Quasi-stationary distributions of a stochastic metapopulation model

J. Math. Biol. (1994) 33:35-70 Journol of

MathemaUcal 8k ogy

© Springer-Verlag 1994

Quasi-stationary distributions of a stochastic metapopulation model

Mats Gyllenberg ~, Dmitrii S. Silvestrov 2

1Department of Mathematics, University of Turku, SF-20500 Turku, Finland 2Department of Mathematics, Lule~t University of Technology, S-95187 Lule~, Sweden

Received 5 April 1993; received in revised form 30 November 1993

Abstract. A stochastic metapopulation model which explicitly considers first order interactions between local populations is constructed. The model takes the spatial arrangement of patches into account and keeps track of which patches are occupied and which are empty. The time-evolution of the metapopulation is governed by a Markov chain with finite state space. We give a detailed description of the long term behaviour of the Markov chain. Many interesting biological issues can be addressed using the model. As an especially important example we discuss the so-called core and satellite species hypothesis in the light of the model.

Key words: Markov chain with finite state space - Discrete renewal theorem Ergodic theorem - Patchy environment - Incidence function - Core-satellite species hypothesis

1 Introduction

Most population models, both deterministic and stochastic, usually assume that all individuals of the population live in the same habitat and interact homogeneously with each other. Models of this type have successfully been used to describe, explain and predict the local dynamics of one or several interacting species.

Natural populations of most species have a spatial structure: There are several geographically distributed habitat patches that can support local populations. Such a population of populations is called a metapopulation. Local populations in a metapopulation are connected by migration. A local population may go extinct due to a catastrophe, while the metapopulation persists. An empty patch may be colonized by migrants from other patches. Extinction and recolonization are the essential features of metapopulation

36 M. Gyllenberg, D. M. Silvestrov

dynamics. Hanski and Gilpin (1991) even characterized the study of metapopulation dynamics as the study of conditions under which these two processes are in balance and the consequence of this balance to associated processes.

There are many important questions that require the metapopulation concept to be appropriately analysed. For instance island biogeography and conservation biology are two important areas where metapopulation dynamics plays a prominent role (see the book edited by Gilpin and Hanski 1991 for many other examples).

The simplest deterministic patch model, that is, a model where the time- evolution of the fraction p( t ) of occupied patches is modelled, was introduced by Levins (1969, 1970) and has the following form:

dp( t ) = cp( t ) (1 - p( t ) ) - e p ( t ) . (1.1)

dt

Here c and e are the colonization and extinction parameters, respectively. A model very closely related to (1.1) is the main land- i s land model

dp( t ) - c(1 - p( t ) ) - e p ( t ) , (1.2)

dt

which is a single-species version of a model by MacArthur and Wilson (1967). In (1.2) colonization is assumed to occur as a result of migration from the mainland and the colonization rate is thus directly proportional to the fraction of empty patches, whereas in (1.1) empty patches are colonized by migrants from the occupied patches and so the colonization rate is assumed to be proportional to the product of the fractions of empty and occupied patches.

The equations (1.1) and (1.2) make several simplifying assumptions. First of all, since they are deterministic, it is tacitly assumed that the number of patches is infinite. Secondly, all patches are assumed to be identical, whereas in nature there is always variation in patch size and quality. Thirdly, the spatial arrangement of the habitat patches is completely ignored: An isolated patch which is located far away from the mainland or the other patches has the same probability of being colonized as a patch close to the mainland or in the middle of a cluster of patches. Moreover, in the Levins model (1.1) the colonization rate depends only on the fraction of occupied patches and not on the specific patches that are occupied. It is clear that in real life an empty patch surrounded by nearby occupied patches is more likely to be colonized than a patch whose neighbouring patches are all empty. Finally, local dynamics is ignored and the models therefore neglect the effects of migration on local dynamics.

To relax Levins's assumptions that all patches are identical and that local dynamics is not affected by migration, Gyllenberg and Hanski (1992) and Hanski and Gyllenberg (1993) introduced a structured metapopulation model. This model incorporated the rescue ef fect (the decreasing extinction rate with increasing fraction of occupied patches) in a natural way. Being

Quasi-stationary distributions 37

deterministic this model was still based on the assumption of a very large number of patches. Since colonization is an essentially stochastic event and since local population densities are low immediately after colonization, the purely deterministic modelling of the colonization event caused problems that have not yet been completely resolved. The models by Gyllenberg and Hanski (1992) and Hanski and Gyllenberg (1993) also neglect the spatial structure of the habitat patches and information about which of the patches are actually occupied.

Stochastic metapopulation models are comparatively rare in the literature. Levins (1970) and Hanski (1982) used a version of(1.1), where c and e were not constant but random variables, thus allowing for temporal variance in the colonization and extinction rates. Nisbet and Gurney (1982, Chap. 6.8) presented a patch model in the form of a stochastic differential equation with white noise. In that model the number of patches was indeed finite but all the other simplifying assumptions of the Levins model were made. Verboom et al. (1991) used a similar stochastic version of Levins's model.

Several authors have considered stochastic population models incorpora- ting migration. Bailey (1968) introduced a simple continuous-time birth, death and migration process, where the individuals were distributed over a one-, two-, or three-dimensional lattice of discrete points, and migration took place between neighbouring sites only. Bailey's (1968) model has subsequently been elaborated and extended by, among others, Adke (1969) and Davis (1970). Kingman (1969) treated a continuous-time Markov chain model of a metapopulation in a finite number of patches. He assumed that the transition probabilities describing arrival of an individual at and departure of an individual from a specific patch depend only on the state of the local population inhabiting that patch. The transition rate describing transfer of an individual from one patch to another depended on both local populations involved. For a review of models of this type - so called simple stepping-stone models see Renshaw (1986). Since these models do not focus upon local extinction and recolonization and, in fact, do not allow for local extinction except in the case where the last surviving individual at a patch moves away, they are not true metapopulation models.

Durrett (1991) considered a continuous-time contact process, that can be interpreted as a metapopulation model with the following assumptions. All local populations have the same extinction rate and they all produce migrants at the same rate. A migrant chooses its new patch at random from the nearest neighbours. If it arrives at an empty patch, this will be colonized and if it arrives at an occupied patch nothing happens. Bramson et al. (1991) investigated a similar contact process where they took variation of patch quality into account by dividing the patches into "good" and "bad" and assuming a higher extinction rate in the bad patches. Both these models assumed that immigration does not affect local extinction and hence they did not incorporate the rescue effect. For more information about applications of the theory of interactin9 particle systems to spatial models i n ecology the readers are referred to the paper by Durrett and Levin (1993) and the references therein.


Hanski (1992) employed a very simple model where the presence/absence of a species in a given patch was modelled as a two state Markov chain with given constant extinction and colonization probabilities. Assuming that these probabilities depend on the size (area) of the patch according to a certain functional form, Hanski (1992) could even from this simple model draw ecologically significant conclusion about for instance the role played by environmental and demographic stochasticity for local extinction.

A concept of fundamental importance in stochastic metapopulation models is that of incidence (of occupancy). The incidence of a given patch is defined by Hanski (1994) (see also Gilpin and Diamond 1981) as the stationary probability that the patch is occupied. Incidence functions describe how the incidence depends on patch size or some other patch characteristic (Diamond 1975). Thus for instance in the two state Markov chain model mentioned above, the incidence J(A) as a function of patch area A is given by

C(A) J(A) - (1.3)

C(A) + E(A)'

where C(A) and E(A) are the colonization and extinction probabilities as functions of area.

Hanski (1994) extended the two state Markov chain model to explicitly take the spatial arrangements of patches and information about which patches are occupied into account. He did so by assuming that the colonization probability C~ of patch i is a function F(M~) of the expected number Mi of migrants arriving to patch i per year. Mi was assumed to depend on the distances to the occupied patches and their areas in the following way:

Mi = fl ~ xje-~a'~Aj. (1.4) j4=i

Here dij is the distance between patch i and patch j, Aj is the area of patch j and x~ is 1 or 0 according to whether patch j is occupied or not. As the specific functional form of F, Hanski (1994) chose

Ci = F(Mi):= M2 + V2, (1.5)

where V is a constant. Hanski (1994) did not formulate or analyse the Markov chain resulting

from the above assumptions. Without justification he assumed that at steady- state the variation of M~ is small and that a good approximation for the incidence is obtained by substituting (1.4) and (1.5) with the mean value for M~ into (1.3).

It is the purpose of this paper to formulate and analyse a general Markov chain patch model that explicitly considers first order interactions between patches. In particular, the spatial arrangement of occupied patches is taken into account. Being a patch model, local dynamics apart from extinction and colonization events are ignored, but the rescue effect is included.


As explained above, the stationary or long term behaviour of the metapopulation is of utmost importance. We shall therefore give a rather precise description of the asymptotics of the model. This description includes expressions for the incidence and the asymptotic distribution of the number of occupied patches.

If the collection of patches contains a 'mainland', that is, a patch where the local extinction probability is zero, then it is clear that the metapopulation will persist (provided the mainland is initially inhabited) and it follows from well-known facts about finite Markov chains that with appropriate assumptions there exists a unique stationary probability distribution towards which the t-step transition probabilities converge from all initial states with the mainland inhabited. If, on the other hand, all local extinction probabilities are positive, then the metapopulation will go extinct in finite time with probability one. In nature one usually observes even in this case a metapopulation that persists for a very long time (longer than the period of study) and one can also observe a convergence towards a distribution that resembles a stationary distribution.

There are essentially two different situations in which convergence towards a so-called quasi-stationary distribution can take place. In the first one the number of patches is very large, so the probability that all local populations will simultaneously go extinct is small and hence the expected life-time of the metapopulation is large. A precise mathematical description of the convergence towards the quasi-stationary distribution involves a limiting procedure, where the number of patches tend to infinity at the same time as time tends to infinity. This procedure will not be carried out in this paper and is left to later investigations. In the other situation, which will be considered in this paper, at least one patch is so large (resembles a mainland) that the extinction probability of its local population is close to zero. If this patch is initially inhabited the expected time to metapopulation extinction is very long and before this extinction takes place, the distribution will come close to a quasi-stationary distribution. Again the exact mathematical formulation depends on a limiting procedure - this time a small parameter e (corresponding to the extinction probability of the local population inhabiting the 'quasi-mainland') tends to zero as time tends to infinity. It turns out that the relation between the speeds at which e tends to zero and time to infinity has a delicate influence on the asymptotic behaviour.

