all patches.

Step I To differentiate contributions of t iming of patch suitabil i ty to

populat ion persis tence, I proposed a model Eq. 1) to d i scount reproduc-

tive success in patch i at t ime t .

wh ere D B (i , t ) is the discounted reprodu ct ive success of patch i a t t ime t

B(i , t ) is th e nominal repro duct ive success in patch i a t t ime t , an d T( i , t ) is

the nu m ber of years tha t pa tch i ha s been unsu i table at t im e t s ince its f irst



I LIC

Time (year)

Fig. 3 Pcriod ic dynamics of rep roductiv e success of Bachm an s Sp arrow in an initial 1-ycar

patch d uring a period of 100ycars (five forest rotations). The dash linc . . ) indicates the

nominal reproductive success, while thc solid line ---- ) refers to the discounted repro-

ductive success. In cach rotation, thc patch was suitable for the first five years. During the

first rotation, the discounted rcproductivc succcss was equal to the nominal reproductive

success. For th e last four rotations, the discountcd reproduction success beca m e smaller an d

smaller .

time of being suitable. I discounted th e nominal rep roduc tive success based

on two facts: (1) T h e longer the uninhabitable t ime period, the less chance

that an individual can survive; (2) Even if an unsuitable patch becomes

suitable later, the chance for the newly suitable patch to be occupied is

lower than a continuously suitable patch (Liu, 1992). I will give an example

to show how this m odel works. F or an init ial l-ye ar patch (Fig. 3), th e first

five years in each rotation were continuously suitable, the remaining 15

years were not ( no offspring could be produced). Th erefore , ther e was nodiscount of reproductive success for initial 1-year patches in the first five

years of the first rotation, while the discounted reproductive success during

th e first five years of the seco nd, the th ird, the fo urth and th e fifth rotation

was 1/1 6, 1/31, 1/ 46 an d 1/6 1 of the nom inal reprodu ctive success

respectively because th e patche s were suitable again afte r being un suitable

for 15 , 30, 45 a nd 60 years.

Step 2 To consider the duration of patch suitability, I added up d is -counted reproductive success (Eq. 2) for the t ime period of interest .

r

C D B , D B ( i , t )0

(2 )

where CDB, is the cumulative discounted reproductive success in patch i ,

DB(i, t) is the discounted reproductive success in patch i at t ime t and m

is the last time interval in simulations.



P R E D I C T I O N S O F E X T I NC T IO N P R O R A R l l . l l I t 5 57

TABLE 1

Relative cumulative reproductive success (R C D B ,) in patches of different agc classes

Init ial agc of patches RC DB ,

(year)

1 339

2 0.024

3 0.0142

4 0.0109

5 0.0077

Matu r e > 80) 1.0000

Step 3 In order to compare sui tabi l i ty of one patch with that of other

patches, I calculated relat ive cumulative discounted reproductive success

with maximum cumulat ive discounted reproduct ive success as a basel ine

(E q. 3).

C D B ,R D C B ,

CDBm,ix

RCDB, is the relat ive cumulat ive discounted reproduct ive success (or

relative suitability) in patch i; CDB,,, is th e maximum cum ula tive dis-

cou nte d reproductive success am ong all patches. T ha t is , CDB,,, , M A X

( C D B , . C D B , , . ., C D B , , . , CD B, , ), where r i s the number of patches.

T h e relat ive abili ty rang es from 0 to 1 As presented in Table 1, a

m atu re patch had relat ive suitabil ity of 1 . Th is is bec aus e i t was always

suitable an d ha d high nom inal reproductive success. A n init ial 1-year patc h

had lower relative suitability because of three reasons: 1) It was suitable

for only five years and unsuitable for f if teen years in each rotation (recall

that each forest rotat ion was 20 years long). (2) Although i ts nominal

reproductive success durin g the first two years of eac h ro tat ion was as high

as th at of a m atu re patch, the nominal reproductive success during th e next

thre e years was just one-third t he nominal repro duct ive success of a m atur e

patc h. (3) All of the nom inal reprodu ctive su ccess was disc ou nted except

for the first five years of the first rotation. An initial 2-, 3-, 4-, and 5-yearpatch had even lower relat ive suitabil i ty (Table 1). Besides the reasons (1)

an d (2) me nt ioned above. the nominal reproduct ive success was discou nted

earl ier because an older ini t ial patch became unsui table sooner .

