Uncertainty in Dynamic Process to Extinction

Kei-ichi Tainaka, Nariyuki Nakagiri, Tomoyuki Sakata and Tomomi Taoa

aDepartment of Systems Engineering, Shizuoka UniversityJyohoku 3-5-1

Hamamatsu, Japan

Abstract: The investigation of perturbation experiments is important not only to forecast the effect of eco-logical management but also to know community interactions. So far, uncertainty has been well known forperturbation experiments: many authors has reported that long-term response of ecosystem to applied pertur-bations has been very difficult to predict. In the present article, however, we report the uncertainty even inshort-term response. We carry out a computer experiment of extinction, and explore whether the so-calledfluctuation enhancementoccurs or not, where the fluctuation enhancement means that there are a variety ofprocesses to the extinction. We apply the contact process (CP) on a two-dimensional lattice in which interac-tion is restricted between adjacent lattice points. We also apply the mean-field simulation (MFS) of CP, wherelong-range interaction is aloud between any pair of lattice points. Computer simulation reveals that for bothCP and MFS, stationary state exhibits the so-calledcritical slowing-downwhich denotes that relaxation timediverges near the extinction. It is also found that the fluctuation enhancement occurs in the case of CP. Becauseof short-range interaction, there are a variety of different processes to extinction.

Keywords: lattice model; ecosystem; uncertainty; perturbation experiments

1 INTRODUCTION

Under various human management, ecosystems re-ceive perturbations or disturbances. The investiga-tion of perturbation experiments is essential to con-serve biospecies [Paine, 1966; May, 1973; Tilmanand Downing, 1994]. The most familiar approachon perturbation experiment is the press perturba-tion, where one or more quantities are altered andheld at higher or lower levels. It is well known thatthe response of an ecosystem to a perturbation con-sists of two parts [Bender, 1984; Yodzis, 1988]; thatis, short- and long-termresponses. The former re-sponse is usually determined by the so-calleddirecteffectwhich is a straightforward to the cause. Onthe other hand, the final stationary state (long-termresponse) is determined not only by direct but alsoby indirect effect. One of the most striking resultsis an indeterminacy (uncertainty) of indirect effect[Yodzis, 1988; Tainaka, 1994; Schmitz, 1997]. Thelong-term response of press perturbation is very dif-ficult to predict. In the present article, however, wereport the uncertainty even in short-term response.

In the field of physics, the dynamics of phase transi-tion has been studied by a number of authors. Con-siderable data on perturbation experiments of phase

transition have been accumulated. There are mainlytwo types of phase transition: first- and second-order transitions. In the latter case, the value ofor-der parametercontinuously changes between bothphases, and we can usually observeenhancement offluctuation[Kubo et al., 1973; Tsuchiya and Horie,1985] which means that there are a variety of pro-cesses for a phase transition. So far, it is thoughtthat the fluctuation enhancement is originated in theso-calledcritical slowing-down: relaxation time di-verges at a stationary state near the phase boundary.On the other hand, extinction in ecosystems can beregarded as a a phase transition, where the order pa-rameter corresponds to the population size (density)of individuals. When the extinction gradually oc-curs, and when the density is slowly decreased, thenwe expect that the critical slowing-down and fluctu-ation enhancement may be observed in this system.In the present paper, we report that the fluctuationenhancement is not always originated in the criticalslowing-down.

When the population size of a species becomessmall, the risk of extinction increases. So far, sev-eral authors have estimated the risk of extinction,using the following stochastic differential equation

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[Lande, 1995]:

:x = Rx(1� x=K) +N(x); (1)

where the dot represents the derivative with respectto the time, the variablex is the population size of aspecies. The first term in the right hand side of (1)denotes the logistic equation which includes con-stant parametersR andK, and the last term in (1)denotes a noise (fluctuation) which usually dependson x. If x becomes small, the noise term becomescritical. We consider that such an approach hasa flaw; namely, the noise termN(x) is arbitrarilygiven by authors; especially, the intensity of noiseis arbitrary, despite it plays an essential role for riskestimation. However, the noise intensity should beautomatically given [Tainaka et al., 2000].

In recent years, lattice models are widely appliedin the field of ecology [Nowak et al., 1994; Haradaand Iwasa, 1994; Durrett and Levin, 1996]. In thepresent paper, we apply the contact process (CP)which has been extensively studied from mathe-matical [Harris, 1974; Liggett, 1985] and physical[Konno, 1994; Marro and Dickman, 1999] aspects.The CP is a lattice version of logistic equation. Con-siderable data on CP have been accumulated, butthey are mainly related to stationary state. However,we focus on dynamic process of CP.

