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1 Paper 195 Disc Ecology Letters, (2001) 4 : 57 63. REPORT. Habitat destruction and extinction in competitive and mutualistic metacommunities Abstract Christopher A. Klausmeier Because habitat loss is a leading cause of extinction, it is important to identify what University of Minnesota, kind of species is most vulnerable. Here, I use algebraic and graphical techniques to Department of Ecology, study metacommunity models of weak competition or locally facultative mutualism in Evolution, and Behavior. which species may coexist within patches. Because a competition colonization trade- Present Address: EAWAG, off is not required for regional coexistence of competitors, poor competitors are often Seestrasse 79, CH-6047. regionally rare and most prone to extinction, in contrast to results from previous Kastanienbaum, Switzerland. models of strongly competitive metapopulations. Metacommunities of mutualists can E-mail: suffer the abrupt extinction of both species as habitat destruction is increased.

2 These highlight the importance of identifying the mechanisms by which species coexist to predict their response to habitat loss. Keywords Habitat destruction, extinction, competition, mutualism, metapopulation, model. Ecology Letters (2001) 4 : 57 63. Ahed Bhed Ched INTRODUCTION extinction in competitive metacommunities (Nee & May Dhed 1992; Tilman et al. 1994, 1997; Neuhauser 1997;. Ref marker Soon after the metapopulation concept was introduced for Klausmeier 1998). Competition is strong in these models;. Fig marker a single species (Levins 1969, 1970), it was extended to they assume severe extinction and colonization competi- Table mar- cover interspecific competition (Levins & Culver 1971). tion so that species cannot coexist within local patches and ker In general, species competing in a patchy environment can form a competitive hierarchy. Regional coexistence is Ref end affect each other in three ways: possible between species that show a competition Ref start (1) by increasing the extinction rate of the other species in colonization trade-off; inferior competitors survive be- patches where they co-occur, cause their greater colonization ability allows them to (2) by decreasing the chance a propagule will successfully colonize empty patches before a superior competitor colonize a patch occupied by the other species and arrives (Skellam 1951; Hutchinson 1951; Hastings 1980.)

3 (3) by decreasing the rate at which propagules are Tilman 1994). Habitat destruction effectively lowers the produced in jointly occupied patches. The first form of colonization rate of all species. Thus, these models predict competition is called extinction competition, while the that the first species driven extinct is usually the best second and third have not been clearly distinguished and competitor/poorest colonist (Nee & May 1992; Tilman et both have been called migration competition (Levins & al. 1994, 1997; see Klausmeier 1998 for exceptions in Culver 1971; Slatkin 1974; Hanski 1983). I will call the communities of three or more species). second form of competition establishment competition and To complement these results, in this paper I develop the third form of competition propagule production and analyse a model of weakly competing metapopula- competition. In their analysis, Levins and Culver (1971) tions in which species can coexist within patches. By tacitly assumed that the species are independently changing parameter values so that species benefit each distributed.

4 This simplifies the mathematics by reducing other, the same equations can be used to model the number of equations, but in general the assumption of metapopulation dynamics of locally facultative mutualists. independence is incorrect (Slatkin 1974). Here, I will As in previous models, species with low colonization show that independence holds under pure propagule ability are most prone to extinction due to habitat loss. production competition when dispersal is global. However, because the competition colonization trade-off In the last decade, metacommunity models have been is not required to allow regional coexistence, this model used to examine the effect of habitat destruction on does not predict the biased extinction of superior #2001 Blackwell Science Ltd/CNRS. Paper 195 Disc 58 Klausmeier competitors. In fact, I argue that if all else is equal, locally dP1. c1 P1 f12 P12 P0 m1 P1 c2 P2 f21 P12 P1. rare competitors will be regionally rare and most prone to dt extinction. Among mutualists, either the poorer colonist m2 P12.

5 Goes extinct first or both species go extinct simultaneu- ously as habitat destruction is increased. dP2. c2 P2 f21 P12 P0 m2 P2 c1 P1 f12 P12 P2. dt THE MODEL m1 P12 1 . Consider an infinite number of identical patches, each capable of supporting viable populations. Each patch may dP12. c1 P1 f12 P12 P2 c2 P2 f21 P12 P1. be empty (proportion P0), occupied by a population of dt either species 1 or species 2 alone (proportions P1 and P2), occupied by populations of both species (proportion P12), m1 m2 P12. or permanently destroyed (proportion D). I assume that P0 1 D P1 P2 P12. species can stably coexist within a patch, so that extinction and establishment are not affected by the presence of the It is useful to consider the occupancy of non-destroyed other species. This is pure propagule production interac- patches; let P' be occupancy proportions conditioned on tion. Patches occupied by species i alone produce the site not being destroyed (P' = P/(17D)). Then propagules at rate ci.

