Sources, sinks and population regulation

Abstract

Animal and plant populations often occupy a variety of local areas and may experience different local birth and death rates in different areas. When this occurs, reproductive surpluses from productive source habitats may maintain populations in sink habitats, where local reproductive succes fails to keep pace with local mortality. For animals with active habitat selection, an equilibrium with both source and sink habitats occupied can be both ecologically and evolutionarily stable. If the surplus population of the source is large and the per capit deficit in the sink is small, only a small fraction of the total population will occur in areas where local reproduction is sufficient to compensate for local mortality. In this sense, the realized niche may be larger than the fundamental niche. Consequently, the particular species assemblage occupying any local study site may consist of a mixture of source and sink populations and may be as much or more influenced by the type and proximity of other habitats as by the resources and other conditions at the site. -Author

Full text

The University of Chicago
Sources, Sinks, and Population Regulation
Author(s): H. Ronald Pulliam
Source:
The American Naturalist,
Vol. 132, No. 5 (Nov., 1988), pp. 652-661
Published by: The University of Chicago Press for The American Society of Naturalists
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Vol. 132, No. 5 The American Naturalist November 1988
SOURCES, SINKS, AND POPULATION REGULATION
H. RONALD PULLIAM
Institute
of
Ecology and
Department of Zoology, University
of
Georgia,
Athens, Georgia
30602
Submitted December
9, 1986; Revised August 3, 1987; Accepted
February 11,
1988
Many
animal and
plant species can regularly
be found in a variety
of
habitats
within a local geographical region. Even so, ecologists often
study population
growth and regulation with little or no attention paid to the
differences
in
birth
and
death rates
that occur in different habitats.
This paper is concerned
with
the
impact of habitat-specific demographic
rates on
population growth
and
regulation.
I argue that, for many populations, a large fraction of the
individuals may regu-
larly occur
in
"sink" habitats, where
within-habitat
reproduction
is insufficient
to
balance local mortality; nevertheless, populations may persist
in such
habitats,
being locally maintained
by continued immigration
from more-productive
"source" areas nearby.
If
this
is commonly
the case for
natural
populations,
I
maintain that
some basic ecological notions concerning niche
size, population
regulation,
and
community structure must
be reconsidered.
Several authors
(Lidicker 1975; Van Horne 1983)
have discussed
the need to
distinguish
between
source and
sink
habitats
in field
studies of
population regula-
tion; however,
most
theoretical treatments (Gadgil 1971; Levin 1976; McMurtie
1978; Vance 1984) of the dynamics of single-species
populations
in spatially
subdivided habitats have
not
explicitly
addressed
the
maintenance of
populations
in
habitats
where
reproduction fails to keep pace with
local
mortality.
Holt
(1985)
considered the
dynamics
of
a food-limited predator
that
occupied
both
a source
habitat
containing prey and a sink habitat
with
no prey. He demonstrated
that
passive dispersal
from the
source can maintain
a population
in the
sink
and
that
the
joint sink
and
source populations can exceed what
could be maintained
in the
source
alone. Furthermore,
he
showed
that
"time-lagged" dispersal back
into
the
source from the sink
can stabilize an otherwise
unstable
predator-prey
interac-
tion. Holt argued, however,
that
passive dispersal
between source and sink
habitats in a temporally constant environment
is usually
selectively disadvanta-
geous, implying
that sink
populations
will
be transient
in
evolutionary
time.
In this
paper, I consider the consequences of active dispersal (i.e., habitat
selection based on
differences
in
habitat quality) on
the
dynamics
of single-species
populations
in
spatially heterogeneous
environments.
I argue
that
active dispersal
from
source habitats can maintain large sink populations and
that such dispersal
may
be evolutionarily
stable.
Am. Nat. 1988. Vol. 132, pp. 652-661.
? 1988
by
The University
of
Chicago. 0003-0147/88/3205-0009$02.00.
All
rights reserved.
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SOURCES, SINKS, AND POPULATION REGULATION 653
BIDE MODELS
One approach to modeling spatially
heterogeneous
populations
is to employ
BIDE models
(Cohen 1969, 1971),
which simultaneously
consider birth
(B), im-
migration
(I), death (D), and emigration
(E). Normally,
in BIDE models, the
parameters
are considered
random
variables
but not spatially
heterogeneous.
