Competitive coexistence in a dynamic landscape

ARTICLE IN PRESS

Theoretical Population Biology 66 (2004) 341–353 www.elsevier.com/locate/ytpbi

Competitive coexistence in a dynamic landscape Manojit Roya,,1, Mercedes Pascuala, Simon A. Levinb a

Department of Ecology and Evolutionary Biology, University of Michigan, Ann Arbor, MI 48109-1048, USA b Department of Ecology and Evolutionary Biology, Princeton University, Princeton, NJ 08544-1003, USA Received 4 December 2003 Available online 6 October 2004

Abstract This paper investigates the effect of a dynamic landscape on the persistence of many interacting species. We develop a multispecies community model with an evolving landscape in which the creation and destruction of habitat are dynamic and local in space. Species interactions are also local involving hierarchical competitive trade-offs. We show that dynamic landscapes can reverse the trend of increasing species richness with higher fragmentation observed in static landscapes. The increase in the species-area exponent from a homogeneous to a fragmented landscape does not occur when dynamics are turned on. Thus, temporal aspects of the processes that generate and destroy habitat appear dominant relative to spatial characteristics. We also demonstrate, however, that temporal and spatial aspects interact to influence the persistence time of individual species, and therefore, rank–abundance curves. Specifically, persistence in the model increases in habitats with faster local turnover because of the presence of dynamic corridors. r 2004 Elsevier Inc. All rights reserved. Keywords: Dynamic landscape; Competition models; Community dynamics; Trade-off models; Diversity

1. Introduction A large body of theory has addressed the relationship between landscape properties and species persistence (Nee and May, 1992; Palmer, 1992; Green, 1994; Tilman et al., 1994; Bascompte and Sole´, 1996; Hraber and Milne, 1997; Keitt, 1997; Neuhauser, 1998; Bolker and Pacala, 1999; Hill and Caswell, 1999; With and King, 1999; Swihart et al., 2001; Urban and Keitt, 2001; Bascompte et al., 2002; Prakash and de Roos, 2002; With, 2002; Shima and Osenberg, 2003). In parallel to these largely theoretical efforts, a growing number of landscape experiments are targeting how habitat fragmentation and corridor effects influence community composition in the long term (Tewksbury et al., 2002; Holt and Debinski, 2003). In these studies, the landscape Corresponding author. Fax: +1-352-392-3704.

E-mail address: roym@ufl.edu (M. Roy). Present address: Department of Zoology, University of Florida, Gainesville, FL 32611-8525, USA. 1

0040-5809/$ - see front matter r 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.tpb.2004.06.012

is assumed to be more or less permanent, or equivalently, to change over time scales far too large to be ecologically relevant. In many instances, however, temporal variations such as patch lifespan are comparable to the lifetime of an individual, in which case species persistence will be significantly influenced by habitat turnover (Hastings, 2003). An example of such a dynamic landscape is found in the rocky intertidal where the gaps that form, expand, and then close within mussel beds provide the habitat for numerous subdominant species (Levin and Paine, 1974; Paine and Levin, 1981; Agur and Deneubourg, 1985). Similar gaps can form in forests as the result of physical disturbance but also the spatial spread of pathogens (Hubbell, 1995). Dynamic landscapes can also result from anthropogenic factors such as human development and habitat restoration activities. Fahrig (1992) demonstrated that in an ephemeral habitat, patch turnover dominates species dynamics far more than patch size and distribution. Keymer et al. (2000) considered simultaneous variation of patch availability, connectivity and lifetime, and

ARTICLE IN PRESS 342

M. Roy et al. / Theoretical Population Biology 66 (2004) 341–353

showed that the extinction characteristics of a locally dispersing population depend on their mutual interactions. For multiple competitive species the scenario is less clear. Studies with permanent (or static) landscapes have shown that, while a single population suffers a higher risk of extinction in a fragmented habitat (Bascompte and Sole´, 1996; Hill and Caswell, 1999; With and King, 1999), low patch connectivity can prevent competitive extinction of weaker populations in a community where many species interact locally, and thus promote diversity (Palmer, 1992; Green, 1994; Lavorel et al., 1995; Pacala and Levin, 1997; Neuhauser, 1998; With, 2002). Similarly, dispersal and patch connectivity play a critical role in models that consider source–sink dynamics (Amarasekare and Nisbet, 2001) in a metacommunity context to investigate the role of fragmentation in local and regional diversity (Mouquet and Loreau, 2002, 2003; see also Amarasekare, 2003, for a review of spatial coexistence mechanisms). Although space is treated implicitly, the ‘landscape’ is still static in the sense that patches themselves do not exhibit birth and death processes. Closer to the notion of dynamic landscapes is the treatment of environments as variable in both space and time (Chesson, 1985, 2000a, b; Snyder and Chesson, 2003). By extending the storage effect from environments that vary in time (Chesson, 1985, 1994) to those that vary in space, Chesson (2000a) has shown that environmental heterogeneity can promote species coexistence. A main mechanism behind the spatial storage effect is the buildup of a covariance between the strength of competition and the response of population parameters to the environment. Interestingly, this effect is weakened or lost when variability is spatio-temporal because populations do not have sufficient time to grow locally in response to a favorable environment, and therefore such a covariance is itself weak or nonexistent (Chesson, 2000a). Although space is treated implicitly and variability is assumed to be uncorrelated in time and space, this finding suggests that the outcome of competition should differ strongly betweeen static and dynamic landscapes. In this paper, we investigate the effect of a dynamic landscape on competitive outcomes and community properties, with a model that treats space explicitly and allows for the development of spatial and temporal correlations in the environment. We compare community properties, including species richness, species–area curves, and rank–abundance curves, between dynamic and static landscapes with similar spatial characteristics. Competitive interactions between species are considered hierarchical and incorporate one of two trade-offs: the competition–fecundity (C–F) trade-off proposed by Tilman (1994) and the competition–survival (C–S) trade-off explored by May and Nowak (1994). Examples