The main mathematical methods used to describe the asymptotic behaviour of our model are based on renewal theory and perturbation results for Markov chains. We prove a theorem on the asymptotic behaviour of limits of solutions to discrete renewal equations, which contains the usual discrete renewal theorem as a special case, and an ergodic theorem for perturbed Markov chains. These general results are then applied to the metapopulation model under consideration.

In Sect. 2 the metapopulation model is derived. In Sect. 3 we recall some basic facts about Markov chains and adapt these to the specific Markov chain describing the metapopulation. In Sect. 4 some simple results on absorption


and ergodicity are given. Sections 5 and 6 form the core of the paper. Here the new general theorems mentioned above and the main results for the metapopulation model are formulated. In Sect. 7 we consider three simple but biologically interesting examples. The biological significance of the model, especially with respect to the core and satellite species hypothesis of Hanski (1982) is discussed in Sect. 8. Complete proofs of all the theorems can be found in the appendix.

2 The model

We consider a collection of n patches that at the discrete time instants t = 0, 1, 2 . . . . can be either occupied or empty. The state of patch i at time t is given by the random indicator variable th(t) that takes on the value 1 if patch i is occupied and 0 if patch i is empty at time t. The state of the metapopulation is described by the vector random process with discrete time 0 ( t )= (t/l(t) . . . . , q,(t)), t = 0, 1 . . . . . The state space of the process 6(t) is the n-dimensional hypercube J ( = { f f = ( X l , . . . , x , ) : xi ~ {0, 1} }. It has 2" states.

The local dynamics is modelled by preassigning the n by n interaction

matr ix Q = (q~i). Here q,, i e {0, 1 . . . . . n} is the probability that, in the absence of migration, the population inhabiting patch i will go extinct in one time-step, qj~ is the probability that patch i will not be colonized in one time-step by a migrant originating from patch j. Typically qj~ depends on at least the distance between the patches i and j and the area of patch j in a way similar to (1.4). Since the qj~ are probabilities we have to assume that 0 < qji < 1. We shall assume that the local extinction processes and the colonization attempts from different local populations are all independent. As a consequence of this independence the conditional probabilities q~(2) for patch i to be empty at moment t + 1 under condition that at moment t the metapopulation was in a state ff = ( x l , . . •, x,) are given by the product

q~(~) = ( I qJ~J, i~{1 . . . . . n } , x e X , j = l

(2.1)

where we have used the convention 0 ° = 1. It should be noted that the functional form (1.5) used by Hanski (1994) is

based on the assumption that there are interactions between migrants and that the different colonization attempts are not independent.

Notice that our model incorporates the rescue effect. The over-all extinction probability of the local population inhabiting patch i may be considerably less than the "internal" extinction probability qii if there are many large occupied patches in the vicinity (many small qji).

Having described the local patch dynamics we can deduce the law govern- ing the time-evolution of the process 0(t) giving the state of the metapopulation. The process fl(t) is a homogeneous Markov chain with state space )( and


t ransi t ion probabil i t ies

P(Y,Y) = [I qi(x) 1 r'( 1 - qi(Y))Y',Y,Y ~ X . (2.2) i = 1

Note that the process q(t) is complete ly determined by the interact ion matr ix Q.

3 Basic properties of Markov chains

In this section we recall some basic definitions f rom the theory of stochastic processes. We then formula te condit ions on absorp t ion and recurrence for the specific M a r k o v chain of our me tapopu la t ion model. These condit ions are impor t an t for the asympto t ic behaviour of the metapopula t ion . F o r more details the reader is referred to Hoel et al. (1972) or any other text on stochastic processes.

T h r o u g h o u t the paper we let t/(t) denote a general M a r k o v chain with state space X. The elements of X are denoted by x, y . . . . . Fo r the specific M a r k o v chain of Sect. 2 we keep the no ta t ion O(t) for the process, Jf for the state space, and 2, )7 . . . . for the elements of J~.

We denote by Px the probabi l i ty measure induced by the M a r k o v chain t/(t) s tar t ing f rom initial state x and by Ex the corresponding expectat ion. Let

r(B) = min{t => 1 :t/(t) ~ B} (3.1)

denote the first hitting time for a set B c X. The hitting probabilities h(x, B, A), x ~ X, A c B c X are defined as follows:

h(x, B, A) = Px{t/(z(B)) ~ A, z(S) < oo } . (3.2)

If the subset A equals B we simply write h(x, B) = h(x, B, B). In the case of a one-point set {y} we simplify the no ta t ion further by writ ing r ({y}) = z(y) and h(x, {y}) = h(x, y).

The quant i ty h(x, y) is the probabi l i ty that state x leads to state y in finite time. If

h(x, y) > 0 , (3.3)

we s imply say that x leads to y. A state y is called recurrent if h(y, y) = 1 and transient if h(y, y) < 1. A set B c X is called recurrent (transient) if all its states are recurrent (transient). A state y is called absorbing if the transi t ion probabi l i ty P(y, y) = 1 or, equivalently, if P(y, z) = 0 for all z e X \ { y } .

A n o n e m p t y set B c X is said to be closed if no state inside B leads to any state outside B, that is, if

h(x, y) = O, x e B, yq~B. (3.4)

A closed set B is called irreducible if x leads to y for all choices of x and y in B. It is wel l -known that if B is an irreducible set of recurrent states, then h(x, y) = 1 for all x and y in B.


A set B c X is said to be ergodic if for all x, y ~ B the t-step transition probabilities P~{t/(t) = y} converge to n(y) > O, with ~y~Bn(y) = 1 as t ~ ~ .

Consider a discrete probability distribution {f(t)}~=o which may be • . 09 t defective, that is, f (t) => O, t = O, 1,. and ~k=Of( ) -----< 1. The greatest com-

mon divisor of all t for which f ( t ) > 0 is called the period of {f(t)}2=o. A distribution with period 1 is said to be aperiodic•

The first return time of a state x is the random variable having as distribution Px {z(x) = t}. The period of the state x is by definition the period of this distribution• A state with period 1 is called aperiodic. All states in an irreducible set have the same period and it therefore makes sense to speak of the period and aperiodicity of an irreducible set.

For a Markov chain with finite state space any irreducible, aperiodic set is recurrent and ergodic.

We now specialize to the Markov chain g/(t). It is clear that the state 0 = (0 . . . . . O) corresponding to metapopulation extinction is an absorbing state. Next, we formulate conditions that provide a natural communicative property of the non-zero states. Among other things these conditions imply that every non-zero state leads to all other states.

We shall throughout the paper assume that the interaction matrix satisfies the following condition:

A: (l) qji > O, j #: i; (2) qu < 1, iE {1 . . . . . n}; (3) For each pair (j, i) of patches, j, i ~ {1 . . . . . n}, there exist an integer m and

m

a chain of indices j = io . . . . . i,, = i such that l--[k= 1(1 -- qi,-li~) > O.

Condition A1 means that no local population is able to colonize another patch in one time-step with probability 1. Condition A2 means that even in the absence of migration (rescue effect) no local population has extinction probability 1. Finally condition A3 means that every local population is able to colonize with a positive probability any other patch either directly or through a chain of patches (stepping-stone dispersal).

Condition A is not the weakest one which implies the asymptotic behaviour of O(t) proved in this paper, but it is a natural and non-restrictive condition.

The extinction probabilities qu, i~ {1 . . . . . n} play a key role for the behaviour of the metapopulation. Consider first the absorption condition

Bo: qu > O, i ~ {1 . . . . . n } .

When Bo holds the set X\{O} consists of transient states and all absorption probabilities h(ff, O) = 1, 2 ~ X'\{O}. Therefore, starting from any state

E X\{O} the system will eventually be absorbed in O. This situation corresponds to a system of patches with no mainland•

If Bo does not hold, then qii = 0 for at least one patch i6 { 1 , . . . , n}. We shall consider the special case where the local population of only one patch


has zero extinction probability:

BI: qll = 0 , a n d q u > 0 f o r i > 1.

Patch 1 plays the role of the mainland. Denote by D~ the set of states ~ = (1, x2 . . . . . x,) with the first coordinate

equal to 1 and arbitrary values of the other coordinates and let Do = Jf\(DlU{0}). The state space )( is thus decomposed into a disjoint union of O1, Do and {0}.

When condition B1 holds the set DI is an irreducible set. Therefore, starting from any state Y ~ D~ the system never exits D1 and never reaches the absorbing state 0. We call condition B~ the ergodic condition and the set D~ the ergodic class.

The set Do consists of transient states and

h(f, D1) + h(Y, 0) = 1 (3.5)

for 2 ~ Do. Therefore, starting from any state ff E Do the system will exit Do in a finite time with probability 1 and either enter the ergodic class D1 or the absorbing state 0. We call the set Do the transient class.

There are similar results if instead of B1 we assume the more general ergodic condition with several extinction probabilities qu = 0, but this more general situation will not be treated in this paper.

4 Asymptotic behaviour of t-step transition probabilities

In this section we formulate two simple results for the asymptotic behaviour of t-step transition probabilities. Both are adaptions of general results for Markov chains with finite state space (see e.g. Feller 1968, Chap. 15) to the specific Markov chain 0(t) under consideration and therefore require no proofs. The first proposition, which is simpler, is concerned with the absorption situation, the second one with the ergodic situation.

Proposition 4.1 Let the absorption condition Bo hold. For all initial states ~2 ~

{01 if 35~ Jr\{0} , (4.1) lirn P~{tT(t ) = 35} = i f 35 = 0 .

We now turn to the ergodic case. Let the ergodic and transient classes D1 and Do be as defined in the preceding section.