In
this paper,
I make the
opposite
assumptions,
namely, that
rates of birth,
death,
immigration,
and emigration
are
deterministic
but may differ
between
habitats.
First,
consider a spatially
distributed
population
with
m subpopulations,
each
occupying
a discrete
habitat or
"compartment."
If
bj
and
dj
are, respectively,
the
number
of births and
the number
of deaths occurring
over
the course
of a year
in
compartment
j, then
the total
number of births
and deaths
during that
year
in
all
compartments
is given,
respectively,
by
m m
j and D=Ld1, (1)
j=1 j=1
since
every
birth
and
every death
takes place in
some
compartment.
Now, let
ijk be the
number
of
individuals
immigrating
from
compartment
k into
compartment
j. Each immigrant
into
j must
come from
one of the other
m - 1
compartments
or come
into j from
outside
the m
compartments
that
constitute
the
ensemble
of interest.
That is, immigration
into compartment
j is given
by
rn m
ij =
L
ik + ijO = ijk
k=1 k=O
where
ijo
represents
immigration
from outside
the ensemble
into compartment
j
and
ijj
is zero.
Similarly,
if
e1k represents
the number
of
emigrants
from
j into
k,
then
m m
ej = ejk + ejO = E elk
k=1 k=O
Note
that
ekj = ijk
for
allj, k
#
0. Finally,
to
complete
the definitions
of the
BIDE
parameters,
let
m m
I
= i
o and E = ekO.
j=1 k=1
The ensemble of all compartments
is said to be in dynamic equilibrium
in
ecological
time when
the
number
of
individuals
(nj) in each and
every compart-
ment
does not
change
from
year
to
year.
This occurs
only
if the number
of
births
plus the number
of immigrants
exactly equals the
number of deaths
plus the
number
of
emigrants
for
every
compartment.
That
is,
bj + ij - dj - ej = (bide)j = O, (2)
for
every
j, and
BIDE = 0. Source and
sink
compartments
can now
be defined
in
terms of the
BIDE parameters.
A source
compartment (or
habitat)
is one
for
which
bj
> dj and ej > ij (3)
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654 THE AMERICAN NATURALIST
"WINTER"
End of "WINTER"
nPnPAn
+FP/3n
DISPERSAL n
n
+,/n
ANNUAL CENSUS "SUMMER" End of
"SUMMER"
(Reproductive Season)
FIG. 1.-An annual
census
is
taken
in each habitat
or
"compartment"
in the spring
at the
initiation
of the breeding season (summer).
Each
individual
breeding in the habitat produces
P
juveniles
that are alive
at the end of
the breeding season.
There is no adult mortality
during
the breeding season;
adults survive
the nonbreeding (winter)
season with probability
PA and
juveniles
survive
with
probability
PJ.
when
(bide)j = 0. A sink compartment
(or habitat)
is one
for which
bj
< dj and ej < ij (4)
when (bide)j = 0.
The above definitions
apply
strictly
for
equilibrium
populations
only.
A more
general
definition
of a source
is a compartment
that,
over
a large period
of
time
(e.g., several generations),
shows no
net
change
in
population
size
but,
nonethe-
less, is a net exporter
of individuals.
Similarly,
a sink is a net importer
of
individuals.
HABITAT-SPECIFIC
DEMOGRAPHICS
To see how
the BIDE parameters
relate to
habitat-specific
survival
probabilities
and per capita birthrates,
consider
a simple
annual
cycle
for
a population
in
a
seasonal environment (see fig. 1). An annual
census
is
taken
in
the
spring
at the
initiation of the breeding
season. Each individual
breeding
in
habitat
1
produces
(on
the
average)
PI
juveniles
that are alive
at the end of the
breeding
season.
There
is no adult mortality during
the
breeding season; adults
survive the
nonbreeding
(winter) season with
probability
PA and
juveniles survive
with
probability
PJ.