2 MODEL AND M ETHOD

We consider a simple model ecosystem which con-tains a single species X. Birth and death processesof X is given by

X+O�! 2X; (2a)

Xm�!O; (2b)

where X denotes an occupied site of biospeciesand O is the vacant site. The reactions (2a) and(2b) mean reproduction and death processes, re-spectively: the parameterm represents the deathrate of X. The reaction (2a) are carried out by twodifferent methods; namely, CP and its mean-fieldsimulation (MFS).

The Simulation method for CP is defined as follows:

1) Initially, we distribute individuals on square lat-tice; each lattice site is either empty (O) or occupiedby an individual (X).

2) The reactions (2) are performed in the followingtwo steps:

(i) First, we perform two-body reaction (2a):Choose one lattice site randomly, and then specifyone of four neighboring sites. If the pair of sites areX and O, then O is changed into X.

(ii) Next, we perform one-body reaction (2b).Choose one lattice point randomly; if the site is oc-cupied by X, the site will become O by a probability(rate)m.

3) Repeat step 2 byL�L times, whereL�L is thetotal number of lattice points. This step is called asMonte Carlo step [Tainaka, 1988.]. We assume thatthe value ofL is 100.

4) Repeat step 3) until the system reaches a station-ary state. Here we employ periodic boundary con-ditions.

Next, we describe the method of mean-field simu-lation (MFS) which alouds the long-ranged interac-tion. Almost all procedures of CP are not changedin MFS, but the step (i) in 2) for CP is changed asfollows:

(ii)’ Two lattice sites are randomly and indepen-dently chosen; if these sites are X and O, then Ois changed into X.

Now we consider the computer experiment of ex-tinction caused by the sudden increase of the valueof m. In other words, we carry out perturbationexperiments of phase transition. It is known thatCP exhibits a phase transition at a critical pointm = mc. The value ofmc ismc � 1:213 on squarelattice. The phase transition resembles the second-order transition; at the critical point, the steady-statedensity continuously changes from non-zero valueto zero. The experiment is performed as follows:Before the perturbation (t < 0), our system staysin a stationary state atm = m1, where the speciesX exists (m1 < mc). After t = 0, the death ratem is suddenly increased, and held atm = m2

(m2 > mc). Because of this perturbation, the sys-tem eventually reaches the extinct phase. We repeatthe same experiment (fromm1 to m2) many time,and record the time dependence of species density.

3 CRITICAL SLOWING -DOWN

In this section, we explore the dynamics in station-ary state to know the initial condition of perturba-tion experiment. In Fig. 1, typical examples ofstationary state for CP is illustrated, where black

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and white denote the lattice sites of X and O, re-spectively. We obtain the averageA(t) and varianceV (t) of densityx(t) which are defined by

A(t) =1

T

Z T

0

x(t)dt; (3a)

V (t) =1

T

Z T

0

[x(t)�A(t)]2dt: (3b)

It is known that the varianceV (t) in stationarystate has been increases with the decrease of steady-state densityA(t) [Marro and Dickman, 1999]. Thevariance diverges near the critical pointmc. Sucha phenomenon is caused by the so-called “criticalslowing-down”(divergence of the relaxation time).The similar result is obtained for MFS.

Fig. 1: Snapshots of typical stationary patterns forCP. The value of annihilation ratem is below thecritical pointmc (mc � 1:2), so that species X cansurvive. (a)m = 0:3, (b)m = 1:0. Black and whitedenote the lattice sites of X and O, respectively.It isworth while nothing that species X goes extinct forthe higher values thanmc.

In the case of MFS, we can prove the criticalslowing-down. If the total sites is infinitely large(L!1), the dynamic equation for MFS becomes

:x = Rx(1� x=K); (4)

where the dot represents the derivative with respectto the timet which is measured by the Monte Carlostep, and the quantitiesR andK are defined by

R = 2�m; K = (2 �m)=2: (5)

The steady-state solution of logistic equation (4) canbe obtained by setting the time derivative in (4) tobe zero. It is well known that non-trivial solution(x = K) for m < 2 is stable, and that phase transi-tion occurs atmc = 2: for m � mc (orm < mc),species cannot (or can) exist. Moreover, the logis-tic equation explains the critical slowing-down nearm = mc. When the differencex(t) � K is muchsmaller than unity, we have from (4) that

x(t)�K / exp(� j mc �m j t): (6)

Hence, we see that the relaxation time1= j mc�m j

diverges in the limitm ! mc. If m = mc, we ob-tain power-law decay. Because of critical slowing-down, the variance in stationary state diverges nearm = mc.

