vol. 160, no. 1 the american naturalist july 2002
Extinction and Introgression in a Community of Partially Cross-Fertile Plant Species
Jean-Baptiste Ferdy1,2,3,* and Fre´de´ric Austerlitz3,4,†
1. Department of Botany and Agricultural Biochemistry, frequency dependence and make extinction possible, even when hy- University of Vermont, Burlington, Vermont 05405; brid individuals have no intrinsic fitness advantage. 2. Conservatoire Botanique National du Bassin Parisien, Muse´um National d’Histoire Naturelle, 61, rue Buffon, F-75 005 Paris, Keywords: hybridization, genetic diversity, model, identity by descent, France; demography. 3. Laboratoire Evolution et Syste´matique, UPRESA 8079 Centre National de la Recherche Scientifique, Universite´ Paris-Sud, Baˆtiment 362, F-91 405 Orsay cedex, France; Hybridization between species or subspecies is a very com- 4. Department of Ecology, Evolution, and Natural Resources, mon phenomenon in plants and, to a smaller extent, in Cook College, Rutgers University, New Brunswick, New Jersey animals (Rhymer and Simberloff 1996). It can happen 08901-8551 naturally or as a consequence of accidental introduction Submitted August 20, 2001; Accepted December 7, 2001 or anthropogenically caused breakdown of natural barri- ers. This can lead, depending on the degree of reproductive isolation between the species, to either the extinction or the complete introgressive subversion of one species. Ex- amples of extinction by hybridization are reported by abstract: We develop a model to study the demography and ge- Levin et al. (1995) in island flora, in which endemic plant netics of an encounter between two partially cross-fertile plant spe- species hybridize with an introduced congener from a cies. We assume prezygotic reproductive isolation between the spe- cies, a common situation when the species differ by their phenology nearby continent. In some cases, hybridization does not or floral traits that cause assortative mating. Three outcomes are yield complete extinction of the endemic species, but after possible: coexistence of both species with minimal introgression; a few generations, the derivative population has a large domination by one species, with the other becoming extinct or sur- fraction of its gene pool derived from the introduced con- viving only through recurrent migration; or domination of the com- gener (Rhymer and Simberloff 1996). munity by hybrid derivatives, with both species surviving but with In such situations, genetic markers provide a tool to a rather high level of introgression between them. The first situation assess the level of introgression experienced by either spe- is reached when interfertility is low, while the third requires high cies. In some cases, the two species meet in a well-defined interfertility to develop. Occurrence of the second situation is ob- contact zone, and genetic markers define a cline across that served with intermediate values of interfertility. Gene flow from contact zone (Harrison and Bogdanowicz 1997; Goodman nearby monospecific populations can prevent both introgression and et al. 1999; Latta and Mitton 1999). The shape and steepness the domination of the community by one species. Conversely, in- creasing the number of loci that determine the reproductive isolation of the cline depend on the speed of diffusion of alleles between species or decreasing the degree of nonadditive interactions from one species into the other (Nichols and Hewitt 1994), (epistasis and/or dominance) between alleles and loci makes intro- and diffusion equations have been developed that yield a gression more likely. We found that hybridization can create positive good approximation of this speed (Barton and Gale 1993). The development of a cline, however, requires that gene * Corresponding author. Present address: Laboratoire Ge´nome, Populations, diversity follows an isolation-by-distance pattern (Nichols Interactions, Centre National de la Recherche Scientifique, Unite´ Mixte de and Hewitt 1994). In contrast, for species that can disperse Recherche 5000, Universite´ de Montpellier II, Montpellier, France; e-mail: for long distances or species with only few populations, a [email protected]. clinal pattern is not expected. Hybridization will lead to † Present address: Laboratoire de Ge´ne´tique et d’Ame´lioration des Arbres Forestiers, INRA, B.P. 45, Pierroton, F-33611 Gazinet cedex, France; e-mail: a few admixed populations rather than a recognizable [email protected]. cline. In this case, the distribution of genotypes can be Am. Nat. 2002. Vol. 160, pp. 74–86. ᭧ 2002 by The University of Chicago. bimodal, with the population comprising few hybrids and 0003-0147/2002/16001-0006$15.00. All rights reserved. being composed of individuals resembling one or the other Hybridization, Introgression, and Extinction 75 of the parent species, or unimodal, with the population boring populations of pure species. Thus, we can determine being then described as a hybrid swarm (Jiggins and Mallet the