patch dynamics and metapopulation theory: the case of successional species

J. theor. Biol. (2001) 209, 333}344doi:10.1006/jtbi.2001.2269, available online at http://www.idealibrary.com on

Patch Dynamics and Metapopulation Theory:the Case of Successional Species

PRIYANGA AMARASEKARE*?A AND HUGH POSSINGHAM*-

*National Center for Ecological Analysis and Synthesis,;niversity of California, Santa Barbara, 735 StateStreet, Suite 300 Santa Barbara, CA 93101-5504, ;.S.A. -Departments of Zoology and Mathematics,

¹he ;niversity of Queensland, St ¸ucia, Q¸D 4072, Australia

(Received on 1 August 2000, Accepted in revised form on 22 January 2001)

We present a mathematical framework that combines extinction}colonization dynamics withthe dynamics of patch succession. We draw an analogy between the epidemiological categor-ization of individuals (infected, susceptible, latent and resistant) and the patch structure ofa spatially heterogeneous landscape (occupied}suitable, empty}suitable, occupied}unsuitableand empty}unsuitable). This approach allows one to consider life-history attributes thatin#uence persistence in patchy environments (e.g., longevity, colonization ability) in concertwith extrinsic processes (e.g., disturbances, succession) that lead to spatial heterogeneity inpatch suitability. It also allows the incorporation of seed banks and other dormant life forms,thus broadening patch occupancy dynamics to include sink habitats. We use the model toinvestigate how equilibrium patch occupancy is in#uenced by four critical parameters: coloniz-ation rate, extinction rate, disturbance frequency and the rate of habitat succession. Thisanalysis leads to general predictions about how the temporal scaling of patch succession andextinction}colonization dynamics in#uences long-term persistence. We apply the model toherbaceous, early-successional species that inhabit open patches created by periodic distur-bances. We predict the minimum disturbance frequency required for viable management ofsuch species in the Florida scrub ecosystem.

( 2001 Academic Press

Introduction

Understanding how populations persist inpatchy environments is a central problem inbasic and applied ecology. Metapopulation the-ory focuses on the dynamics of patch occupancyas a function of extinctions and colonizations(Levins, 1969, 1970; Gilpin & Hanski, 1991; Day& Possingham, 1995; Hanski & Gilpin, 1997). An

-Present address: Department of Ecology and Evolution,The University of Chicago, 1101 East 57th Street, Chicago,IL 60637-1573, U.S.A.AAuthor to whom correspondence should be addressed.

e-mail: [email protected]

0022}5193/01/070333#12 $35.00/0

implicit assumption in this framework is thatpatches themselves do not change in terms oftheir quality or suitability. The only distinctionbetween patches is whether they are occupied orempty. All empty patches are considered suitablefor colonization. This framework emphasizes thelife-history attributes of species that in#uence ex-tinction}colonization dynamics (e.g., longevity,dispersal ability). It does not consider spatialheterogeneity in patch suitability.

Patch dynamics theory on the other hand,focuses on changes in the state of patches them-selves (Picket & White 1985; Levin et al., 1993;Wu & Loucks, 1995). Patches are de"ned in

( 2001 Academic Press

FIG. 1. State transitions between the four patch types fora successional species. Note that the resistant patches con-sist of late-successional vegetation and hence di$cult tocolonize for early-successional species. Any individuals thatcolonize resistant patches will stay dormant until a distur-bance occurs. Hence e

L<b

Land b

L;b

I, making b

LRI

a vanishingly small term.

334 P. AMARASEKARE AND H. POSSINGHAM

terms of whether they are suitable or unsuitablefor colonization. For example, succession, litteraccumulation and invasions by natural enemiesmay cause patches to become unsuitable.Disturbances may reverse these biotic pro-cesses and make patches suitable for coloniz-ation. This framework emphasizes the role ofspatial heterogeneity in extinction}colonizationdynamics.

Here, we present a mathematical frameworkthat combines both approaches. This broad per-spective is useful for several reasons. First, itintroduces an element of spatial heterogeneity tothe patchy but spatially homogeneous land-scapes envisioned in metapopulation theory.Second, it emphasizes the temporal scaling ofpatch dynamics with population dynamics ofthe species that occupy patches. This scalingrelationship is key to understanding how thedynamics of patch suitability (e.g., succession anddisturbance) in#uence extinction}colonizationdynamics.

Our approach is two fold. First, we use theframework to make general predictions abouthow life history and habitat characteristics act inconcert to determine long-term habitat occu-pancy. Second, we apply the theory to a speci"cbiological example: herbaceous, early suc-cessional plant species that inhabit gaps main-tained by periodic "re in the Rosemary phase ofthe Florida sand pine scrub. We are particularlyinterested in predicting the minimum disturbancefrequency required for long-term persistence atan acceptable level of patch occupancy.

The Model

In classical metapopulation theory the land-scape consists of only two types of patches: occu-pied patches and suitable but empty patches(Levins, 1969, 1970; Levins & Culver, 1971; Slat-kin, 1974; Hastings, 1980). In patch dynamicstheory the landscape also consists of two types ofpatches: those that are suitable for colonizationand those that are unsuitable (Pickett & White,1985; Levin et al., 1993; Wu & Loucks, 1995).Merging the two perspectives leads to a land-scape characterized by four types of patches:occupied}suitable, unoccupied}suitable, occu-pied}unsuitable and unoccupied}unsuitable.