Proposition 4.2 Let the ergodic condition B1 hold. For all initial states ~ ~ D1

if 35e D 1 , ,-~limoo i f 35 e Do u {0} (4.2)

where n(29), 35 e D1 are stationary probabilities for the states in the ergodic class D1. The probability vector {n(35): 35 e D1 } is the unique solution of the following


system of linear equations

~z(#) = ~ rc(Y)P(£, 37), # 6 D~, ~ rc(£) = 1 . (4.3)

The probability h(Y, D1) that the process starting at £ ~ Do enters the ergodic class satisfies 0 < h( f;, D1) < l for all 2 ~ Do. The vector {h(ff, D1):x 6 Do} is the unique solution of the following system of linear equations

h(X, D1) = ~ P(x , i ) + ~ P(~Y,#)h(#,Di), ~ e D o . (4.4) ~ D 1 # ~ D o

For all initial states ~ ~ Do

(~(~,Dt)~(#) i f # ~ D ~ ,

lim P~{O(t) = #} = if y e Do , (4.5)

' ~ ~ [ h ( 2 , O) if # = 0 ,

where the stationary probabilities ~(#) and the probabilities h(2, D1) are given by (4.3) and (4.4), respectively.

The validity of Proposition 4.2 depends in an essential way on condition A. For instance the uniqueness of solutions of (4.4) and (4.3) is a consequence of condition A. The asymptotic behaviour of t-step transition probabilities depends on periodicity properties of the ergodic class D1 and it follows from condition A that the class D1 is aperiodic.

Proposition 4.2 can be used to derive expressions for biologically important quantities. We shall be content with expressions for the incidence of the i th patch (the stationary probability that this patch is occupied) and the stationary distribution of the number of occupied patches. These quantities are defined by

Ji = ~ ~r07) (4.6) y l = I

and

p(k) = ~ To(#), Yl + "'" +Yn = k

respectively.

Corollary 4.3 Let the ergodic condition B1 hold. Then

lim P~{q,(t) = 1} = {Ji i f £ e Da , t - ~ h(~, D1)J i i f "2 ~ D o ,

and

. . . + . . ( t ) = k} = f p(k) [h(x, D1)p(k )

i f ff~D1 , i f #: ~ Do .

(4.7)

(4.8)

lim P~{~/1 (t) + (4.9) t - + o o


5 Renewal and ergodic theorems

Proposition 4.1 showed that if there is no mainland the metapopulation will certainly go extinct. But as mentioned in the introduction one can in nature still observe a convergence towards a quasi-stationary distribution. To give a precise mathematical formulation of this phenomenon, we consider our Markov chain as a perturbation of another Markov chain satisfying the ergodic condition B1. The perturbation depends on a small parameter which is of the same order as the extinction probability of patch 1 (the 'quasi-mainland').

Let Q = !qji) be an interaction matrix defining the Markov chain q(t) and let Q ( ) = (q~i ~) be interaction matrices defining Markov chains O(~)(t). We assume that the interaction probabilities depend on the parameter e in such a way that the following condition of asymptotic ergodicity is satisfied.

C: (1) q(~) * ' = qji + eqii + o(e) for e e (0, 1], i , j E {1 . . . . . n}, where 0 N qji < o0, (2) Condition A holds for all matrices Q(~), e e (0, 1] and Q; (3) The ergodic condition B1 holds for the limiting matrix Q.

We are interested in the large time behaviour of the metapopulation. The exact notion of 'largeness' turns out to be crucial for the results. As long as the quasi-mainland remains occupied we can expect the system to behave as in the ergodic situation. When the local population on the quasi-mainland has gone extinct the quasi-mainland can either be recolonized or the whole metapopulation goes extinct. The number of such recolonizations of the quasi- mainland before metapopulation extinction is a random variable with a distribution that resembles the geometrical distribution. As the expected life-time of the quasi-mainland population is of the order 1/e the expected life-time of the metapopulation is of the same order and we anticipate that the asymptotic behaviour depends on whether a 'large' time is considerably less than or greater than 1/e.

We shall investigate the behaviour of the t-step transition probabilities P~{q(~)(t) = y} as e-- , 0 and t - - , oo simultaneously. To make the above intuitive ideas precise we assume that time t = t~ is also a function of the parameter e such that the following condition is satisfied

D: t ~ oo a s e - + 0 , and e t ~ - + a s e ~ 0 , w h e r e 0 N s N ov .

Let as before the stationary probabilities n(y) and the probabilities h(2, D1) be given by (4.3) and (4.4), respectively. Let P(~)(2, y) be the transition probabilities of the perturbed system. We shall prove in the appendix that under condition C, P(~)(2~, y) can be expanded as

P(~)(2, y) = P(X, 37) + ~P(2, 37) + o(e), (5.1)

for all 2, 37 ~ D~ w Do. If Y E D 1 and 37 ~ Do w {0}, then P(£, 37) = 0 and

P(x, 37) = 1--[ ~,(x)l-",(1 - ~,(x))y,c)~(x), (5.2) i ¢ 1


where

qj, q~, i , 1, 0~(2) = 1~ q j l q l l - (5.3) j=l=l j=t=l

Finally, define the constant 2 by

2 = ~ n()~)(/3(2,0)+ ~ P(2, )5)h(y, 0 ) ) . (5.4) .~eDt ~eDo

We are now ready to formulate the main result of the paper.

Theorem 5.1 Let conditions C and D hold. Then

(i) For any initial state 2 ~ D 1

I~ -asT"c(y) i f y e D 1 ,

lim P~{0(")(t~) = y} = i f 37 e Do , (5.5) e ~ 0

( 1 - - e -a* i f ; = O ,

(ii) For any initial state 2 ~ Do

t~ -~Sh(?2, D1)n(y) i f y e D 1 ,

lim P~{0(~)(t,) = 37} = i f )7 ~ Do , (5.6)

e~o { 1 - - e - a S h ( 2 , D1) i f y = O .

From Theorem 5.1 we immediately get expressions for the incidence and the asymptotic distribution of the number of occupied patches.

Corollary 5.2 Let conditions C and D hold. For any initial state 2 ~ D1 we have

lim P~{tff)(t~) = 1} = e-ZSJi, i e {1 . . . . . n}, (5.7) ~--*0

and

re-aSp(k), if ke {1 . . . . . n} (5.8) lim P~{t/~*)(t~) + . . . + t/(~)(t~) = k} = (1 - e -~s if k = 0 , g-*O

where Ji and p(k) are given by (4.6) and (4.7), respectively.

Theorem 5.1 calls for some explanation. As expected the parameter s has a decisive influence on the asymptotic behaviour.

When s = 0 the time t~ grows slowly compared with I/t, that is, t, is asymptotically considerably less (in fact on a different time scale) than the expected life time of the quasi-mainland population. We therefore expect the metapopulation to behave as a system in the ergodic situation. Indeed, in this case the factor e -as takes on the value 1 and the limiting expressions in (5.5) and (5.6) coincide with limiting expressions in (4.2) and (4.5), respectively.

When s = oo the time t~ is also on a different time scale than the expected life time of the quasi-mainland population but this time it is much larger. Now the factor e-as vanishes and consequently the limiting expressions in (5.5) and (5.6) coincide with the limiting expression in (4.1) for the absorption case.


The intermediate quasi-stationary case 0 < s < ~ is the most interesting. It corresponds to the situation when the rate of growth of t~ and 1/e are of the same order, in other words, extinction of the quasi-mainland population happens on the time scale of t,. The asymptotic behaviour of the metapopulation therefore exhibits components typical of both the ergodic and the absorption situations. This is reflected in the formulae through the factor e -zs depending on the perturbation coefficient oh,.

As a first step towards a complete understanding of the factor e ~s we notice that

P~{z(~)(~) = oo } 2 = lim , (5.9)

~ o ~(~)

where the first hitting time ~(~)(ff) is as defined by (3.1) but with 0(t) replaced by 0(~)(t). Formula (5.9) holds for all ~ e DI and the value of 2 is independent of the choice of 2 e DI. This means that asymptotically 2e7c(2) is the probability that an initial state 2 in the ergodic class DI will eventually lead to the absorption state. As 2s = 2et~ holds asymptotically, we infer that the larger the probability that a state in D1 leads to 0, the more the system resembles a metapopulation in the absorption situation. In the next section we shall derive an expression for the distribution of the metapopulation extinction time that will shed more light over the meaning of the constant 2.

Theorem 5.1 is a corollary of the following more general ergodic theorem for perturbed Markov chains.

Theorem 5.3 Let for each e e [0, 1], t/(~)(t) be a homogeneous Markov chain with finite state space X and transition probabilities P(~)(x, y) satisfying

P(~)(x, y) = P(°)(x, y) + eP(x, y) + o(~), x, y e X . (5.10)

Assume that the state space is decomposed as the disjoint union X = D ] w D o w D , where

(a) D1 ~ 0 is an irreducible, aperiodic set, (b) Do is a transient set, (c) D is a closed set

with respect to the Markov chain t/(°)(t). Let 7z(x), x c Dx be the stationary probabilities of tl ~°) restricted to D1 and let

condition D hold. Then

(i) For x ~ D1

{ ; - ~ ( y ) i f y ~ D 1 , lim Px {tl (~)(t~) = y} = (5.11) ~ o i f y e D o ,

(ii) For x ~ Do

{ho(°)(x, D l w D , D1)e-~S~(y ) i f y e D , , lim P~{q(~)(t~) = y} = (5.12) ~ o i f y e D o ,


where

and

xeD1 y s D o

P(x, D) = ~ P(x, y). (5.14) y ~ D

It is well-known that for all x, y e X the probabili ty (~) .

zxr(t) .= Px{rl(~)(t) = y} (5.15)

satisfies the discrete renewal equation

t (e) (~) (e) , . (e)

k) fx , (k) , t = O, 1, z~y(t) = + . . . . gx,(t) ~ (5.16) zyytr -- k = O

where f x , ( t ) ' = Px{~(~)(y) = t} (5.17)

and (~)

gxy(t) '= P~{~I")(y) > t, t/(~)(t) = y} . (5.18)

To prove Theorem 5.3 we therefore need to investigate the asymptotic behaviour of limits of discrete renewal equations. The appropriate result is given by the following theorem which is a discrete analogue of the corresponding renewal theorem given by Silvestrov (1979, 1980) for the case of continuous time renewal equations.