of the C–F trade-off can be found in plant communities between seed size and numbers: large-seeded plants are usually stronger competitors than the smaller-seeded ones, but small seeds are produced in large abundance (Tilman, 1994; Levin and Muller-Landau, 2000). The C–S trade-off, on the other hand, can occur for example if competitively superior species are more susceptible to disturbance, herbivory or predation (Paine, 1966; Crawley, 1997; Chave et al., 2002). In epidemics when a single host is co-infected by multiple strains, dominant strains have higher virulence resulting in faster death of the host and therefore themselves (May and Nowak, 1994). Our model is spatially explicit and individual-based, with a limited interaction range for both landscape and species dynamics. Although multiple trophic levels are not considered here, a similar modeling framework could incorporate the effect of predators in a spatial context. Related studies involving spatially explicit models with static landscapes for simple predator–prey (Swihart et al., 2001; Prakash and de Roos, 2002) and food web (Keitt, 1997) dynamics have demonstrated that patchiness and heterogeneity can promote species coexistence and hence diversity. By generating static and dynamic landscapes of similar geometry in the spatial model, we can examine how temporal aspects of the habitat in which species compete modify the relationship between fragmentation and community properties. We demonstrate that the inclusion of temporal changes can drastically alter this relationship, causing in our model a reversal of the trend of increasing species richness with increasing fragmentation. Furthermore, the power-law exponent of the species–area curve becomes close to that generated in a homogeneous landscape, lacking patches of unsuitable habitat. Thus, in agreement with single population studies, our results underscore the importance of temporal aspects of habitat generation and destruction. We also show, however, that temporal and spatial aspects can interact to determine the persistence time of individual species, and the associated rank–abundance curves. Specifically, persistence time of individual species increases for habitats with higher turnover. This counter-intuitive outcome results from changes in habitat connectivity, but interestingly, can only be observed when the landscape changes in time. Another effect of dynamic landscapes is the possibility of complete extinction (of all species in the community), while habitat is still available. For the C–S community, this result is easily explained with a mean-field analysis that assumes the existence of habitat patches but does not take into consideration the explicit arrangement of those patches. Mean-field models have the advantage of analytical tractability, and have been used for example by Tilman (1994) to explain competition of multiple species in a spatial context. This treatment of space is

ARTICLE IN PRESS M. Roy et al. / Theoretical Population Biology 66 (2004) 341–353

limited, however, to the existence of different sites and specifically ignores the explicit distribution of species in the landscape and the locality of interactions whose outcomes depend on patterns of clustering and dispersion. It therefore provides a basis for determining the effect of explicit space and associated clustering patterns. We end with an interpretation of our results in the context of previous theory for uncorrelated environments and implicit space, and with some implications for recent efforts on metacommunity dynamics.

2. The model Our model combines landscape and species dynamics. The idea is that patches of habitat in which the successional species compete, reproduce, and disperse are continuously being created and destroyed. In nature, habitat destruction and creation could result for example from the economic development of natural habitat and from the restoration of developed land, respectively. Landscape dynamics could also result from the disturbance and recovery of a dominant species whose gaps in distribution generate available habitat for other species. Gaps in a mussel bed or forest provide well-known examples. The spatial propagation of disturbance and recovery in a mussel bed has been modeled in a spatial lattice by Guichard et al. (2003). We incorporate this spatially explicit and stochastic model of disturbance in our landscape dynamics. Its main feature is that both habitat creation and destruction are local processes whose spatial propagation depends on the state of neighboring sites. Specifically, space in the model is a square grid in which each site represents a habitat patch. Neighborhoods consist of the eight nearest-neighboring sites. At any given time, only some sites in the lattice are available for colonization by a suite of successional species, forming the ‘gaps’ in which competitive interactions occur. We refer to these sites as suitable to first describe the competitive dynamics. Each successional species is denoted by an intrinsic mortality d drawn from a real number between (0,1) (endpoints excepted), and species competition is governed by a hierarchical C–S trade-off consisting of the following rules (Buttel et al., 2002): 1. A species i occupying a suitable site k dies at a rate di : 2. Invasion of a randomly chosen neighboring site m, if suitable, occurs as follows: the species at k reproduces at a rate unity, and sends offspring to m. If m is empty, or occupied by a species j with dj odi ; the offspring establishes on site m. 3. There is a slow mutation/immigration process, by which a new species arises at a random suitable site,

343

^ with a per capita rate 1=ðxNÞ where x^ is the equilibrium fraction of suitable sites. The mutation/immigration events (rule 3) account for a slow external driving at a rate reduced in proportion to the grid area over which the dynamics operate. This reduction guarantees that the rate of this process is independent of the system size. The hierarchical C–F trade-off is similarly implemented by assigning a fecundity b between (0,1) to each species, and altering rules 1 and 2 as follows: 1. Species i occupying a suitable site k dies at a constant rate d0 : 2. Invasion of a randomly chosen neighboring site m, if suitable, occurs as follows: the species at k reproduces at a rate bi ; and sends offspring to m. If m is empty, or occupied by a species j with bj 4bi ; the offspring establishes on site m. For meaningful comparison between the two trade-offs, we choose d0 (=0.21 in our model) such that in an undisturbed habitat the fraction of colonized space P (¼ i pi ; where pi is the relative abundance of species i) for the C–F community with this d0 matches that for the C–S community (Chave et al., 2002). It has been shown that the two communities behave similarly in a homogeneous landscape both with global and local dispersal under this condition. Consider now a dominant species that has the potential to fully monopolize space in the absence of disturbance, and whose dynamics provide the changing landscape in which successional species compete for space. Thus, sites in the landscape are in one of three possible states: ‘dominant’ (a patch occupied by a dominant species), ‘unstable’ (a patch that has been recently disturbed) and ‘empty’ (unoccupied). The ‘empty’ sites provide the gaps or suitable habitat for the successional species and for the competitive dynamics described above. The ‘unstable’ state propagates the disturbance to neighboring patches whose position adjacent to a recently disturbed site makes them more susceptible to disturbance. In the case of the mussel bed, for example, newly created gaps have unstable edges whose individuals need to rebuild the byssal threads attaching them to the substrate (Guichard et al., 2003). Similarly, the infected state propagates disease locally in susceptible-infective-recovered models (Rhodes et al., 1997), and burning trees propagate fire locally in forest fire models (Clar et al., 1994). Consideration of a third state could also be appropriate for landscape models of developed and restored land, to describe the process of restoration as dependent on the proximity of similar activities. Landscape models with three states differ in important ways from more typical models based on dynamic percolation, which have only two states (e.g. Keymer et al.,