conditions under which one of the pure species or a 2000). hybrid form will invade the community, the conditions The aim of this work is to provide a model for the under which both species will coexist, and the level of processes leading to extinction of or introgression in hy- genetic introgression from one species into the other. bridizing populations and to describe the links between these processes and classical population genetic outcomes. To our knowledge, there has been only one attempt to Material and Methods model such a situation (Huxel 1999). Based on previous General Description of the Model work on Wright’s shifting balance theory (Crow et al. 1990; Kondrashov 1992), Huxel’s effort is a model in which the We assume that we have a community of constant size N difference between two species is determined by a single composed of hermaphroditic, self-fertile, diploid individ- biallelic locus, and it focuses on the probability of invasion uals belonging to two pure plant species, 0 and 1, and to of an initially rare (introduced) species. Models developed various hybrid forms. We assume that reproductive isolation for studying sympatric speciation (Kondrashov and Shpak between these two species is determined by nL independent 1998; Kondrashov and Kondrashov 1999) also yield some biallelic loci, with the two alleles denoted as ϩ and Ϫ. Based predictions about the outcome of a community consisting on the proportion of ϩ alleles an individual possesses, we p ϩ of several partially cross-fertile phenotypes. However, these definen 2n L01/(21 phenotypes P,PnL)1 , …,P , with the models concern only isolated populations. Moreover, in convention that the phenotypePk/(2nL) corresponds to the car- ϩ all these models, the fate of molecular markers that could rier of k alleles. Thus, phenotypes P0 and P1 are those of ≤ ≤ Ϫ introgress from one species to the other is not assessed. species 0 and 1, and phenotypes Pi for 1/(2n L) i 1
Here we develop a general deterministic model for the 1/(2n L) are hybrids in which the proportion of ancestry study of the transient dynamics and equilibrium states of from species 0 decreases when i increases (and, conversely, the demography and genetics within an admixed com- the proportion of ancestry from species 1 increases with munity that emerges from the encounter of two partially i). For the sake of simplicity, we will denote an individual interfertile plant species still linked through migration with of phenotype Pi as being of type i. Reproduction is pan- populations from their own species. We can thus inves- mictic among individuals of identical phenotype but is tigate the conditions leading to the extinction of one of partly prevented by prezygotic reproductive isolation be- the two species and how genetic differentiation between tween individuals of different phenotypes. The n pheno- the two species evolves under hybridization pressure; this types present in the community therefore define n ad- allows us to predict when meaningful introgression will mixture classes. The relative probability of an individual occur. of type j fertilizing an individual of type i is denoted hij We study the case in which reproductive isolation is and is computed as prezygotic. There are several biological situations in which p Ϫ F Ϫ F1Ϫe Ϫ this occurs. This is the case, for instance, for two interfertile hij 1 i j (1 h) i, j {0, … , 1}, (1) species with different phenologies. Mating chances would be reduced between early- and late-flowering individuals, where h is the interfertility between species 0 and 1 and and hybrids with intermediate phenology would primarily e is the degree of nonadditive interaction between the al- mate among themselves (Husband and Schemske 2000). leles and loci that determine the reproductive isolation Traits such as flower color (Smithson and Macnair 1997; between species 0 and 1. The casee p 0 therefore cor- Jones and Reithel 2001) or plant height (Peakall and Han- responds to the fully additive case. When e increases, re- del 1993) that cause assortative mating in flowering species productive isolation between two classes increases faster p with nonspecialized pollinators should create the same sort with the number of classes that separate them. If n L of situation. 1, e is the degree of dominance between the alleles of the We study the impact of several parameters on the tran- unique loci that determines reproductive isolation between sient dynamics and final equilibrium reached by the com- the two species. In the general case, e encapsulates both munity, starting with a situation in which the community dominance between the alleles of a loci and epistasis be- is composed only of the two pure species, that is, when tween loci. The calculation of hij when e and nL vary is the species meet for the first time. The parameters that we illustrated in figure 1. model are the level of interfertility between the two species, Once we know the probability that any types i and j the number of loci controlling this interfertility, the degree cross, we must determine the probability distribution of of nonadditive interaction between alleles (dominance) and the type of the offspring produced by that cross. Our model loci (epistasis), and the level of gene flow from the neigh- for the genetic basis of reproductive isolation with nL bial- 76 The American Naturalist
