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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
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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
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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.
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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
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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
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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.
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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
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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
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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
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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.
Harris: R.B., Maguire, L.A. and Shaffer, M.L., 1987. Sample sizc for minimum viable
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
environmcnts . N ature , 344: 150- 153.
Pulliam, H .R ., 1988. So urces, sinks, and population regulation. Am . Nat. , 132: 652-661.
Pulliam, H .R . and Da niclso n, B.J., 1991. Sou rce, sink, and habitat selection: a landscape
perspectiv e on po pulation dynamics. A m . Nat., 137: S50-S66.
Pulliam, H.R., Dunning, Jr . , J .B. and Liu, J . , 1992. Population dynamics in complexlandscapes: a case study. Ecol. Appl., 2: 165-177.
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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
grizzly be ar Ursus arctos) . Ph.D. Disser ta t ion Du ke Univers ity Du rham NC.
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.
Shaffer M.L. an d Samson F.R. 1985. Popu lation size a nd cxtinction: A note on de te rmin-
ing critical population sizes. A m . Na t. 125: 144-152.
Wiens J.A. 1976. Population response s to patchy e nvironm ents. A nn . Rev. Ecol. Syst. 7:
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