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Incomplete mixing promotes species coexistence in a lottery model with permanent spatial heterogeneity
ARTICLE IN PRESS
Theoretical Population Biology 64 (2003) 359–368
http://www.elsevier.com/locate/ytpbi
Incomplete mixing promotes species coexistence in a lottery model with permanent spatial heterogeneity Soyoka Muko and Yoh Iwasa Mathematical Biology Laboratory, Department of Biology, Faculty of Sciences, Kyushu University, 6-10-1 Hakozaki, Higashi-ku, Fukuoka-shi, Fukuoka-ken 812-8581, Japan Received 2 February 2001
Abstract For many marine organisms, the population dynamics in multiple habitats are affected by migration of planktonic larvae. We herein examine the effect of incomplete larval mixing on the condition for species coexistence. The system consists of two heterogeneous habitats, each composed of a number of sites occupied by adults of two species. Larvae produced in a habitat form a pool and migrate to the pool of the other habitat. When an adult dies, the vacant site becomes occupied by an individual randomly chosen from the larval pool. We study (1) the invasibility of a inferior species which has no advantage in either habitats, (2) the dynamics when larval migration and competition among adults are symmetric between habitats, and (3) the case with unidirectional migration. The coexistence of competitors is more likely to occur when larval migration is weak. r 2003 Elsevier Inc. All rights reserved. Keywords: Localized dispersal; Recruitment; Unidirectional flow; Spatial heterogeneity
1. Introduction In many marine invertebrates, the sedentary adults occupy a portion of a rocky substrate and produce planktonic larvae which are subsequently dispersed. Larvae produced in different habitats are mixed in a pelagic pool and then settle back to vacant substrates in the habitats. As a consequence, larvae produced in a favorable habitat may be transported to other locations that are not very favorable for the species. If one species dominates a highly productive habitat, the larvae produced there may be carried to other habitats, where they settle, grow, and possibly drive out the competitors. Hence, we need to consider the population dynamics, including both sedentary adults in different habitats and larvae in the pelagic pool, in order to understand the coexistence of multiple species in such a patchy environment. For marine organisms, permanent spatial variations between different habitats in environmental factors, such as water depth, substrate quality, water current,
Corresponding author. Fax: +81-92-642-2645. E-mail address: [email protected] (S. Muko). 0040-5809/03/$ - see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0040-5809(03)00085-6
irradiance, and exposure to predators, are often very important in maintaining the species diversity. The importance of spatial heterogeneity has been demonstrated in many different taxa, such as terrestrial plants (Reynolds et al., 1997; Guo, 1998), fresh water macrophytes (French and Chambers, 1996), ascidians (Stoner, 1992), and mussels (Hunt and Scheibling, 1998). However, most theoretical studies on the coexistence of multiple competitors focus on the role of spatial heterogeneity that is transient rather than permanent (e.g. Levin, 1974; Hasting, 1980; Chesson and Warner, 1981; Shmida and Ellner, 1984; Comins and Noble, 1985; Tilman, 1994; Shigesada and Kawasaki, 1997). For example, in the standard lottery model studied by Chesson and Warner (1981), either of the two species stays at each site throughout its life and creates spatial heterogeneity. However, the spatial heterogeneity in this sense is not permanent, because there is no basic difference in the quality between sites. The sites currently occupied by one species will later become occupied by the other. In other examples, Comins and Noble (1985) analyzed the effect of transient spatial heterogeneity in the lottery model, in which the patch quality varies randomly but the long-term statistical distribution of conditions is the same between sites.
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S. Muko, Y. Iwasa / Theoretical Population Biology 64 (2003) 359–368
In contrast, there have been relatively fewer studies on permanent heterogeneous environments (e.g., Shigesada and Roughgarden, 1982; Holt, 1997). Chesson (1985) noted that the spatial heterogeneity in adult mortality between habitats could promote the coexistence in a specific example of lottery model, in which adults can survive only in some patches, such suitable patches being different between two species. Recently, Chesson (2000) has developed a general model, which includes a variety of competitive interactions, to facilitate the study of the mechanisms of species coexistence. Roughgarden and her colleagues studied the competition of species within multiple heterogeneous habitats where are connected by a pelagic pool of larvae (Roughgarden and Iwasa, 1986; Iwasa and Roughgarden, 1986; Possingham and Roughgarden, 1990; Alexander and Roughgarden, 1996). Their models have a complex structure in which each habitat might have unfilled space and in which the larvae have their own dynamics. Muko and Iwasa (2000) studied the lottery competition of two species with permanent spatial heterogeneity, which is more tractable mathematically than the previous studies. The model consists of several habitats for sedentary adults, which may differ in mortality and reproductive rate. The larvae produced in different habitats are mixed completely in a larval pool and then settle back to vacant space immediately after such space is created by the death of adults. Muko and Iwasa show that, in addition to two equilibria of single species, there can be up to n 1 internal equilibria in which stable coexistence can occur, if there are n habitats for adults. The location of these internal equilibria and their local stability can be calculated from a single function indicating species difference in reproductive success. In general, if the ratio of adult mortality of the two species varies between habitats, the coexistence is likely to occur, while between-habitat variation in the ratio of larval production rate does not promote coexistence. This conclusion contrasts with the role of temporal variation in the standard lottery model proposed by Chesson and Warner (1981), in which the between-year variation of the relative reproductive rate promotes coexistence but the variation of relative mortality does not. However, recent research has provided evidence that dispersal of planktonic larvae is sometimes localized by physical conditions such as water current and wind (e.g., Sammarco and Andrews, 1988; Roughgarden et al., 1988; Black et al., 1991; McQuaid and Phillips, 2000), or by the behavior of larvae (e.g., Olson, 1985; Stoner, 1992; Carlon and Olson, 1993). Larvae are more likely to stay at the natal site than to disperse to other habitats (Petersen and Svane, 1995; Swearer et al., 1999; Carballo, 2000). If the water current is unidirectional, larvae produced in a habitat are carried to another site, but the larval migration in the opposite direction is
negligible (Olson, 1985; Stoner, 1992; Horvath et al., 1996). Hence the larvae produced in different habitats might not be mixed completely. We should consider how larval retention affects the condition of species coexistence. The present paper deals with cases of incomplete mixing of larvae among the heterogeneous habitats. This is a sequel to Muko and Iwasa (2000) which assumed the complete mixing of larvae. Here, we consider a separate larval pool for each adult habitat and assume that a certain number of larvae are exchanged between those pools. To simplify the analysis, we concentrate on the case of two habitats. First, we show the invasibility of an inferior species which has no advantage in either habitat. Second we consider symmetric competition in which species 1 has a high reproductive success in habitat 1 and species 2 has an equally high reproductive success in habitat 2. Finally, we study the case of asymmetric larval migration, in which the water current carries the larvae produced in habitat 1 to habitat 2, but larvae from habitat 2 do not settle in habitat 1.