We introduce a patch nomenclature that isdrawn from epidemiological theory (May & An-derson, 1979; Anderson & May, 1991). Lawtonet al. (1994) and Nee (1994) noted a direct anal-ogy between the metapopulation and epi-demiological approaches: occupied patches arethe equivalent of infected individuals (I), andempty patches, the susceptible individuals (S).Here we extend this analogy. We use an early-successional species as an example.

For an early successional species empty, un-suitable patches are those that are dominated bylate successional species. Such patches are theequivalent of individuals that are resistant orbecome temporarily immune to the disease (typeR; Fig. 1). Disturbances such as "re will convertresistant patches to empty but suitable patches(type S). These patches are now susceptible tocolonization by the early successional species.Once colonized, they become infected patches(type I).

Over time, patches occupied by the earlysuccessional species will be invaded by late-successional species. Litter accumulation andshading by these competitive dominants willmake the patch unsuitable for seedling recruit-ment (Rees & Paynter, 1997). Individuals will

PATCH DYNAMICS AND METAPOPULATION THEORY 335

senesce and eventually above-ground plant partswill die, giving the appearance of local extinction(Noble & Slatyer, 1981). However, the patch maycontain a seedbank or some other dormant formof the life cycle (Zammit & Zedler, 1993). Suchpatches could also be thought of as refuges frompredators or pathogens where the plant can sur-vive but cannot reproduce. Pursuing the epi-demiological analogy, a patch of this sort wouldbe equivalent to a latent individual (type L) thatis infected but cannot transmit the disease (cf.Grenfell & Harwood, 1997). Latent patches areessentially sink populations in time or space. Dis-turbances that reverse the successional process(e.g., "re) can bring the seeds out of dormancyand revert latent patches to the occupied, suitablepatches (type I). Extinction of seed banks or otherdormant life stages from latent patches will inturn give rise to resistant patches. Fire can com-plete the cycle by converting resistant patches toempty but suitable patches (Fig. 1). Note thatonly infected patches can create latent or infectedpatches, and that the rate at which patches be-come suitable or unsuitable is independent ofwhether or not it is occupied. In other words,patch occupancy is a function of the species'biology (e.g., fecundity, longevity, colonizationability) while patch availability is a function ofprocesses extrinsic to the species (e.g., distur-bance, little accumulation, invasion by competi-tive dominants, and predation).

We use a system of di!erential equations torepresent the dynamics of the four patch-typesystem:

dIdt

"bISI!e

II#f¸!gI,

dSdt

"eII!b

ISI#fR!gS,

R*2"

bL(e

I( f#e

L)#g (e

L!f ))!b

I( f#e

L)AbLA

f

f#gB#( f#eL)B#Jb2!4ac

2b2L

f,

d¸dt

"gI!f¸!eL¸#b

LRI,

dRdt

"gS!fR#eL¸!b

LRI. (1)

¸*2"1!I*

2!S*

2!R*

2,

wherea"b

IbL,

b"bI( f#e

L)#b

L(e

I#g)!b

IbLA

ff#gB

Here f is the disturbance frequency and g,the rate of habitat succession. Quantities e

Iand e

Lrepresent local extinction rates, and b

Iand b

Lthe per patch colonization rates of

infected and latent patches, respectively.As in epidemiological theory (May & Anderson,1979; Anderson & May, 1991; Grenfell &Harwood, 1997), the total number of patches inthe system is assumed to be constant such thatI#S#¸#R"P. Alternatively, I, S, ¸ andR can be thought of as the frequency of eachpatch type in the landscape in which caseI#S#¸#R"1.

The Levins metapopulation model (Levins,1969, 1970) arises as a special case of eqn (1) whenwe ignore the distinction between suitable andunsuitable patches.

Equation (1) has two "xed points, a trivial"xed point with the successional speciesabsent:

I*1"0, ¸*

1"0,

S*1"

ff#g

, R*1"

gf#g

(2)

and an internal "xed point with the successionalspecies present in both active and dormantstages:

I*2"

!b#Jb2!4ac2a

,

S*2"

b#2bIbL A

ff#gB!Jb2!4ac

2a,


and

c"( f#eL) AeI!b

IAf

f#gBB#g AeL!b

LAf

f#gBB. (3)

From eqn (3) it can be seen that I*2'0 if c(0.

By transforming c, we get the following criterionfor a positive internal "xed point:

!

(bI!e

I)

( f#g) A f 2#f AeL#gbL!(e

I#e

L)

bI!e

IB

!

geL(g#e

I)

bI!e

IB(0. (4a)

In the absence of succession, disturbance orother factors that in#uence patch suitability, eqn(4a) simpli"es to b

I!e

I'0, the persistence cri-

terion of the classical Levins metapopulationmodel. For instance, when g"0 and e

L"0,

patches are always suitable and hence latent andresistant patches do not exist. When g'0 how-ever, persistence additionally requires positivityof the second-order polynomial in f. This willoccur as long as

f'!AeL2#g

bL!(e

I#e

L)

2(bI!e

I) B

#SAeL2#g

bL!(e

I#e

L)

2(bI!e

I) B

2#

geL(g#e

I)

bI!e

I

.

