Habitat Fragmentation and Biodiversity Collapse in Neutral Communities

Ecological Complexity 1 (2004) 65–75

Habitat fragmentation and biodiversity collapse in neutral communities

Ricard V. Solé a,b,∗, David Alonso a,c, Joan Saldaña d

a ICREA-Complex Systems Lab, Universitat Pompeu Fabra, Dr Aiguader 80 08003 Barcelona, Spain b Santa Fe Institute, 1399 Hyde Park Road, New Mexico 87501, USA c Department of Ecology, Facultat d’Ecologia, Diagonal, Barcelona, Spain d Departament d’Informàtica i Matemàtica Aplicada, Universitat de Girona, 17071 Girona, Spain Received 3 November 2003; received in revised form 5 December 2003; accepted 7 December 2003

Abstract

Models of habitat fragmentation have mainly explored the effects on a few-species ecologies or on a hierarchical community of competitors. These models reveal that, under different conditions, ecosystem response can involve sharp changes when some given thresholds are reached. However, perturbations, recruitment limitation and other causes may prevent competitive hierarchies from actually operating in natural conditions: the process of competitive exclusion underlying hierarchies could not be a determinant factor structuring communities. Here we explore both spatially-implicit and spatially-explicit metapopulation models for a competitive community, where the colonization-extinction dynamics takes place through neutral interactions. Here species interactions are not hierarchical at all but are somehow ecologically equivalent and just compete for space and resources through recruitment limitation. Our analysis shows the existence of a common destruction threshold for all species: whenever habitat loss reaches certain value a sudden biodiversity collapse takes place. Furthermore, the model is able to reproduce species-rank distributions and its spatially explicit counterpart predicts also species–area laws obtained from recent studies on rainforest plots. It is also discussed the relevance of percolation thresholds in limiting diversity once the landscape is broken into many patches. © 2004 Elsevier B.V. All rights reserved.

Keywords: Habitat fragmentation; Biodiversity collapse; Neutral communities

1. Introduction be very small islands isolated from each other by crop land and pasture. Fragmentation and loss of habitat are When native vegetation is cleared (usually for agri- recognized as the greatest existing threat to biodiver- culture or other kinds of intensive exploitation) habi- sity. Fragmentation is occurring, to various degrees, tats which were once continuous become divided into in nearly all of the major habitat types found through- separate fragments (Hanski, 1999; Pimm, 1991). Af- out the world. Rainforests, which comprise 6% of the ter intensive clearing, the separate fragments tend to world’s land area and which contain at least 50% of the world’s total species, are being cleared and frag- ∗ Corresponding author. mented at a rate far exceeding all other types of habi- E-mail address: [email protected] (R.V. Sole).´ tat. Human-caused habitat fragmentation precipitates