Step 4 A nominal population size in a patch was discounted according to

relative suitability of the patch (Eq. 4).

D P S , R C D B , P O P, 4



iscounted initial population size

Fig. 4. There was a significant correlation between extinction probabilitics and discountcd

initial p opulatio n sizes of Bachman s Sparrow in patchy environments. T he relatio nship was

described by a logarithm equation, Y 0.6229-0.5518 log(X) , w here Y is the extinction

probability and X is the discounted initial population size.

where DPS, is the discounted populat ion s ize in patch i ; POP, is the

nominal population size in patch i Relative suitability can tell how manyindividuals in poo rer patch es a re equivalent to a certain n um be r of individ-

uals in patches with the highest suitabil i ty. For example, because relat ive

suitability of initial 1-year patches was 0.0339, 100 individuals in initial

I -year pa tches w ere equivalent to 3.39 individuals in m atu re patch es.

Step 5. A total discounted populat ion s ize (DPS) was summation of the

discounted population sizes in al l patches (Eq. 5).

t

D P S D P S ,I

wher e n is the number of patches.

R E S UL T S A F T E R D I S C O U N T I N G

Nominal init ial population sizes presented in Fig. 2 were discountedaccording to eq ua tio ns (1)-(5). Fig. 4 shows that extinction rates Y

decreased with the logari thm of discounted init ial population sizes X

Y 0.6229 0.5518 log (X ), Sp earm an rank correlat ion coefficient r

.9636, n I I P 0.01). This indicates that the discounting models

worked very well, because before the discounting there was n o co rrelat ion

between the nominal ini t ial population sizes and extinction probabil i t ies

(Fig. 2). T h e results suggest that init ial p opu lation sizes alone could no t tell

extinction probabilities if individuals were initially distributed in various



I KLDIC I O N OF E X I INC ION I KOBAHILITIbb 9

patch types . T h e problem was solved throug h in corpo rat ion of patch

characteristics with nominal initial population sizes.

DISCUSSIONS AND CONCLUSIONS

T h e seedin g process of in i tia l populat ion s in t he s imulat ions was s imilar

to the in t roduc t ion or re in t roduc t ion of popula t ions . The d iscount ing

models sugges t some management s t ra teg ies . In ordcr to main ta in lower

extinct ion probabil it ies , it is be t ter to place individuals in m at ur e patch es a t

th e very beginning because they were s table an d h ad high quality for the

sparrows reproduc t ion . Th e spar rows would have much lowcr chances tosuccessfully keep persistent populations if individuals were seeded in the

ephemera l cyc l ic pa tches . When breeding pa tches became unsui tab le , no

more offspr ing were produced if the adults could not f ind sui tablc patches

nearby . Even though the ephemera l pa tches were su i tab le aga in a f te r a

long time, they had very low likelihood to be occupied because of insuffi-

cient number of offspring.

The discounting models may be valuable in predic t ing extinct ion proba-

bi l i t ies of other species in heterogeneous environments af ter some modif i-cat ions . The patch sui tabi l i ty in my models was measured by reproductive

success . Although the discounting models performed sat is factor i ly in this

paper , o ther pa tch charac te r is t ics may be poten t ia l ly impor tan t to o ther

species. Fo r exam ple, if mortality differs in various pa tch es, th e discou nting

models should incorpora te i t . A no the r impor tan t fac tor to cons ider may be

patch posi t ion, which par t ia l ly determines the success of locat ing sui table

patches (H arr ison e t a l ., 1988) .Many s tudies have shown effects of patchy environments on populat ion

dynamics and community s tructure (Kareiva , 1990; Hassel l e t a l . , 1991) .