(4b)

In the presence of succession, metapopulationpersistence requires that the disturbance frequencyf exceeds the minimum speci"ed by eqn (4b).

Note that I*#S*"f/( f#g) and ¸*#R*"g/( f#g). This is because the transition be-tween suitable and unsuitable patches is indepen-dent of whether patches are occupied or empty.Hence it is always true that the fraction ofsuitable patches is f/( f#g) and the fraction ofunsuitable patches is g/( f#g). Moreover,0(I*

2)f/( f#g) necessarily implies 0)S*

2(

f/( f#g), 0)R*2(g/( f#g), and 0(¸*

2)

g/( f#g).

Local stability of the two "xed points deter-mines whether successional species can invadeand persist in the landscape. An unstable trivial"xed point implies invasibility, while a stableinternal "xed point implies long-term persistence.

We "rst investigate whether the successionalspecies can invade when rare. Invasion will suc-ceed if the dominant eigenvalue of the Jacobianmatrix is positive when evaluated at the trivial"xed point.

The Jacobian of eqn (1) with ¸"1!I!S!R is

!eI!f!g#b

IS* b

II*!f !f

eI!b

IS* !g!b

II* f

!eL!b

LR* g!e

L!e

L!f!b

LI*

.

The eigenvalues of the Jacobian evaluated at(I*

1"0, S*

1"f/( f#g), ¸*

1"0, R*

1"g/( f#g))

are: !f!g, (!x!Jx2!4y)/2 and (!x#Jx2!4y)/2 where

x"!bIA

ff#gB#e

I#e

L#f#g

and

y"( f#eL) AeI!b

IAf

f#gBB#g AeL!b

LAf

f#gBB.Note that x2'4y for all positive values of b

I,

bL, f and g. All eigenvalues are therefore real, and

the transition from stability to instability in-volves a zero real root (Gurney & Nisbet, 1998).Hence, the system de"ned by eqn (1) does notexhibit oscillatory instability.

The dominant eigenvalue is (!x#Jx2!4y)/2, and its sign is determined by themagnitude of y. For instance, when y'0 thedominant eigenvalue is negative and the trivial"xed point [I*

1"0, S*

1"f/( f#g), ¸*

1"0,

R*1"g/( f#g)] is stable to invasion by the suc-

cessional species. When y(0, the dominanteigenvalue is positive and the successional speciescan invade when rare. The invasion criterion is


therefore

bIA

ff#gB#b

LAg

f#gB

'eIP

eLf AeI#g!b

IAf

f#gBB. (5)

A few manipulations reveal that the invasioncriterion [eqn (5)] is the same as the persistencecriterion derived previously [eqn (4a)]. Condi-tions that allow the successional species tomaintain a non-zero abundance also ensure itsinvasion and spread.

The invasion criterion is a necessary but notsu$cient condition for long-term persistence. Wenext investigate the local stability of the internal"xed point.

The eigenvalues of the Jacobian are the rootsof the characteristic equation

j3#A1j2#A

2j#A

3"0,

where

A1"e

I#e

L#2( f#g)#I*

2(b

I#b

L)!b

IS*2,

A2"( f#g)2#e

I(e

L#2f#g)#e

L( f#2g)

!bIbLI*2S*2!(b

I(e

L#2f#g))S*

2

!bL

f R*2#b

IbLI*22

#(bI(e

L#2f#g)#b

L(e

I#f#2g))I*

2

and

A3"( f#g) (b

IbLI*22

#(bI( f#e

L)#b

L(g#e

I)

!bIbLS*2)I*

2#e

I( f#e

L)#ge

L

!bIS*2( f#e

L)!b

LfR*

2).

The Routh}Hurwitz criteria for the stability ofthe equilibrium are A

1'0, A

3'0 and

A1A

2!A

3'0 (May, 1974; Gurney & Nisbet,

1998). As mentioned previously, the transition

from stability to instability involves a zero realroot rather than a complex root with zero realparts. Since there is no oscillatory instabilityinvolved, the important criterion for stability isA

3'0 (Gurney & Nisbet, 1998).A little manipulation reveals that A

3'0 if

I*2'

!b@#Jb@2!4a@c@2a@

,

where

a@"bIbL,

b@"bI( f#e

L)#b

L(g#e

I)!b

IbLS*2

and

c@"( f#eL)(e

I!b

IS*2)#ge

L!b

LfR*

2. (6)

But, I*2"(!b#Jb2!4ac)/2a with a, b and

c de"ned as in eqn (3). Also S*2(f/( f#g) and

R*2(g/( f#g). Comparing eqns (3) and (6) re-

veals that b(b@ and c(c@. Hence, the inequalityin eqn (6) is always satis"ed. The internal "xedpoint is stable when it exists.

In summary, when the internal "xed point ispositive (persistence), the trivial "xed point be-comes unstable, i.e., the successional species caninvade when rare. The internal "xed point, when-positive, is asymptotically stable.