ARTICLE IN PRESS 344

M. Roy et al. / Theoretical Population Biology 66 (2004) 341–353

2000). We return to the differences between two and three state landscape models in the Discussion. The rules for landscape dynamics are as follows: 4. An unstable site becomes empty at rate 1. 5. A dominant site reoccupies with probability d a randomly chosen neighbor, if empty. 6. A dominant site becomes unstable with probability c if at least one neighboring site is unstable. 7. A new disturbance hits a random dominant site making it unstable at a rate 1=N; where N is the grid size. In the numerical implementation of the dynamics, asynchronous updating is utilized and the rates of the (Poisson) processes specify probabilities for the associated events to happen. The conditions specified by the ‘if’ statements in rules 5 and 6 are checked at rate 1, and the parameters c and d are therefore directly defined as probabilities. Rule 4 makes the unstable state short-lived (relative to other processes), becoming suitable at a rate normalized to one. Rule 7 implements the slow process of new disturbances such that a single event occurs on average for each time step, regardless of the size of the grid. An alternate interpretation of the states in the model is from the perspective of habitat availability for the successional species. In this case, ‘empty’ refers to suitable habitat, ‘dominant’ to unsuitable habitat, and ‘unstable’ to a state of transition that locally generates suitable habitat. Henceforth we adopt this terminology. The probabilities c and d above then govern the creation and destruction of suitable habitat, and are interpreted in this way hereafter. Landscape dynamics affect species dynamics (a dominant individual can take over a suitable site which may or may not be colonized), but not vice versa. A static landscape is considered to be the one for which changes in habitat availability and fragmentation with time are considered extremely slow or absent, so that the landscape effectively appears permanent to the species in it. For our simulation purposes, we generate a series of landscape snapshots through rules 4–7 for different values of c; d: Species dynamics then occur through rules 1–3 within the suitable sites on each of these frozen snapshots. A dynamic landscape, on the other hand, is one in which the landscape and community processes occur on comparable time scales. Notice that for a given pair of parameters c and d, dynamic and static landscapes generated in this way exhibit similar amounts and spatial pattern of suitable habitat for the community dynamics. In all spatial simulations, the grid size is N ¼ 400  400 unless noted otherwise, boundary conditions are periodic, and transients are discarded. We begin with a mean-field analysis (MF) of the two models. In this analysis, the explicit configuration of the lattice is ignored. Instead, mean-field models consider

that individual sites are well-mixed and therefore, that neighboring sites are a random sample from the whole population of sites. The mean-field approximation will be used to establish: (1) the requirement of a trade-off for coexistence (trade-offs that are then assumed in the spatial lattice models); (2) the existence of thresholds in mortality and fecundity, respectively, for the C–S and C–F trade-offs; (3) the possibility of total extinction before habitat availability goes to zero for the C–S community; and (4) the dependence of species richness on the dynamics of the landscape. These results, in particular (4), serve then as a basis for comparison with those of the spatially explicit simulations.

3. Mean field equations We analyze a MF description of the model that ignores the two slow processes of new disturbances and mutation/immigration. These MF equations are given by dx ¼ ð1  x  yÞ  dxy; dt

(1)

dy ¼ dxy  cy½1  ðx þ yÞ8 ; (2) dt ! ! i i1 X X dpi ¼ bi p i x  pj  ðdi þ dyÞpi  pi bj pj ; dt j¼1 j¼1 (3) where x; y denote the fractions of sites that are suitable and unsuitable, respectively, and therefore ð1  x  yÞ corresponds to the fraction of sites that are in a state of transition and generate suitable habitat. Equations similar to (1) and (2) were first proposed for a spatial system by Durrett and Levin (2000). The first term in (1) describes the generation of suitable habitat; the second term, the loss of those sites as they become unstable. The second term in (2) describes the loss of unsuitable habitat as sites move into the transition state, at a rate determined by the probability of finding at least one neighboring site that is in a state of transition. Because sites are considered well-mixed, this probability is equal to the probability of finding at least one site in a state of transition in eight neighbors chosen at random from anywhere in the lattice. Eq. (3) is adapted from Tilman (1994): the species are arranged in a simple competitive hierarchy in which the lower labeled species (denoted by i) are the better competitors. The parameters bi and di correspond to the fecundity and mortality of species i. The terms on the r.h.s. represent, respectively, the colonization of suitable patches minus those occupied by superior competitors and species i itself, the mortality enhanced by patch destruction, and the competitive loss to superior species.

ARTICLE IN PRESS M. Roy et al. / Theoretical Population Biology 66 (2004) 341–353

1

δmax

x

1

345

0.5

0 1

0.5

0 1 1 0.5 d

0

0

1 0.5

0.5 c

d

0 0

0.5 c

(b)

(a)

Fig. 1. Results for the MF dynamics. (a) Fraction of available habitat x^ as a function of habitat creation and destruction (turnover) c and d. (b) Threshold mortality dmax as a function of c and d, from Eq. (4) (with b0 ¼ 1).

^ yÞ ^ for Eqs. (1)–(2) have a locally stable equilibrium ðx; all c; d in (0,1) (see Fig. 1a) (Guichard et al., 2003). Assuming bi ¼ b0 (¼ 1 in rule 5), that is, the same fecundity for all species, it is easy to show that species 1; 2; . . . ; n can coexist if and only if there is a hierarchy of mortality dn odn1 o od2 od1 (May and Nowak, 1994). Thus, the C–S trade-off (rules 5 and 6 of the spatial model) is required for coexistence. It follows that there is a threshold mortality dmax ; dmax ¼ b0 x^ 

1  x^ ; d 1 þ x^

(4)

such that any species with dXdmax will go extinct in the landscape (see Appendix). The threshold dmax decreases with decreasing x^ and increasing d, i.e., as the habitat becomes more and more unstable. Hence, with increasing instability, the poorest survivor (best competitor) is the first to go extinct, followed by the next species in the hierarchy, and so on. Total extinction eventually occurs when dmax ¼ 0: Fig. 1b shows dmax against c; d for b0 ¼ 1; with two well separated domains of total extinction and species coexistence ðdmax 40Þ: For a static habitat, ^ This implies d ¼ 0; and Eq. (4) reduces to dmax ¼ b0 x: ^ dmax 40; and therefore coexistence, as long as x40: The value of the threshold is next used to establish how the number of species varies with the dynamics of the landscape. If mutants/immigrants have d drawn from a uniform distribution, the total number SðtÞ of species grows logarithmically with time t (May and Nowak, 1994), SðtÞ / logðdmax tÞ:

(5)

Thus, species richness at any given time is larger for landscapes with larger x^ (and therefore higher c and lower d values). In a similar manner, a C–F community can be explored by assuming di ¼ d0 in Eq. (3) (same mortality for all species). Coexistence occurs if and only if there is a hierarchy in fecundity such that bn 4bn1 4 4b2 4b1 (May and Nowak, 1994). The

existence of a minimum threshold bmin follows, with   1 1  x^ bmin ¼ d0 þ 1 ; (6) x^ d þ x^ such that a species with bpbmin will not survive (see Appendix). Total extinction requires bmin ! 1; which can happen only if x^ ! 0 or equivalently d ! 1: In other words, unlike in a C–S community, some species always persist in a dynamic landscape no matter how ephemeral (as long as do1). The behavior of SðtÞ with landscape changes is qualitatively similar to that of the C–S community. Thus, in the simple MF scenario, a C–F community exhibits better persistence than a C–S community that can even go totally extinct, because of the presence of fugitive species with high fecundity in the former. We expect broadly similar patterns of behavior in communities with long-range interactions. However, for species interacting locally, habitat fragmentation will affect these outcomes. In particular, we are interested in the patterns of species richness with fragmentation. The MF model is further limited by the unrealistic assumption of an infinite number of sites. As a result, S increases without bound as time increases (Eq. (5)). S also fails to exhibit the power-law dependence with area of available habitat typically observed in real communities (Buttel et al., 2002). As we shall see, the spatially explicit model accounts for both of these shortcomings.