denote byckl,i the proportion of offspring derived from a cross between an individual of type k and one of type l that will be of type i for all k, l, and i in{0, … , 1} . These
probabilities correspond to Doebeli’s (1997)qij ’s and are computed using his equation (12), which becomes with our notations
p ckl,iklprob (g)prob (h)b(g, h, i), (2) g,h{0,…,1}
where probk(g) is the probability for an individual of type g to have an intragenomic overlap of k andb(g, h, i) is the probability for the mating of two parents with intrage- nomic overlap g and h to produce an individual of type i. The intragenomic overlap of an individual is simply the number of loci in this individual at which both alleles are ϩ alleles. The probability probk(g) is given by equation (14) in Doebeli (1997) and becomes
n 2n p LLZ probk (g) [(2ngLL ) (2nk )]
Ϫ Ϫ Ϫ # n LLLLL2ng n 2ng 2ns Ϫ Ϫ .(3) s{0, …, kϪ2g} ()()2nsLLL k 4ng 2ns
Similarly,b(g, h, i) is given by equation (10) in Doebeli (1997) and becomes
Ϫ ϩ Ϫ p (2nkLL4ng) (2nl LL4nh) b(g, h, i) Ϫ ϩ []2niLLL(2ng 2nh)
(2nkLLϪ4ng)ϩ(2nl LLϪ4nh) 1 # .(4) ()2
We assume that local reproduction within the com- munity as described above generates a fraction1 Ϫ m of the offspring that will constitute the next generation. The remaining m fraction comes by migration from two large
Figure 1: Relative probability (hij) for an individual of type j to fertilize neighboring monospecific populations,PP01∗∗ and , of spe- an individual of type i, with level of nonadditivitye p 0 , 0.5, and 0.9. cies 0 and 1, respectively;PP∗∗ and send migrants to the Top, reproductive isolation between the two species is determined by one 01 locus. Only three admixture classes exist (the two species, 0 and 1, and community but do not receive any migrants from it or the hybrids withFi Ϫ jF p 1/2 ). Bottom, reproductive isolation between from each other. The genetic composition ofPP01∗∗ and the two species is determined by two loci. Five admixture classes now therefore remains constant over time and identical to what exist. it was initially among individuals of P0 and of P1. Hybridization and migration within the community de- lelic loci corresponds to the approximate hypergeometric termine changes over time in the frequency of the n phe- phenotypic model for diploid individuals (Barton 1992; notypes. These two processes will also affect the inbreeding Doebeli 1997; Shpak and Kondrashov 1999). As pointed level for genes other than those that determine reproduc- out by Shpak and Kondrashov (1999), the model is ap- tive isolation (such as neutral molecular markers) both proximate since it assumes that all diploid genotypes within among individuals of the same phenotype and among in- a type are equiprobable, but it is a good approximation to dividuals of different phenotypes. In the following sections, the exact model that is extremely computer intensive. We we describe the iterative procedure we used to estimate Hybridization, Introgression, and Extinction 77
p p p (at each generation) the frequency of each phenotype and wheredkl1 ifk l andd kl 0 otherwise. Thus, the the probabilities of identity by descent (IBD, i.e., the prob- frequencies of the different types in the community at time ability that two genes share a common ancestor gene) t ϩ 1 are within the community; this allows us to estimate the level of inbreeding within the community. All parameters of the rkl,i kl model are recalled in table 1. x ϩ p , i {0, … , 1}, (7) i,t 1 s
Demographic Model where s is once again a normalization factor chosen so p thati{0,…,1}xi,tϩ1 1 .
At any generation t, the frequencies of the n phenotypes The transient dynamics ofxi,t were obtained by numerical p of individuals in the community will be denotedxi,t , with iteration of these equations, starting fromx 0, 0 0.495 , p p p ( i {0,…,1}xi,t 1. Initially, the community contains no hy- x 1, 0 0.505, andxi,0 0 , withi 0, 1 . We therefore brids, the pure species 0 and 1 being in frequencyx 0, 0 and start from a situation in which the two species have about ϩ p xx1, 0, respectively, with 0, 0x 1, 01 . We denote by x 0∗ the same frequency and no hybrid form is present. The andx 1∗ the proportion of individuals of species 0 and 1, slight initial numerical advantage given to species 1 pre- ∗∗ p p respectively, arriving in the community fromPP01 and vents us from staying atx 0, t x 1, t 1/2 , which is an ϩ p (x 01∗∗x 1 ). These proportions are assumed to be con- equilibrium state but might be unstable. As analytical so- stant over time. Here we assume that both external pop- lutions are unavailable for this system, we used the same p p ulations contribute equally, thusx 