1476-945X/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.ecocom.2003.12.003 66 R.V. Solé et al. / Ecological Complexity 1 (2004) 65–75 biodiversity decline because it destroys species, dis- presence of a trade-off. Using this approximation, rupts community interactions, and interrupts evolu- Stone derived a simple relation between the√ number of tionary processes (Ehrlich and Ehrlich, 1981; Pimm extinct species and fragmentation: E = D/α where et al., 1998; Leakey and Lewin, 1995; Levin, 1999). α is the ratio between the abundance of two successive How does habitat fragmentation affect biodiversity species used to set the initial distribution of abun- in rich ecosystems? The answer to this question will dances. Stone’s work reveals that very small amounts depend on the type of interactions displayed by the of habitat destruction have a huge detrimental effect species under consideration. Previous multispecies on coral reefs: a destruction of only 10% of the habitat models of habitat fragmentation have assumed the can trigger the loss of up to half the total species. From presence of a hierarchical organization of competitive Tilman’s et al. work, the effects on a forest ecosys- interactions. The best known example is Tilman’s tem would be less important, and the same amount of model, defined as (Tilman, 1994; Tilman et al., 1994; destruction would only remove 5% of the species. Tilman and Kareiva, 1997; Kareiva and Wennengren, The competition–colonization trade-off seems to 1995): be fundamental among competing species: the ability   to persist on a site, i.e., to be a good competitor, ver- i i−1 dpi   sus the ability to get to a new site, i.e. to be a good = cipi 1−D− pj − mipi − cjpipj dt colonizer. As a mechanism of coexistence was termed j=1 j=1 fugitive coexistence (Fisher et al., 1951; Skellam, (1) 1951). Since these two pioneering studies this trade-off has been explored in several forms (MacArthur and where pi (i = 1,... ,n) is the fraction of the occupied Wilson, 1967; Tilman, 1982, 1994; Stone, 1995; habitat for the i-th competitor and D the fraction of Crowley and McLetchie, 2002)—see also Hanski destroyed patches. Species specific colonization and (1999) and reference therein. But, to what extent are extinction rates are represented by ci and mi, respec- actually hierarchies structuring natural competitive tively. The single (non-zero) equilibrium state for this communities? system is given by p∗ = (p∗,... ,p∗), where 1 n It is important to remark here that the existence i−1 of competitive hierarchies has been clearly shown to ∗ mi cj ∗ pi = 1 − D − − 1 + pj , structure competitive communities under experimen- ci ci j=1 tal conditions. When there is an only limiting resource, i = 1,... ,n (2) poorer competitors are displaced by superior competitors as the resource is depleted (Tilman, 1982). Here there is a hierarchical order from the first (best) However, some recent studies point to the evidence competitor to the last (worst) one. Extinction is as- that competitive hierarchies are not always at work sumed to be constant for all species and the coloniza- under natural conditions. Rather than competitive lo- i− tion rates are chosen as ci = m/(1 − q)2 1, where q cal exclusion under a strict hierarchy, studies on tree the abundance of the best competitor and i is the order diversity in neotropical forests seem to indicate that label in the competitive hierarchy. This choice assumes recruitment limitation appears to be a major source of a trade-off: the best competitor (i = 1) is the worst col- control of tree diversity (Hubbell et al., 1999; Tilman, onizer. Furthermore, such a choice allows to recover a 1999). In other words, the winners of local compe- geometric distribution of species-abundances. One of tition are not necessarily the best (local) competitors the most important consequences of this model is the but those that happened to colonize that particular site presence of a debt: a cascade of extinctions that occur first. Such priority effects have been reported also in generations after fragmentation (Tilman et al., 1994). other systems (Barkai and McQuaid, 1988). Further analysis by Lewi Stone expanded the pre- Furthermore, a community of ecologically equiv- vious approach to coral reefs (Stone, 1995). By using alent species rather than a hierarchically structured some studies on coral reefs in Israel, where the best competitive community is the key assumption of a colonizer has the largest abundance, he correlated theory supported by extensive field research mainly mortality rates with colonization, thus assuming the in rainforest that, as E.O. Wilson says, “could be one R.V. Solé et al. / Ecological Complexity 1 (2004) 65–75 67 of the most important contributions to ecology of the had seeds dispersed by gravity or by ballistic dis- past half century” (Hubbell, 2001). Thus, in this paper, persal. Thus, interactions between dispersal and the instead of considering a hierarchy of competitors, we ability to persist can affect spatial structure in popula- use a metapopulation model where species are eco- tions in ways that could alter local diversity and com- logically equivalent in their colonization ability. In a petitive interactions at the community level. Such a metapopulation context such an equivalence regarding scenario is known to provide an essentially unlimited the fraction of colonized habitat and extinction thresh- diversity. olds is achieved by an equal colonization-to-extinction ratio (ci/mi) (Hanski, 1999). We are considering a community where species may just colonize empty 2. Mean field model sites and share the same ci/mi ratio. This constancy can be seen as another possible expression of the fun- A spatially-implicit, mean field (MF) metapopula- damental competition–colonization trade-off. The bet- tion model of such recruitment limitation process can ter colonizer a species is, the worst ability to persist on be easily formulated by means of a generalization of a site (high extinction rate) it must have. Our model is Levins model (Levins, 1969) as: as general as that of Tilman (1994), but instead of as-   S suming a strict competitive hierarchy it assumes strict dpi   = fi(p) = cipi 1 − D − pj − φ(ci)pi, neutral competitive interactions. dt Given that neutrality has been shown to account for j=1 important regularities of community structure, such as i = 1,... ,S, (3) species