We next investigate how the four key para-meters (colonization rate, extinction rate, distur-bance frequency and rate of habitat succession)in#uence equilibrium patch occupancy. The func-tional form of I*

2reveals that it depends only on

the ratios f/bI, g/b

I, e

I/b

I, e

L/b

Iand b

L/b

I. We

use 1/bIas the time scale and analyse I*

2in terms

of these scaled parameters. This approach allows usto emphasize the temporal scaling of patch dyna-mics with extinction}colonization dynamics.

Figure 2 summarizes the main results of thisanalysis. Panels 2a and 2b illustrate how thescaling of disturbance frequency and per patchcolonization rate of occupied, suitable patches( f/b

I) in#uences equilibrium patch occupancy

(I*2). Note that the colonization rate depends on

the life history attributes of the species such as

FIG. 2. Change in equilibrium patch occupancy (I*2) with ratios of key parameters. Panel (a) depicts the relationship

between I*2

and f/bIfor g/b

I(1 (solid curve) and g/b

I'1 (dashed curve) when e

I/b

I"0. Note that when gP0 or fPR,

I*2P1!e

I/b

I, the equilibrium fraction of occupied patches in the Levins model. Panel (b) illustrates the same set of

relationships when eI/b

I'0. Panels (c) and (d) depict how I*

2changes with g/b

Ifor e

I/b

I"0 and e

I/b

I'0, respectively. In

each panel, the solid curve is for f/bI'1 and the dashed curve, f/b

I(1. Other parameter values are as follows: b

L/b

I"0.001,

eL/b

I"0.5.


fecundity and dispersal, while disturbance fre-quency is determined by processes extrinsic to thespecies' biology. The key result here is the asymp-totic relationship between I*

2and f/b

I. In other

words, increasing the disturbance frequency rela-tive to the colonization rate yields diminishingreturns on I*

2. The exact nature of this relation-

ship depends on how the rate of succession of thehabitat scales with the colonization rate (g/b

I).

For instance, when succession rate is low relativeto colonization rate, the species will establisheven at very low values of f/b

I. On the other

hand, when the rate of habitat succession is fasterthan the species' ability to colonize new patches,then there is a threshold value of f/b

Ibelow which

the species will not be able to establish itself[compare the two curves in Fig. 2(a)]. Thisthreshold can be calculated using the persistence

criterion in eqn (4):

AfbIBcr

"!AeL

2bI

#gbL!(e

I#e

L)

2bI(b

I!e

I) B

#SAeL

2bI

#gbL!(e

I#e

L)

2bI(b

I!e

I) B

2#

geL(g#e

I)

b2I(b

I!e

I).

(7)

The qualitative nature of the relationship be-tween I*

2and f/b

Idoes not depend on the scaling

relationship between extinction rate and the col-onization rate. For example, species that arehighly vagile or long-lived (e

I/b

IP0), exhibit the

same asymptotic relationship as do species thatare relatively sedentary or short-lived [e

I/b

I'0;


compare Figs 2(a) and (b)]. There is a quantitat-ive di!erence in that the former will occupya higher proportion of the habitat at equilibriumcompared to the latter for all values of f/b

I.

Figures 2(c) and (d) illustrate how the scaling ofhabitat succession rate with colonization rate in-#uences equilibrium patch occupancy. Equilib-rium patch occupancy declines as the rate ofsuccession increases relative to colonization rate.This decline is faster than linear when distur-bance frequency is low relative to the coloniz-ation rate. It is roughly linear when disturbancefrequency is high [Fig. 2(c)]. As before, the scal-ing of e

Iwith b

Idoes not change the qualitative

nature of the relationship between I*2

and g/bI. It

merely lowers the equilibrium patch occupancyfor relatively sedentary or short-lived species (cf.Figs 2(c) and (d)].

These results can be compared with predic-tions of classical metapopulation theory (Levins,1969, 1970; Gilpin & Hanski, 1991). Metapopula-tion theory predicts that a species will persist inthe landscape as long as the colonization rateexceeds the extinction rate (e

I/b

I(1). In our

model where patches may change from beingsuitable to unsuitable on a time scale comparableto extinction}colonization dynamics, persistencewill additionally depend on the (i) net rate atwhich suitable patches arise in the landscaperelative to the species' colonization ability, and(ii) longevity of dormant stages of the life cyclerelative to the disturbance frequency.

Biological Examples

The framework we have presented captures theessence of some patchily distributed communi-ties. For example, the Florida scrub is a shrub-dominated habitat that is subject to periodicburns (Quintana-Ascencio & Menges, 1996;Menges & Kimmich, 1996). The dominant shrub,Florida rosemary (Ceratiola ericoides) is elimi-nated almost completely by "re, and recoversslowly from a soil seed bank. Herbaceous peren-nials such as the endemic Eryngeum cuneifoliumare restricted to the open patches created by "res.This is largely due to below-ground competitionand allelopathy from the dominant shrubs(Menges & Kimmich, 1996). The herbs increasein abundance when openings are created by "res.