Thus,
the
expected
number of
individuals
alive
at
the end of
the winter
and
just
before
spring
dispersal
is given
by
n1(t + 1) = PAnl(t) + Pj,lini(t) = X1n,.
If there were
only
one compartment
(habitat),
X1
for
a small
population
would
be
the
finite rate of increase for
the
population.
In
a multi-compartment
model,
the
Aj's
indicate
which
compartments
are sources
and
which
are
sinks.
For a simple
example, consider
two habitats that
do not differ
in either adult or
juvenile
survival
probabilities
but that do differ
in per capita reproductive
success. If
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SOURCES, SINKS, AND POPULATION
REGULATION 655
habitat
1
is a source and habitat 2 a sink, then by
definition,
\1 =PA + PJ1I > I (5a)
and
A2 =PA + PJ2 < l (Sb)
The
finite rate of
increase
for a multi-compartment
model
depends on the
fraction
of the
population in each habitat.
This, in turn,
depends on how individuals
distribute
themselves
among available
habitats at
the time of
dispersal,
before the
onset
of
breeding (fig. 1).
Before
considering how habitat
dispersal
between source and sink
habitats
influences
population
regulation, I briefly
discuss population regulation in a
source habitat
in the
absence of a sink
habitat. To do this,
I must
specify
the
nature
of density
dependence in the source habitat.
For the model discussed
below, the
critical feature
of density
dependence is that some
individuals
in
the
source
habitat
do predictably better
than
others
in
terms of fitness.
Simple
as-
sumptions
reflecting this
feature are
that the number
of
breeding sites
is limited
and that
some
individuals
obtain
breeding sites and
others do
not.
A
more
general,
and
more
realistic,
model of the
distribution
of
habitat
quality
is
discussed
briefly
in
the
next
section.
I assume that there are
only n'
breeding
sites available
in
the
source
habitat.
If
the
total population
size is N and no other
breeding
sites are available,
N - n
individuals either
stay
in the
source
habitat
as nonbreeding
"floaters" or
migrate
to
nearby sink habitats.
In
either
case, they
fail
to
reproduce
but
survive
with the
same
probability
(PA) as do breeding
individuals.
(The qualitative
features
of
the
model
are
unchanged by
the
assumption
that
nonbreeding
individuals survive
with
higher
or
lower
probability
than
breeders.)
Since the
average
reproductive
suc-
cess of an individual
securing
a breeding
site is 1, the
average reproductive
success for
the entire
population
is given
by
p
I ~if
N n? ,i
13(N) ={1iN'n,(6)
(nrI/N)3 if N > n.
Thus, according
to the definition
of a source habitat
(eq. 5a), the
population
increases
when rare and continues to
grow
at
the rate
X1
= PA + Pj41 until
all
breeding
sites are
occupied. The population
will be regulated
when
X(N) = PA + (niIN)Pjpl = 1
or
N* = nPj13i/(l
- PA) (7)
Again, from the definition of
a source,
Pj431/(I
- PA) is
greater
than
one;
thus,
the
equilibrium
population
density (N*) exceeds the number of
breeding
sites
(ni),
implying
the existence of
a nonbreeding surplus.
Assume that,
adjacent to the source habitat,
is a large sink
habitat,
where
breeding
sites
are abundant but
of
poor
quality.
According
to the definition of
a
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656 THE AMERICAN
NATURALIST
sink
(eq. 5b), 12 < (1 - PA)/PJ; thus,
the sink
population declines and eventually
disappears altogether in the
absence of
immigration from
the source. Individuals
unable to find a breeding site in the
source emigrate to
the sink because a poor-
quality breeding site
is better than none
at
all. If the
source
is saturated
and there
are sufficient
breeding sites
in the
sink,
the entire
nonbreeding surplus
from
the
source emigrates,
yielding
an increase
in the
growth
rate of the
total
population:
X(N) = (n1/N)X1 + (n2/N)X2
= (n/N)X1
+ X2(N- n)/N (8)
= \2 + (niiN)Pj(PI -22).