relative abundance distributions and species including habitat fragmentation. Here the extinction richness, and given also that it has been tested in very rate mi has been replaced (last term) by mi = φ(ci) different natural ecosystems (Hubbell, 2001), the con- which gives the functional form of the trade-off. This clusions, in particular, with regard to the consequences trade-off is chosen to rend all species ecologically of habitat loss, drawn from our approach might even equivalent in their colonization ability, where as in be more realistic and valid that those drawn form pre- Hanski (1999), this equivalent ability means the same vious work. ability to colonize and persist if they were in isola- In the second part of the paper, we expand the model tion. So, we assume the same c/m ratio for all species. into a spatially-explicit context. Interestingly, the Equivalently, we can write the functional form of the spatially-explicit model supports the spatially-implicit trade-off then as: derived results and is able to produce realistic species- abundance and species–area curves (SAR). In the φ(c) = αc (4) spatially-explicit model species are located on a We assume α<1, which means that no species squared lattice and colonize just empty neighboring would go to extinction in isolation. Therefore, such sites. Better colonizers have greater extinction rates trade-off implies that high (low) colonization rates and tend to persist less on a site. There is a fair are linked to high (low) local extinction rates, and can amount of evidence from many plant communities be seen as a particular expression of the fundamental for a trade-off between competitive strength, i.e, the competition–colonization (persistence ability versus ability to persist, and dispersal, i.e, colonization abil- colonization ability) trade-off (Lehman and Tilman, ity (Lehman and Tilman, 1997). In forests limited 1997). by light, the trees with the greatest ability to tolerate As a result of this choice, it is easy to see that the shade are often competitively superior but severely total fraction of occupied patches P = pi will limited by dispersal. Dispersal limitation may be criti- i evolve in time as: cal to understanding species coexistence and commu- dP nity dynamics in tropical forests. It has been observed =(1 − D − P − α)P (5) dt that some tree species exhibits densely aggregated spatial distributions independent of habitat structure. where = i pi(t) ci/P(t) is the average colo- Nearly all species exhibiting pronounced aggregation nization rate at time t, and it is used that the average 68 R.V. Solé et al. / Ecological Complexity 1 (2004) 65–75 extinction rate is < φ(c) >= αunder the as- uals so long as all individuals obey the same interac- sumption of a linear trade-off given by (4). Hence, tion rules”. In particular, notice that under such a strict Eq. (5) has a unique positive stationary value, namely, equivalence, our model predict as well a biodiversity ∗ P = 1 − D − α, which is globally asymptotically collapse at the same threshold Dc = 1 − α, where stable for any initial condition P(0)>0. This implies α = mi/ci. that the main result of this mean-field model (where Note that, when φ(c) is nonlinear, a model where spatial correlations are not taken into account) is that species just colonize empty sites like ours (3) has no the global population will experience a collapse at a positive equilibrium solution different from the one critical threshold of habitat destruction Dc = 1 − α with only one competitor, namely, the best competi- ∗ since, for this value of D, P = 0. And thus a diversity tor, i.e, that one with the lowest ratio φ(ci)/ci.In collapse will occur at Dc, instead that at increasing this case, Eq. (3) do not support diversity. The un- values for D, as it was shown by Tilman et al. (1994). derlying hierarchy that can be seen now as a ranking Eq. (3), endowed with a positive initial condition of species after colonization ability (the ratio mi/ci) p0 = (p0,... ,p0 ) 1 S , has as a unique stable equilibrium leads to the extinction of all species but one, the best p∗ which is given by competitor. Once this point is reached the known crit- ∞ ical threshold of habitat destruction for monospecific ∗ 0 D = pi = pi exp ci (1 − D − α − P(t)) dt , metapopulations remains true but now reads c 0 1 − mini{φ(ci)/ci}. For instance, if φ(c) = αc2, the i = ,... ,S 1 (6) best competitor, i.e., the one which survives, will√ be the one with the lowest ci. However, if φ(c) = α c, P(t) P0 = with a solution of (5) with initial condition then the survival will be the competitor with the high- p0. However, such an equilibrium is not asymptot- i i est ci. D p∗ = p0 0 and, from (6), i i . In other words, model using some approximations. Let us consider the lin- p∗ (3) has an infinite set of steady states given by the earized system solutions compatible with the set of conditions S ∗ ∗ ∗ dpi ∂f i(p ) pi = 1 − D − α − pj ,i= 1,... ,S (7) = pj,i= 1,... ,S, (8) dt ∂pj j=i j=1 L = When P0 = 1 − D − α, it follows from (3) that, for where the elements of the Jacobian matrix ij ∗ D<1 − α and P0 P∗), all ∂f i(p )/∂pj are given by: p (t)   fractions of occupied habitat i increase (decrease) S p∗ t →∞  ∗ ∗ monotonously to i given by (6) as since L = ci 1 − D − α − p − p (9) P(t) < P∗ P(t) > P∗ t> D ≥ ii i j ( ) for all 0. Instead, for j=1 1−α, all fractions of occupied habitat tends uniformly to0ast →∞and, hence, diversity collapse occurs. (with i = 1,... ,S) and, for i = j, It is important to mention that the existence of a ∗ Lij =−ci pi . (10) non-trivial equilibrium of coexistence depends cru- cially on the assumption of the ecological equivalence Now let us assume that all populations start at t = 0 in the way we have defined, i.e., as a linear trade-off with the same, small values function φ(c). We could have even defined a more pi(0) = 1 − D − α, i = 1,... ,S. (11) strict version of ecological equivalence, as in Hubbell (2001), as an equivalence in per capita rates among Close to the fixed point p∗ = (0,... ,0), the linear all individuals of every species. Even such an strict approximation (8) reads (using the linear trade-off): ecological equivalence, as Hubbell (2001) states “per- dpi/dt = ci(1−D−α)pi which has a solution: pi(t) = mits complex ecological interactions among individ- exp(ci[1 − D − α]t). R.V. Solé et al. / Ecological Complexity 1 (2004) 65–75 69