4. Results: spatial model Fig. 2 shows the changes in the landscape as the creation and destruction parameters c and d are varied. The equilibrium fraction of suitable habitat x^ is zero for cp0:3 and increases afterwards with c. By contrast, x^ is less sensitive to changes in d. This pattern is a characteristic feature of the model chosen for the landscape dynamics, and allows us later to examine the effects of habitat connectivity separately from those of habitat availability.

ARTICLE IN PRESS M. Roy et al. / Theoretical Population Biology 66 (2004) 341–353

346

1

flarge

x

1

0.5

0 1

0.5

0 1 1 0.5 d

0.1

0.1

1 0.5

0.5 c

(a)

d

0.1

0.1

0.5 c

(b)

Fig. 2. Simulation results for the landscape dynamics, given by rules 1–4. (a) Equilibrium fraction x^ of suitable habitat as a function of c and d. (b) Largest cluster fraction f large as a function of c and d. In all simulations, 10,000 transients are discarded and data are recorded every 10 time steps for the next 20,000 steps.

Patch-connectedness can only be meaningfully defined based on the interaction range of the species (for instance, a species with gap-crossing ability will view physically segregated patches as ‘functionally connected’, With, 2002). Thus in our dynamic landscape, suitable sites form clusters, or connected habitat segments, of all shapes and sizes via eight-neighbor connectedness. Simulations show that these clusters appear and disappear continuously in space and time. Fig. 2b shows the proportion f large of suitable habitat x^ contributing to the largest cluster, as c and d are varied. This quantity gives a rough measure of patch connectivity. It displays an abrupt shift from low to high values across c 0:6 similar to percolation (Fig. 2b). This observation allows us to describe the fragmentation and connectedness of the landscape relative to the percolation-based characterization typically used in landscape ecology (With, 2002). In percolation-type neutral landscape models (NLMs), which have only two states, the sharp transition from fragmented to connected landscapes occurs at a critical density of the habitat. A landscape is therefore called fragmented on one side of this phase transition, and connected on the other. A fully connected landscape in an NLM is also known as homogeneous as the available habitat occupies space completely. Our three state model has been shown to display a percolation-type transition in the connectedness of clusters (Guichard et al., 2003). This transition has been studied in detail for a similar three state predator–prey model (Roy et al., 2003), demonstrating both similarities and differences with static percolation. As expected from eight-neighbor percolation (Guichard et al., 2003), when the suitable habitat crosses the critical density of 0.4, our landscape undergoes a transition from fragmented (with only isolated clusters) to largely connected (compare a and b, Fig. 2). For c above a critical value, the plateau in f large reflects the existence of a ‘spanning cluster’, extending from one boundary to another, in the system. We call this landscape connected (although there still exist a

small number of isolated clusters and the available habitat is not fully connected). It is to be noted that for large turnover rates d, the values of f large in this plateau are almost one: over 90% patches are indeed connected in the range 0:7pcp1; 0:4pdp1: This is also the range over which the available habitat x^ is maximum. Intuitively, fast patch turnover d, combined with high patch creation rate c, generates a dynamic corridor thereby maintaining the high level of connectivity. By contrast to NLMs, however, when f large ¼ 1; the landscape is fully connected but not homogeneous, because the available habitat x^ does not completely fill up space. Instead, the suitable habitat is entirely composed of a large and typically ramified cluster of connected patches. We call this landscape with both suitable and unsuitable habitat ‘heterogeneous’. Another important difference between percolation-type models and ours is that for connected landscapes (high c), we still observe variation in the degree of connectedness as turnover rate d goes from low to high (Fig. 2b), and as available habitat remains almost constant (Fig. 2a). This variation allows us to investigate later the response of species persistence in landscapes of different connectedness but effectively constant habitat availability. We use the parameter range 0:4pcp1 and 0:1pdp1; ^ for which x40; in all our following simulations. Fig. 3 compares, for a C–S community, both the mortality threshold dmax and the average species count S between dynamic (a,c) and static (b,d) landscapes for different values of c and d. The parameters c and d for the static landscapes should be interpreted as ‘‘geometric parameters’’ instead of probabilities associated with the dynamics, because each ðc; dÞ pair creates a unique spatial arrangement of patches with x^ and patch connectivity similar to those of the corresponding dynamic landscape. This comparison demonstrates that the temporal changes of the landscape modify signifi which become cantly the patterns of dmax and S; qualitatively similar to those of the MF dynamics. No noticeable abrupt shift in community behavior occurs

ARTICLE IN PRESS M. Roy et al. / Theoretical Population Biology 66 (2004) 341–353

1 δmax

δmax

1 0.5 0 1 d

0.1

0.4

1

0.5

0.7 c

d

0.1

0.4

0.7 c

0.1

0.4

0.7 c

0.1

0.4

0.7 c

(b)

(a) 150

800

75

400

S

S

0.5 0 1

1

0.5

1 1

1 1

1

0.5 0.1

d

0.4

1

0.5

0.7 c

d

(d)

(c) 150

400

75

200

S

S

347

0 1 0.5 d

0 1

1 0.1

0.4

(e)

1

0.5

0.7 c

d

(f)

Fig. 3. Simulation results for the species counts and mortality threshold of the C–S community (a–d). In (a) and (c), dmax and the average species count S versus the parameters c and d for the dynamic landscape. In (b) and (d), similar quantities for the corresponding static landscape. In (e) and (f), species count S for the C–F community in dynamic and static landscapes, respectively. (Unlike the MF results, SðtÞ in these finite systems fluctuates around a steady mean, Buttel et al., 2002).

across a plane c 0:6 (near the threshold) in either the dynamic or static landscape. Not surprisingly, for any combination of c and d both dmax and S are much higher in the static landscape than in the dynamic one, because of the stable environment in the former. More interestingly, the patterns of dmax and S with c and d, and therefore with similar patterns of fragmentation and amount of available habitat, vary from static to dynamic landscapes. First, only in the dynamic case is total extinction possible (dmax ¼ S ¼ 0). Second, notice that the surfaces in Figs. 3(c,d) show almost opposite trends of variation. In the dynamic landscapes, S decreases steadily in the coexistence regime with decreasing c and x^ (Figs. 3c and 2a), implying a gradual loss of diversity starting again with the best competitors and going downward along the competitive hierarchy (since dmax decreases too), in agreement with Eq. (5). By contrast, in the static landscapes, S increases with decreasing x^ (except for the dip for low c and d values when x^ ! 0; Figs. 3d and 2a). Together with the slow decline of dmax (Fig. 3b), this pattern implies that while the best competitors are still going extinct, the weaker ones are doing better as the increase in fragmentation reduces competition. This effect of fragmentation is completely absent in the dynamic case. Also, dmax and S appear largely insensitive to d for cX0:7; since the spatial properties of the landscape do not vary much with d alone (see Figs. 2a,b). Both species count S and the threshold dmax decrease slightly with increasing d for a