The total population
equilibrates
when
X(N) = 1; and,
according
to
equation
(8),
N* = Pjn(13I - 12)1(1 - PA - PJ
P2)- (9)
A relatively
simple way to determine
the equilibrium
populations
that
will
inhabit the
source and sink habitats
under this model is to note
that,
since
the
annual
census is
taken
after the emigration
of the
reproductive
surplus,
nr
individ-
uals remain in the
source and n,(Xl
- 1) immigrate.
Therefore,
in terms
of the
BIDE model,
i2l = n^(PA
+ PJ1PI
- 1).
The local
reproduction
and survival
in
the
sink
is supplemented
by
this
immigration,
so that
n2(t
+ 1) = (PA + PJP2)n2(t) + i2l = X2n2(t)
+ ^(XI - 1).
At
equilibrium,
n* = i2l/(1 - K2), or
n*2 ^
(XI - 1)/(1 - A2). (10)
Notice that
XI - 1
is the
per capita
reproductive surplus
in the
source
and 1 - X2
is the per capita
reproductive
deficit
in
the sink.
If
there are many
habitats,
the total
population
reaches
an equilibrium
when
the
total surplus
in
all source habitats
equals the total deficit
in
all sink habitats.
That
is,
ml ml m2 m12
Z
ej = > n>KA
- 1) = >
nk(1 - ik) = k
j=1 k=lI k= 1
where there are
ml source
habitats and
m2
sink habitats.
ECOLOGICAL AND EVOLUTIONARY STABILITY
In
the
preceding
analysis,
I calculated
the
equilibrium
population sizes
in
source
and sink
habitats
without addressing the stability of this
equilibrium. A local-
stability analysis
involves finding
the
slope (b) of K(N) evaluated
at
the equilib-
rium
population
size
N*. If - bN is less than one, the
equilibrium is locally
stable
and
approached
monotonically (Maynard Smith
1968). The
rate of increase for the
combined
source-sink
population
is given
by equation
(8). Differentiating,
one
obtains
dA(N)/dN = -hPj(I - P2)/N2.
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SOURCES, SINKS, AND POPULATION REGULATION 657
()D
u) I ,li ( 1
) ' Bl3n (n)
Ow
0
010
1 2 3 4 5 6 7 8 9 10
NUMBER
(n) OF INDIVIDUALS
BREEDING ON S ITE
FIG. 2.-The reproductive
success of each individual breeding
in a particular
habitat
depends on the number of other individuals
in that
habitat. n)(n),
the expected
reproductive
success of the individual occupying
the jth-best breeding site in habitat 1 when there are a
total of n individuals breeding there;
r3(n), the average
reproductive
success in the
habitat.
Noting
the equilibrium population
size given in
equation
(9), the value of
- bN
is
easily seen to be 1 - PA - P432 or 1 - K2. Since habitat
2 is a sink, K2 is less
than
1, and, therefore,
- bN is also less than one. As for the case of a source
habitat
with no sink, 32 equals zero and - bN equals 1 - PA. Thus,
with or without a sink
habitat, the equilibrium
population size is locally stable. For the simple
cases
analyzed above, there is only one nonzero equilibrium,
and this equilibrium
is
approached
monotonically
from any positive initial population
size.
A different question of stability concerns the evolutionary
stability of the
dispersal rule that determines
the proportion of individuals
in each habitat.
Holt
(1985)
argued that passive dispersal
between a source and a sink is
evolutionarily
unstable. Two essential differences between the current
model and that of
HIolt
are passive versus active dispersal
and unequal versus equal fitnesses
within
a
habitat. In my model, individuals
choose to leave the source whenever
their
expected reproductive
success is higher in the sink. This never happens
in
Holt's
model because all
individuals
in the source have equal fitness
and the mean
fitness
in the source
never drops to less than one. Since the mean
fitness in the sink
is
always less than one, it always pays for individuals not to immigrate
and the
evolutionarily
stable strategy
is no
dispersal.
In my model, when the local population
in the source exceeds the number
of
breeding
sites available, it pays all surplus individuals to emigrate because they
can achieve
a higher fitness by doing
so. The habitat-selection
rule built into
the
model is a special case of the more general rule "never occupy
a poorer
breeding
site when a better one
is still available." Assuming no habitat-specific
differences
in survival probability,
this is the evolutionarily
stable habitat-selection
rule
because no individual can do better
by changing
habitats
(see Pulliam and
Caraco
1984; Pulliam
1989).