2. Model We consider two species indexed by s ðs ¼ 1; 2Þ living in multiple habitats for sessile adults, indexed by i ¼ 1; 2; y; n: These habitats may differ in substrate quality, water current, disturbance rate, and nutrient conditions. It is assumed that all the sites in adult habitats are occupied most of time, except for brief periods between the death of an individual and the subsequent settlement of larvae. Hence, if the fraction of sites occupied by species 1 in habitat i is denoted by xi ; the fraction of sites occupied by species 2 is 1 xi for i ¼ 1; 2; y; n: Let Lsi be the number of planktonic larvae of species s in the pelagic larval pool near habitat i: The time unit is one year and larval period is much shorter than a year. The dynamics of xi are determined both by the loss of adults and by the recruitment of planktonic larvae: xi ðt þ 1Þ ¼ ð1 d1i Þxi þ fd1i xi þ d2i ð1 xi Þg L1i ; i ¼ 1; 2; y; n; L1i þ L2i
ð1aÞ
where dsi is the annual mortality of species s at site i ð0pdsi p1Þ: In Eq. (1a), the suffix t is abbreviated in the right-hand side. The first term of the right-hand side indicates the fraction of adults surviving to the next year. The second term indicates the recruitment of larvae and is the fraction of vacant sites created by the death of adults of the two species multiplied by the proportion of the species in the larval pool. A fraction of the planktonic larvae produced in habitat i move to habitat j and become part of the larval pool for subsequent settlement. This fraction is denoted by mji :
ARTICLE IN PRESS S. Muko, Y. Iwasa / Theoretical Population Biology 64 (2003) 359–368
The planktonic larvae that are ready for settlement in habitat i are ! X X L1i ¼ 1 mki b1i xi þ mij b1j xj ; ð1bÞ jai
kai
L2i ¼ 1
X
! mki b2i ð1 xi Þ
kai
þ
X
mij b2j ð1 xj Þ:
ð1cÞ
jai
bsi is the reproductive rate, defined as the total number of larvae produced by adults in habitat i in one year, if the habitat is occupied only by species s: It increases with the area and the productivity of the habitat. The first term of the r.h.s. indicates migration out of the local pool to other pools, and the second one represents the immigration from the other pools to the local pool. Muko and Iwasa (2000) analyzed the case with mji ¼ 1=n ði; j ¼ 1; 2; y; nÞ; in which larvae produced in different habitats are mixed completely in a common larval pool. For simplicity of analysis, we concentrate on the case of two habitats in this paper. To examine the effect of larval migration on the condition for coexistence and the outcome of competition between two species, we can compare two extreme cases: one without larval migration between habitats, and one with complete mixing in a single pelagic pool. In an isolated habitat, the fraction of species 1 in habitat 1 is Dx1 ¼ x1 ðt þ 1Þ x1 ðtÞ ¼
x1 ð1 x1 Þ b b d11 d21 11 21 : b11 x1 þ b21 ð1 x1 Þ d11 d21
ð2Þ
Since 1=dsi indicates the mean lifetime of an adult, bsi =dsi is the expected number of larvae produced in its lifetime ðs ¼ 1; 2Þ: The sign of Eq. (2) depends on the difference of the lifetime reproductive success between species in habitat 1. Eq. (2) implies that species 1 will eventually occupy habitat 1 (x1 ðtÞ-1 as t-N) if species 1 is superior to species 2 ðb11 =d11 4b21 =d21 Þ: More than one species cannot stably coexist within an isolated habitat. If the whole system includes two distinct patches, coexistence is possible when one species wins in habitat 1 but the other wins in habitat 2 (b11 =d11 4b21 =d21 and b12 =d12 ob22 =d22 ; or b11 =d11 o b21 =d21 and b12 =d12 4b22 =d22 ). In the other extreme, the larvae produced in two habitats are completely mixed in a single pool (Muko and Iwasa, 2000). In such a case, in addition to two single-species equilibria, there may be an equilibrium including two species. According to Muko and Iwasa (2000), the condition for stability of the two-species
361
equilibrium is given by b11 b21 d22 b22 b12 4 ; d11 d21 d21 d22 d12 b22 b12 d11 b11 b21 4 : d22 d12 d12 d11 d21
ð3aÞ ð3bÞ
There are four possible cases: (I) If Eqs. (3a) and (3b) hold, the two species stably coexist. (II) If Eq. (3a) holds but Eq. (3b) does not, species 1 eliminates species 2. (III) If Eq. (3b) holds but Eq. (3a) does not, species 2 drives out species 1. (IV) If neither Eq. (3a) nor Eq. (3b) holds, there is a coexistence equilibrium, but it is unstable while the two single-species equilibria are locally stable. The outcome of the competition between two species depends on the initial abundance—whichever species has a large initial occupancy will drive out the competitor. This is the situation called bistability. In the case without migration, two species are completely segregated in different habitats and have no contact. The migration between habitats could cause coexistence in the sense that all species are present in all habitats (Levin, 1974). In the following, we study the effect of larval migration at an intermediate level on the conditions for coexistence. Here we concentrate on three solvable cases that assume different types of competitive ability in order to examine how the competitive outcome depends on the magnitude and pattern of migration. 3. One species is inferior in both habitats If one species is inferior to the other in both habitats, it is eliminated from each habitat in the absence of larval migration between habitats. We might conjecture that the inferior species declines monotonically from any abundance to extinction even if larval migration occurs. However, we can disprove this by a counter-example illustrated in Fig. 1. Here, species 1 is inferior to species 2 in both habitats (b11 =d11 ob21 =d21 and b12 =d12 ob22 =d22 ), but the total fraction of species 1 increases in the initial phase (Fig. 1). This is possible if habitat 1 is more favorable for reproduction of both species than habitat 2 (bs1 4bs2 for s ¼ 1; 2) and if the initial fraction of inferior species 1 is higher in favorable habitat 1 but lower in unfavorable habitat 2. In the end, species 1 starts to decrease and gradually approaches zero after a long time. The dynamics of the fraction of inferior species show unexpected behavior if habitats are coupled by larval migration. To examine the final state, we consider the local stability of the equilibrium ð0; 0Þ: The dynamics near the origin are given by x1 ðt þ 1Þ ¼ ð1 d11 Þx1 þ d21