At a given time step T , the probability of species i i.e., the time at which the linear system reaches the under the linear approximation will be given by: maximum available population allowed by the nonlinear model. Strictly that equilibrium should be reached eci[1−D−α]T Pi(T) = ,i= 1,... ,S. (12) asymptotically as time tends to infinity. However, that S cj[1−D−α]T j=1 e time can be approximately computed using the properties of the geometric series. It can be easily shown c = i/S Using the ordering i and using that that (16) can be written as S γT/S γT ( − ) S ∗ ( γT/S)j = e 1 e γT − e γT/S (13) ∗ e 1 1 − e pj(T ) = ∗ = 1 − D − α. j=1 1 − e−γT /S j=1 where γ = 1 − D − α, we obtain Hence, for γT ∗/S close to zero and recalling that γ = (i−1)γT/S Pi(T) = Ze ,i= 1,... ,S. (14) 1 − D − α it follows ( − D − α)2 Here Z is the normalization factor: 1 ( γT ∗ − ) ≈ 1 . T ∗ e 1 S (17) 1 − eγT/S Z = (15) T ∗ 1 − eγT So, can be reasonably estimated from (17) as long as the number S of species is large enough to guar- The last equations give a geometric distribution of antee that γT ∗/S is close to zero. For instance, for abundances, in agreement with field observations α = 0.5, S = 150, = 10−3 and D = 0, one obtains from virgin habitats (Fisher et al., 1943; May, 1975) T ∗ = 4.129 with γT ∗/S = 0.014. In Fig. 1 we plot To restate our result in ecological terms: the pro- the exact, computed through the numerical integration cess of ecological succession, i.e., the assembly of a of system (3) and the approximate (14) distributions. competitive community from scratch (pi(0) = We can see an excellent agreement between them. 1 − D − α) onto an empty area under the rules of The behavior of this mean field model under habitat recruitment limitation and ecological equivalence destruction is summarized in Fig. 2. Here we can see gives rise to a geometric distribution of abundances. (Fig. 2a) that species extinctions start to occur close to Alternatively, due to priority effects any other ini- Dc and that no extinction occurs below the threshold. tial distribution of abundances will not ever tend to Note that, according to Eq. (5), the characteristic time a the geometric distribution at equilibrium unless a strong perturbation resets the process of ecological succession again. Moreover, it is interesting to note that any geometric species rank distribution can -2 be rewritten as a relative abundance distribution of 0.012 10 the form p(x) ≈ x−β with β = 1. Thus, the same power-law scaling exponent has been obtained before in previous similar models where species compete for Probability P(k) 10-3 space in a somehow neutral way (Engen and Lande, 100 101 102 1996; Solé et al., 2000; McKane et al., 2000; Pueyo, 0.007 species rank k 2003). Probability P(k) To finish, now we need to estimate a reasonable exact value T ∗ in order to obtain some bound to the exact Linear distribution. Since there is no obvious characteristic ∗ 0.002 time scale to be used, we can estimate T by using 0 50 100 150 the condition species rank k S ∗ Fig. 1. Species-rank distribution from the mean field model (open pj(T ) = 1 − D − α (16) circles) and from the linear approximation (line). A geometric j=1 series is obtained (showed in log-log scale in the inset). 70 R.V. Solé et al. / Ecological Complexity 1 (2004) 65–75