01∗∗x 1/2 . procedure to determine numerically the equilibrium At each generation t, we compute the frequencies of situations. paternal contribution to the offspring of each class of in- dividuals, i.e., the probabilities qkl for an individual of type k to be fertilized by an individual of type l, for all k, l Dynamics of Probabilities of Identity by Descent (IBD) {0, … , 1} according to the rules described in the previous We consider here a neutral locus not implicated in repro- subsection: ductive isolation. At any generation t, we denote by fij,t the probability of identity by descent (IBD) for two genes hx q p kl l,t for k, l {0, … , 1}, (5) belonging to two different individuals of types i and j kl ∗∗ w sk (i, j {0,0,…,1,1}). We also denote byfii,t the prob- ability of IBD for the two alleles of an individual of type ∗∗ where the sk’s are normalization factors such that i (i {0,0,…,1,1} ) for that neutral locus. Note that p l qkl 1 for allk {0, … , 1} . within each admixture class i, there are two different prob- w The total number (rkl,i ) of offspring of type i produced abilities of IBD: within and among individuals (fii,t and by all individuals of type k as a result of fertilization by fii,t , respectively). These parameters are assigned an initial individuals of type l will be value at the beginning of each simulation. For each sub- sequent generation, we compute the probabilities of IBD p Ϫ rkl,ik(1 m)xqc,tklkl,i for k {0, … , 1} from those of the previous generation using the method ∗∗∗,(6) {r p mx d c for k {0 , 1 } that follows. kl,i k kl kl,i
Table 1: Parameters and main variables of the model Symbol Definition N Size of community
nL Number of loci determining reproductive isolation between the two species E Degree of nonadditivity between the alleles and loci that determine reproductive isolation H Interfertility of the two pure species
ckl, i Probability that a cross between phenotypes k and l yields an offspring with phenotype i
xx01∗∗, Frequencies of phenotypes 0 and 1 in the neighboring populations M Migration rate from surrounding populations
xi, t Frequency of phenotype i at time t
hij Interfertility (see above) between phenotypes i and j
Fij, t Pairwise genetic differentiation between phenotypes i and j at time t
FIP, t Departure from panmixia in the community at time t Note: Variables that appear only in “Material and Methods” are not presented here. 78 The American Naturalist
At each generation t, we must compute the probability iterative methods have proven to be a much faster way
pi,kl for an individual of type i to have a mother of type compared with simulations to obtain results on transient k and a father of type l (i {0, … , 1} and k, l dynamics of diversity and differentiation, for which it is ∗∗ {0,0,…,1,1}) and the probabilitypi,k for a gene in an very difficult to obtain analytical results. To assess the level individual of type i at timet ϩ 1 to have been in an in- of introgression between the two species in the community,
dividual of type k at time t. Thepi,kl ’s are deduced from we also computed the pairwise differentiation between
therkl,i ’s: phenotypes (Fij,t ). For each pair of phenotypes i and j,
rkl,i ϩ Ϫ p p ,(8) [( fii,tjjf ,tij)/2] f ,t i,kl F p .(14) rgh,i ij,t Ϫ g,h 1 fij,t
andpi,k is Finally, we computed the departure from panmixia p 1 ϩ (F ) within the community at each generation: pi,ki(p ,klp i,lk). (9) IP,t l 2 f Ϫ f The probability of IBD for the two alleles of an indi- F p I,t P, t ,(15) ϩ w IP,t Ϫ vidual of type i att 1(fii,tϩ1) is then computed as 1 fP, t
w p ϩ d fii,tϩ1 pfi,kl kl,tipf,kk kk,t ,(10) wherefI,t is the average probability of IBD between two k,l k k(l homologous genes of the same individual,
and the probability of IBD att ϩ 1 for two alleles be- n p w longing to two different individuals of types i and j is fI,tixf,tii,t ,(16) ip1
p ϩ d fij,tϩ1 ppfi,kj,lkl,tippf,kj,kkk,t ,(11) k k,l andfP, t is the average probability of IBD for two genes k(l carried by individuals of two different phenotypes:
d wherefkk,t is the within-phenotype probability of IBD after xxfi,tj,tij,t drift. For the phenotypes within the community, since we i(j f p .(17) P, t sample two genes that have ancestors in the same admix- xxi,tj,t i(j ture class of sizeNxk,t , these ancestors have a probability
1/(2Nxk,tk)1/(2of being the same, a probabilityNx ,t) of being two distinct alleles of a single individual, and a prob- Ϫ ability1 (1/Nxk,t) of being in two different individuals. The term FIP is a good indicator of the level of intro- d Thus,fkk,t is calculated as gression. Indeed, if it is high, then different admixture classes in the population are differentiated, and they there-
fore do not introgress much. Conversely, if FIP is low, either 1 f w 1 d p ϩϩk,t Ϫ different admixture classes are not very different because fkk,tkk1 f ,t 2Nxk,tk2Nx,tk() Nx ,t they exchange a lot of genes or the community is domi- nated by a single admixture class and reproduction is then for k {0, … , 1}. (12) panmictic.