ð1 m21 Þb11 x1 þ m12 b12 x2 ð1 m21 Þb21 þ m12 b22
þ ½higher order terms ;
ð4aÞ
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analysis, we further assume the symmetric conditions for mortality and reproductive rate as
x1+x2
1.2 1.0 x1
0.8 0.6 0.4
x2
0.2 0 0
20
40
60
80
100
time t Fig. 1. A numerical example of the trajectory of species 1 when species 1 has no advantage in either habitat. The horizontal axis is the time step. The bold curve indicates the total fraction of species 1 in the whole system. The broken curve is x1 ; the fraction of species 1 in habitat 1, and the solid curve is x2 ; that in habitat 2. Parameters are d11 ¼ d21 ¼ d12 ¼ d22 ¼ 0:1; b11 ¼ 2:1; b21 ¼ 2:0; b12 ¼ 1:1; b22 ¼ 1:0; m12 ¼ m21 ¼ 0:4; x1 ð0Þ ¼ 0:9; and x2 ð0Þ ¼ 0:1:
ð1 m12 Þb12 x2 þ m21 b11 x1 x2 ðt þ 1Þ ¼ ð1 d12 Þx2 þ d22 ð1 m12 Þb22 þ m21 b21 þ ½higher order terms :
d11 ¼ d22 d;
d12 ¼ d21 ¼ pd;
b11 ¼ b22 b;
b12 ¼ b21 ¼ qb;
ð5Þ
where d is the mortality of species 1 in habitat 1 and also that of species 2 in habitat 2. The mortality of species 1 in habitat 2 and that of species 2 in habitat 1 are the same but are different from d; denoted by pd: Parameter implies a factor for ‘‘cross’’-habitat mortality and 0opo1=d holds because 04pdo1: In a similar way, b is the reproductive rate of species 1 in habitat 1 and that of species 2 in habitat 2; qb is the reproductive rate of species 1 in habitat 2 and that of species 2 in habitat 1. Parameter q is a factor for the ‘‘cross’’-habitat reproductive rate. The abundance of species 1 in habitat 1 and habitat 2, denoted by x1 and x2 ; respectively, follows the dynamics given as x1 ðt þ 1Þ ¼ ð1 dÞx1 þ dfx1 þ pð1 x1 Þg x1 þ aqx2 ; fx1 þ qð1 x1 Þg þ afqx2 þ ð1 x2 Þg
ð6aÞ
ð4bÞ
The matrix of the linearized dynamics, M; is a positive matrix (see Appendix A). According to Perron–Frobenius’s Theorem (e.g. Karlin and Taylor, 1975), both eigenvalues are real, and the larger one is positive when the corresponding eigenvector is also positive. The local stability of the dynamics around ð0; 0Þ is equivalent to the condition for the larger (or dominant) eigenvalue of the matrix being less than 1. Since this is rewritten in terms of the condition for a quadratic equation of characteristic equation for the matrix, the stability condition can be rewritten as TrðMÞ 2o0; and 1 TrðMÞ þ DetðMÞ40: From the assumptions, d21 b11 od11 b21 ; d22 b12 od12 b22 ; D1 40; and D2 40; and the calculation in Appendix A, it can be shown that both of the inequalities for stability hold. With TrðMÞ40; we can conclude that both eigenvalues are positive and less than 1. Then the origin ð0; 0Þ is stable. This implies that if species 1 cannot be maintained in the absence of migration, it cannot invade the population dominated by the competitor at any values of larval migration (m12 and m21 ).
4. Symmetric competition and migration To study the dynamics in more detail, let us concentrate on the case in which the migration rate from one habitat to another is the same as that in the opposite direction ðm21 ¼ m12 ¼ mÞ: For simplicity of
x2 ðt þ 1Þ ¼ ð1 pdÞx2 þ dfpx2 þ ð1 x2 Þg ax1 þ qx2 ; afx1 þ qð1 x1 Þg þ fqx2 þ ð1 x2 Þg
ð6bÞ
where a ¼ m=ð1 mÞð0oap1Þ; which indicates the ratio of the immigration rate to the emigration rate. a ¼ 1 holds when larvae are completely mixed, and a becomes small as migration between the habitats decreases. These are the two-dimensional discrete time dynamics of x1 and x2 : 4.1. The equilibria We denote the abundance of adults at a coexistence equilibrium by symbols with a hat ðxˆ i ; s ¼ 1; 2; i ¼ 1; 2Þ: By setting xi ðt þ 1Þ ¼ xi ðtÞ ¼ xˆ i and using the calculation in Appendix B, we obtain three types of equilibria. There are always two trivial equilibria—one with species 1 only, ðxˆ 1 ; xˆ 2 Þ ¼ ð1; 1Þ; and the other with species 2 only, ðxˆ 1 ; xˆ 2 Þ ¼ ð0; 0Þ: In addition to these single-species equilibria, the system has one equilibrium indicating the coexistence of two species, ðxˆ 1 ; xˆ 2 Þ ¼ ðx ; 1 x Þ with qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðp qÞ 2apq þ ðp qÞ2 þ 4a2 pq : ð7Þ x ¼ 2fp q þ að1 pqÞg If migration rate is a special value, i.e., a ¼ a ¼ ðp qÞ=ð1 pqÞ; there is a line of equilibria, i. e., x1 ¼ x2 :
ARTICLE IN PRESS S. Muko, Y. Iwasa / Theoretical Population Biology 64 (2003) 359–368
4.2. Invasibility condition
q
ð8Þ
If Eq. (8) is violated, one of the eigenvalues is greater than 1, and the species can invade the habitats occupied by the competitor. Hence, the sufficient invasibility condition is ðp qÞfðp qÞ þ að1 pqÞgo0:
ð9Þ
Similarly, the condition for the successful invasion of species 2 into a system dominated by species 1 can be derived from Eq. (9). Two equilibria with a single species, ð1; 1Þ and ð0; 0Þ; are stable or unstable simultaneously. According to numerical analysis, the coexistence equilibrium is stable when both single-species equilibria are unstable, and the coexistence equilibrium is unstable when both single-species equilibria are stable.