100 by (11), species that have higher colonization rates are T=100 occupying larger areas (see (14)). That is why they 80 T=500 T=1500 suffer ﬁrst from habitat destruction. As in Tilman et al. 60 T=3000 (1994), habitat destruction affects ﬁrst those species 40 occupying larger areas. However, in contrast to previous models, the point to stress here is that a clearly

Extinct species 20 deﬁned common extinction threshold arises. (A) 0 5.0

4.0 3. Spatially-explicit model 3.0 2.0 In the this section, we consider the introduction Entropy of spatially explicit effects, which are known to 1.0 play a leading role (Dytham, 1994; Dythman, 1995; (B) 0.0 Bascomplte and Solé, 1996; Bascompte and Solé, 0.8 1998; Solé et al., 1996). Mean ﬁeld models are often 0.6 challenged by their spatially explicit counterparts. In particular, it has been shown that percolation pro- 0.4

cesses can alter the presence and location of extinc- 0.2 tion thresholds (Bascomplte and Solé, 1996). Besides, finite-size effects will lead to a slow stochastic extinc- 0.0 tion of species unless immigration is introduced (Solé 0.0 0.2 0.4 0.6 0.8 1.0 et al., 2000; McKane et al., 2000; Hubbell, 2001). The (C) Habitat destruction D spatially explicit model is constructed using a stochas- Fig. 2. Results obtained from the mean field model. Here we tic cellular automaton (Durrett and Levin, 1994; show: (A) the fraction extinct species; (B) the entropy, as defined Bascomplte and Solé, 1996) on a two-dimensional by (15) and (C) the average colonization rate. lattice of L × L discrete patches and periodic bound- ary conditions. Each site on the lattice can be either to extinction is propportional to 1/mi, and, therefore, unoccupied or occupied by one of S possible species. the loss of most persistent species can be very long. Actually the model can be accurately formalized as an Once D>Dc, however, extinctions sharply raise. A interacting particle system as in Durrett (1999), where different metric is provided by the entropy defined as it is also included a multispecies neutral competition model. However, our model assumes local coloniza- S ∗ ∗ tion of just neighboring sites as in Keymer et al. H(D) =− Pj log(Pj ) (18) (2000). Moreover, although it has been stated that such j=1 neutral competition, where births are just possible on where the stationary probabilities are: empty sites, precludes the system from stable species ∗ coexistence (Neuhauser, 1992), time to fixation of just pj P∗ ≡ ,j= ,... ,S, one species can be very long when the system is large. j − D − α 1 (19) 1 In addition, external immigration prevents to reach (see Fig. 2b). It first slightly increases (as the species monodominance and, provided low, could be added to occupying larger areas start to decrease in population our analysis without changing our conclusions —see and the system becomes more homogeneous) and then also Hubbell (2001), Alonso (2004). it rapidly falls off as species are lost. The average colonization shows a steady decrease for D

mi j −→ 0. (20)