Theorem 5.4 Let for each e ~ [0, 1] {f~)(t)}~=o be a discrete probability distribution that may be defective and {g(~)(t)}~=o a sequence of real numbers satis-

fy ing the following conditions.

E: (1) {f(°)(t)}ff= o is aperiodic, (2) f(°)(0) < 1, (3) f ( o ) . = ~ = o f ( O ) ( t ) > O,

F: (1) f(~)(t) ~ f(°)( t) as e --* O, t = O, 1 . . . .

oo (2) m~) '= ~ = o t f ~ ° ( t ) ~ m~°):= ~t=ot f~°)( t ) < oo as s ~ O.

G: (1) g(°(t) --* g(°)(t) as e ~ 0, t = 0, 1 . . . . (2) Ug(~)][:= ~ , ~ o ]g(~)(t)l ~ ]/g(°)]L'= ~,Zolg~°)(t)] < ~ as ~ ~ O.

Le t t~ be a sequence of positive integers such that there exists an ~ E [0, ~ ] such that

H: t~ ~ c~ and t~(1 -f (~)) ~ ~ as e --+ O, where f(~): = ~=of(e) ( t ) . Then for each e small enough the renewal equation

z~)(t) = g~)(t) + ~ z~)(t -- k)f(~)(k), t = 0, 1 . . . . . (5.19) k = 0

Quasi -s ta t ionary dis t r ibut ions 49

has a unique solution and g(O)

z¢~)(t~) -~ e-~/ , : ' m (o)

where

as e ~ 0 , (5.20)

g(O):= ~ g(O)(t). (5.21) t = 0

The renewal Theo rem 5.4 can be applied to the special case x = y in (5.16). The general case can then be obta ined using the following corollary.

Corol lary 5.5 Let the conditions of Theorem 5.4 be satisfied and let the sequences ~t°(t) o f real numbers and the discrete probability distributions ff~)(t) satisfy

lim l i m ~ o max [Ot~)(t)[ = O, (5.22) N ~ o o t ~ N

f f ' ) ( t ) ~ f f ° ) ( t ) as e ~ 0, t = 0, 1 . . . . . (5.23)

;,,,:_- (5.24) t = O t = O

Let for each sufficiently small e, z(~)(t) be the unique solution to the renewal equation (5.19). Then the sequence 2~)(t) defined by the explicit formula

satisfies

~(~)(t):= O(~)(t) + ~ z(~)(t -- k)ff~)(k), t = O, 1 . . . . , (5.25) k = O

e-~/m(°) as e ~ 0 . (5.26)

Theo rem 5.4 contains the s tandard discrete renewal theorem (Feller 1968, p. 3 3 0 ) ( t a k e {f~)(t)}~=o = {f~°)(t)}~=o, {9~)(t)};~=o = {9(°~(t)}~=0 for all e ~ (0, 1]) as a special case. Not ice that in the extreme cases ~ = 0 and ~ = the behav iour ofz~)(t~) as t~ ~ ~ is the same as the behaviour of the solution z(°)(t~) of the limiting equat ion as t~ ~ ~ , but that this is not t rue in the in termediate cases 0 < ~ < oo.

As a ma t t e r of fact, we will not need the full s t rength of Theo rem 5.4. It is easily seen that the probabi l i ty (~) gxy(t) defined in (5.18) is equal to

(0 gxy(t) = 6tobxy , (5.27)

where 6 is the Kronecke r symbol, and thus in the appl icat ion to our meta- (e)

popula t ion model the forcing term gxy(t) in the renewal equat ion (5.16) is in fact independent of e.

Theo rem 5.4 and Corol la ry 5.5 are p roved in the appendix.


6 The limiting distribution for the time to metapopulation extinction and conditional stationary distributions

We consider the random variable z(~)(0) = min{t > 1:0(~)(t) = 0} which is by definition the metapopulation extinction time for the Q(~)-system.

Under condition C, ~(~)(0) grows to ~ as e -~ 0. We shall show that the order of this growth is e-1. Actually we do more and show that under condition C the metapopulation extinction time normalized by e has a limiting distribution. We also give an explicit formula for this distribution.

The following relationship between certain random events,

U_ {0<~)(t) = ~9} = {v<~)(0) > t} , (6.1) ~ : ~ 0

is obvious as are the corresponding relationships between the probabilities

P{O(~)(t) = y} = P{O(~)(t) = y, z(~)(O) > t} , (6.2) and

P{V/(')(t) = iT, z(~)(O) > t} = P{z(~)(O) > t} . (6.3) y~ Dl ~ Do

The following theorem which gives an expression for the limiting distribution of the metapopulation extinction time is a corollary of Theorem 5.1 and relationship (6.3).

Theorem 6.1 Let condition C hold and let 2 be given by (5.4). Then for all ue(0, ~)

fe -zu i f )c E D 1 , (6.4) lim P~{ez(~)(0) > u} = (e_Z,h(y~, D1) i f ~2 ~ Do • ~ 0

Notice that in the case u = 0, 2 E D~ the right hand side of (6.4) is trivially equal to 1.

When the initial state ~ is in the ergodic class D1 the limiting distribution is exponential. When 2 ~ Do, it is the distribution of the random variable Z~z which is a product of two independent random variables Z~ taking on the values 1 and 0 with probabilities h(~, D1) and h(2, O) = 1 - h(2, Da), respectively, and g~ which is exponentially distributed with parameter 2.

Taking u = s (the parameter in condition D) in Theorem 6.1 and recalling that asymptotically t~ = s/e, we see from (6.4) that the correction factors e -z~ and e - ~ h ( 2 , D1) occurring in formulae (5.5) and (5.6), respectively, are the probabilities that metapopulation extinction has not yet taken place. This observation suggests that the probabilities n(y) may be interpreted as condi- tonal stationary probabilities, given that metapopulation extinction has not yet occurred. To formulate this intuitive idea in a precise way we introduce the conditional probabilities

P~{q(~)(t~) = )~[z(~)(O) > t~} n~{q(O(t~) = 37, "c(~)(O) > t~} = P~{~(~)(0) > t~} (6.5)

We have the following result.


Theorem 6.2 Let conditions C and D hold with 0 < s < ~ . Then for any initial state ,2 ~ Da w Do

e ~ O ~u

I f Y: ~ D1, then (6.6) holds in the case s = O, too.

i f y e D 1 , (6.6)

i f y E D o .

Notice that Theorem 6.2 also covers the situation where all matrices Q(~) coincide with Q. In this case Theorem 6.2 describes the asymptotic behaviour of t-step conditional probabilities P~{Vl(t ) = ylr(0) > t} when t ~ ~ . This special case could alternatively have been derived from standard ergodic theory (cf. Feller 1968).

From Theorem 6.2 we immediately obtain the following characterization of the incidence and the distribution of the number of occupied patches under the condition that the metapopulation has not yet gone extinct.

Corollary 6.3 Le t conditions C and D hold with 0 < s < ~ . Then fo r any initial state ~ ~ D1 w Do

l i m P ~ { ~ i ( t ~ ) = l l z ( ~ ) ( O ) > t ~ } = J i , i~{1 . . . . . n}, (6.7) e ~ O

and

lim P~{t/~)(t~) + . . . + tl~,~)(t~) = k]z~)(0) > t~} = p(k), k ~ {1 . . . . ,n} e - - * O

(6.8)

where Ji and p(k) are given by (4.6) and (4.7), respectively. I f Y ~ D1, then (6.7) and (6.8) hold in the case s = O, too.

Theorem 6.2 is a corollary to Theorems 5.t and 6.1 and the proof is given in the appendix.

Notice that we have explicitly excluded the case s = ~ from the results of this section. The question about the exact asymptotic behaviour in this case is still open. We have been able to prove that the statements of Theorem 6.2 and Corollary 6.3 remain true if et~ ~ ~ but (e2 + o(e))t~ ~ 0 as e --. 0, where o(e) is, as in condition C, the remainder term in the Taylor expansion of q~) j •

As the case s = ~ corresponds to the situation where t~ is much larger (on a different time scale) than the expected metapopulation extinction time, it is, although mathematically interesting, of minor biological significance and we have therefore refrained from giving a detailed discussion of this case.

7 Some simple examples

In a system with n patches the ergodic class Da has 2 "-1 elements. To determine the quasi-stationary distribution ~()7) one therefore has to solve a system of 2"- * linear equations. When the number of patches is large this is of course a practically impossible task. In this section we shall investigate some systems with only three patches. The resulting four-dimensional linear


system for the quasi-stationary distribution can readily be solved. Even these simple systems make some biologically interesting predictions.

Let dij be the distance between the patches i andj . The matrix d = (dij) is obviously a symmetric square matrix with zero diagonal elements. Let A~ be the area of patch i. We assume that the entries qij of the interaction matrix Q,

t ha t is the probabilities that patch j is not colonized in one time-step by a migrant originating from patch i, are given by the formula

qij: = exp( -- e - ' d ~ A i ) , i , j ~ {1 . . . . . n} . (7.1)

The formula (7.1) is based on the plausible assumptions that the extinction probability (in the absence of rescue effect) decreases with increasing patch size and that the probability that a migrant from patch i successfully colonizes patch j is positively correlated with the area of patch i and negatively correlated with the distance between the two patches. The parameter a is a measure of how bad the individuals are at migrating long distances. In the extreme case a = 0 the probability of successful colonization does not depend on the distance d~j at all, whereas for large values of a this probability rapidly

decreases with increasing distance. In our first example we consider a patch constellation with three patches of

equal area. The quasi-mainland (patch 1) is quite far away from the two other patches (the 'islands') which are located close to each other. Specifically we choose as the distance matrix

[0 1 1 t d = ] i 0 0.1 (7.2)

1 0.1 0

and as the area vector

A = ( 1 1 1). (7.3)

Substituting d and A given by (7.2) and (7.3) into (7.1) one obtains the interaction matrix Q. One then obtains the transition matrix from (2.1) and (2.2) and the quasi-stationary distribution ~(3~) can be solved from (4.3). Finally the distribution {p(k)}3= 1 of the number of occupied patches under the condition that the metapopulation has not yet gone extinct is obtained from (4.7) (see Corollary 6.3). The results for different values of the migration parameter a are shown in Fig. la-f.