1
(a)
0
1
p
1
p
q cross-habitat reproductive rate
Let us now concentrate on the condition for the successful invasion of species 1 into a system dominated by species 2. This is the same as the condition for the corresponding terminal equilibrium ðxˆ 1 ; xˆ 2 Þ ¼ ð0; 0Þ to become unstable. According to the calculation in Appendix C, the Jacobi matrix for the dynamics linearized around ð0; 0Þ; M; is a positive matrix. The conditions for the local stability of ð0; 0Þ are TrðMÞ 2o0 and 1 TrðMÞ þ DetðMÞ40: If both inequalities hold, both eigenvalues of matrix M are less than 1 in magnitude. Thus, the equilibrium ð0; 0Þ is stable and species 1 cannot invade the habitats occupied by species 2. After some arithmetic, as shown in Appendix C, the stability condition of the equilibrium ð0; 0Þ is given by ðp qÞfðp qÞ þ að1 pqÞg40:
363
(b)
1
0 q
4.3. Effects of larval migration rate 1
Fig. 2 illustrates the parameter regions in which the coexistence equilibrium is stable on a ðp; qÞ-plane. Figs. 2a–c are three cases with different values of the relative migration rate a: The open regions are the parameters for which the coexistence equilibrium ðx ; 1 x Þ is stable, while the shaded regions are those in which the coexistence equilibrium is unstable. The regions stable for the coexistence equilibrium increase as a becomes small, implying that the coexistence of two species is more likely to occur when the migration of larvae between the two habitats is weak than when such migration is strong. When a ¼ 1; larvae produced in different habitats are completely mixed in a common larval pool (Muko and Iwasa, 2000) (Fig. 2a). The stability condition given by Eq. (3) becomes ð1 pÞðp qÞo0:
ð10Þ
In this region, the coexistence equilibrium is stable irrespective of the relative migration rate a: In the region
(c)
0
1 cross-habitat mortality
p
Fig. 2. Phase plane of the dynamics for three different relative migration rates a: Open regions indicate that the coexistence equilibrium is stable. Shaded regions indicate that the coexistence equilibrium is unstable. The horizontal axis is the cross-habitat mortality rate p; and the vertical axis is the cross-habitat reproductive rate q: The solid curve indicates the boundaries of Eq. (9), which has an intersection point with horizontal axis at ð0; aÞ: (a) a ¼ 1 (complete mixing case). (b) a ¼ 0:5: (c) a ¼ 0:17:
in which Eq. (10) does not hold, the stability of the coexistence equilibrium depends on the value of a: Let a ¼ ðp qÞ=ð1 pqÞ: If a4a ; the coexistence
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equilibrium is unstable. Fig. 3a illustrates the dynamics of Eq. (6) for this case. The point ðx1 ; x2 Þ would move and converge either to ð1; 1Þ or to ð0; 0Þ; implying that species 1 wins in the system and species 2 disappears, or species 2 wins and drives out species 1, respectively. The competitive outcome depends on the initial condition. In 1.0
0.5
(a)
0
0.5
1.0
contrast, Fig. 3c shows the trajectories of Eq. (1) for aoa ; in which the coexistence equilibrium is stable. Starting from any initial condition, the system converges to ðx ; 1 x Þ; which is globally stable. In between these two cases, we have a ¼ a ; in which the dynamics would converge to a line of equilibria. Fig. 3b illustrates such a case. Fig. 3 suggests that the line of x1 þ x2 ¼ 1 might be a trajectory of the dynamics for all the parameters. In fact, from Eqs. (6a) and (6b), we can show that x1 ðt þ 1Þ þ x2 ðt þ 1Þ ¼ 1 holds if x1 ðtÞ þ x2 ðtÞ ¼ 1: The internal equilibrium ðx ; 1 x Þ with Eq. (7) is on this line. When the internal equilibrium is unstable (a saddle), the line x1 þ x2 ¼ 1 is a separatrix. The system converges to ð1; 1Þ if the initial condition is x1 þ x2 41; while it converges to ð0; 0Þ if the initial condition is x1 þ x2 o1: Biologically, this implies that the species with the greater total abundance will win in the end. These results illustrate that the coexistence of species is more likely to occur when the rate of larval migration between habitats is low than when it is high.
1.0
species 1 in habitat 2 x2
5. Unidirectional migration
0.5
(b)
0
0.5
1.0
1.0
Since larvae are carried away by the water current, the migration of the larvae between habitats may be strongly asymmetric. In an extreme case, the water current is unidirectional during the critical period during which the larval migration is determined. The larvae produced in habitat 1 are transported to habitat 2 and become mixed with the larvae produced in habitat 2—but there is almost no successful migration of larvae from habitat 2 into habitat 1 ðm21 ¼ m; m12 ¼ 0Þ: The dynamics of the abundance of species 1 in two habitats are given by x1 ðt þ 1Þ ¼ ð1 d11 Þx1 þ fd11 x1 þ d21 ð1 x1 Þg b11 x1 ; b11 x1 þ b21 ð1 x1 Þ
0.5
ð11aÞ
x2 ðt þ 1Þ ¼ ð1 d12 Þx2 þ fd12 x2 þ d22 ð1 x2 Þg mb11 x1 þ b12 x2 : mfb11 x1 þ b21 ð1 x1 Þg þ fb12 x2 þ b22 ð1 x2 Þg ð11bÞ (c)
0
0.5 species 1 in habitat 1 x1
1.0
Fig. 3. Trajectories of the dynamics of Eq. (3). The horizontal axis is x1 ; the fraction of species 1 in habitat 1, and the vertical axis is x2 ; that in habitat 2. (a) a ¼ 0:67: The coexistence equilibrium ðxˆ 1 ; xˆ 2 Þ ¼ ð0:47; 0:53Þ is unstable. Two single-species equilibria, ð0; 0Þ and ð1; 1Þ; are stable. The broken line indicates the separatrix that is the boundary between two regions with different competitive outcomes. (b) a ¼ a ¼ 0:5: There is a line of equilibria which is stable. (c) a ¼ 0:25: The coexistence equilibrium ð0:60; 0:40Þ is stable. Other parameters are d ¼ 0:2; p ¼ 0:8; and q ¼ 0:5:
Without loss of generality, we assume b11 =d11 ob21 =d21 ; implying species 1 becomes extinct in habitat 1 (otherwise we can switch the names of the two species). After the convergence of x1 ðtÞ-0; as t-N; we have the equation for species 1 in habitat 2 as follows: x2 ðt þ 1Þ ¼ ð1 d12 Þx2 þ fd12 x2 þ d22 ð1 x2 Þg b12 x2 : mb21 þ fb12 x2 þ b22 ð1 x2 Þg
ð12Þ
ARTICLE IN PRESS S. Muko, Y. Iwasa / Theoretical Population Biology 64 (2003) 359–368
This equation may have a positive equilibrium. If d12 þ d22 b12 =ðmb21 þ b22 Þ40; x2 increases when it is small, and the species 1 can increase and be maintained in the system. The equilibrium fraction of species 1 in habitat 2 is xˆ 2 ¼ 1 m
d12 b21 : d22 b12 d12 b22
ð13Þ
For this to satisfy 0oxˆ 2 o1; we have the following two conditions: b12 b22 4 ; d12 d22 b b b21 : mo 12 22 d12 d22 d22
ð14aÞ
ð14bÞ
According to the calculation in Appendix D, we can prove that the derivative of the right-hand side of Eq. (12) estimated at x2 ¼ xˆ 2 satisfies 0odx2 ðt þ 1Þ= dx2 o1: From these results, we can conclude that when Eq. (14) holds, the origin is unstable and there is the equilibrium ð0; xˆ 2 Þ with Eq. (13), which is locally stable. If instead either one of the two inequalities in Eq. (14) is violated, there is no equilibrium with positive abundance of species 1 in habitat 2, and ð0; 0Þ is locally stable. Eq. (14a) implies that species 1, which is defeated in habitat 1, can win in habitat 2 without migration ðm ¼ 0Þ: Under Eq. (14a) and b11 =d11 ob21 =d21 ; two species can exist if there is no migration. Eq. (14b) indicates that, for the two species to coexist with migration, the migration rate should not be very high when there is larval migration. Hence, we can again conclude that the coexistence becomes more difficult for large larval migration.