fixed c, as the stronger competitors are progressively eliminated in the more ephemeral habitats. Figs. 3(e,f) show the corresponding results for species counts in the C–F community. The patterns are for the most part similar to those obtained for the C–S community, although the trend reversal with decreasing x^ occurs within a restricted range (c40:7). In this range, the number of species in static landscapes exhibits a slow decrease with increasing c (the connectivity rises slowly, see Figs. 3f and 2b). By contrast, species count increases rapidly for the corresponding dynamic landscapes (Fig. 3e), as observed for the C–S community and for the MF models. Outside this range, the comparison is uninformative since dynamic landscapes cannot sustain the community and static landscapes predictably lose species rapidly for large fragmentation. The species–area curves produced by our model follow for both trade-offs the typical power-law form,  SðNÞ / N z;

(7)

 relating the number of species SðNÞ to the landscape area N (May, 1975). We compare species–area curves between dynamic and static landscapes, but also between these heterogeneous landscapes and a homogeneous one in which the whole lattice is composed of suitable habitat. Results show that the temporal dynamics make the landscape more similar to the homogeneous one, effectively erasing the effect of habitat structure on the exponent of the species–area

ARTICLE IN PRESS M. Roy et al. / Theoretical Population Biology 66 (2004) 341–353

348

 curve. Fig. 4 shows species count SðNÞ for four different 2 grid sizes N ¼ L and for the C–S community on a log–log scale (for c ¼ d ¼ 1). Although the patches are

static landscape (z = 0.46) homogeneous landscape (z = 0.34) dynamic landscape (z = 0.31)

S

102

101

103

104

N

105

Fig. 4. Species–area curves are shown on a log–log scale for a C–S community (Eq. (7)). Data points for the dynamic and static landscapes are shown (o and + respectively) as well as those for an equivalent homogeneous landscape (*). The straight line segments give least-squares fits with slopes z ¼ 0:31; 0.46 and 0.34, respectively. (N ¼ L2 with L ¼ 50; 100; 200; 400).

not homogeneously distributed on the grid, the suitable ^ scales trivially (with an exponent close to habitat xN unity) with N. It follows that species–area curves ^ instead of N would yield the same obtained for xN exponent z. We choose to plot these curves using the size of the whole grid because it is intuitively closer to the sampling techniques applied in field studies. The static landscape shows a higher exponent (z ¼ 0:46) than the homogeneous landscape (0.34, in agreement with Buttel et al., 2002) (Fig. 4). The faster accumulation of species promoted by heterogeneity is lost, however, when the landscape dynamics are turned on (z ¼ 0:31). Complete (100%) habitat suitability in both space and time is the reason why the line for the homogeneous landscape lies appreciably above that for the dynamic landscape. Computed exponent z for other values of c and d, and for the C–F community, give qualitatively similar patterns (not shown), with estimates for the static landscape (0:21pzp0:72) always higher than those for the dynamic landscape (0pzp0:32). The patterns of persistence of individual species show an interplay between connectivity and the temporal changes of the landscape. Specifically, an effect of connectivity on persistence is only seen when the landscape is dynamic. To examine the effect of connectivity for landscapes with similar levels of suitable habitat, we now fix c ¼ 1 and let only d vary. From top to bottom in Fig. 5, d increases and therefore the turnover of sites also increases. The ordinate corresponds to time and the abscissa identifies the mortality

Fig. 5. The time evolution of species abundances is shown for a C–S community in pseudocolor plots, with dark and bright (yellow) regions representing zero and maximum abundance, respectively. (a)–(d) correspond to a dynamic landscape, for c ¼ 1 and d ¼ 0:1; 0:4; 0:7; 1 from top to bottom. The identity of the species (mortality d) is shown along the abscissa and time, along the ordinate. (e)–(h) are the corresponding plots in a static landscape.

ARTICLE IN PRESS M. Roy et al. / Theoretical Population Biology 66 (2004) 341–353 100

100 (static landscape)

(dynamic landscape) -2

-2

10

10-4

p

p

10

10-6 10-8

349

10-4 10-6

0

50

100

150

10-8

0

R

100

200

300

R

Fig. 6. Rank–abundance (R–p) curves for a C–S community, in a dynamic and a static landscape. The values of c and d are the same as those of Fig. 5. The continuous, dashed, dashed–dotted and dotted lines correspond to d ¼ 0:1; 0:4; 0:7 and 1 respectively (c ¼ 1). The curves are generated as follows: an R–p curve at each time step is obtained from Fig. 5 by sorting p horizontally in the descending order and assigning a rank R to each species (starting with R ¼ 1 for the species with largest p and going upward with decreasing p). These R–p plots are then averaged over time, so that each p in the plot represents the time-averaged abundance for a given R. This method is different from averaging p for a given d first, and then assigning rank R (the processes of sorting and averaging are not interchangeable, at least for the dynamic plots of Fig. 5 in which p for a given d fluctuates considerably with time). The latter method does not yield an accurate R–p curve because of the accumulation of a large number of species  over time, resulting in high values of R far exceeding the species count S:

(or identity) of the species. The colors correspond to abundance levels. Thus, a continuous vertical line indicates continuous persistence, and a discontinuous one, the extinction of a particular species in the hierarchy. Panels on the right correspond to static landscapes for the C–S community (Figs. 5e–h). Other than a change in the mortality threshold and therefore in the total number of species, these panels show no substantial differences in the persistence of species. The variation in connectivity, albeit small (see Fig. 2b), has no apparent effect on persistence. By contrast, the panels on the left for the corresponding dynamic landscapes show substantial effects on species persistence (Figs. 5a–d). Surprisingly, persistence appears to increase, in particular for the best competitors, with higher turnover of the sites. An explanation for this counter-intuitive pattern lies in the fact that, as d increases, more and more sites are added to the single largest cluster, thereby increasing landscape connectivity (see Fig. 2b), which in turn sustains a dynamic corridor even in the presence of high patch turnover. This corridor facilitates species persistence through local replacement/colonization of neighboring patches. We have estimated T av ; which is defined as the average, over both time and d; of all intervals T during which the ^ abundances remain above 1% of xN: T av increases steadily with d (not shown here), confirming the above observations. Since the species count S decreases with increasing d (see Fig. 3c), the community has a characteristic low diversity but high persistence in an ephemeral habitat. Results for the C–F community are qualitatively similar. From the patterns in Fig. 5, one would expect the rank–abundance (R–p) curves to vary significantly with landscape parameters. Figs. 6(a,b) show, on log-linear scale, the time-averaged R–p plots for a C–S community in the dynamic and static landscape, respectively. The

curves have the familiar S-shapes (Hubbell, 2001; Chave et al., 2002), with a linear segment (except for large d in 6a) indicating a log-linear relationship. Fig. 6a exhibits a strong dependence on d both in the cut-off value (S decreases with increasing d) and also in the shape of the curve, such that the log-linear nature is lost as d ! 1: The static plot (Fig. 6b), on the other hand, changes little with the value of d, except for a slight decrease of the cut-off as expected. Similar results were obtained for the C–F community. As turnover increases, the communities are increasingly dominated by fewer species with higher abundances. These low diversity communities result from a landscape that is more highly connected, in which a dynamic corridor allows the dominant competitors to persist in spite of a highly ephemeral landscape.