From Fig. 1 we see that the distribution of the number of occupied patches is distinctly unimodal for most values of the migration parameter. In a metapopulation with efficient migrators (small value of a) all three patches are inhabited with high probability, whereas if the individuals are bad at migrating long distances (large value of a) the distribution {p(k)}~=l is skewed towards k = 1. Only for few intermediate values close to a -- 3 the distribution of the number of occupied patches is slightly bimodal.

Consider now an assemblage of species inhabiting the given three patch system. Different species have different vital parameters. In our simple model


O

I 0,8 0,5 I ~

0,3 0,6 0,4 0,2 0,4 0,2 0,2 0,1

0 0 0 t 2 3 1 2 3 1 2 3

(a) a=0 (b) a=l (c) a=2

o 0 i iL 0,10,2' 0 , 3 . 0 , 4 - 0 ' ' ~ ' 0,4 0,6 0 , 2 0 L 0,8-

1 2 3 1 2 3 1 2 3

(d) a=3 (e) a--4 (f) a=5

Number of occupied patches

Fig. 1. The quasi-stationary probability distributions of the number of occupied patches for different values of the migration parameter a. d and A are given by (7.2) and (7.3), respectively

there is only one vital parameter, a, which roughly describes the non-success of migration. In the assemblage of species the parameter a has a certain distribution. If for instance a is uniformly distributed over [0, 6] the fraction of species present at different sites has the typical bimodal core-satellite distribution (cf. Hanski 1982) given in Fig. 2. Notice that when applying Corollary 6.3 to an assemblage of species we have to condition on all species still being present in the study region.

Let us now consider a system with patches of different size. Of the two islands one is half the size of the quasi-mainland and 5000 times larger than the other island, which is located very near the larger island. We take

/0 1 1 J d = [1 0 0.001 (7.4)

1 0.001 0

and

A = ( 1 0 5 0.001). (7.5)

The distribution of the number of occupied patches was calculated as in the previous example and the results are shown in Fig. 3a-e.


I~ 0,5. u,_, 0,4. 0 I:= 0,3.

.0 0,2.

0,1. LT., O.

I 2 3


Fig. 2. The quasi-stationary fraction of species present at different number of patches in a species assemblage with a uniformly distributed over [0, 6]. d and A are as in Fig. 1

o

Li O LI 011 0,8. 0,5 0,5

0 , 6 ' 0,4 0,4 0 , 4 . 0,3 0,3

0,2 0,2 0,2. 0,1 0,1

0 0 0 1 2 3 1 2 3 1 2 3

(a) a=3 (b) a=7 (¢) a=7.5

o[, L 0,6- 0,8 0,5 0,6 0,4

0,3 0,4 0,2 O, I 0,2 -

0 0- 1 2 3 1 2 3

(d) a=8 (e) a=9



This time the distribution of the number of occupied patches is distinctly bimodal for a wide range of parameter values. Only in the case of extremely bad or extremely good migrators the distribution is skewed towards low or high occupancy, but even then the situation with exactly two occupied patches


is the least likely. A species assemblage will therefore always have the core- satellite distribution. For a species with a in the range [7.2, 7.53 the probabilities that one or three patches are occupied are both close to 0.5, whereas the probability of two patches being occupied is almost zero. Such a species will therefore every now and then switch from the core state (all patches occupied) to the satellite state (only one patch occupied).

Our third example shows that our model also can predict a unimodal distribution {p(k)}~= 1 with the peak at k = 2. We take the distance matrix d as in (7.4) and

A = (1 1 0.001). (7.6)

The system is similar to the previous example but with less discrepancy in patch sizes. The distributions of the number of occupied patches are depicted in Fig. 4a -d for different values of a.

We notice that for a wide range of a values the probability that exactly two patches are occupied is relatively high and that it is never the smallest probability as in Fig. 3. For a close to 1 it is even the largest probability. In this

o"LIII °it m o,6 • 0,4-

0, 1111 0,2 - O, O-

1 2 3 1 2 3

(a) a=O (b) a=l

°i1 0,5 - ~ ~ q

0,5 0,4- 0,4

0,3 - 0,3 O,Z - 0,2 0,1 - O,

0- 1 2 3 1 2 3

(c) a=l.5 (d) a=2




case the core-satellite distribution is very unlikely, it requires a species assemblage consisting of only extremely good and extremely bad migrators and no mediocre ones. On the other hand, species having a values in the range where the {p(k)}~= 1 is rather uniform will quite frequently switch not only between the core and satellite states, but between all numbers of occupied patches.

8 Discussion

In this paper we have presented a Markov chain model for a metapopulation. The model is completely characterized by the interaction matrix Q, the entries of which describe all first order interactions between pairs of local populations. The model thus takes the spatial distribution of patches explicitly into account and keeps track of which patches are occupied and which are empty. It seems impossible to achieve this with the aid of deterministic models in the form of differential equations. The model also allows for variable patch size but this can also be incorporated in deterministic ordinary or partial differential equation models, see Hanski and Gyllenberg (1993). Since the two most prominent features of metapopulation dynamics - local extinction and recolonization - are stochastic events, it seems natural to model the dynamics using a stochastic model, at least when the number of patches is relatively small.

The long term behaviour of the metapopulation, in particular the asymptotic expressions for the incidence and the distribution of the number of occupied patches, is naturally of great interest. If there is no mainland or immigration of individuals from outside the study region, then the metapopulation will eventually go extinct.. We have therefore given a detailed description of the asymptotic behav~our of the metapopulation under the condition that metapopulation extinction has not yet taken place. We have related this conditional asymptotic behaviour to the asymptotic behaviour for the extinction probability of one patch - the so-called quasi-mainland - tending to zero and to the limiting case where the extinction probability of this patch is zero.

All information on the conditional asymptotic behaviour of the meta- populations is contained in the quasi-stationary distribution re. Unfortunately one needs to solve a system of 2"-1 linear equations to find the explicit expression for re, a task which with an increasing number n of patches rapidly becomes impossible even for the most efficient super computer. On the other hand, important asymptotic quantities like the incidence and the distribution of the number of occupied patches can readily be obtained by simulation methods without calculating re.

When the number of patches is very large a different limiting procedure, where the number of patches tends to infinity, seems more adequate for describing the asymptotics of the metapopulation. Thus the model investigated in this paper is most appropriate for systems with a moderate number of patches.


A celebrated hypothesis in metapopulation dynamics is the core and satellite species hypothesis of Hanski (1982) which predicts that many species assemblages have a bimodal patch occupancy distribution, that is, at any given time most species are either present in most patches or occupy only a small fraction of the patches. For a brief review of some very different explanations of the core-satellite distribution, see the paper by Hanski and Gyllenberg (1993).

Hanski (1982) was the first to propose metapopulation dynamics as an explanation to the observed bimodal distribution. His explanation was based on a Levins type model for identical patches with the rescue effect included and temporal intraspecific variance in the colonization parameter. He also assumed immigration from outside the study region. In addition to the bimodal core-satellite distribution his model predicted core-satellite switching.

Hanski and Gyllenberg (1993) investigated a structured deterministic metapopulation model with rescue effect which suggested two other mechanisms creating the bimodal core-satellite distribution. Being based on a deterministic model neither of these mecli~lfisms assumed intraspecific variation in the colonization parameter. In the case of identical patches, migration from outside the study region was required to produce the core-satellite distribution but if the patches were of different sizes the bimodal distribution could occur even without a mainland. The mechanisms assumed interspecific variance in the colonization parameter and they did not predict core-satellite switchings, except for a narrow range of values of the colonization parameter.

The simple three-patch model considered in Sect. 7 suggests a new explanation for the core-satellite distribution. The examples show that spatial heterogeneity may produce the typical bimodal distribution. Recall that all models mentioned above assume that migrants choose their new patch at random, in other words, the spatial structure of the patches in ignored.

The assumption of colonization success in the model of Sect. 7 is similar in spirit to the assumption made by Hanski (1994) (compare the expressions (1.4) and (7.1)). The main difference is that we assume independence of different colonization attempts, whereas Hanski (1994) assumed interaction between immigrants.

The first example in Sect. 7 showed that with three identical patches heterogeneously distributed in space, the model predicted that, unless the migration parameter belonged to a narrow range of intermediate values, the distribution of the number of occupied patches is distinctly skewed towards either full occupancy or occupancy of only one patch. In other words, most species are either present in all patches or only in one patch and core-satellite switchings are rather unlikely. As such switchings are rare in nature (Lawton and May 1983; Gaston and Lawton 1989), this is a good description of many real species assemblages. The reason for the predicted behaviour is simple. In the case of relatively successful migrants (small a), the island populations will time after time be rescued by migrants from the quasi-mainland and so full occupancy will be the most likely state. On the other hand, if successful


migration from the quasi-mainland to the islands is difficult (large a), then typically only the quasi-mainland will be occupied.

In the second example in Sect. 7 we considered patches of very different sizes, where an extremely small patch was located very close to a patch almost as large as the quasi-mainland. In this case the model predicted frequent core-satellite switchings. The reason is the small distance between the islands. As long as the bigger island remains occupied, its local population will with a probability very close to one rescue the small island-population. But when the population on the big island goes extinct, the population on the small island will face the same fate, since without the rescue effect its extinction probability is very large. Therefore the system will switch between full occupancy and occupancy of the quasi-mainland only.

Finally, in the third example, there are two medium sized patches and a small one with a relative location as in the previous example. The population on the larger patches have a relatively low extinction probability, but they are not sufficiently large to rescue the population on the small patch. As a consequence, for many values of a the most likely number of occupied patches is the intermediate value k = 2.

The three-patch model of Sect. 7 is admittedly oversimplified, but it clearly demonstrates that different spatial arrangements of patches can produce different distributions of occupancy and that heterogeneity in the spatial distribution of patches combined with the rescue effect can explain the bimodal core-satellite distribution. It is clear that a model with a larger number of patches can exhibit all the different behaviour of the three-patch model and much more complicated behaviour in addition.