We assumed that the external populationsPP01∗∗ and were sufficiently large so that they do not experience any detectable level of genetic drift. Therefore, we have Results
d p ∗∗ fkk,tkkf ,t for k {0 , 1 }. (13) In the following, the community is assumed to be com- posed ofN p 100 individuals. The initial probability of This provides us with iterative equations to compute all IBD between genes of individuals of the same species was the probabilities of IBD over time. In other cases (Aus- set at 0.9, while that between genes of individuals of dif- terlitz et al. 1997, 2000; Le Corre and Kremer 1998), these ferent species was 0.1. Hybridization, Introgression, and Extinction 79
Hybridization Can Yield Three Different Dynamics sharper (fig. 2E). The community rapidly becomes com-
posed mainly of hybrid (P1/2) individuals. The large ma-
We first consider that a single locus determines the re- jority of P0 and P1 individuals that remain in the community p productive isolation between the two species (n L 1 ) so is the result of crosses among individuals of P1/2 because p that the community comprisesn 3 admixture classes; migration is overpowered by hybridization. Thus, both P0 we give a description of the transient dynamics assuming and P1 experience strong introgression (fig. 2F). Simulta- p p m 0.05 ande 0 (which is the fully codominant case neously, the genetic distance between P0 andP1∗ remains here, as interfertility is determined by a single locus), with high compared with the differentiation between P0 and P1; interfertility (h) being set at 0.01, 0.1, or 0.75 and initial this occurs because P0 receives gene flow directly from P1 p p conditions beingx 0, 00.495 ,x 1, 0 0.505 . Figure 2 il- and only indirectly fromP1∗ and because genetic drift in- lustrates the three different types of dynamics that are duced by equation (12) is occurring within each admixture possible depending on the balance between migration and class. hybridization. In fact, in all three situations, the less common species When hybridization is weak in comparison with mi- experiences more interspecific crosses than the more com- gration (h p 0.01 , situation 1), the rare hybridization mon species. A higher proportion of its offspring are there- events yield a decrease in the initial frequencies of the two fore hybrids, yielding a fitness disadvantage for the less pure species (fig. 2A), but these hybridization events are frequent species. In other words, the hybridization process overwhelmed by the arrival of migrants. An equilibrium creates positive frequency dependence. This explains the state is reached within 30 generations, by which time the existence of an asymmetric equilibrium in which the in- two pure species predominate in the community at equal itially rare species is driven close to extinction (situation frequencies. Because of low interfertility, hybrids remain 2). When interfertility is low, however, positive frequency rare (!10%) throughout the process, and gene flow from dependence is weak and migration can rescue the initially hybrids to pure species (generated by backcrosses and rare species (situation 1). Because interfertility is deter- crosses between hybrids) is limited. The differentiation mined by one locus, one-fourth of the crosses between between the two species and their respective pool of mi- individuals of P1/2 generates individuals in P0. Therefore, grants (FF0-0∗∗ and 1-1 in fig. 2B) remains low, but in spite when interfertility is very high, the initially rare species of the weak degree of hybridization, the differentiation can be rescued by the crosses among hybrids, but both between the two species within the community (F0-1 in fig. species are then highly introgressed (situation 3). 2B) is divided by two over the first 50 generations. When interfertility is higher (h p 0.1 , situation 2), the Equilibrium Situation community experiences a rapid increase in the frequency of hybrids over the first few generations (fig. 2C), followed The same three types of dynamics can occur when the by a large number of backcrosses between P1/2 and the two reproductive barrier between the two species is determined pure species. Because of its slightly lower initial frequency, by two or more loci. The stability of the three situations species 0 experiences more interspecific crosses than spe- we described nevertheless varies with the number of loci cies 1. This amplifies the initial numerical superiority of (nL) and the type of interaction between them. Figure 3 species 1 until it becomes predominant in the community. presents which one of the three equilibrium situations that Eventually, species 0 almost disappears, remaining only is observed when the reproductive isolation is determined because of recurrent migration fromP0∗ (see fig. 2C). Both by one, two, four, or eight loci and when the interfertility species experience introgression as a result of the rapid (h) and the degree of nonadditivity (e) vary. increase in hybrid frequency over the first 20 generations; As we have seen in the previous section, the asymmetric the differentiation between these two pure species (F0-1)is equilibrium (situation 2) is stable for intermediate values reduced by more than sixfold (fig. 2D) and remains low of h. This range of h for which situation 2 is stable becomes until, after 150 generations, species 1 has recovered from smaller when nL increases and when e decreases (cf. fig. its initial decrease in frequency. P1 individuals then mate 3A–3D). Similarly, as nL increases or e decreases, the range predominantly among themselves, and P0 individuals are of values of h over which situation 3 is stable increases, mainly immigrants, so differentiation between P0 and P1 and that over which situation 1 is stable decreases. There- begins to increase again. However, while P0 recovers its fore, increasing the number of loci that determine the initial genetic composition, P1 is closer to P0 andP0∗ at isolation barrier between the two species or decreasing the equilibrium than it was initially; slight introgression has degree of nonadditivity between them both decrease the occurred (see fig. 2D). chance that two hybridizing species coexist and keep their When interfertility is very high (h p 0.75 , situation 3), genetic specificity. the initial decrease in frequency of species 0 and 1 is much This is further illustrated by figures 3E–3H, which in- 80 The American Naturalist
Figure 2: Changes through time in the frequency (xi) of each typei {0, 1/2, 1} of individual in the community and the pairwise differentiation p p coefficients Fij between individuals of type i and j. Species are initially at frequencyx150.505 andx 0.495 , with set values for initial within- and among-type diversity (see text for details). The proportion of external migrants ism p 0.05 . Interfertility is, respectively, low (h p 0.01 , A, B), intermediate (h p 0.1 , C, D), or high (h p 0.75 , E, F).
dicate FIP, the departure from panmixia in the community, probability that the two species coexist while remaining
asafunctionofh and e and for the same four values of nonintrogressed decreases when nL and h increase or when
nL; FIP decreases when e decreases and when h increases. e decreases.