As time since "re increases the shrubs start tospread into the open patches, leading to dramaticreductions in survival and fecundity ofE. cuneifolium and other gap-dependent species(Abrahamson et al., 1984; Johnson & Abraham-son, 1990; Menges & Kohfeldt, 1995). Coexist-ence of herbs and shrubs thus requires frequent"res relative to the longevity of the herbs and therate at which the dominant shrubs colonize openpatches (Menges & Kimmich, 1996).

A similar situation occurs in the "re-proneshrublands and woodlands of south-eastern Aus-tralia (Keith & Bradstock, 1994; Gill et al., 1995;Keith, 1996). Dominant shrubs that are seroti-nous, obligate seeders decline when "res are fre-quent relative to juvenile longevity. In contrast,legumes that have buried, dormant seeds arestimulated to germinate by "re (Bradstock et al.,1998). A third example comes from the wetlandprairies of the Willamette valley in westernOregon (Pendergrass et al., 1999). Prairie plantspecies such as the endangered Bradshaw's pars-ley (¸omatium bradshawii) are adapted to survivefrequent disturbances created by fall season "res.In the absence of natural and anthropogenic "res,woody species encroach and threaten to displaceprairie natives such as ¸. bradshawii (Pendergrasset al., 1999).

As the above examples suggest, periodic distur-bances are necessary for the persistence of earlysuccessional species such as the Florida scrubendemics and the prairie plant species of westernOregon. Proper management of such species re-quires knowledge of the minimum disturbancefrequency that allows long-term persistence.We use the theory developed above to addressthis problem for the herbaceous endemicE. cuneifolium. We take two approaches. First, wecompute the minimum "re frequency f

prequired

for long-term persistence (I*2'0). Second, we

derive the minimum "re frequency fxrequired for

maintaining the population at a prescribed levelof occupancy (I*

2'x, x"0.1, 0.2, etc.). We set

x"0.1 for illustrative purposes. For species suchas E. cuneifolium that have restricted ranges, 10%occupancy provides a more realistic criterion forpersistence in a stochastic environment. Theopen patches colonized by E. cuneifolium areeventually crowded out by invading shrubs.Hence opportunities for natural extinction of

FIG. 3. Minimum "re frequency required for persistence [ fp, (a)] and 10% habitat occupancy [ f

0.1, 3(b)] as a function of

seed bank longevity (eL) and colonization rate of empty, suitable patches (b

I). Parameter values are based on available

information: eI"0, b

L"0 and g"0.025}0.05. When e

L'0, both f

pand f

0.1decline rapidly with b

Iregardless of the

magnitude of eL. This qualitative relationship holds for the observed range of g values. There is a quantitative di!erence

between two measures of "re frequency such that fp(f

0.1. Panel (c) and (d) depict this discrepancy when e

L"0.00001 for

g"0.025 [panel (c)] and g"0.05 [panel (d)]. Panels (e) and ( f ) depict the discrepancy when eL"0.1 for g"0.025 [panel (e)]

and g"0.05 [panel (f )]. The di!erence between fp

and f0.1

is greatest when eLP0 and decreases at higher values of e

L.


these patches are not realized (i.e., eI"0). Colon-

ization rate of latent patches bL

could conserva-tively be assumed to be zero. Data on time since"re and E. cuneifolium mortality due to encroach-ment by shrubs (Menges & Kimmich, 1996)

suggests above-ground extirpation 20}40 yearspost-"re, yielding a maturation rate g in therange 0.025}0.05 per year. We were unable to "ndpublished information on the rate at which openpatches are colonized by E. cuneifolium (b

I) and


the longevity of the seed bank (1/eL). These are

typically the parameters that are hard to estimatein the "eld. We explore the sensitivity of f

pand

fx

to variation in bIand e

L.

Figure 3 gives the minimum "re frequency re-quired for population persistence ( f

p) as a func-

tion of bI

and eL. When seed bank longevity is

essentially in"nite (i.e., eLP0), the persistence

criterion simpli"es to bIfp'0, suggesting that

when colonization rate is non-zero, even very low"re frequencies can ensure persistence [Figs. 3(a),(c) and (d)]. When the seed bank has "nite lon-gevity (e

L'0) but the colonization rate is low

(i.e., below 0.001), fp

increases by an order ofmagnitude or more [Figs 3(e) and (f )]. However,as b

Iincreases f

pdeclines rapidly, regardless of

the magnitude of eL.

The minimum "re frequency required for 10%habitat occupancy ( f

0.1) exhibits a qualitatively

similar relationship with bI

and eL

[Fig. 3(b)].When e

L'0, f

0.1declines rapidly with b

I. As

with fp, this relationship is not a!ected by "nite

values of the seed bank longevity [Figs 3(e) and(f )]. When e

LP0, long-term habitat occupancy

requires fx'(I*

2g)/(1!I*

2). This means that if the

seed bank is very long-lived, f0.1

'g/9 is su$-cient to guarantee 10% habitat occupancy [Figs3(c) and (d)].