9 Appendix: proofs of the theorems

Proof of Theorem 5.4 and Corollary 5.5

Note that conditions E2 and F imply that

f(~)'= ~ f(~)(t) ~ f(o) as E ~ 0 , (9.1) t = O

and hence the condition E holds with {f(°)}2°= o replaced by ¢ c(~)~oo t v st=o, for all e e [0, eo] provided e0 is small enough. We shall use this observation without reference in the sequel. In particular it follows that for all e e [0, Co] the discrete renewal equation (5.19) has a unique solution given by

z(~)(t) = i g(~)(t -- k)u(~)(k), t = 0, 1 . . . . . (9.2) k = 0

where

u(~)(t) = ~ f(~)*~(t), t = 0, 1 . . . . (9.3) k = 0


is the resolvent or renewal function for the distributionf(~)(t) a n d f (~)*k(t) is the k-fold convolution of the distribution f(~)(t).

The asserted asymptotic behaviour of z(~)(t~) will follow easily from the following proposition on the asymptotic behaviour of the corresponding resolvents.

P r o p o s i t i o n 9.1 Let conditions E, F and H hold. Then

u(~)(t~) -~ e - ~ 1 re(o) as e ~ 0 , (9.4)

where u(~)(t) is the resolvent of f(~)(t).

Notice that Theorem 5.4 reduces to Proposition 9.1 in the special case 9(~)(t) = 6to for all e.

Proposition 9.1 will be proved through a series of lemmas. An extra difficulty is caused by the fact that the distributions f(~)(t) are possibly defective. We therefore start by proving Proposition 9.1 for proper distribution and then turn to the general case after a normalization of the distributions. We introduce the following normalized (proper) distributions and their corresponding expectation:

f(~)(t) f(~)(t):- f(~) , (9.5)

_ m (~) rh(~): = ~ tf(~)(t) f (~) . (9.6)

t = 0

It is obvious that if the distributionsf (~) satisfy conditions E and F, then the same is true of the normalized distributionsf(~)(t).

In the sequel we denote quantities associated with the distributionsf(~)(t) by putting a bar over the symbol for the corresponding quantity associated with the possibly defective distributions f(~)(t). Thus, for instance, the resolvent off(~)(t) is denoted by ~i(~)(t).

Observe that if all the distributions are proper (f(~) = 1), then the constant c~ in condition H is necessarily equal to zero. The following lemma and its corollary therefore yield Proposition 9.1 in the case f (~) = 1.

L e m m a 9.2 Let conditions E and F hold. Then for eo small enough

sup ]ti(~)(t) -- 1/rh(~)l -* 0 as t -~ ~ . (9.7) E~[O,eo]

Lemma 9.2 can be proved by methods analogous to those used by Silvestrov (1979) to prove a corresponding result for continuous time processes. Alterna- tively the proof could be based on a result by Kalashikov (1978), who proved that under conditions that follow from E and F, the following estimate for the resolvent a(~)(t) off(~)(t) holds:

[ti(e)(t)-1/m(~)] ~ a(~)( t - l + k_tr~> ~>--kf(e)(r)) " (9.8)


Anichkin (1985) gave explicit expressions for the constants a (~) in (9.8) in the fo rm of functionals of the dis tr ibut ions f( ')(t). F r o m these expressions it easily follows that under condi t ions E and F

sup a (~) < ~ (9.9) ee[0,e0]

and hence (9.7) holds.

Corol lary 9.3 Let the assumptions of Lemma 9.2 be satisfied. Then for any sequence t~ tendin9 to infinity as e tends to zero, one has

~(~)(t,) ~ 1/n~ (°) a s e ~ 0 , (9.10)

where ~i(~)(t) is the resolvent o f f(~)(t).

Proo f This follows immedia te ly f rom L e m m a 9.2 and condi t ion F2. [ ]

Let KI "), i = 1, 2 . . . . be independent identically dis tr ibuted r a n d o m variables with dis t r ibut ion {f(~)(t)}~=o satisfying condi t ions E and F and let

z~ ) = 0, (9.11) z~ ~)=tc] ~)+ . . . +~c~ ~ ) , k = l , 2 . . . . .

and v(')(t) = max{k :z~ ') < t}, t = 0, 1 . . . . . (9.12)

Let ~I "), i = 1, 2 . . . . be independent identically dis tr ibuted r a n d o m variables with dis tr ibut ion {f(~)(t)}?~=o and define f~") and ~7(')(t) analogously. I t is wel l -known tha t the expecta t ions of v(')(t) and ~(~)(t) can be obta ined as functionals of the resolvent; more precisely, we have

Ev(~)(t) = ~ u(~)(k), t = 0, 1 . . . . . (9.13) k=O

Eg(~)(t) = ~ ff(~)(k), t = 0, 1 . . . . . (9.14) k=O

Conversely, we get the following expressions for the resolvents:

u(')(t) = E(v(~)(t) - v ( ' ) ( t - 1)), t = 1, 2 . . . . . (9.15)

~i(~)(t) = E(q(")(t) - q(')(t - 1)), t = 1, 2 . . . . . (9.16)

We shall also m a k e use of the r a n d o m variable v (~) with geometr ic dis t r ibut ion

P{v (") = m} = (f(~))m(1 - f (~)) , m = 0, 1 . . . . . (9.17)

The next s imple l e m m a is a var iant of the wel l -known fact that the exponent ia l dis t r ibut ion is a limit of geometr ic distributions.

L e m m a 9.4 Let condition H hold. Then for all x > 0

p f } ( - ~ > x ~ e -~x (9.18)

a s e - * 0 .


P r o o f This is an immedia te consequence of the definition (9.17) of v (~) and H. [ ]

The next l e m m a expresses the distr ibution of ¢~)(t) in terms of the distribu- t ion of the process ~(~)(t) and the distr ibution of ¢~) given by (9.17).

Lemma9 .5 For all O = to < h < " '" < t i n < oo and O = ko < kl < " '" < k,,, m = 1, 2 . . . . we have

P{v(~)(tr) >-_ k , r E {1 . . . . . m}} = P{v (~) >= k , , } P { ~ ) ( 6 ) >= kr, r e {1 . . . . . m}}

= P{rrfln{v(~),¢~)(tr)} >kr, r e {1 . . . . . m}}, (9.19)

where v (~) is a random variable independent o f the process ¢~) and distributed according to (9.17).

P r o o f The identi ty (9.19) is a consequence of the following chain of identities:

P { v ( ~ ) ( t r ) > k , , r e { 1 , . . = . , m}} =ri/Ck.--" (~) <=tr, r e {1 . . . . . m}}

= r ~ o < ~ }

P" (~) {1, ,m} (a ~ Z k , < tr, r e = . . "Ckm < 0 0 }

= ( f (E))kmp{~) < tr, r e {1 . . . . . m}} (9.20)

= P { v (~) >= km}P{~(~)(6) >-_ k,, r e {1 . . . . . m}}

= P{min{v(~),v~")(t~)} > k,, r e {1 . . . . . m}}. [ ]

L e m m a 9.6 Le t conditions E and F hold. Then for any sequence t~ tending to infinity as e --* 0 and any integer r one has

~(~)(t~ + s) 1 - ) . - -

te rn (°)

in probabili ty as e ~ O.

(9.21)

The p roof of L e m m a 9.6, which is a slight general izat ion of the ordinary discrete renewal theorem, is complete ly analogous with the p roof given by Silvestrov (1979, 1980) for the case of cont inuous time. It is therefore omitted.

L e m m a 9.7 Le t conditions E and F hold. Then

lim sup E(~(a(t) - ¢~)(t - 1))" < oo (9.22) ~--*0 t

f o r all positive integers r.

P r o o f Let p(~)(t) = P{f(k ~) 4 = t, k = 0, 1 . . . . } . (9.23)

With this definition and the definitions of Ck ~) and ¢~) it is obvious that

P{v-(~)(t) -- v-(~)(t -- 1) = m} ~p(~)(t 1) if m ~0~

= [(1 -- p(~)(t -- 1))(fi(~)(O)) m- 1(1 --f(~)(O)) if m _-> 1.

(9.24)


As a c o n s e q u e n c e we get the fo l l owing e s t i m a t e

E(~(~)(t) -- ~(')(t -- 1))" =

a n d so

m~(1 - p(~)(t -- 1))(f( ' )(O)) m- 1(1 --f(~)(O)) m = l

< ~, m~(f(~)(O)) m-1 m = l

(9.25)

o o

l i ra sup E (~(~)(t) - - q(~)(t - 1)) r < l im ~ mr(f(~)(O)) m- 1

e ~ 0 t e - * 0 m = l

= ~ m r ( f ( o ) ( o ) ) m - 1 < oo (9.26) m = l

w h i c h p r o v e s the a s se r t ion .

P r o o f o f P r o p o s i t i o n 9.1. I t fo l lows f r o m L e m m a 9.5 t h a t

v(~)(t~) - - v(~)(t~ -- 1) = (~(~)(t~) - ~(")(t~ - 1))Z{v (~) > F(~)(t~)}

+ ( v ( ' - ~( ' ( to - 1))

× z { ~ ( ~ ) ( t ~ - 1) __< v ( ' < ~( ' ( t~)}

+ 0Z{v (~) < 9(~)(t~ - 1)} (9.27)

in d i s t r i b u t i o n . D e n o t e the first n o n z e r o t e r m on the r igh t h a n d s ide of (9.27) b y c~ a n d the

l a t t e r n o n z e r o t e r m b y fie. W e shal l s h o w t h a t Efl~ -~ 0 as ~ - 0 a n d so

l im u(~)(t~) = l im E(v(~)(tA - vt~)(t~ - 1)) = l im E ~ . (9.28) e - - + 0 e ---~ 0 e - * 0

U s i n g L e m m a s 9.6 a n d 9.7 we get

f v ( ~ ) ( t ~ - 1) v (~) 9(~)(t~)'~ EZ{V(~)(t~ -- 1) < v (~) < ~7(~)(t~)} = "~zZ~ t~ < - - <

= = t~ t~ c J

~ P __< p~ < -- 0 , (9.29)

w h e r e p~ is an e x p o n e n t i a l l y d i s t r i b u t e d (wi th p a r a m e t e r c~) r a n d o m var iab le . I t fo l lows f r o m (9.29) a n d L e m m a 9.6 t h a t

fl~ ~ 0 in p r o b a b i l i t y as e ~ 0 (9.30)

S ince by de f in i t i on

fl~ < ~(~)(t) - 9(~)(t - 1) . (9.31)

we ge t f r o m L e m m a 9.7 t h a t

l im Efl~ < ~ (9.32) e ~ O

for all pos i t i ve in tege r s r. C o m b i n i n g (9.30) a n d (9.32) we get Efl~ --, 0 as e ~ 0 a n d so (9.28) ho lds .