4 RESULTS OF PERTURBATION EXPERI -MENTS

4.1 Results of MFS

Perturbation experiment is repeatedly performedfrom m1 to m2, wherem2 > mc andm1 < mc.We prepareN kinds of initial patterns (ensembles)which are in stationary state atm = m1; each ofthem has the densityxi(0) (i = 1; 2 : : :N ). Thevalue ofxi(0) is almost equivalent to the steady-state density atm = m1. We obtain the dynamicsxi(t) for t > 0 in order to calculate the ensembleaverageA(t) and the varianceV (t) which are de-fined by

A(t) =1

N

Xi

xi(t); (7a)

V (t) =1

N

Xi

[xi(t)�A(t)]2: (7b)

In both methods of CP and MFS, the stationarystate near the critical pointmc exhibits the criticalslowing-down. Since the variance in stationary stateincreases with the decrease of steady-state density,it is expected thatfluctuation enhancement[Kuboet al., 1973; Tsuchiya and Horie, 1985] (extreme in-crease ofV (t)) occurs in the transient state of dy-namic process.517

We carry out perturbation experiments of extinctionin mean-field simulation (MFS).In Fig. 2, a typicalresult of perturbation experiment is displayed; thetime dependencies of both average densityA(t) andthe varianceV (t) are plotted. It is found from com-puter simulation that the fluctuation enhancementnever takes place for various values ofm1 andm2;especially whenm2 takes a value much larger thanmc, the value ofV (t) rapidly decreases with time.

Fig. 2: The result of perturbation experiment in themean-field limit. The time dependencies of both av-erageA(t) and varianceV (t) defined by (7a) and(7b) are displayed, where the value ofV (t) is mul-tiplied by 200. At timet = 0, the annihilation ratem is jumped from 1.0 to 2.1. We repeat the sim-ilar experiment 100 times (N = 100) with differ-ent initial patterns. The system has104 lattice sites(L =100 2).

In the below, we give an explanation for the fact thatfluctuation enhancement never occurs in the mean-field limit. From (7b), we have

V (t) =1

N[Xi

xi(t)2�

1

N(Xi

xi(t))2]

=1

N2

Xi>j

[xi(t)� xj(t)]2:

It follows that

_V =2

N2

Xi>j

(xi � xj)( _xi � _xj): (8)

The dynamics fort > 0 is assumed to be determinedby the logistic equation (4), where both parametersR andK are negative (m2 > 2). Let xi(t) andxj(t) be two solutions of (4) whose initial valuesare slightly different from each other. Here we putxi(0) > xj(0). From (4), we have

:xi�

:xj = R(xi � xj)[1� (xi + xj)=A]; (9)

Note thatxi(t) 6= xj(t) for t > 0 because of theuniqueness of the solution to (4) andxi(t) > xj(t)for all t > 0. Hence, the right-hand side of (9) is al-ways negative, and the differencexi(t) � xj(t) de-creases as the time proceeds. It is therefore proved

from (8) that the varianceV (t) always decreases.Strictly speaking, this proof is insufficient, sincethe dynamics for a finite value ofL is describedby some stochastic differential equation, such as(1). Nevertheless, the logistic equation (4) well ex-plains that i) critical slowing-down occurs, and ii)enhancement of fluctuation does not occur.

4.2 Result of CP

The basic equation of CP forL!1 becomes;

:x = (PXO + POX)�mx (10a)

wherePij is the probability density finding a statei at a site and a statej at a nearest neighbor of theformer site (i; j =X,O). Note the difference fromthe conditional probability; the relations

Pij = Pji; �jPij = Pi (10b)

Unfortunately, (9) cannot be solved; nevertheless, itis known that the phase transition occurs atm = mc

(mc � 1:2 for d = 2).

We carry out similar perturbation experiment forCP: the value of annihilation ratem is jumped fromm1 to m2 (m1 < mc andm2 > mc). In Fig.3, the time dependencies of both average densityA(t) and the varianceV (t) are plotted, where weputm1 = 0:3 andm2 = 1:3. It is found from Fig. 3that the fluctuation enhancement takes place. Eventhough the average density decreases, the varianceV (t) increases at intermediate stage of phase tran-sition. Note that such a phenomenon is not alwaysobserved for all experiments. Fig. 4 illustrates thetime dependencies of bothA(t) andV (t) for thecases ofm2 = 2:0 and 5.0. As the value ofm2 in-creases, the enhancement of fluctuation disappears.