Although both fp

and f0.1

exhibit the samequalitative relationship with colonization rateand seed bank longevity, there is a quantitativedi!erence. For all values of b

Iand e

L, "re fre-

quency needed for 10% habitat occupancy isgreater than that required for non-zero habitatoccupancy. This is to be expected since theformer is a more stringent requirement forpersistence than the latter. This discrepancy isillustrated in Figs. 3(c}f ) for the two cases, e

LP0

and eL'0. The di!erence between f

pand f

0.1appears to be greater when the seed bank isrelatively long-lived (i.e., e

LP0). For example,

when eLis small the "re frequency may need to be

an order of magnitude higher for 10% occupancycompared to non-zero occupancy.

As can be seen from the above analyses, the "refrequency required for long-term persistencedeclines rapidly with increasing rates of coloniz-ation. Seed bank longevity does not a!ect thequalitative nature of this relationship. It doeshowever have a strong quantitative e!ect on "re

frequency when colonization rates are very small(i.e., below 1

1000generations). Management for

a prescribed level of habitat occupancy f0.1

re-quires greater "re frequency than that requiredfor non-zero occupancy ( f

p), but provides for

a more realistic management criterion.

Discussion

We have presented a model that combineselements of both metapopulation and patchdynamics. We have analysed the model using theepidemiological framework. The model leads topredictions about equilibrium patch occupancyas a function of both the species' biology as wellas extrinsic processes such as disturbance andsuccession.

This approach of combining the dynamics ofpatch suitability with extinction}colonizationdynamics is important for several reasons. First,persistence of many species cannot be under-stood within a simple metapopulation frame-work that de"nes patches solely in terms ofoccupancy. For example, for most herbivorousinsects a habitat patch consists of a particularhost plant or a patch of host plants. If the hostplant is an annual or a short-lived perennial,then the dynamics of patch turnover have tobe considered in concert with the extinction}colonization dynamics of the herbivore.

A framework that considers both patch dy-namics and extinction}colonization dynamics isessential to understanding how diversity is main-tained in successional habitats or those subject tofrequent disturbance. In classical metapopulationtheory, disturbances are considered only in termsof their impact on the extinction rates of localpopulations (e.g., Hastings, 1980). Our approachallows one to separate the role of disturbances(e.g., "res, droughts, earthquakes) on species'extinction rates from that due to life history char-acteristics (e.g., longevity). It also providesa straightforward way of incorporating dormantlife cycle stages into the patch occupancy frame-work. There is little or no theory that investigatessuccessional dynamics (but see Caswell & Cohen,1991; Pacala & Rees, 1998). The epidemiologicalapproach allows us to explore the interactionbetween seed accumulation (colonization) andseed dormancy (treated as equivalent to the


occupation of a latent host), an issue that iscentral to the dynamics of early plant succession.It also allows us to make predictions about howextrinsic processes such as disturbances scalewith life history characteristics such as longevity,dormancy and dispersal ability.

The model has the potential to provide broadmanagement guidelines for species that live indisturbance-prone habitats. Model predictionsmay be particularly useful in "re management,and the analytical theory developed here pro-vides a benchmark for more detailed spatiallyexplicit modelling. One result that is particularlyrelevant is the asymptotic relationship betweenthe disturbance to colonization ratio ( f/b

I) and

the equilibrium patch occupancy. This relation-ship allows one to determine a cost-e!ective "remanagement strategy by knowing only threeparameters: colonization rate of the focal species,successional or maturation rate of the habitat(e.g., the rate at which early successional speciesare replaced by late successional species), and thenatural disturbance frequency. Since the qualitat-ive nature of the predictions involving these threeparameters are not altered by how the extinctionrate scales with the colonization rate, the man-agement implications may be applicable to spe-cies with widely di!ering life-history strategies.

One natural community that these predictionsare likely to apply to is the Florida scrub eco-system discussed previously. The "re frequencyis su$ciently low that the dominant shrubs caninvade and displace herbaceous local endemicssuch as E. cuneifolium from open patches. Burn-ing of sites on a more frequent basis has beensuggested as a management strategy forE. cuneifolium (Menges & Kimmich, 1996). Weused the model to predict the critical "refrequency required to maintain E. cuneifoliumpopulations at or around 10% occupancy. A sen-sitivity analysis of the model to the two unknownparameters, colonization rate and seed bank lon-gevity, suggests the former may have a greaterin#uence on persistence than the latter. For in-stance, seed bank longevity has a strong e!ect on"re frequency when colonization rate is below0.001 per year, but no discernible e!ect whencolonization is in the range 0.001}1 per year. If E.cuneifolium has an intrinsically low rate of seed(or pollen) dispersal, or if the distance between

occupied and vacant patches is too large relativeto the species' colonization ability, then seedbank longevity should be taken into account.Otherwise, the rate at which E. cuneifolium colon-izes empty patches is likely to be the informationmost relevant for management. It should benoted that the model is spatially implicit anddoes not consider such factors as patch size andcon"guration, and spatial scale of disturbance(local as opposed to global). An important futuredirection is to examine the robustness of modelpredictions when these aspects of spatial struc-ture are taken into account. The analytical theorydeveloped here provides the basis for interpretingresults from a more detailed spatially explicitmodel.