To analyse the asymptot ic behaviour if Ect~ we first note that

fv ~ f(~(t~) ~ ~ = (V~)(t~) - V~)(t~ - 1 ) ) Z ~ - >

t~ J

(v t~ 1 -- c )

it} + (V~)(t~) - V~)(t~ - 1))Z V~ t~) < (9.33)

= n~(o)

for any c ~ (0, 1). +

Denote the former term on the right hand side of(9.33) by a~c and the latter by ~ . It follows from Corol lary 9.3 and Lemma 9.6 that a~ tends to zero in probabil i ty as ~ ~ 0 and from Lemma 9.7 that E(a~)2 < ~ and hence

E ~ - ~ 0 a s e ~ 0 , 0 < c < l . (9.34)

Using the independence of v ~ and V~)(t), t = 0 1 . . . . and Corol lary 9.3 and L e m m a 9.7 we get

fv (~ 1 - c ) lim E ~ + = lim E(V~)(t~) - V~)(t~ - 1)) E ~ - = > e ~ O e ~ O ( t~

_~) ( v (~) 1 -- c ) = !irn ° u,, P ~ > - - f f ~

1 e ~ (9.35) ~(o)

Using (9.34) and (9.35) we now get

l i m E a ~ = < l i m E ( ~ + + a S ) = l i m l i m E ~ + = l i m 1 e _ ~ ~= 1 _ . . . .

(9.36)

In an entirely similar way one can show the opposite relation

lim E ~ > 1 ~-----6 = ~-?65 e - ~ " (9.37)

It now follows from (9.28), (9.36) and (9.37) that

lim u ~ ) ( t ~ ) = 1 o ~ o f i e - ~ (9.38)

Finally we note that if ~ = oo, then the right hand sides of (9.4) and (9.38) are both zero and the assertion holds. If c~ ~ [0, ~ ), then by (9.1) and condit ion H we have f~o) = 1 and hence rfi ~°~ = m ~°~ and again the right hand sides of (9.4) and (9.38) coincide.

This completes the proof. [ ]


Proof o f Theorem 5.4 We split the solution (9.2) of the renewal equation (5.19) into two parts as follows:

N te

z(")(t~) = ~ g(~)(k)u(~)(t~ - k) + ~ g(~)(k)u(~)(t~- k) . (9.39) k = 0 k = N + l

Denote the first term on the right hand side of (9.39) by v~N and the second by w~N. It follows from condition G and Lemma 9.7 that

lim lim [W~NI : 0 . ( 9 . 4 0 ) N~o~ e ~ 0

Since condition H is satisfied by the sequence t~ it is also satisfied by t~ - k for any fixed nonnegative integer k. Proposition 9.1 therefore yields that

u(~)(t~ -- k) ~ - ~ e .... (9.41)

as e --, 0 for all k = 0, 1 . . . . . It follows from (9.41) and condition G that

N 1 lira v,N = ~ g(°)(k) ~ e - ~ . (9.42) ~ 0 k = 0

Finally we infer from (9.39), (9.40) and (9.42) that the assertion (5.20) holds. This completes the proof. []

Proof o f Corollary 5.5 We split the expression (5.25) for ~)(t~) into three terms:

2(°(t~) = O(°(t~) + v~N + weN, (9.43)

where

and

N

VeN:~- E Z(e)(te - - k)f)~)(k) ( 9 . 4 4 )

k = 0

t~

w~u:= ~ z (" ) ( t~ -k ) f f~ ' (k ) . (9.45) k=N+l

It follows from condition G, Lemma 9.7 and the representation (9.2) of z(~)(t) that

lim maxlz¢~)(t)l < ~ . (9.46) e ~ O t

We now find as in the proof of Theorem 5.4 that (5.23), (5.24) and (9.46) imply

lim lim I w~NI = 0 , (9.47)

whereas G2 and (9.42) yield

lim lim IV=NI = g<o) 1 N ~ e ~ 0 m - ~ e - ~ .

(9.48)


Finally, (5.22) implies that

lim IO(~)(t~)] < lim max I0(~)1 ~ 0 e ~ O e-*O k >=N

(9.49)

as e ~ 0. The assertion (5.26) now follows from (9.47), (9.48) and (9.49). This completes the proof. []

Proof of Theorem 5.3. The idea of the proof is to first apply the renewal Theorem 5.4 to the possibly defective probability distributions f(~)~(t) and sequences g~)~(t) given by (5.17) and (5.18). Corollary 5.5 will then give the corresponding result for z~)y(t) in the general case x + y. We therefore have to verify that the assumptions of Theorem 5.4 are satisfied by these distributions and sequences. As pointed out at the end of Sect. 5 (5.27) the sequence g~)~(t) given by (5.t8) is independent of e and therefore condition G holds automatically.

Recall the definitions (3.1) and (3.2) of the first hitting time and hitting probabilities. For a Markov chain ~/(~)(t) the corresponding first hitting time and hitting probability will be denoted by z(~)(B) and h(~)(x, B, A), respectively. We will actually need the following more general hitting probability:

h~')(x, B, A):= Px{r/(~)(z(~)(B)) ~ A, z(~)(B) = t} (9.50)

for A c B c X, t = 0, 1 . . . . . Then

h(~)(x, B, A) = ~ h(9(x, B, A) . t = 0

(9.51)

We also introduce the moments

h(~)(k)(x, B, A):= ~ tkh~)(x, B, A) (9.52) t = O

for all nonnegative integers k. It is clear that hl~)(°)(x, B, A) = hl+)(.x, B, A). For any real valued functjon g on X we define the expectations

~(~) (B) 1

E(+)(x, B, A):= Ex t = 0

g(rl(~)(t))Z{rl(~)(r(~)(B)) e A, z(~)(B) < oo } . (9.53)

If in any of formulas (9.50), (9.52), (9.53) the sets A and B coincide, we omit the third argument on the left hand side. If A is a one point set A = {w} we simply write w as the third argument.

We have the following result for the behaviour of the hitting probabilities and their moments and of the expectations (9.53) as e ~ 0.

Lemmas 9.8 Let for each e ~ [0, 1], q(~)(t) be a homogeneous Markov chain with (e) finite state space X and transition probabilities P (x, y) ~ P(°)(x, y) as e ~ O.

Then for every A c B ~ X, x ~ X such that h(°)(x, B) = 1 and every nonnegative


integer k one has

h(k~)(x, B, A) ~ htk°)(x, B, A) as e ~ O, (9.54)

h(~)(x, B, A) ~ h(°)(x, B, A) as e ~ 0 , (9.55)

h(')(k)(x, B, A) --, h(°)(k)(x, B, A) < oo as e --* 0 , (9.56)

E(')(x, B, A) ~ E(°)(x, B, A) < oe as e ~ 0 . (9.57)

The proof of Lemma 9.8 is a straightforward modification of the proof of a similar result for the more general case of a Markov chain with countable state space (Silvestrov 1974, pp. 210, 273-274) and is therefore omitted.

Corollary 9.9 For every x , y ~ X the probability distributions fxy(t).=(~) • P~{z(")(y) = t} satisfy the condition F of Theorem 5.4.

Proof. This is obvious s incef~( t ) = h(t°(x, y). []

Lemma 9.10 The distributions f~)(t) satisfy the condition E for all x ~ Ot and all e small enough.

Proof. Since (") fx~(O) = P~{z(~)(x) = 0} condition E2 is trivially satisfied. Since (o) t ~ o D1 is by assumption an aperiodic, irreducible set, {f~x ( )} = is aperiodic and

f(o) ~ = h(°)(x, x) = 1 for all x e D1. It now follows from condition F proved in the preceding corollary, that E1 and E3 hold for all sufficiently small e. []

Our next task is to show that the sequence t, has the property H. Our first result in this direction is the following lemma

Lemma 9.11 For all x, y ~ D 1 one has

1 -r(~):~y --. ~2~y as e ~ 0 , (9.58) where

2~,:= ; ~ )~(~DP(z 'v )+ ~ P(z,v)h(°)(v, DlwD, D) (9.59) z v~Do

and z(~)(y)- 1

(~) "=E~ Z X{r/(~)(t) = z}. (9.60) P xyz t = O

Proof. For y e D 1 one has

1 _:(Oj~y = px{z(~)(y) = ~ }

zED1 t=O vEDowD

×(1-h(~)(v , Dx)+ ~ h(~)(v, Dl, w ) (1 - f : ) y ) ) w~D1

E (~) = Pxyz Y'. P(~)(z, v)(1 - h(~)(v, D1) z~D1 v~DouD

+ E (e) P~y~ E P(~)(z,v) E h(~)( v, D,,w)(1 --f(g)y). z~D1 v~Dow D waD1

(9.61)


Obvious ly p(~) xyz = E(~)(x, Y) (9.62)

for g(v) = Z{v = z}. It therefore follows f rom L e m m a 9.8(c) that

(e) (o) P xyz a s e ~ 0 . P xyz ~ (9.63)

Since the set DI is irreducible, P(°)(z, v) = 0 for z ~ D1, v e Do u D and so

P(')(z, v) = eP(z, v) + o(e) (9.64)

for all zeD~ , v e D o u D . F r o m L e m m a 9.8(a) it follows that

{~ i f v ~ D (9.65) h ( O ( v ' D l ) - * h ( ° ) ( v ' D 1 ) = - - h ( ° ) ( v , D I ~ D , D ) if v~Do

as e ~ 0. It follows f rom (9.63), (9.64) and (9.65) that the first term on the right hand side of (9.61) is of order

( v) + ~ P(z, v)h(°)(v, D1 uD, D)] (9.66) P(z,

z e D 1 \ v E D r e D o /

as e --, 0, whereas the latter term is of order

E (o) P~,z (eP(z, v) + o(e)) ~ h(")(v, D1, w)(1 -~w,,f(~)~ = o(e) (9.67) z ~ D 1 w e D I

as ~ ~ 0, because ~wyf(~) --*f~r ) = I as e --, 0 for w , y ~ D l . This proves the assertion.