5. Discussion We have shown that a dynamic landscape can significantly affect community properties, including species counts, species–area curves, and rank–abundance curves. In particular, temporal changes appear to override the effect of spatial properties such as habitat fragmentation. Specifically, the expected trend of increasing species numbers with fragmentation, as available habitat makes a transition from connected to fragmented, can be completely reversed if the landscapes we are comparing are dynamic instead of static ones. Our results on species–area curves confirm the previous observation (Buttel et al., 2002) that local interactions and explicit space are important to generate community properties consistent with those observed in nature. However, the slope of these curves depend strongly on the dynamics of the landscape, which can reduce the faster accumulation of species produced by spatial

ARTICLE IN PRESS 350

M. Roy et al. / Theoretical Population Biology 66 (2004) 341–353

heterogeneity. Also, total extinction of an entire community becomes possible in dynamic landscapes even before the complete loss of suitable habitat. Although temporal changes of the landscape appear dominant, they can also interact with spatial properties to influence the persistence of individual species. We have shown that landscapes with similar amounts of suitable habitat, but different rates of patch turnover, differ significantly in persistence time and rank–abundance curves. Persistence time increases with patch turnover, a counter-intuitive result driven by higher connectivity. A highly connected landscape provides a dynamic corridor, thereby allowing species to persist longer even though the species count is reduced. Interestingly, however, differences in connectivity alone fail to generate these patterns in a static landscape. In hierarchical communities with long-range interactions, weaker species with high fecundity can escape competitive extinction by colonizing empty patches wherever available. For species interacting locally, competition is reduced in a fragmented habitat. Weaker competitors with high longevity do better in such a landscape than those with high fecundity who cannot colonize distant patches. These relative differences disappear in a dynamic landscape. These observations raise interesting implications for real systems. In an evolving landscape, a C–F trade-off is likely to facilitate the coexistence of competing species with long-range dispersal. On the other hand, a C–S trade-off would facilitate coexistence of locally interacting species in a highly fragmented landscape. In a connected landscape with fast patch turnover, a community is expected to be low on diversity but high on persistence, regardless of the trade-off. Real communities are of course more complicated, with dispersal ranges lying somewhere between the two extremes considered here. Future work should address this issue by exploring different types of dispersal ranges (Chave et al., 2002). Moreover, the trade-off models studied here include a strict hierarchy that is known to support many species, unlike the more realistic case of a continuous type tradeoff function that would limit diversity (Geritz et al., 1999; Adler and Mosquera, 2000). Such a scenario can be incorporated in our trade-off models by appropriately modifying the invasion rules 2 and 20 (in the model description), such that invasion occurs stochastically at a rate proportional to di  dj and bj  bi ; respectively. Also, it is important to know to what extent the conclusions presented here depend on the specific model for the dynamics of the landscape. We have considered that both habitat destruction and regeneration are local processes that propagate in space as a function of the neighboring state of the landscape. We have also considered a three-state model, motivated by the treatment of physical or biological disturbances that

create gaps in an otherwise dominant species (e.g. mussel bed). As a result, the likelihood of disturbance over space is not fully specified by the distribution of the susceptible state, but is also influenced by the recent history of local disturbance. In this sense, consideration of a third state would also appear sensible for landscape models of developed and restored habitat. The process of restoration would then depend locally on the presence of similar nearby activities, and not simply on the presence of neighboring natural habitat. The addition of a third state has important consequences for the dynamics of the system (reviewed in Pascual and Guichard, submitted). For example, three-state systems of the type used here do not exhibit a sharp transition in the density of types (suitable and unsuitable habitat) as the landscape goes from fragmented to connected (Roy et al., 2003). Two-state systems such as the SI contact process (Levin and Durrett, 1996) and static percolation, do exhibit such a transition. Other differences were already outlined in our description of the landscape dynamics in Section 4. It would be interesting to see if our conclusions are modified in landscapes generated by dynamic percolation (Keymer et al., 2000). Similarly, it would also be interesting to consider three-state systems such as the forest-fire model that differ from ours in their temporal separation of time scales and exhibit selforganized criticality (Pascual et al., 2002). Finally, dynamical processes generating fractal landscapes (Saupe, 1988; Palmer, 1992; With, 2002) would provide comparisons across different types of habitat connectivity as the landscape evolves in time. Our results can be interpreted in light of previous theory on coexistence in spatially varying environments (Chesson, 1985, 2000a, b; Muko and Iwasa, 2000; Snyder and Chesson, 2003). One of the main mechanisms promoting coexistence in such environments is the spatial storage effect, operating primarily through the generation of a covariance between competition and the environment. Spatio-temporal variability was shown to weaken this effect (Chesson, 2000a), suggesting that the outcome of competition should differ significantly between dynamic and static landscapes. However, because variability is assumed to be uncorrelated in space and time, and the models underlying these results treat space implicitly, this conclusion does not immediately follow. In particular, the process of habitat destruction and regeneration can generate important spatial and temporal correlations, as well as spatiotemporal ones such as those influencing persistence times through the above-described dynamic corridors. Nevertheless, we can informally relate our findings to the idea of a weakened covariance between environment and competition, by noting that space in our model is heterogeneous in terms of patch connectivity. The thought process consists of viewing a patch as composed of a focal site and surrounding neighbors of suitable