6. Discussion The analyses in the present paper suggest that coexistence results when one species is superior to the competitor in one habitat but inferior in the other, and that two species are more likely to coexist when the larvae produced from different habitats are mixed at a smaller rate. We can compare the conditions for the coexistence of two species between the case without migration ðm12 ¼ m21 ¼ 0Þ and the case with migration ðm12 40; m21 40Þ: When habitats are isolated, the competitive outcome in each habitat concludes with the elimination of one species by the other. Coexistence is possible only when two species are completely segregated in different habitats. This situation implies the species coexistence at the regional scale. In contrast, if larvae produced in different habitats are mixed partially or completely, the system shows complex
365
behaviors. If the coexistence equilibrium is stable, the two species are observed in each local habitat. However, if the coexistence equilibrium is unstable while the two single-species equilibria are locally stable, the outcome of the competition depends on the initial abundance. The whole system is dominated by either one of the two species depending on the initial abundance. In symmetric competition and symmetric migration of the two species, the regions where the system is bistable become wider as the migration rate increases while the regions of coexistence become narrower (see Fig. 2). Bistability is a characteristic behavior of the complete mixed lottery model with two heterogeneous habitats, because the corresponding model with a sufficiently low rate of larval migration does not show bistability. Following Muko and Iwasa (2000), the mechanism of bistability can be explained in the symmetric case with two habitats as follows. When the inequality in Eq. (10) is violated, each species has a shorter life in the habitat where it can enjoy greater lifetime reproductive success. In such a situation, when species 1 dominates, it holds that the sites in habitat 2 are more effectively than those in habitat 1, resulting in a low average reproductive success of rare species 2 which cannot do well in habitat 1. Similarly, when species 2 dominates the system, species 1 cannot increase because most of the vacant spaces are created in habitat 2. In either case, the lifetime reproductive success of the resident species is higher than the invader species, and bistability occurs. This condition is met when reproductive rate bsi varies between sites and between species in such a way to reverse the order of longevity (Proposition 5 in Muko and Iwasa, 2000). The bistability is hence caused by larval settlement to an unproductive habitat that is not suitable for the species. This effect disappears when the rate of migration between the two habitats is at a low level. In symmetric competition, each species is adapted to its respective habitat and can do well there even if its initial density is low. Sufficiently low migration can weaken the interspecific competition at the regional scale and thus promote the species coexistence. The effect of dispersal on the condition of species coexistence has been studied intensively. Levin (1974) reported that limited migration between two heterogeneous habitats tends to work to maintain species diversity. In his model, the habitats were initially homogeneous but became heterogeneous because one species happened to dominate a part of habitats and the other species occupied the other. The dispersal of individuals from the occupied area to the unoccupied one prevented the competitive exclusion of one species. The difference in dispersal abilities between the competitors has been emphasized as another mechanism which promotes species coexistence, especially when there is hierarchical competition among species (e.g., Levin, 1974; Shmida and Ellner, 1984; Comins and Noble,
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1985; Nee and May, 1992; Tilman, 1994; Shigesada and Kawasaki, 1997). An inferior competitor can coexist with a superior one if it has a higher colonization rate and can preempt the bare space. However, these theoretical studies focused on the role of transient spatial heterogeneity created by biological activities within homogeneous habitats. There have been relatively fewer studies on permanent spatial variation in habitat quality (e.g., Chesson, 1985, 2000; Roughgarden and Iwasa, 1986; Holt, 1997; Muko and Iwasa, 2000), although its importance has been documented empirically in many different taxa, especially in marine organisms (e.g., Stoner, 1992; Hunt and Scheibling, 1998). Recently, Chesson (2000) examined spatial variation in the environment in a general way by classifying the mechanisms promoting coexistence into spatial storage effect and competitive nonlinearlity. He argued that the limited dispersal between heterogeneous habitats substantially broadens the range of opportunities for coexistence compared with the cases in which there is complete mixing of individuals within the spatiotemporal variable habitats. The present paper is a realistic case employing Chesson’s model. We modeled the lottery competition in a system consisting of spatially heterogeneous but temporally homogeneous habitats, and showed that sufficiently low migration promotes species coexistence. The retention of larvae in the local pool, which has been reported recently by empirical studies, might work to maintain the species diversity in heterogeneous environments. Although many theoretical studies on spatially structured habitats have often assumed symmetric migration, the condition of coexistence may be sensitive to the directionality of migration between habitats. Some theoretical studies have addressed permanent spatial heterogeneity in metapopulation dynamics by including a favorable patch (source population) and an unfavorable one (sink population) connected by migration. The sink population can be maintained by continuous immigration from the favorable source patch (Pulliam, 1988). In the present paper, an asymmetric migration case shows that the species which is adapted to an upstream habitat does not experience competitive exclusion. However, the species, which is adapted to a downstream habitat, but not to an upstream one, may be excluded if the migration is very intense. The effect of a low rate of migration between habitats on the maintenance of species diversity has an important implication. In addition to climatic change, land use change, and environmental pollution, one of the most threatening anthropogenetic processes is the invasion of habitats by exogenous species, which subsequently spread in the new habitat, driving the endemic fauna and flora to extinction (Mooney and Drake, 1989). This is caused by an enhanced rate of migration of organisms
by human activities. As suggested by the simple model studied in the present paper, a larger opportunity of migration between regions and between continents often results in a decline in total biodiversity. This demonstrates the importance of the theoretical study of multiple species systems in heterogeneous environments.
Acknowledgments This work was supported in part by Grants-in-Aid from the Japan Society for the Promotion of Science to S.M. and to Y.I. We greatly appreciate Professor Peter Chesson for his very important comments on a previous version of this paper. We also thank the following people for their useful comments: T. Namba, A. Sasaki, N. Shigesada, and T. Takegaki.
Appendix A The matrix of the linearized dynamics is 8 d21 ð1 m21 Þb11 > > < 1 d11 þ D1 M¼ > d22 m21 b11 > : D2
9 d21 m12 b12 > > = D1 ; d22 ð1 m12 Þb12 > > ; 1 d12 þ D2
ðA:1Þ where D1 ¼ ð1 m21 Þb21 þ m12 b22 and D2 ¼ ð1 m12 Þ b22 þ m21 b21 : From the assumptions, d21 b11 od11 b21 ; d22 b12 od12 b22 ; D1 40; and D2 40; we have TrðMÞ 2 1 ½fðd21 b11 d11 b21 Þð1 m21 Þ d11 m12 b22 g ¼ D1 D2 þ fðd22 b12 d12 b22 Þð1 m12 Þ d12 m21 b21 g o0
ðA:2aÞ
1 TrðMÞ þ DetðMÞ 1 ¼ ½ðd21 b11 d11 b21 Þðd22 b12 d12 b22 Þð1 m21 Þ D1 D2 ð1 m12 Þ ðd21 b11 d11 b21 Þð1 m21 Þd12 m21 b21 ðd22 b12 d12 b22 Þð1 m12 Þd11 m12 b22 þ ðd11 b22 d12 b21 d21 b12 d22 b11 Þm12 m21 40:
ðA:2bÞ
With TrðMÞ40; we can conclude that both eigenvalues are positive and less than 1.