L e m m a 9.12 For all y ~ D1 one has

t~(1--f(~)~ ~ (o)o v yy ! ~ Amyy o --

[]

/~s as e ~ 0 . (9.68)

n(y)

Proof. The s ta t ionary distr ibution n of t/(°)(t) restricted to D1 is given by

1 n(y) - ._(0) (9.69)

gftyy

and for y, z ~ D1 one has

co) _ re(z) (9.70) pyyz n(y) "

It therefore follows f rom (9.58) and (9.59) with x = y e D1 that

1 --((~) (0)2 (9.71) , y y ~ ~ , m y y

with 2 defined as in (5.13). The assertion (9.68) follows f rom (9.71). [ ]

Completion of Proof of Theorem 5.3 By Corol la ry 9.9 and L e m m a 9.10 and the {fyy(t)}~=o and {gyy(t)}~=o satisfy the c o m m e n t in the beginning of the p roof (~) oo (~) o0

assumpt ions E, F, and G of Theorem 5.4 for x e D1. F r o m L e m m a 9.12 it


follows that condition H holds with

~ m (o) (~ ~-- .~..~yy .

By (5.27)

g(~) = ~to y y

and therefore g(o) = 1 .

(9.72), (9.74) and the renewal Theorem 5.4 now imply that

Py {t/(~)(t~) y} (~) = = Zyy(t~) ~ e-ZSn(y) as e ~ 0 . (9.75)

and (i) is proved in the case x = y e D1. The case x, y e Dx, x 4 = y and x e Do, y e D1 fits exactly into the framework

of Corollary 5.5 with f~y)(t) =f~y)(t) and so (i) is proved for all x , y e D1. Assume now that x e Do, y e D1. Since

f(o) = h(O)(x, D1 wD, D1) (9.76) xy

we can again use Corollary 5.5 and together with Corollary 9.9 this yields (ii) in the case y e D1.

Finally we consider the case y e Do. Since Do is a transient set for the Markov chain t/(°)(t) we h a v e f ( f ) < 1 and sincef~y ) ~f(o)~yy as e ~ 0 we get t~(1 -f(~)) ~ ~ as ~ ~ 0. The renewal Theorem 5.4 therefore yields that y y

(i) holds and applying Corollary 5.5 in the same way as above we infer that (ii) holds, too.

This completes the proof of the theorem. []

Proof of Theorem 5.1. The idea of the proof is to apply the general ergodic Theorem 5.3 to the sequence fl(~)(t) of Markov chains describing the dynamics of the metapopulation.

According to condition A the decomposition X = D1 w D o w {0} into the disjoint union of the ergodic class, the transient class and the absorption state, described in Sect. 3 satisfies the conditions (a), (b), and (c) of the Theorem 5.3 with respect to the Markov chain fl(°)(t). Next we check that the transition probabilities P(~)()L t 5) have an expansion of the type (5.10). Using condition C and the expressions (2.1) and (2.2) for the transition probabilities P(Y, 15) = P(°)()L 15) we find

= I - (qj~ + ~qj~ + o ( e ) ) ~ i = 1 \ j = 1 j = l

1 - ~. qjl + eP(~,15) + o(~) (9.77) j = l

= p(o)(f , 15) + e p ( f , V) + o (e ) ,

in other words, the expansion (5.10) holds. To calculate the constant 2 in formula (5.4) we need an explicit expression

for the first order coefficients/3(£, 15) for )~ e D1 and 15 ~ Do w {0}. Since xl = 1 and Ya = 0 for f ~ D1 and 15e Dow{0} a straightforward calculation yields

(9.72)

(9.73).

(9.74)


tha t for such 2 and 35, P(2, 35) has the form (5.2) with gli(2) and q1(37) given by (5.3). The def ini t ion (5.4) of 2 agrees exact ly with the one given by (5.1 3) and so T h e o r e m 5.3 can be appl ied.

This comple tes the proof. [ ]

Proof of Theorem 6.1. Theo rem 6.1 is an immedia te coro l l a ry of Theorem 5.1.

Proof of Theorem 6.2. F o r 0 < s < ~ and 2 E D~ w Do, 37 ~ D~ we have using Theo rems 5.1 and 6.1

l im P~{q(~)(t~) = 371z(~)(0) > t~} e - * O

= lim P~{q(~)(t,) = y} = e-~zc(y)h(~, D1) = n(y) (9.78) ~ o P~{zt~)(0) > t~} e-ZSh(2, D1)

as asserted. If 37 6 Do, then by Theorem 5.1 lim~_~ oP~{tT(~)(t~) = 37} = 0 and so the express ion in (9.78) is 0 (the d e n o m i n a t o r still tends to the same l imit as before, which is different f rom zero). [ ]

Acknowledgement. Mats Gyllenberg thanks Ilkka Hanski and Timo Koski for many illuminating discussions. The work was partially supported by The Bank of Sweden Tercentenary Foundation, The Swedish Council for Forestry and Agricultural Research, The Swedish Natural Science Research Council, The Carl Trygger Foundation, and the Swedish Cancer Foundation.

References

Adke, S. R.: A birth, death and migration process. J. Appl. Prob. 6, 687-691 (1969) Anichkin, S. A.: Rate of convergence in Blackwell's theorem. In: Problems of Stability of

Stochastic Processes, pp. 4-9, VNIISI, Moscow 1985 Bailey, N. T. J.: Stochastic birth, death and migration processes for spatially distributed

populations. Biometrika 55, 189-198 (1968) Davis, A. W.: Some generalisations of Bailey's birth, death and migration model. Adv. Appl.

Prob. 2, 83-109 (1970) Diamond, J. M.: Assembly of species communities. In: Ecology and Evolution of

Communities, Cody, M. L., Diamond, J. M. (eds.), pp. 342 444. Cambridge, Mass. Harvard University press 1975

Durrett, R.: Stochastic models of growth and competition. In: Proceedings of the Inter- national Congress of Mathematicians, Kyoto, 1990, Satake, I. (ed.), Vol. II, pp. 1049-1056, Springer Berlin-Heidelberg New York 1991

Durrett, R., Levin, S. A.: Stochastic spatial models: a user's guide to ecological applications. Trans R Soc. (to be published)

Feller, W.: An Introduction to Probability Theory and its Applications, Vol. I, New York, Wiley 1968

Gaston, K. J., Lawton, J. H.: Insect herbivores on bracken do not support the core-satellite hypothesis. Am. Nat. 134, 761 777 (1989)

Gilpin, M. E., Diamond, J. M.: Immigration and extinction probabilities for individual species: relation to incidence functions and species colonization curves. Proc. Nat. Acad. Sci. 78, 392-396 (1981)

Gilpin, M. E., Hanski, I.: Metapopulation Dynamics. New York, Academic Press 1991 Gyllenberg, M., Hanski, I.: Single-species metapopulation dynamic: a structured model.

Theor. Popul. Biol. 42, 35 61 (1992)


Hanski, I.: Dynamics of regional distribution: the core and satellite species hypothesis. Oikos 38, 210-221 (1982)

Hanski, I.: Inferences from ecological incidence functions. Am. Nat. 139, 657 662 (1992) Hanski, I.: A practical model of metapopulation dynamics. J. Anim. Ecol. 63, 151-162 (1994) Hanski, I, Gilpin, M. E.: Metapopulation dynamics: brief history and conceptual domain.

In: Metapopulation Dynamics, Gilpin, M. E., Hanski I. (eds.), pp. 3 16. New York, Academic Press. 1991

Hanski, I., Gyllenberg, M.: Two general metapopulation models and the core-satellite species hypothesis. Am. Nat. 142, 17-41 (1993)

Hoel, P. G., Port, S. C., Stone, C. J.: Introduction to Stochastic Processes. Boston, Houghton Mifflin Company 1972

Kalashikov, V. V.: Qualitative Analysis of the Behaviour of Complex Systems by the Method of Test Functions. Moscow, Nauka Publishers 1978

Kingman, J. F. C.: Markov population processes. J. Appl. Prob. 6, 1 18 (1969) Lawton, J. H., May, R. M.: The birds of Selbourne. Nature 306, 732-733 (1983) Levins, R.: Some demographic and genetic consequences of environmental heterogeneity for

biological control. Bull. Entomol. Soe. Am. 15, 237 240 (1969) Levins, R.: Extinction. In: Some Mathematical Problems in Biology, Gerstenhaber M. (ed.),

pp. 77-107. Providence, RI American Mathematical Society, 1970 MacArthur, R. H., Wilson, E. O.: The Theory of Island Biogeography. Princeton, Princeton

University Press, 1967 Nisbet, R. M., Gurney, W. S. C.: Modelling Fluctuating Populations. Chichester, Wiley

1982 Renshaw, E.: A survey of stepping-stone models in population dynamics. Adv. Appl. Prob.

18, 581-627 (1986) Silvestrov, D. S.: Limit Theorems for Compound Random Functions. Kiev, Kiev University

Press 1974 Silvestrov, D. S.: The renewal theorem in a series scheme. I. Theory. Probab. Math. Stat. 18,

155-172 (1979) Silvestrov, D. S.: The renewal theorem in a series scheme. II. Theory. Probab. Math. Stat. 19,

113-130 (1980) Verboom, J., Lankester, K., and Metz, J. A. J.: Linking local and regional dynamics in

stochastic metapopulation models. In: Metapopulation Dynamics Gilpin, M. E. Hanski I. (eds.), pp. 39-55. New York, Academic Press 1991

Quasi-stationary distributions of a stochastic metapopulation model

Documents

Transcript of Quasi-stationary distributions of a stochastic metapopulation model