A more general application
of an evolutionarily stable habitat
rule that
also
results in stable occupancy
of sink
habitatsi
isillustrated
in
figure
2. In this
figure,
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658 THE AMERICAN
NATURALIST
Ill(n) is the
expected
breeding
success of the
individual
using
the best
breeding
site
in
habitat 1, and I3PI(n)
is the
expected success of
the individual
using
the
poorest site occupied when there
are n individuals
in habitat
1 (the source).
Assuming
that
individuals never
occupy a poorer
site when a better one is still
available, habitat
2 (the sink)
will not be occupied as long
as P11N(N)
> 121 (1). That
is to say,
the sink
habitat will not be occupied as long as all N members of
the
population
can enjoy greater
reproductive
success in the source. However, if
p21(1) exceeds 131N(N)
before the
average reproductive
success in the source
reaches
one, the
sink will
be occupied and the habitat distribution
will
be evolu-
tionarily
stable. Of
course, the relative numbers
of
individuals
in the
source
and
sink
habitats
depend
on details of how
reproductive
success changes
with
crowd-
ing in each habitat. If
good breeding
sites
in the
source
are rare and
poor
sites
in
the
sink
are relatively
common,
a large population
may
occur
in the
sink.
IMPLICATIONS
Sink
habitats
may
support very
large populations
despite the
obvious fact
that
the sink
population
would eventually
disappear
without
continued
immigration.
Consider the
simple situation
in
which each year
i
individuals
are
released into
a
habitat
where local reproduction
is incapable of
keeping up with
local mortality.
The equilibrium
population
maintained in this sink
habitat
would be iI(1 - A).
Thus, if no individuals
survived the
winter (K = 0), only the i recently
released
individuals
would be censused each year. If adults
survived
winter with probabil-
ity
?/2,
2i individuals
would be censused each year. If, in addition, each adult
produced
an average of
0.4 juveniles that survived
to the
following spring,
the
equilibrium
population
would be 10
times i,
even
though the population
could
not
be maintained
without an annual
subsidy.
In
some
circumstances, only a small
fraction of
the
population
may be
breeding
in
a source
habitat.
Figure 3 shows
the fraction of
the
equilibrium population
in
source habitat based on the assumptions of the model developed above and
calculated
according
to
equation (10). Clearly,
if the
reproductive
surplus of
the
source is
large and the
reproductive
deficit of the sink
is small, a great majority
of
the
population may occur in the
sink habitat.
For example,
with a per capita
source
surplus of 1.0 and a sink
deficit of 0.1, less than 10% of
the
population
occurs in habitats where
reproductive success is sufficient to balance annual
mortality.
The
concept of
niche.-Joseph Grinnell is often
credited with
introducing
the
niche
concept into
ecology. James et al. defined the
Grinnellian niche as "the
range
of
values of
environmental
factors that are
necessary and
sufficient to
allow
a species to carry out
its
life
history"; under normal
conditions,
"the species is
expected
to occupy
a geographic
region that is
directly
congruent with the distri-
bution
of its
niche" (1984, p. 18). Though James et
al. suggested that a species
with
limited
dispersal
may not occur
in some areas
where its
niche is found,
they
clearly implied
that the
species will
not occur
where
its niche is absent.
A sink
habitat
is by definition an area where
factors are not
sufficient
for
a species to
carry
out its life
history,
but as discussed above, some species may be more
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SOURCES, SINKS, AND POPULATION REGULATION 659
1.5
00
vo
1t0 _ p* 0.60 p*=0.50
a) a. C\/
> X<
X D 0.5 / / / / * =~~~~~~~P
0.25
-~o 0.5
p -
0.10
0.5 1.0 1.5 2.0
Source
Per Capita
Reproductive Surplus
(X1
-1
)
FIG. 3.-The equilibrium
proportion
(p*) of
the
population
in the source
habitat
depends
on both the
per capita surplus
in
the
source and the per
capita deficit in
the sink. A large
proportion of the
population may occur
in the
sink habitat if
the source
surplus is large
and
the sink
deficit
is small.
common in
sink
habitats
than in the
source
habitats on which sink
populations
depend.