Interestingly, the sensitivity analysis of theparameterized model for E. cuneifolium suggestsan overwhelming role for colonization in persist-ence. Similar results have been obtained for patchoccupancy models in which species coexist viaa trade-o! between competition and dispersal(Tilman et al., 1994, 1997). These parallels beg animportant question: is this result an artifact of thepatch occupancy models themselves or is therean underlying ecological phenomenon?

Patch occupancy models, by de"nition, put anoverwhelming emphasis on extinction}coloniz-ation dynamics. Hence it is possible that theimportant role of colonization in persistence isa direct consequence of model structure and as-sumptions. However, there is some biological jus-ti"cation for the role of colonization in speciespersistence and coexistence (Hanski et al., 1994;Tilman et al., 1994, 1997; Lei & Hanski, 1998).Fugitive species in general, and early successionalspecies in particular, are able to persist becausethey can preempt superior competitors in time orspace (Harper, 1961; Bazzaz, 1979; Paine, 1979;Brown, 1982; Silander & Antonovics, 1982).A seed bank may allow in situ recolonization ofa previously occupied site, but the species willnevertheless be displaced once a superior com-petitor arrives. The key to persistence lies in theability to colonize more sites, and faster, thansuperior competitors. Clearly, the fugitive lifestyleputs a high premium on colonization ability.

This approach of combining patch dynamicsand metapopulation dynamics using the epi-demiological framework could be extended to


other types of species interactions. For example,the four-patch framework can be interpreted asa mutualistic interaction between an obligate(e.g., a plant that requires animal pollination orseed dispersal) and a facultative mutualist (a gen-eralist pollinator or seed disperser). If the obligatemutualist has a density-independent colonizationrate and the facultative mutualist has a density-dependent colonization rate, then the criterionfor the persistence of both species can be derivedwith a slight modi"cation of parameters. We willpresent a detailed analysis of the mutualismmodel in a subsequent paper.

This research was conducted as part of the workinggroup on Population Management at the NationalCenter for Ecological Analysis and Synthesis(NCEAS), a center funded by NSF DEB-9421535,University of California, Santa Barbara and the stateof California. A postdoctoral fellowship from NCEASsupported P.A. during preparation of the manuscript.We thank J. Moore, W. Murdoch, K. Shea D. Yu andthree anonymous reviewers for helpful comments onthe manuscript.

REFERENCES

ABRAHAMSON, W. G., JOHNSON, A. F., LAYNE, J. N. & PER-

ONI, P. A. (1984). Vegetation of the Archibold BiologicalStation, Florida: an example of the Southern Lake WalesRidge. Fl. Scientist 47, 209}250.

ANDERSON, R. & MAY, R. M. (1991). Infectious Diseasesof Humans: Dynamics and Control. New York: OxfordUniversity Press.

BAZZAZ, F. A. (1979). The physiological ecology ofplant succession. Annual Review of Ecol. Syst. 10,351}371.

BRADSTOCK, R. A., BEDWARD, M., KENNY, B. J. & SCOTT, J.(1998). Spatially-explicit simulation of the e!ect of pre-scribed burning on "re regimes and plant extinctions inshrublands typical of south-eastern Australia. Biol.Conserv. 86, 83}95.

BROWN, K. M. (1982). Resource overlap and competition inpond snails: an experimental analysis. Ecology 63,412}422.

CASWELL, H. & COHEN, J. E. (1991). Disturbance, inter-speci"c interaction and diversity in metapopulations. In:Metapopulation dynamics (Gilpin, M., Hanski, I., eds),pp. 193}218. San Diego, CA: Academic Press.

DAY, J. R. & POSSINGHAM, H. P. (1995). A sotchastic meta-population model with variability in patch size and posi-tion. ¹heor. Popul. Biol. 48, 333}360.

GILPIN, M. & HANSKI, I. (1991). Metapopulation Dynamics:Brief History and Conceptual Domain. London: AcademicPress.

GILL, A. M., BURROWS, N. D. & BRADSTOCK, R. A. (1995).Fire modelling and "re weather in an Australian desert.CA¸M Sci. Suppl. 4, 29}34.

GRENFELL, B. & HARWOOD, J. (1997). Metapopulation dyna-mics of infectious diseases.¹rends Ecol.Evolution 12,395}399.

GURNEY, W. S. C. & NISBET, R. N. (1998). EcologicalDynamics. Oxford: Oxford University Press.

HANSKI, I., KUUSSAARI, M. & NIEMINEN, M. (1994). Meta-population structure and migration in the butter#yMelitaea cinixa. Ecology 75, 747}762.

HANSKI, I. & GILPIN, M. (1997). Metapopulation Biology:Ecology, Genetics and Evolution. London: Academic Press.

HARPER, J. L. (1961). Approaches to the study of plantcompetition. In: Mechanisms in Biological competition(Milthrope, F. L., ed.), pp. 1}39. Symposium No. 15, So-ciety for Experimental Biology. Cambridge: CambridgeUniversity Press.

HASTINGS, A. (1980). Disturbance, coexistence, history andcompetition for space. ¹heor. Popul. Biol. 18, 363}373.

HORN, H. S. & MACARTHUR, R. H. (1972). Competitionamong fugitive species in a harlequin environment.Ecology 53, 749}752.