Appendix B We denote the abundance of adults and larvae at a coexistence equilibrium by symbols with a hat (xˆ i and Lˆ s ; s ¼ 1; 2; i ¼ 1; 2). By setting xi ðt þ 1Þ ¼ xi ðtÞ ¼ xˆ i ;
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Eq. (6a) and (6b) are rewritten as ðp qÞxˆ 1 ð1 xˆ 1 Þ axˆ 1 ð1 xˆ 2 Þ þ apqð1 xˆ 1 Þxˆ 2 ¼ 0; ðp qÞxˆ 2 ð1 xˆ 2 Þ axˆ 1 ð1 xˆ 2 Þ þ apqð1 xˆ 1 Þxˆ 2 ¼ 0;
367
1 TrðMÞ þ DetðMÞ ðB:1aÞ
¼
ðB:1bÞ
pd d þ qþa
pd þ
qd aq þ 1
apqd ad
40: q þ a aq þ 1
ðC:2bÞ
If p ¼ q; Eqs. (B.1a) and (B.1b) are the same and are equal to xˆ 1 ð1 xˆ 2 Þ ¼ p2 ð1 xˆ 1 Þxˆ 2 ; which corresponds to a curve on the ðx1 ; x2 Þ-plane. Any point on this curve is the equilibrium of Eq. (6) because it satisfies Eq. (B.1). This gives the ‘‘curve of equilibria’’ of the dynamics. If paq; Eq. (B.2) gives xˆ 1 ¼ xˆ 2 ; or xˆ 1 ¼ 1 xˆ 2 : First ˆ Eqs. (B.1a) and we consider the case with xˆ 1 ¼ xˆ 2 ¼ x: (B.1b) are the same and are rewritten as
If both inequalities hold, both eigenvalues of matrix M are less than 1 in magnitude. Then the equilibrium ð0; 0Þ is stable and species 1 cannot invade the population occupied by species 2. In contrast, if Eq. (C.2a) holds, but Eq. (C.2b) is violated, there is one eigenvalue exceeding 1 and the other is less than 1. If Eq. (C.2b) holds but Eq. (C.2a) is violated, both eigenvalues of matrix M exceed 1. In both of these cases, species 1 succeeds in spreading throughout the system occupied by species 2. After some arithmetic, the stability condition Eq. (C.2b) is rewritten as
ˆ xÞfp ˆ xð1 q að1 pqÞg ¼ 0:
ðp qÞfðp qÞ þ að1 pqÞg40:
respectively. Subtracting Eq. (B.1b) from Eq. (B.1a), we obtain ðp qÞðxˆ 1 xˆ 2 Þð1 xˆ 1 xˆ 2 Þ ¼ 0:
ðB:2Þ
ðB:3Þ
Let a ¼ ðp qÞ=ð1 pqÞ: When a ¼ a ; all the points satisfying are equilibria of the dynamics, Eq. (6). Hence, x1 ¼ x2 is the line of equilibria. If aaa ; the dynamics have only two equilibria: the one with species 1 only ðxˆ 1 ; xˆ 2 Þ ¼ ð1; 1Þ; and another with species 2 only ðxˆ 1 ; xˆ 2 Þ ¼ ð0; 0Þ: In the case of xˆ 1 ¼ 1 xˆ 2 ¼ x ; Eqs. (B.1a) and (B.1b) are the same and are rewritten as ax2 ðp qÞx ð1 x Þ þ apqð1 x Þ2 ¼ 0; which can be solved as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðp qÞ 2apq þ ðp qÞ2 þ 4a2 pq x ¼ : ðB:4Þ 2fp q þ að1 pqÞg x satisfies 0ox o1 for p ð0opo1=dÞ; any q; and a ð0oap1Þ:
Since the relative migration rate a is smaller than 1, a should satisfy pq oao1: 1 pq
The Jacobi matrix for the dynamics linearized around ð0; 0Þ is 2 3 pd apqd 1dþ 6 7 qþa qþa 7; ðC:1Þ M¼6 4 ad qd 5 1 pd þ aq þ 1 aq þ 1 which is a positive matrix. The conditions for the local stability of the matrix are pd TrðMÞ 2 ¼ d þ qþa qd þ pd þ o0; ðC:2aÞ aq þ 1
ðC:4Þ
This inequality is satisfied either for 1oqop or for 1opoq: From Eq. (C.4), the left hand side of Eq. (C.2a) can be written as pð1 pqÞ TrðMÞ 2od ð1 þ pÞ þ qð1 pqÞ þ ðp qÞ qð1 pqÞ þ ; qðp qÞ þ ð1 pqÞ ¼
Appendix C
ðC:3Þ
pq ; 1q
o0;
ðC:5Þ
which is negative for the parameter region which satisfies Eq. (C.4). Therefore, the stability condition (C.2a) holds when the stability condition (C.2b) is satisfied. Thus, the stability of the equilibrium ð0; 0Þ is determined by the sign of the left-hand side of Eq. (C.3). If Eq. (C.3) is violated, one of the eigenvalues is greater than 1, and the species can invade the population occupied by the competitor.
Appendix D By setting x2 ðt þ 1Þ ¼ x2 ðtÞ ¼ xˆ 2 ; the dynamics of x2 given by Eq. (12) in the text becomes d12 ¼ fd12 xˆ 2 þ d22 ð1 xˆ 2 Þgb12 =D; where D¼ mb21 þ b12 xˆ 2 þ b22 ð1 xˆ 2 Þ; at equilibrium.
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The derivative of the right-hand side of Eq. (12) at x2 ¼ xˆ 2 is dx2 ðt þ 1Þ 1 dx2 b xˆ 2 ¼ d12 þ ðd12 d22 Þ 12 þ fd12 xˆ 2 þ d22 ð1 xˆ 2 Þg D b12 xˆ 2 ðb12 b22 Þ 1 ; D D xˆ 2 ðD:1Þ ¼ ðd22 b12 þ d12 b22 Þ; o0 D because b12 =d12 4b22 =d22 : Subtracting the numerator of Eq. (D.1) from the denominator of Eq. (D.1), we obtain D ðd22 b12 þ d12 b22 Þxˆ 2 ¼ b12 xˆ 2 þ ðd22 b12 d12 b22 Þxˆ 2 þ
d22 b12 ð1 xˆ 2 Þ; d12
40 because b12 =d12 4b22 =d22 : Hence1odx2 ðt þ 1Þ=dx2 1 when x2 ¼ xˆ 2 : Therefore the derivative of the right-hand side of Eq. (12) estimated at x2 ¼ xˆ 2 satisfies 0odx2 ðt þ 1Þ= dx2 o1: This implies that x2 ðtÞ converges to xˆ 2 without oscillation.