Hutchinson's (1958) particularly
influential
formulation of the niche
concept
differentiated between the fundamental
niche
and the
realized niche. Hutchinson
argued
that
the
realized
niche
of most
species would
be smaller than the
funda-
mental
niche
as a result of
interspecific
competition.
I have argued
in
this
paper
that
reproductive
surpluses from
productive
sources may immigrate
into
and
maintain
populations
in
population
sinks.
If
this is
commonly
true in
nature,
many
species
occur
where
conditions are not sufficient to
maintain a population
without
continued
immigration.
Thus, in
such
cases, it
can be said
that the
fundamental
niche is smaller than
the realized niche.
Species conservation.-Given that a species may
commonly
occur and suc-
cessfully
breed
in
sink
habitats,
an investigator could
easily be misled
about
the
habitat
requirements of
a species. Furthermore,
autecological
studies of
popula-
tions
in
sink
habitats
may yield
little information
on
the
factors
regulating
popula-
tion
size if
population size in the sink is determined
largely
by the size and
proximity of
sources.
Population-management
decisions based on studies in sink
habitats could lead
to undesirable
results. For example, 90% of a population might
occur in one
habitat.
On the basis of the
relative
abundance and
breeding
status
of
individuals
in
this
habitat,
one
might
conclude that
destruction of a nearby
alternative habitat
would have relatively
little
impact on the
population.
However, if
the
former
habitat
were
a sink and the
alternative a source, destruction of
a relatively small
habitat
could
lead to
local population
extinction.
Community
structure.-What is a sink habitat
for
one
species
may
be a source
for
other
species.
Thus,
a "community" may
be a mixture of
populations,
some of
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660 THE AMERICAN
NATURALIST
which are self-maintaining and some of which
are not. Attempts to understand
phenomena such as the local coexistence of species
should, therefore, begin
with
a determination of the extent to which the persistence
of populations depends
on
continued immigration.
Many attempts to
understand
community
structure have focused
on resource
partitioning and the local diversity of food
types.
The diversity
and
relative
abundance of the organisms in any particular
habitat may depend
as
much
on
the
regional diversity of habitats as on
the
diversity
of resources locally available.
In
extreme cases, the local assemblage of species
may be an artifact
of the
type
and
proximity of neighboring habitats
and have little to do
with the resources
and
conditions
at
a study
site. This is not to
imply
that local studies of
the
mechanisms
of
population regulation
and
species
coexistence
are
unnecessary,
but rather
that
they need to be done
in
concert
with "landscape" studies of
the
availability
of
habitat types on a regional basis.
My goal is to draw attention to some
of the
implications
of
habitat-specific
demographic rates.
In
many ways, they may
be
ecologically
more
important
than
the
age-specific demographic
rates
that have received
so much attention
in
the
ecological and evolutionary literature.
SUMMARY
Animal and
plant populations
often
occupy
a variety
of
local areas
and
may
experience different local birth and death
rates
in
different areas. When this
occurs, reproductive surpluses
from
productive
source habitats
may
maintain
populations
in
sink
habitats,
where local
reproductive
success fails to
keep pace
with
local mortality. For animals
with
active
habitat selection, an equilibrium
with
both
source and
sink
habitats occupied can
be both
ecologically
and
evolutionarily
stable.
If
the
surplus population
of the
source
is
large
and
the
per capita
deficit
in
the
sink is small, only a small fraction of
the
total population will occur
in
areas
where
local reproduction
is sufficient to
compensate for local mortality.
In
this
sense,
the
realized niche
may
be
larger
than the
fundamental niche. Consequently,
the
particular species assemblage occupying
any
local
study
site
may consist
of a
mixture of source
and
sink
populations
and may be as much or more influenced by
the
type
and
proximity
of other habitats as
by
the resources and
other conditions
at the site.
ACKNOWLEDGMENTS
I wish to acknowledge
the
assistance
of
G. Reynolds
and
J. Nelms
in
the
preparation of
the
manuscript and the financial
support of the National Science
Foundation (BSR-8415770).
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