ARTICLE IN PRESS M. Roy et al. / Theoretical Population Biology 66 (2004) 341–353

habitat. Competition itself is viewed as two sequential processes, the dispersal of propagules to a neighboring site and the ability to outcompete the resident species in that site. In a static landscape, there is clearly a covariance between the environment and competition, since interspecific competition will be stronger in more highly connected patches. Similarly, weakly connected patches will favor the local growth of subdominant species, since a larger fraction of neighboring sites will tend to be free from species higher up in the competitive hierarchy. A dynamic landscape will then tend to destroy such a covariance. Another spatial mechanism known to promote coexistence is also relevant here: the spatial storage effect can be reinforced by local dispersal, in a process explained by the buildup of a covariance between the local growth rate and the local relative density of a species (Snyder and Chesson, 2003). We expect this mechanism to operate in our static landscapes and to weaken for the dynamic case. There is also a large and expanding body of literature on metacommunities with source–sink dynamics (Wilson, 1992; Holt, 1993; Amarasekare and Nisbet, 2001; Yu and Wilson, 2001; Levine and Rees, 2002; Mouquet and Loreau, 2002, 2003). Mouquet and Loreau (2003) show that source–sink models incorporating niche differences produce community patterns similar to those obtained under neutrality (Hubbell, 2001). They have therefore reiterated a call for comparative studies of competition in metacommunities models that would produce testable hypotheses capable of differentiating among underlying mechanisms. Our results indicate that the dynamics of the patches themselves may introduce important modifications of the community patterns under consideration. Furthermore, the treatment of space as implicit may not capture patterns of persistence and related rank–abundance curves, when landscape dynamics involve local processes of habitat destruction and regeneration. Finally, among the landscape ecology metrics in use today, spatial metrics such as ‘dominance’ (for habitat availability), ‘contagion’ and ‘fractal dimension’ (for fragmentation) enjoy a ‘‘C’’ status (‘‘ready for field tests and implementation’’); on the other hand, temporal metrics such as ‘recovery time’, ‘landcover transition matrix’ and ‘diffusion rates’ are given status ‘‘A’’ (‘‘requiring further conceptual development’’) (Frohn, 1998). Determining critical temporal parameters will require a better understanding of the interplay of spatial and temporal landscape processes.

Acknowledgments We thank Fre´de´ric Guichard for useful discussions, and Peter Chesson and three anonymous reviewers for insightful comments and suggestions on a previous draft

351

of the manuscript. We are also pleased to acknowledge the support by the James S. McDonnell Foundation through a Centennial Fellowship to MP, and by the David and Lucile Packard Foundation under Award 8910-48190 to SAL.

Appendix A.1. Eqs. (4,6) Species equilibrium for Eq. (3) occurs at  i1   bj di þ d y^ X p^ i ¼ x^   p^ j 1 þ : bi bi j¼1

(8)

With bi ¼ b0 for the C–S community, and y^ ¼ ð1  ^ ^ from Eq. (1), the equilibrium abundance of xÞ=ð1 þ d xÞ species 1 (the best competitor) is given by   1 1  x^ d1 þ 1 p^ 1 ¼ x^  : (9) b0 d þ x^ It persists ðp^ 1 40Þ if d1 odmax ; where dmax ¼ b0 x^ 

1  x^ : d 1 þ x^

Eq. (4) follows because of the hierarchy of d: Similarly, with di ¼ d0 and b0 ¼ b1 for a C–F community, we can rewrite Eq. (9) as   1 1  x^ d0 þ 1 p^ 1 ¼ x^  ; b1 d þ x^ which gives the condition b1 4bmin for persistence, with   1 1  x^ bmin ¼ d0 þ 1 : x^ d þ x^ Eq. (6) follows because of the hierarchy of b:

References Adler, F.R., Mosquera, J., 2000. Is space necessary? Interference competition and limits to biodiversity. Ecology 81, 3226–3232. Agur, Z., Deneubourg, J.L., 1985. The effect of environmental disturbances on the dynamics of marine intertidal populations. Theor. Popul. Biol. 27, 75–90. Amarasekare, P., 2003. Competitive coexistence in spatially structured environments: a synthesis. Ecol. Lett. 6, 1109–1122. Amarasekare, P., Nisbet, R.M., 2001. Spatial heterogeneity, sourcesink dynamics, and the local coexistence of competing species. Am. Nat. 158, 572–584. Bascompte, J., Sole´, R.V., 1996. Habitat fragmentation and extinction thresholds in spatially explicit models. J. Animal Ecol. 65, 465–473. Bascompte, J., Possingham, H., Roughgarden, J., 2002. Patchy populations in stochastic environments: critical number of patches for persistence. Am. Nat. 159, 128–137. Bolker, B.M., Pacala, S.W., 1999. Spatial moment equations for plant competition: understanding spatial strategies and the advantage of short dispersal. Am. Nat. 153, 575–602.

ARTICLE IN PRESS 352

M. Roy et al. / Theoretical Population Biology 66 (2004) 341–353

Buttel, L., Durrett, R., Levin, S.A., 2002. Competition and species packing in patchy environments. Theor. Popul. Biol. 61, 265–276. Chave, J., Muller-Landau, H.C., Levin, S.A., 2002. Comparing classical community models: theoretical consequences for patterns of diversity. Am. Nat. 159, 1–23. Chesson, P.L., 1985. Coexistence of competitors in spatially and temporally varying environments: a look at the combined effects of different sorts of variability. Theor. Popul. Biol. 28, 263–287. Chesson, P.L., 1994. Multispecies competition in variable environments. Theor. Popul. Biol. 45, 227–276. Chesson, P.L., 2000a. General theory of competitive coexistence in spatially-varying environments. Theor. Popul. Biol. 58, 211–237. Chesson, P.L., 2000b. Mechanisms of maintenance of species diversity. Ann. Rev. Ecol. Syst. 31, 343–366. Clar, S., Drossel, B., Schwabl, S., 1994. Scaling laws and simulation results for the self-organized critical forest-fire model. Phys. Rev. E 50, 1009–1018. Crawley, M.J., 1997. Plant Ecology, second ed. Blackwell Science, Oxford. Durrett, R., Levin, S.A., 2000. Lessons on pattern formation from planet WATOR. J. Theor. Biol. 205, 201–214. Fahrig, L., 1992. Relative importance of spatial and temporal scales in a patchy environment. Theor. Popul. Biol. 41, 300–314. Frohn, R.C., 1998. Remote Sensing for Landscape Ecology. Lewis Publishers, Boca Raton, FL. Geritz, S.A.H., van der Meijden, E., Metz, J.A.J., 1999. Evolutionary dynamics of seed size and seedling competitive ability. Theor. Popul. Biol. 55, 324–343. Green, D.G., 1994. Simulation studies of connectivity and complexity in landscapes and ecosystems. Pacific Conserv. Biol. 3, 194–200. Guichard, F., Halpin, P., Allison, G., Lubchenco, J., Menge, B., 2003. Mussel disturbance dynamics: signatures of oceanographic forcing from local interactions. Am. Nat. 161, 889–904. Hastings, A., 2003. Metapopulation persistence with age-dependent disturbance or succession. Science 301, 1525–1526. Hill, M.F., Caswell, H., 1999. Habitat fragmentation and extinction thresholds on fractal landscapes. Ecol. Lett. 2, 121–127. Holt, R.D., 1993. Ecology at the mesoscale: the influence of regional processes on local communities. In: Ricklefs, R.E., Schluter, D. (Eds.), Species Diversity in Ecological Communities: Historical and Geographical Perspectives. University of Chicago Press, Chicago, pp. 77–88. Holt, R.D., Debinski, D.M., 2003. Reflections on landscape experiments and ecological theory: tools for the study of habitat fragmentation. In: Bradshaw, G.A., Marquet, P.A. (Eds.), Ecological Studies 162. Springer-Verlag, Berlin, pp. 201–223. Hraber, P.T., Milne, B.T., 1997. Community assembly in a model ecosystem. Ecol. Modelling 103, 267–285. Hubbell, S.P., 1995. Towards a theory of biodiversity and biogeography on continuous landscapes. In: Carmichael, G.R., Folk, G.E., Schnoor, J.L. (Eds.), Preparing for Global Change: a Midwestern Perspective. SPB Academic Publishing, Amsterdam, pp. 171–199. Hubbell, S.P., 2001. A Unified Neutral Theory of Biogeography. Princeton University Press, Princeton, NJ. Keitt, T.H., 1997. Stability and complexity on a lattice: coexistence of species in an individual-based food web model. Ecol. Model. 102, 243–258. Keymer, J.E., Marquet, P.A., Velasco-Herna´ndez, J.X., Levin, S.A., 2000. Extinction thresholds and metapopulation persistence in dynamic landscapes. Am. Nat. 156, 478–494. Lavorel, S., Gardner, R.H., O’Neill, R.V., 1995. Dispersal of annual plants in hierarchically structured landscapes. Landscape Ecol. 10, 277–289. Levin, S.A., Durrett, R., 1996. From individuals to epidemics. Proc. Roy. Soc. Lond. Ser. B 351, 1615–1621.