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I.A., Gilpin, M.E. (Eds.), Meta-population Biology: Ecology, Genetics and Evolution. Academic Press, San Diego, pp. 149–164. Horvath, T.G., Lamberti, G.A., Lodge, D.M., Perry, W.L., 1996. Zebra mussel dispersal in lake-stream systems: source-sink dynamics? J. N. Am. Benthol. Soc. 15, 564–575. Hunt, H.L., Scheibling, R.E., 1998. Effects of whelk predation on mussel assemblages in tidepools and on emergent rock on a waveexposed rocky shore in Nova Scotia, Canada. J. Exp. Mar. Biol. Ecol. 226, 87–113. Iwasa, Y., Roughgarden, J., 1986. Interspecific competition among metapopulations with space-limited subpopulations. Theor. Popul. Biol. 30, 194–214. Karlin, S., Taylor, H.M., 1975. A First Course in Stochastic Processes. 2nd Edition. Academic Press, London, p. 557. Levin, S.A., 1974. Dispersion and population interactions. Amer. Nat. 108, 207–228. McQuaid, C.D., Phillips, T.E., 2000. Limited wind-driven dispersal of intertidal mussel larvae: in situ evidence from the plankton and the spread of the invasive species Mytilus galloprovincialis in South Africa. Mar. Ecol. Progr. Ser. 201, 211–220. Mooney, H.A., Drake, J.A., 1989. Biological invasions: a SCOPE program, overview. In: Drake, J.A., Mooney, H.A., di Castri, F., Groves, R.H., Kruger, F.J., Rejmanek, M., Williamson, M. (Eds.), Biological Invasions: a Global Perspective, SCOPE, Vol. 37. Wiley, New York, pp. 50–491. 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. Anim. Ecol. 61, 37–40. Olson, R.R., 1985. The consequences of short-distance larval dispersal in a sessile marine invertebrate. Ecology 66, 30–39. Petersen, J.K., Svane, I., 1995. Larval dispersal in the ascidian CionaIntestinalis—evidence for a closed population. J. Exp. Mar. Biol. Ecol. 186, 89–102. Possingham, H.P., Roughgarden, J., 1990. Spatial populationdynamics of a marine organism with a complex life-cycle. Ecology 71, 973–985. Pulliam, H.R., 1988. Sources, sinks, and population regulation. Amer. Nat. 132, 652–661. Reynolds, H.L., Hungate, B.A., Chapin, F.S., DAntonio, C.M., 1997. Soil heterogeneity and plant competition in an annual grassland. Ecology 78, 2076–2090. Roughgarden, J., Iwasa, Y., 1986. Dynamics of a metapopulation with space-limited subpopulations. Theor. Popul. Biol. 29, 235–261. Roughgarden, J., Gaines, R., Possingham, H., 1988. Recruitment dynamics in complex life cycles. Science 241, 1460–1466. Sammarco, P.W., Andrews, J.C., 1988. Localized dispersal and recruitment in great barrier reef corals: the helix experiment. Science 239, 1422–1424. Shigesada, N., Kawasaki, K., 1997. Biological Invasions: Theory and Practice. Oxford University Press, Oxford. Shigesada, N., Roughgarden, J., 1982. The role of rapid dispersal in the population dynamics of competition. Theor. Popul. Biol. 21, 253–373. Shmida, A., Ellner, S., 1984. Coexistence of plant species with similar niches. Vegetatio 58, 29–55. Stoner, D.S., 1992. Vertical-distribution of a colonial ascidian on a coral-reef—the roles of larval dispersal and life-history variation. Amer. Nat. 139, 802–824. Swearer, S.E., Caselle, J.E., Lea, D.W., Warner, R.R., 1999. Larval retention and recruitment in an island population of a coral-reef fish. Nature 402, 799–802. Tilman, D., 1994. Competition and biodiversity in spatially structured habitats. Ecology 75, 2–16.
Theoretical Population Biology 64 (2003) 359–368
http://www.elsevier.com/locate/ytpbi
Incomplete mixing promotes species coexistence in a lottery model with permanent spatial heterogeneity Soyoka Muko and Yoh Iwasa Mathematical Biology Laboratory, Department of Biology, Faculty of Sciences, Kyushu University, 6-10-1 Hakozaki, Higashi-ku, Fukuoka-shi, Fukuoka-ken 812-8581, Japan Received 2 February 2001
Abstract For many marine organisms, the population dynamics in multiple habitats are affected by migration of planktonic larvae. We herein examine the effect of incomplete larval mixing on the condition for species coexistence. The system consists of two heterogeneous habitats, each composed of a number of sites occupied by adults of two species. Larvae produced in a habitat form a pool and migrate to the pool of the other habitat. When an adult dies, the vacant site becomes occupied by an individual randomly chosen from the larval pool. We study (1) the invasibility of a inferior species which has no advantage in either habitats, (2) the dynamics when larval migration and competition among adults are symmetric between habitats, and (3) the case with unidirectional migration. The coexistence of competitors is more likely to occur when larval migration is weak. r 2003 Elsevier Inc. All rights reserved. Keywords: Localized dispersal; Recruitment; Unidirectional flow; Spatial heterogeneity
1. Introduction In many marine invertebrates, the sedentary adults occupy a portion of a rocky substrate and produce planktonic larvae which are subsequently dispersed. Larvae produced in different habitats are mixed in a pelagic pool and then settle back to vacant substrates in the habitats. As a consequence, larvae produced in a favorable habitat may be transported to other locations that are not very favorable for the species. If one species dominates a highly productive habitat, the larvae produced there may be carried to other habitats, where they settle, grow, and possibly drive out the competitors. Hence, we need to consider the population dynamics, including both sedentary adults in different habitats and larvae in the pelagic pool, in order to understand the coexistence of multiple species in such a patchy environment. For marine organisms, permanent spatial variations between different habitats in environmental factors, such as water depth, substrate quality, water current,
Corresponding author. Fax: +81-92-642-2645. E-mail address: [email protected] (S. Muko). 0040-5809/03/$ - see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0040-5809(03)00085-6
irradiance, and exposure to predators, are often very important in maintaining the species diversity. The importance of spatial heterogeneity has been demonstrated in many different taxa, such as terrestrial plants (Reynolds et al., 1997; Guo, 1998), fresh water macrophytes (French and Chambers, 1996), ascidians (Stoner, 1992), and mussels (Hunt and Scheibling, 1998). However, most theoretical studies on the coexistence of multiple competitors focus on the role of spatial heterogeneity that is transient rather than permanent (e.g. Levin, 1974; Hasting, 1980; Chesson and Warner, 1981; Shmida and Ellner, 1984; Comins and Noble, 1985; Tilman, 1994; Shigesada and Kawasaki, 1997). For example, in the standard lottery model studied by Chesson and Warner (1981), either of the two species stays at each site throughout its life and creates spatial heterogeneity. However, the spatial heterogeneity in this sense is not permanent, because there is no basic difference in the quality between sites. The sites currently occupied by one species will later become occupied by the other. In other examples, Comins and Noble (1985) analyzed the effect of transient spatial heterogeneity in the lottery model, in which the patch quality varies randomly but the long-term statistical distribution of conditions is the same between sites.