2. Colonization. Any free site can undergo colonization. rk 0−→j. (21)

where j is any potentially colonizing species from the neighborhood of the k site. Colonization rates depend on the colonization pressure caused by neighboring occupied sites. Assuming that the k Fig. 3. Snapshot of the metapopulation model for L = 100, site is free, colonization rate of this site at time t S = 150 and a linear trade-off with α = 0.25. Here the gray can be computed as: scale indicates the values of the colonization rate displayed by the specific species at each patch. S (k) rk = cj × ρj (22) j=1 snapshot of a L = 100,S = 150 and α = 0.25 sys- (k) tem is shown in Fig. 3. Here gray scales just indicate where ρ is the local density, so the number of j different species. We can see a patchy distribution of neighbors of the k site that are occupied by species most species (some others have very scattered). j divided by the total number of neighboring sites, The patterns of species-abundance (consistent with z (z = 4, so von Neumann neighborhood was used the mean field model) and the species–area relations in our simulations). Whenever a colonization event follow the observed patterns reported from field stud- is to occur at that rate in k site, the particular colo- ies (Hubbell et al., 1999). These results are shown nizing species is elected according to the probabil- in Fig. 4, where the geometric distribution of species ities given by the different non-zero terms in (22). abundances is obvious from the species-rank distribu- The particular simulation strategy is detailed else- tion. The scaling law for the species–area relation is where (Alonso and McKane, 2002). The algorithm is the same as the one observed in the Barro Colorado based on the rejection method described in Press et al. 50 Ha plot (with an exponent z = 0.71 ± 0.01; see (1992). It is interesting to mention that by changing Hubbell et al. (1999), Brokaw and Busing (2000)). local density by global densities in (22), this formal In summary, the imposed ecological equivalence model definition ensures the converge of averages through a simple trade-off together with the rules of values over replicas of stochastic simulations to the local colonization are enough to recover two rele- mean field patch occupancy model (Eq. (3)). vant, well-defined characteristics of real rainforests A first test of the interest of this model involves (compare with Hubbell et al. (1999)). the analysis of the statistical regularities displayed by The effects of habitat destruction are, as expected the virgin, non-destroyed habitat. A first observation from the mean field model, very important. In a previ- is that after a decline in the number of species (which ous study on habitat fragmentation in spatially-explicit can be easily balanced by introducing immigration) models, it was shown that random destruction of the system settles down into a high-diversity state. A patches leads to a critical point where sudden breaking 72 R.V. Solé et al. / Ecological Complexity 1 (2004) 65–75

0.04 into many small sub-areas promotes a rapid decay of the observed global population (Bascomplte and 102 0.71 Solé, 1996; Bascompte and Solé, 1998; Solé and 0.03 Bascompte, 2004). This occurs at the percolation 1 ∗ 10 value D ≈ 0.41 (i.e. when the proportion of avail- ∗ 0.02 able sites is 1 − D ≈ 0.59) where it can be shown

Number of species 0 10 0 1 2 3 4 that the largest available patch cannot longer be con- Abundance 10 10 10 10 10 nected and splits into many sub-patches. The same 0.01 Area effect has been found in our system. In Fig. 5(a),we show the biodiversity decay linked to the presence 0 0 50 100 150 of a percolation phenomenon in the spatially-explicit Species rank landscape. As we can see, there is a steady decay in the number of species present as D increases, with a Fig. 4. Species-rank distribution in the spatially-explicit metapop- D∗ ulation model. A geometric distribution is obtained, as reported rapid decay as is approached. from ﬁeld studies in rainforest plots. Inset: species–area curve A different view of the decay can be obtained by for the same system; the exponent for the power law S ∝ Az plotting the Shannon entropy (Fig. 5(c)). We can see (z = 0.71) is indicated. that it ﬁrst increases as the best colonizers become less and less abundant (and the species-abundance distribution becomes more uniform). At D∗ the effective