Levin, S.A., Muller-Landau, H.C., 2000. The evolution of dispersal and seed-size in plant colonies. Evol. Ecol. Res. 2, 409–435. Levin, S.A., Paine, R.B., 1974. Disturbance, patch formation, and community structure. PNAS 71, 2744–2747. Levine, J.M., Rees, M., 2002. Coexistence and relative abundance in annual plant assemblages: the roles of competition and colonization. Am. Nat. 160, 452–467. May, R.M., 1975. Patterns of species abundance and diversity. In: Cody, M.L., Diamond, J.M. (Eds.), Ecology and Evolution of Communities. Belknap, Cambridge, MA, pp. 81–120. May, R.M., Nowak, M.A., 1994. Superinfection, metapopulation dynamics, and the evolution of diversity. J. Theor. Biol. 170, 95–114. Mouquet, N., Loreau, M., 2002. Coexistence in metacommunities: the regional similarity hypothesis. Am. Nat. 159, 420–426. Mouquet, N., Loreau, M., 2003. Community patterns in source-sink metacommunities. Am. Nat. 162, 544–557. Muko, S., Iwasa, Y., 2000. Species coexistence by permanent spatial heterogeneity in a lottery model. Theor. Popul. Biol. 57, 273–284. Nee, S., May, R.M., 1992. Dynamics of metapopulations: habitat destruction and competitive coexistence. J. Animal Ecol. 61, 37–40. Neuhauser, C., 1998. Habitat destruction and competitive coexistence in spatially explicit models with local interactions. J. Theor. Biol. 193, 445–463. Pacala, S.W., Levin, S.A., 1997. Biologically generated patterns and the coexistence of competing species. In: Tilman, D., Kareiva, P. (Eds.), Spatial Ecology: The Role of Space in Population Dynamics and Interspecific Interactions. Princeton University Press, Princeton, pp. 204–232. Paine, R.T., 1966. Food web complexity and species diversity. Am. Nat. 100, 65–75. Paine, R.T., Levin, S.A., 1981. Intertidal landscapes: disturbance and the dynamics of pattern. Ecol. Monogr. 51, 145–178. Palmer, M.W., 1992. The coexistence of species in fractal landscapes. Am. Nat. 139, 375–397. Pascual, M., Guichard, F., (in review). Criticality in spatial ecological systems with disturbance. Pascual, M., Roy, M., Guichard, F., Flierl, G., 2002. Cluster size distributions: signature of self-organization in spatial ecologies. Philos. Trans. Roy. Soc. 357, 657–666. Prakash, S., de Roos, A.M., 2002. Habitat destruction in a simple predator–prey patch model: how predators enhance prey persistence and abundance. Theor. Popul. Biol. 62, 231–249. Rhodes, B., Jensen, H.J., Anderson, R.M., 1997. On the critical behavior of simple epidemics. Proc. Roy. Soc. Lond. Ser. B 264, 1639–1646. Roy, M., Pascual, M., Franc, A., 2003. Broad scaling region in a spatial ecological system. Complexity 8, 19–27. Saupe, D., 1988. Algorithms for random fractals. In: Petigen, H.-O., Saupe, D. (Eds.), The Science of Fractal Images. Springer, New York, pp. 71–113. Shima, J., Osenberg, C.W., 2003. Cryptic density dependence: effects of spatio-temporal variation among reef fish populations. Ecology 84, 46–52. Snyder, R.E., Chesson, P.L., 2003. Local dispersal can facilitate coexistence in the presence of permanent spatial heterogeneity. Ecol. Lett. 6, 301–309. Swihart, R.K., Feng, Z., Slade, N.A., Mason, D.M., Gehring, T.M., 2001. Effects of habitat destruction and resource supplementation in a predator–prey metapopulation model. J. Theor. Biol. 210, 287–303. Tewksbury, J.J., Levey, D.J., Haddad, N.M., Sargent, S., Orrock, J.L., Weldon, A., Danielson, B.J., Brinkerhoff, J., Damschen, E.I., Townsend, P., 2002. Corridors affect plants, animals, and their interactions in fragmented landscapes. PNAS 99, 12923–12926.

ARTICLE IN PRESS M. Roy et al. / Theoretical Population Biology 66 (2004) 341–353 Tilman, D., 1994. Competition and biodiversity in spatially structured habitats. Ecology 75, 2–16. Tilman, D., May, R.M., Lehman, C.L., Nowak, M.A., 1994. Habitat destruction and the extinction debt. Nature 371, 65–66. Urban, D., Keitt, T., 2001. Landscape connectivity: a graph-theoretic perspective. Ecology 82, 1205–1218. Wilson, D.S., 1992. Complex interactions in metacommunities, with implications for biodiversity and higher levels of selection. Ecology 73, 1984–2000.

353

With, K.A., 2002. Using percolation theory to assess landscape connectivity and effects of habitat fragmentation. In: Gutzwiller, K.J. (Ed.), Applying Landscape Ecology in Biological Conservation. Springer-Verlag, New York, pp. 105–130. With, K.A., King, A.W., 1999. Extinction thresholds for species in fractal landscapes. Conserv. Biol. 13, 314–326. Yu, D.W., Wilson, H.B., 2001. The competition–colonization trade-off is dead; long live the competition–colonization trade-off. Am. Nat. 158, 49–63.