6 Assuming that propagule production 0. dP1. is proportional to the number of individuals in a patch, c1 1 D P 0 1 f12 P 0 12 P 0 0 m1 P 0 1 c2 1 D . locally abundant species will have a higher c than locally dt rare species if all else is equal. Patches occupied by both P 0 2 f21 P 0 12 P 0 1 m2 P 0 12. species produce species i propagules at rate fijci. Thus fij 0. measures how much species j changes species i's dP2 Ahed c2 1 D P 0 2 f21 P 0 12 P 0 0 m2 P 0 2 c1 1 D . propagule production where they locally co-occur. For dt Bhed example, suppose species 1 has a density of 1000 Ched individuals per patch when alone and 800 individuals P 0 1 f12 P 0 12 P 0 2 m1 P 0 12 2 Dhed per patch when co-occuring with species 2, and species 2 Ref marker has a density of 100 individuals per patch when alone and 0. dP12 Fig marker c1 1 D P 0 1 f12 P 0 12 P 0 2 c2 1 D Table mar- 60 individuals per patch when co-occuring with species 1. dt Then f12 = and f21 = , and if individuals of both ker P 0 2 f21 P 0 12 P 0 1 m1 m2 P 0 12 Ref end species create propagules at the same rate, then c1 = 10c2.

7 Propagules are dispersed globally. A propagule of species Ref start P0 0 1 P1 0 P2 0 P 0 12. i which lands on a patch without species i already present successfully colonizes that patch. Species i goes extinct If D = 0, (1) and (2) are identical and P = P'. Let within patches at rate mi. Figure 1 summarizes the p'1 = P'1+P'12 and p'2 = P'2+P'12 so that p'i represents transitions between states. the total proportion of non-destroyed patches occupied by These assumptions result in the following equations: species i: dp 0 1. c1 1 D P 0 1 f12 P 0 12 1 p0 1 m2 p 0 1. dt dp 0 2. c2 1 D P 0 2 f21 P 0 12 1 p0 2 m2 p 0 2 3 . dt Equations (3) are not a closed system since P'1, P'2, and P'12 still appear. To close (3), we note that the distributions of species 1 and 2 become independent as t tends to infinity (see Appendix). Independence means that the probability of finding species 1 in a patch is not affected by the presence or absence of species 2. Species become independent because competition occurs only through the globally-dispersed propagule pool.

8 Since Figure 1 State transition rates. species become independent asymptotically, I assume that #2001 Blackwell Science Ltd/CNRS. Paper 195 Disc Habitat loss and extinction in metacommunities 59. they begin independent. By independence, P'1 = p'1(17p'2), P'2 = p'2(17p'1), and P'12 = p'1p'2. Sub- stituting into (3), dp 0 1. c1 1 D p 0 1 1 p 0 2 f12 p 0 1 p 0 2 1 p0 1 m1 p 0 1. dt dp 0 2. c2 1 D p 0 2 1 p 0 1 f21 p 0 1 p 0 2 1 p0 2 m2 p 0 2. dt (4). Equations (4) will be the model I investigate. MODEL ANALYSIS. No interaction When species do not interact f12 = f21 = 1. In this trivial case, (4) reduces to two separate single-species Levins' Figure 2 Phase plane diagram with isoclines for two equal equations: competitors. Parameter values are: c1/m1 = 2, c2/m2 = 2, f12 = , f21 = , D = 0. Closed (open) circles denote stable dp 0 1. c1 1 D p 0 1 1 p 01 m1 p 0 1 (unstable) equilibria. dt dp 0 2 1 dp 0 i Ahed c2 1 D p 0 2 1 p 02 m2 p 0 2 5 ci 1 D 1 p^0 j fij p 0 j mi Bhed dt p 0 i dt Ched The equilibrium density of species i is Dhed mj mj mi ci 1 D fij fij mi Ref marker p^01 1 6 cj 1 D cj 1 D.

9 Fig marker ci 1 D ;. (9) Table mar- so species i persists when so species i invades species j when ker mi Ref end D<1 7 ci 1. ci 1 D > m m 10 Ref start mi fij 1 cj 1 j D cj 1 j D : Since the right hand side of (10) approaches 1/fij as the Competition c effective colonization rate (mjj )/(17D) approaches infinity, In competitive metapopulations, species densities it is impossible for species j to regionally exclude species i if are lower in patches where they coexist, so fij 5 1. ci 1. It can be shown that there is only one feasible two- 1 D > : 11 . mi fij species equilibrium (see isoclines in Fig 2). The algebraic expression for this equilibrium is unwieldy. When (11) is met for both species, local coexistence Invasion criteria can be used to determine the outcome translates directly into regional coexistence. of metapopulation competition: when species i can Figure 3(A, B) uses (10) to determine the outcome of invade a monoculture of species j, but not vice- competition between two species.

10 Figure 3(A) shows versa, species i outcompetes species j; when both equal competitors which halve each other's abundance in species can invade monocultures of the other, the two patches where they co-occur. Figure 3(B) shows the species coexist. regional outcome when f124f21 and therefore species 1 is The monoculture equilibrium of species j is competitively superior to species 2 within patches. In all mj cases, regional coexistence is assured when the coloniza- p^0j 1 8 tion rates of both species are sufficiently high. cj 1 D : Figure 3 (A, B) can be used to graphically determine the The growth rate of species i when rare is effect of habitat destruction. Equations (4) show that as D. #2001 Blackwell Science Ltd/CNRS. Paper 195 Disc 60 Klausmeier Figure 3 Species persistence and com- petitive outcome as determined by the ratio of colonization to mortality rates and habitat loss. (A) Competition between equal competitors; (B) compe- tition with species 1 competitively superior to species 2; (C) patch occu- pancy of two competitors as a function of habitat destruction.