80.0

60.0

40.0

Present species 20.0

0.0 (A) 0.0 0.2 0.4 0.6 0.8 1.0 0.8 3.0 (C)

0.6 2.0

Entropy 1.0 0.4 0.0 0.0 0.5 1.0 D 0.2 Average colonization

0.0 0.0 0.2 0.4 0.6 0.8 1.0 (B) Habitat destruction D

Fig. 5. Effects of habitat destruction on the spatial metapopulation model; here L = 50,S = 150 and α = 0.25. (A) number of species for different D values; (B) average colonization rate: it shows a decay due to the extinction of best colonizators. In (C) the Shannon entropy ∗ is shown. The arrow indicates the percolation threshold D = 0.41 and the dashed line the mean field threshold Dc = 0.75. R.V. Solé et al. / Ecological Complexity 1 (2004) 65–75 73 loss of diversity reduces the value of the entropy which ical models and to the right species–area relations drops rapidly. As it was shown by the MF model, the reported from field studies in rainforests. average colonization rate also decreases (Fig. 5(b)). 4. Under fragmentation, the system response is As D∗ is approached, the average colonization rate strongly influenced by percolation thresholds: goes down. This indicates in fact that the rapid de- once the critical value D∗ = 0.41 is reached, the crease in available habitat favours species with lower fragmentation of the system into many patches colonization rates in detriment to better dispersors but leads to diversity collapse. The best colonizers are less persistent species. the first to get extinct. This model result should be qualified in the sense that real ecosystems are not isolated and, as a consequence, extinct good 4. Discussion colonizers are most prone to be rescued by immigration from the region where the system is In this paper we have analyzed the properties of a embedded. simple model of metapopulation dynamics involving recruitment limitation and ecological equivalence. A further comment to highlight is related to the geo- By recruitment limitation we meant that species can metric rank-abundance distribution we have obtained. just colonize empty space. Instead of considering a It is formally equivalent to a power-low relative abun- p(x) ≈ x−β β = hierarchical model of species competition, we have dance distribution , with 1. This dis- explored the consequences of recruitment limitation tribution has been derived under different approaches when habitat is destroyed and fragmented. Ecological from sound first principles by making different as- equivalence is achieved by a simple trade-off where sumptions. Tilman (1994) assumes it and finds the colonization is a linear function of extinction. The ba- functional form of colonization rates that, by allowing sic results obtained from our study are as given further. coexistence, is compatible with it. He finds that the higher the competition rank a species has in the hier- 1. The mean-field model of recruitment limitation archy, the lower its colonization rate is. Other similar introduces an infinite number of steady states (in- trade-offs have also found to be responsible for geo- stead of a single attractor). This is an important fac- metric like distributions in competitive communities tor in the path-dependence shown by the stochastic (Chave et al., 2002). Instead of trade-offs, Engen model. In particular and as a consequence of such and Lande (1996), Solé et al. (2000), McKane et al. path dependence, the system can be maintained far (2000), and Pueyo (2003) give the minimum require- away from the expected geometric (or log-series) ments for geometric distribution to arise from simple distributions unless strong perturbations under- dynamic population models. Pueyo (2003) even tests mine priority effects at least from time to time. these predictions using an exceptional amount of data 2. Under this scenario, the mean-field model predicts (almost a million of identified individual cells from that biodiversity will experience a collapse close phytoplankton samples) and finds high statistical sig- −β to the critical density Dc = 1 − α,0<α<1. nificance for the scaling p(x) ≈ x with β = 1 This is very different from previous predictions, within the abundance interval 10 Population dynamics models diversity loss. Our model should provide a better un- generating species-abundance distributions of the gamma type. J. Theor. Biol. 178, 331–355. derstanding of how competitive communities are orga- Fisher, R., Corbet, A., Williams, C., 1943. A theoretical distribution nized and how fragile they are when habitat loss takes for the apparence abundance of different species. J. Anim. place. Ecol. 12, 42–58. R.V. Solé et al. / Ecological Complexity 1 (2004) 65–75 75

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