A high value of FIP indicates a Wahlund effect (i.e., a deficit If the reproductive isolation between the two species is in heterozygotes in a population in which mating is not determined by a single locus, the community comprises
panmictic) as a result of the coexistence of two nonintro- three admixture classes (P0,P1/2, and P1). The interfertility Ϫ 1Ϫe Ϫ gressed species. A low value of FIP would, on the contrary, between P1/2 and the two pure species is 1 (1/2) (1 indicate either introgression of the two species or the dom- h). If the reproductive isolation between the two species inance in the community of a single species, in which case is determined by two loci instead, the community com- reproduction within the community would be almost pan- prises five admixture classes (the same three classes as
mictic. The decrease in FIP therefore indicates that the before plus P1/4 and P3/4). The interfertility between P0 and Hybridization, Introgression, and Extinction 81
Ϫ 1Ϫe Ϫ P1/4 (or between P1 and P3/4) is1 (1/4) (1 h) . Crosses
between P0 and P1/4 (P1 and P3/4) are therefore easier than
those between the pure species and P1/2 (see fig. 1), and the existence of these two additional classes facilitates gene
flow between P0 and P1. More generally, increasing nL in- creases the number of stepping stones between the two species and makes introgression more likely. Furthermore, the interfertility coefficients we just computed increase when h increases and when e decreases, so increasing h or decreasing e also makes introgression more likely.
Both nL and e interact with migration rate (m). This is illustrated by figures 4A–4C, which show the stability of the three types of equilibria as a function of h and m and for three values of e when the reproductive isolation be- tween the two species is determined by two loci. When e p 0, the range over which situation 2 is stable is of very small size (see fig. 4A). In contrast, when the interaction between loci is mainly nonadditive (e p 0.9 , fig. 4C), the three types of equilibria can be stable over a wider range of values. For example, in figure 4C,forh p 0.25 ,asm increases, the stable equilibrium is first asymmetric (sit- uation 2), then symmetric with domination by the hybrids (situation 3), and finally symmetric with domination by the two pure species (situation 1). When the number of
loci (nL) is increased to eight (fig. 5), situation 2 is observed on a very restricted portion of the parameter set. Again it is with the highest level of nonadditivity that it has the highest probability to be observed. Concerning genetic introgression between the two spe-
cies, for strong nonadditivity (fig. 4F), FIP decreases with h and increases with m; hybridization and migration have opposite effects on the degree of departure from panmixia in the community. Conversely, for weak nonadditivity (fig.
4D), FIP still increases with m but remains almost un- changed when h varies, ifm ! 0.25 . In this situation, sur- prisingly, the risk of introgression or extinction of one species depends more on the migration rate than on the interfertility. If reproductive isolation is determined by eight loci instead of two, this situation prevails for almost any value of e (see fig. 5).
Discussion Our model predicts three possible outcomes for the sym- patric encounter of two species that can hybridize, in gen- Figure 3: Stable steady states (left column) and FIP (right column)asa function of the interfertility between species 0 and 1 (h) and the non- eral agreement with documented cases from the literature additivity between the loci that determine reproductive isolation (e)for (Rhymer and Simberloff 1996): situation 1, coexistence of various numbers of loci (nL) implicated in reproductive isolation (A and the two species at equal frequencies, with few hybrids and p p p p E,;nLL1 B and F,;n 2 C and G,;n L4 D and H,n L8 ). Equi- moderate genetic introgression; situation 2, invasion by librium state 1 corresponds to coexistence of the two species without one species, with little genetic introgression from the other, introgression; equilibrium state 2 corresponds to extinction of a species; equilibrium state 3 corresponds to coexistence of the species with intro- which is retained at low frequency or goes extinct; and situation 3, domination of the community by hybrids, with gression; FIP quantifies departure from panmixia in the community, with values close to 1 indicating a strong deficit in heterozygotes. the two species being completely introgressed. 82 The American Naturalist
Figure 4: Stable steady states (first row) and FIP (second row) as a function of interfertility between the two species (h) and migration rate (m)for various levels of nonadditivity (A and D,;e p 0 B and E,;e p 0.5 C and F,e p 0.9 ). The reproductive isolation between species 0 and 1 is determined by two loci. Equilibrium state 1 corresponds to coexistence of the two species without introgression; equilibrium state 2 corresponds to extinction of a species; equilibrium state 3 corresponds to coexistence of the species with introgression; FIP quantifies departure from panmixia in the community, with values close to 1 indicating a strong deficit in heterozygotes.