Predicting the effects , however, is very challenging because patches are

dif ferent in quality (Pull iam a nd Dan ielson, 1991). Th is problem would

beco me s im pler if var ious patch types could b e co mp arable o n t he bas is of

a common index. As discussed above, the discounting models f i rs t consid-

e r ed the pa tch va r ia t ions and then made the pa tches equ iva len t to each

other so tha t a l l pa tches were compared accord ing to a s ing le measure(re la t ive sui tabi l i ty) . I t seems promising that the discounting approach

would be beneficial to forecasting ecological consequences of patchy envi-

ronments .

In sum mary, when init ia l individuals of B achm an s Sp arro w we re pu t in

several types of patches in heterogeneous environments, nominal initial

populat ion s izes a lone could not predic t ext inct ion probabil i t ies . Through

integrating patch characteristics (patch suitabili ty, t iming of patch suitabil-

i ty , and durat ion of patch sui tabi l i ty) , the discounting models discounted



nominal initial population sizes in various patch es. T h e extinction probabil-ities were then significantly correlated with the discounted population

sizes. It is hopcd that the discounting models would be applicable to

predicting extinction rates of some other species in patchy environments.The discounting approach would also be useful for examining impacts of

patchy environments on population dynamics and community structure.

A C K N O W L E D G E M E N T S

For their helpful discussions and comments, I am very grateful to D r. H .

Ronald Pull iam, Dr. Eugene P. Odum, Dr. J . Barny Dunning, Jr . , Mr.Stuar t W hipple, Ms. The lma Richardson, M r. Joe Ferris and M r. Re n M.

Borgella, Jr . (University of Geo rgia), D r. M ark L. S haffer (T h e W ilderness

Society), Mr. Richard B. Harris (University of Montana), Dr. Jianguo Wu

(Cornell University), Dr. Yegang Wu (Oak Ridge National Laboratory),

D r. Lynn A M aguire (D u k e University), and Dr. S. Harrison (University of

California a t Davis) . T h e research was supp orted by D ep art m en t of Energy

grant DE-FG09-89E R6088 1 to H . Ronald Pulliam and a n O du m Ecological

Research G rant to the au thor.

R E F E R E N C E S

Haggcrty, T.M., 1986. Reproduction ecology of Bachman's Sparrows (Airnophiln uestic~nlis

in Ccntral Arkansas. Ph.D. Disscrtation, University of Arkansas. Fayetteville.

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population estimation. Conscrv. Biol., 1: 72-76.

Harrison, S.. Murphy, D.D. and Ehrlich. P.R., 1988. Distr ibution of thc Bay checkerspotbutterfly, Euphydnns cdithu bnyerzsis: evidence for a metapopulation model. Am Nat.,

132: 360-382.

Hassell, M.P., Comins, H.N. and May, R.M., 1991. Spatial structurc and chaos in insect

population dynamics. Nature, 353: 255-258.

Hu tchin son, G.E ., 1978. A n In trodu ction to Population Ecology. Y a k University Prcss, New

Haven.

Karciva. P., 1990. Population dynamics in spatially complcx environments: theory and data.

Phil. Trans. R. Soc. Lond. (B), 330: 175-190.

Levin, S.A., 1976. Population dynamic modcls in hcterogcneous environments. Ann. Rev.Ecol. Syst., 7: 287-310.

Liu, J., 1992. A spatially-explicit model for ecological economics of species conservation in

complcx forest landscapes. Ph.D . Dissertation, University of Ge orgia, A thens .

Pacala, S.W.. Hassell, M P and May, R.M., 1990. Ho st-parasitoid associatio ns in patchy

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PREDICTIONS OF EXTINCTION PR O B B IL IT I~S 6

Shaffer M.L. 1978. D eterm inin g minim um viable pop ulat ion sizes: A case study of the

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Shaffer M.L. 1981. Minim um pop ulat ion sizes for species con serv atio n. BioS cicncc 31:

131-134.Sha ffer M.L. 1990. Pop ulatio n viability analysis. C on scw . Biol. 4: 39-40.

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Wiens J.A. 1976. Population response s to patchy e nvironm ents. A nn . Rev. Ecol. Syst. 7:

81-120.

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