JOHNSON, A. F. & ABRAHAMSON, W. G. (1990). A note onthe "re response of species in rosemary scrubs on thesourthern Lake Wales Ridge. Florida Scientist 2, 138}143.

KEITH, D. A. (1996). Fire-driven extinction of plant popu-lations: a synthesis of theory and review of evidence fromAustralian vegetation. Proceedings of the ¸innean Societyof New South =ales. vol. 116, pp. 37}78.

KEITH, D. A. & BRADSTOCK, R. A. (1994). Fire and competi-tion in Australian heath: a conceptual model and "eldinvestigations. J. <egetation Sci. 5, 347}354.

LAWTON, J. H., NEE, S., LETCHER, A. J. & HARVEY, P. J.(1994). Animal distributions: patterns and processes.In: (Edwards, P. J., May, R. M. & Webb, N. R., eds),pp. 41}58. ¸arge Scale Ecology and Conservation Biology.Oxford: Blackwell.

LEI, G. & HANSKI, I. (1998). Spatial dynamics of two com-peting parasitoids in a host metapopulation. J. AnimalEcol. 67, 422}433.

LEVIN, S. A., STEELE, J. H. & POWELL, T. M. (1993). PatchDynamics. New York: Springer.

LEVINS, R. (1969). Some demographic and genetic conse-quences of environmental heterogeneity for biological con-trol. Bull. Entomol. Soc. Amer. 15, 237}240.

LEVINS, R. (1970). Extinction. In: Some Mathematical Prob-lems in Biology. (Gerstenhaber, M., ed). Providence, RI:American Mathematical Society.

LEVINS, R. & CULVER, D. (1971). Regional coexistence ofspecies and competition between rare species. Proc. NatlAcad. Sci. USA 68, 1246}1248.

MAY, R. M. (1974). Stability and Complexity in Model Eco-systems. Princeton, NJ: Princeton University Press.

MAY, R. M. & ANDERSON, R. (1979). Population biology ofinfectious diseases: II. Nature (¸ondon) 280, 455}461.

MENGES, E. S. & KIMMICH, J. (1996). Microhabitat and timesince "re: e!ects on demography of Eryngium cuneifolium(Apiaceae), a Florida scrub endemic plant. Amer. J. Bot.83, 185}191.

MENGES, E. S. & KOHFELDT, N. (1995). Life history strat-egies of Florida scrub plants in relation to "re. Bull. ¹orreyBot. Club 122, 282}297.

NEE, S. (1994). How populations persist. Nature (¸ondon)367, 123}124.

NEE, S. & MAY, R. M. (1992). Dynamics of metapopulations:habitat destruction and competitive coexistence. J. Anim.Ecol. 61, 37}40.


NOBLE, I. R. & SLATYER, R. O. (1981). Concepts and modelsof succession in the vascular plant communities subjectto recurrent "re. In: Fire and the Australian Biota (Gill,A. M., Graves, R. H. & Noble, I. R., eds). Aust. Acad. Sci.Canberra.

PACALA, S. W. & REES, M. (1998). Models suggesting "eldexperiments to test two hypotheses explaining successionaldiversity. Amer. Naturalists 152, 729}737.

PAINE, R. T. (1979). Disaster, catastrophe and local persist-ence of the sea palm Postelsia palmaeformis. Science 205,685}687.

PENDERGRASS, K. L., MILLER, P. M., KAUFFMAN, J. B.& KAYE, T. N. (1999). The role of prescribed burning inmaintenance of an endangered plant species, ¸omatiumbradshawii. Ecol. Appl. 9, 1420}1429.

PICKETT, S. T. A. & WHITE, P. S. (1985). ¹he Ecologyof Natural Disturbances and Patch Dynamics. New York:Academic Press.

QUINTANA-ASCENCIO, R. F. & MENGES, E. S. (1996). Infer-ring metapopulation dynamics from patch level incidenceof Florida scrub plants. Conservation Biol. 10, 1210}1219.

REES, M. & PAYNTER, Q. (1997). Biological control ofScotch broom: modelling the determinants of abundanceand the potential impact of introduced insect herbivores.J. Appl. Ecol. 34, 1203}1221.

SILANDER, J. A. & ANTONOVICS, J. (1982). A perturbationapproach to the analysis of interspeci"c interactions ina coastal plant community. Nature 298, 557}560.

SLATKIN, M. (1974). Competition and regional coexistence.Ecology 55, 128}134.

TILMAN, D., MAY, R. M., LEHMAN, C. L. & NOWAK, M. A.(1994). Habitat destruction and the extinction debt. Nature(¸ondon) 371, 65}66.

TILMAN, D., LEHMAN, C. L. & YIN, C. (1997). Habitatdestruction, dispersal and deterministic extinction in com-petitive communities. Amer. Naturalist 149, 407}435.

WU, J. & LOUCKS, O. L. (1995). From balance of natureto hierarchical patch dynamics. Quart. Rev. Biol. 70,439}466.

ZAMMIT, C. A. & ZEDLER, P. H. (1993). Size structure andseed production in even-aged populations of of Ceanothus-Greggii in mixed chaparral. J. Ecol. 81, 499}511.

patch dynamics and metapopulation theory: the case of successional species

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