Situation 1 corresponds clearly to the bimodal hybrid mechanism destabilizes the coexistence of the two species zone described by Jiggins and Mallet (2000), while situ- at equal frequencies with or without introgression. The ation 3 would correspond to a unimodal hybrid swarm. only stable equilibrium is then asymmetric with strongly These authors also describe an intermediate distribution uneven frequencies, with one species predominating in the in which all forms are equally frequent; this would cor- community and the other being maintained at a frequency respond to a limiting case between situations 1 and 3. that is determined by the rate of recurrent immigration. Communities corresponding to situation 2 would not be Of course, this is only true as long as we remain in a considered as parts of a hybrid zone because one of the deterministic model. In a stochastic model, the instability two species is nearly extinct; they are not described in of the symmetric equilibrium should remain, but the less Jiggins and Mallet (2000). common species will just more likely be the one that goes Situations 1 and 3 are easily explained: when interfer- to extinction. Another important result of our model is tility (h) is low, the two species coexist because they rarely that transition dynamics can be very slow to develop (see hybridize (situation 1), but when h is high, they largely fig. 2), so the observation of many individuals of both introgress, yielding a community dominated by hybrids, species in the community at a given time does not nec- with the two species being re-created in equal quantities essarily mean that their coexistence will be the final out- by hybrid disjunction and backcrosses (situation 3). Sit- come for that community. uation 2 is less intuitive; in this situation, hybridization is Besides its theoretical interest, our work thus also pro- not strong enough to allow the hybrids to invade the com- vides useful insights into conservation strategies, provided munity, and migration is too weak to rescue the rare spe- that the impact of the different parameters is studied thor- cies. In that case, the changes in the frequencies of the oughly. What can already be said is that a good strategy different phenotypes in the community are driven by pos- is to reinforce the migration from neighboring mono- itive frequency dependence. The more frequent species in specific populations. We have shown that this will shift the the community is advantaged because it experiences more community from a situation in which either one species intraspecific crosses in proportion to the rarer species. This or a hybrid form dominates to a situation in which both Hybridization, Introgression, and Extinction 83
Figure 5: Stable steady states (first row) and FIP (second row) as a function of interfertility between the two species (h) and migration rate (m)for various levels of nonadditivity (A and D,;e p 0 B and E,;e p 0.5 C and F,e p 0.9 ). The reproductive isolation between species 0 and 1 is determined by eight loci. Equilibrium state 1 corresponds to coexistence of the two species without introgression; equilibrium state 2 corresponds to extinction of a species; equilibrium state 3 corresponds to coexistence of the species with introgression; FIP quantifies departure from panmixia in the community, with values close to 1 indicating a strong deficit in heterozygotes.
species can coexist while keeping most of their genetic different kinds of pollen. Without pollen limitation, strong integrity. Because reinforcing migration can be a more competition occurs between the pollen grains that arrive easily accomplished conservation strategy than others (e.g., on a stigma; as a result, conspecific pollen is always fa- manual removal of hybrids, prevention of crossing), it is vored. In that case, both species are always maintained. of particular interest to wildlife managers. Our model therefore indicates that hybridization be- Positive frequency dependence has long been known to tween two species with some degree of reproductive iso- provoke the extinction of rare variants and, therefore, to lation between them and pollen limitation generates pos- reduce polymorphism (Kimura and Ohta 1971). Recently, itive frequency dependence. Positive frequency-dependent Gyllenberg and Hemminki (1999) have shown that in a selection has also been shown to co-occur with assortative system in which two populations of individuals of the same mating in some experimental systems (Smithson and Mac- species are subject to local positive density dependence nair 1997). The existence of positive frequency dependence and a degree of competition between patches, both sym- makes the transitions between the different equilibria very metric and asymmetric equilibria are possible, depending abrupt; a very small change in a parameter such as m or on the parameter values. Our model is somewhat analo- h is sufficient to make the community switch from one gous to theirs, with migration between patches being re- type of equilibrium to another. placed by hybridization between species. Our prediction of three possible equilibria seems to hold In fact, positive frequency dependence operates for all for any number of loci implicated in reproductive isolation parameter values, provided that the interfertility (h)is10. and for most levels of nonadditive interaction between When h is weak, frequency dependence is overpowered by alleles and loci. However, increasing the number of loci migration (situation 1); conversely, when h is high, it is or decreasing the level of nonadditivity between them overpowered by hybridization (situation 2). The fact that makes invasion of the community by hybrid forms much this frequency dependence yields the extinction of one easier. Indeed, the number of intermediate phenotypes species in some cases is a consequence of pollen limitation between the two species