Fig. 3: Perturbation experiment for CP. The valueof m is jumped from 0.3 to 1.3 (N = 100 andL = 1002). Both averageA(t) and varianceV (t)are depicted against time, where the value ofV (t)is increased by a factor of 70.

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Fig. 4: The same as Fig. 3, but the annihilation ratem is jumped from 0.3 tom2; (a) m2 = 2 and (b)m2 = 5.

The fluctuation enhancement is not caused by thecritical slowing-down, since it is not observed inthe mean-field limit. We associate the enhancementwith the cluster formation of particles: due to theclumping behavior, the variation may become rich.This cause is consistent with the results of Figs. 2-4. To explore the degree of clumping, we obtain theratio

RXX � PXX=x2 (11)

in stationary state [Tainaka, 1994]. When the distri-bution of particles is just random, we haveRXX =1: WhenRXX > 1 (RXX < 1), they distributecontagiously (uniformly). The basic equation (10)in stationary state becomesPXO = mK=2. Onthe other hand, (11) leads toPXX = K � PXO.From (11), we haveRXX / K�1. It is well knownfor CP that near the critical point (m ! mc), thesteady-state densityK satisfiesK / (mc � m)� ;where� is a positive constant. It follows that

RXX / (mc �m)�� : (12)

Whenm approachesmc, the degree of clumping ofX becomes rapidly high (RXX ! 1). The en-hancement ofV (t) may be caused by the clumpingbehavior (12) of particles. Ifm2 is not so large,the system more or less stays in stationary state; thevariance thus increases as density decreases.

5 CONCLUSIONS

Study on the process to biospecies extinction is veryimportant for conservation biology. In the present

article, we study perturbation experiments of phasetransition (extinction), and report the uncertaintyoriginated in critical phenomenon. The annihilationratem of particle is suddenly increased fromm1 tom2, wherem1 < mc andm2 > mc. The densityx of particle (order parameter) is thus changed froma positive value to zero. We estimate the fluctuationin population size (density) which is automaticallygenerated. Simulation is carried out by two differ-ent methods: CP and its mean-field version. In thelatter case, the fluctuation enhancement does not oc-cur. On the other hand, when the method of CP isapplied, and whenm2 takes a value near the crit-ical pointmc, then the fluctuation enhancement indynamic processes takes place (Fig. 3).

We can propose an experiment of cultivation of bac-teria. The cultivation are carried out on two kinds ofnutrient media; that is, agar and fluid media. To theformer model, we apply the basic contact process(CP) in which the growth process (1a) occurs withina short range: offspring are reproduced at a neigh-boring site of mother. On the other hand, in the fluidmedium, solution is always stirred, so that offspringare randomly distributed in the medium. Thus thismedium corresponds to the mean-field simulation(MFS) of CP. The reaction (1b) means the deathprocesses of bacteria In actual experiment, (1b) isrealized under UV light or radioactive rays. The pa-rameterm represents the intensity of light (rays).We carry out the experiment of extinction caused bythe sudden increase of the value ofm. Our resultspredict that there are a variety of different processesto extinction in the case of agar medium (CP).

So far, the enhancement of fluctuation has been ob-served in some physical systems [Kubo et al., 1973;Tsuchiya and Horie, 1985]. However, our systemhas distinct properties never seen in the previousworks:

i) The fluctuation enhancement is not always origi-nated in the critical slowing-down: it is not observedin the mean-field limit (Fig. 2).

ii) Previously, the order parameter was increasedfrom zero to a positive value, whereas the presentpaper reports the just opposite case: the order pa-rameter (density) is decreased. If the order param-eter would increase, the varianceV (t) often in-creased; examples are the exponential growth andlogistic equation.

Our results suggest that for the fluctuation enhance-ment, not only the critical slowing-down but alsothe clumping behavior of particles are essential. The

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latter property has an important meaning in ecology,since almost all species spatially form a clumpeddistribution. Especially, when a species becomesendangered, the degree of clumping usually in-creases. This is due to the inherent nature of biolog-ical species: a offspring is produced in the neigh-borhood of parents. In the system (2), we regard Xas a biospecies, andm as the death rate. Even ifthe densityx approaches zero, the conditional prob-ability PXX=x never vanishes (12). The fluctuationenhancement ecologically means that there are a va-riety of processes to extinction of a species. It is noteasy to predict the extinction process of species.

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