increases with the number of loci in our model, i.e., the mother plant cannot choose between that determine reproductive isolation, and this facilitates 84 The American Naturalist the reproduction among admixture classes. Nonadditivity pendent on the levels of gene flow and reproductive can limit this phenomenon because it reduces interfertility isolation. between close phenotypes (and, in fact, reduces positive As we have already pointed out, the prezygotic isolation frequency dependence; see above). If many loci determine model that we have chosen here is realistic in some cases. reproductive isolation and if nonadditivity is weak, either If flowering periods only partially overlap (Husband and migration is strong enough to counterbalance hybridiza- Schemske 2000) or if the species differ by floral traits that tion and situation 1 will be observed or it is not and cause assortative mating (Peakall and Handel 1993; Smith- situation 3 will develop. Invasion by one species (situation son and Macnair 1997; Jones and Reithel 2001), the degree 2) becomes nearly impossible under these circumstances. of reproductive isolation between two forms would in- This result clearly shows the necessity of obtaining ex- crease with the number of alleles by which they differ, as perimental data on the genetic architecture of reproductive in our model. However, it is also known that in some barriers between species. The few existing studies on this species the mechanism of isolation is partially or totally point have yielded contrasting results; in some cases, many postzygotic, with the hybrids being less viable and/or less quantitative trait loci have been found (Rieseberg et al. fertile than either pure species (Rieseberg et al. 1999; 1999), whereas in others, only a few loci under strong Campbell and Waser 2001; Johnston et al. 2001). It is likely nonadditivity seem to govern isolation between closely re- that this would yield similar results. lated species (Orr and Irving 2001). If reproductive isolation were only postzygotic, poorly Kondrashov and Shpak (1998) and Kondrashov and viable or fertile hybrids would harm neither species and Kondrashov (1999) have also shown that reproductive iso- would therefore allow them to coexist. Highly viable and lation must be rather high in order to maintain two sep- fertile hybrids would, on the contrary, easily invade the arate species in the absence of immigration. They had to community and displace the pure species. Hybrids of in- assume that only individuals with moderately different termediate vigor would probably help the initially more phenotypes were cross-fertile and, in particular, that in- frequent species to invade, as in our case. Thus, we would dividuals belonging to the two pure species were com- also have the three situations. Whereas coexistence of the pletely unable to cross. Thus, in their model, if the species two pure species occurs in a limited part of the parameter had been isolated for some time before meeting again as space in our model (especially for the high number of loci we assumed here, the absence of an intermediate form involved), the conjunction of pre- and postzygotic isola- would prevent any hybridization. We have shown here that tion would make the coexistence of the two species pos- the two species can coexist even if they are partially cross- sible for a wider range of the parameters. A more precise fertile, provided that there is at least a small amount of investigation of the impact of postzygotic isolation with immigration from neighboring monospecific populations or without concomitant prezygotic isolation should be per- and that reproductive isolation is sufficient. In fact, in formed in the future. some situations, migration is more important in deter- This study emphasizes that several parameters will de- mining the fate of the community than is the interfertility termine the persistence or lack thereof of two distinct species between the two species. Our results show also that even and their level of genetic introgression. Some of these pa- if the species remain separated, as far as the genes impli- rameters are purely genetic (number of loci implicated in cated in reproductive isolation are concerned, they ex- reproductive isolation, dominance, and epistasis), whereas perience some introgression for the genes that are not others are more ecological (e.g., migration) even if they can implicated. Thus, neutral markers (and adaptive genes not also be partially genetically determined. Ignoring one of implicated in reproductive isolation) are still exchanged. these parameters might yield completely different results Also, when one species invades the community, it acquires in terms of the dynamics. In particular, one must not forget some of the genetic diversity from the other species in the that it is unlikely that a community is completely isolated process. Thus, partial cross-fertility still has an impact on from the rest of the world, and this fact is of major im- the species, even in the absence or the low frequency of portance for the outcome. The more logical model to deal hybrid forms. with these issues in future works might be a metapopu- Our results can be compared with a clinal situation in lation model, with the two species and hybrid forms mov- which the hybrid zone is maintained by a balance between ing across the landscape. selection and isolation (Barton and Shpak 2000). In that case, the width of the cline increases when reproductive isolation decreases. However, the two species are always Acknowledgments maintained because of disruptive selection. We have shown here that they could also be maintained without any se- We thank B. Godelle, J. Molofsky, and I. Till-Botraud for lective pressure, but this coexistence becomes largely de- their helpful comments. We also thank K. Ritland, J. 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