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In contrast, there have been relatively fewer studies on permanent heterogeneous environments (e.g., Shigesada and Roughgarden, 1982; Holt, 1997). Chesson (1985) noted that the spatial heterogeneity in adult mortality between habitats could promote the coexistence in a specific example of lottery model, in which adults can survive only in some patches, such suitable patches being different between two species. Recently, Chesson (2000) has developed a general model, which includes a variety of competitive interactions, to facilitate the study of the mechanisms of species coexistence. Roughgarden and her colleagues studied the competition of species within multiple heterogeneous habitats where are connected by a pelagic pool of larvae (Roughgarden and Iwasa, 1986; Iwasa and Roughgarden, 1986; Possingham and Roughgarden, 1990; Alexander and Roughgarden, 1996). Their models have a complex structure in which each habitat might have unfilled space and in which the larvae have their own dynamics. Muko and Iwasa (2000) studied the lottery competition of two species with permanent spatial heterogeneity, which is more tractable mathematically than the previous studies. The model consists of several habitats for sedentary adults, which may differ in mortality and reproductive rate. The larvae produced in different habitats are mixed completely in a larval pool and then settle back to vacant space immediately after such space is created by the death of adults. Muko and Iwasa show that, in addition to two equilibria of single species, there can be up to n 1 internal equilibria in which stable coexistence can occur, if there are n habitats for adults. The location of these internal equilibria and their local stability can be calculated from a single function indicating species difference in reproductive success. In general, if the ratio of adult mortality of the two species varies between habitats, the coexistence is likely to occur, while between-habitat variation in the ratio of larval production rate does not promote coexistence. This conclusion contrasts with the role of temporal variation in the standard lottery model proposed by Chesson and Warner (1981), in which the between-year variation of the relative reproductive rate promotes coexistence but the variation of relative mortality does not. However, recent research has provided evidence that dispersal of planktonic larvae is sometimes localized by physical conditions such as water current and wind (e.g., Sammarco and Andrews, 1988; Roughgarden et al., 1988; Black et al., 1991; McQuaid and Phillips, 2000), or by the behavior of larvae (e.g., Olson, 1985; Stoner, 1992; Carlon and Olson, 1993). Larvae are more likely to stay at the natal site than to disperse to other habitats (Petersen and Svane, 1995; Swearer et al., 1999; Carballo, 2000). If the water current is unidirectional, larvae produced in a habitat are carried to another site, but the larval migration in the opposite direction is
negligible (Olson, 1985; Stoner, 1992; Horvath et al., 1996). Hence the larvae produced in different habitats might not be mixed completely. We should consider how larval retention affects the condition of species coexistence. The present paper deals with cases of incomplete mixing of larvae among the heterogeneous habitats. This is a sequel to Muko and Iwasa (2000) which assumed the complete mixing of larvae. Here, we consider a separate larval pool for each adult habitat and assume that a certain number of larvae are exchanged between those pools. To simplify the analysis, we concentrate on the case of two habitats. First, we show the invasibility of an inferior species which has no advantage in either habitat. Second we consider symmetric competition in which species 1 has a high reproductive success in habitat 1 and species 2 has an equally high reproductive success in habitat 2. Finally, we study the case of asymmetric larval migration, in which the water current carries the larvae produced in habitat 1 to habitat 2, but larvae from habitat 2 do not settle in habitat 1.
2. Model We consider two species indexed by s ðs ¼ 1; 2Þ living in multiple habitats for sessile adults, indexed by i ¼ 1; 2; y; n: These habitats may differ in substrate quality, water current, disturbance rate, and nutrient conditions. It is assumed that all the sites in adult habitats are occupied most of time, except for brief periods between the death of an individual and the subsequent settlement of larvae. Hence, if the fraction of sites occupied by species 1 in habitat i is denoted by xi ; the fraction of sites occupied by species 2 is 1 xi for i ¼ 1; 2; y; n: Let Lsi be the number of planktonic larvae of species s in the pelagic larval pool near habitat i: The time unit is one year and larval period is much shorter than a year. The dynamics of xi are determined both by the loss of adults and by the recruitment of planktonic larvae: xi ðt þ 1Þ ¼ ð1 d1i Þxi þ fd1i xi þ d2i ð1 xi Þg L1i ; i ¼ 1; 2; y; n; L1i þ L2i
ð1aÞ
where dsi is the annual mortality of species s at site i ð0pdsi p1Þ: In Eq. (1a), the suffix t is abbreviated in the right-hand side. The first term of the right-hand side indicates the fraction of adults surviving to the next year. The second term indicates the recruitment of larvae and is the fraction of vacant sites created by the death of adults of the two species multiplied by the proportion of the species in the larval pool. A fraction of the planktonic larvae produced in habitat i move to habitat j and become part of the larval pool for subsequent settlement. This fraction is denoted by mji :
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The planktonic larvae that are ready for settlement in habitat i are ! X X L1i ¼ 1 mki b1i xi þ mij b1j xj ; ð1bÞ jai
kai
L2i ¼ 1
X
! mki b2i ð1 xi Þ
kai
þ
X
mij b2j ð1 xj Þ:
ð1cÞ
jai
bsi is the reproductive rate, defined as the total number of larvae produced by adults in habitat i in one year, if the habitat is occupied only by species s: It increases with the area and the productivity of the habitat. The first term of the r.h.s. indicates migration out of the local pool to other pools, and the second one represents the immigration from the other pools to the local pool. Muko and Iwasa (2000) analyzed the case with mji ¼ 1=n ði; j ¼ 1; 2; y; nÞ; in which larvae produced in different habitats are mixed completely in a common larval pool. For simplicity of analysis, we concentrate on the case of two habitats in this paper. To examine the effect of larval migration on the condition for coexistence and the outcome of competition between two species, we can compare two extreme cases: one without larval migration between habitats, and one with complete mixing in a single pelagic pool. In an isolated habitat, the fraction of species 1 in habitat 1 is Dx1 ¼ x1 ðt þ 1Þ x1 ðtÞ ¼
x1 ð1 x1 Þ b b d11 d21 11 21 : b11 x1 þ b21 ð1 x1 Þ d11 d21
ð2Þ
Since 1=dsi indicates the mean lifetime of an adult, bsi =dsi is the expected number of larvae produced in its lifetime ðs ¼ 1; 2Þ: The sign of Eq. (2) depends on the difference of the lifetime reproductive success between species in habitat 1. Eq. (2) implies that species 1 will eventually occupy habitat 1 (x1 ðtÞ-1 as t-N) if species 1 is superior to species 2 ðb11 =d11 4b21 =d21 Þ: More than one species cannot stably coexist within an isolated habitat. If the whole system includes two distinct patches, coexistence is possible when one species wins in habitat 1 but the other wins in habitat 2 (b11 =d11 4b21 =d21 and b12 =d12 ob22 =d22 ; or b11 =d11 o b21 =d21 and b12 =d12 4b22 =d22 ). In the other extreme, the larvae produced in two habitats are completely mixed in a single pool (Muko and Iwasa, 2000). In such a case, in addition to two single-species equilibria, there may be an equilibrium including two species. According to Muko and Iwasa (2000), the condition for stability of the two-species
361
equilibrium is given by b11 b21 d22 b22 b12 4 ; d11 d21 d21 d22 d12 b22 b12 d11 b11 b21 4 : d22 d12 d12 d11 d21
ð3aÞ ð3bÞ
There are four possible cases: (I) If Eqs. (3a) and (3b) hold, the two species stably coexist. (II) If Eq. (3a) holds but Eq. (3b) does not, species 1 eliminates species 2. (III) If Eq. (3b) holds but Eq. (3a) does not, species 2 drives out species 1. (IV) If neither Eq. (3a) nor Eq. (3b) holds, there is a coexistence equilibrium, but it is unstable while the two single-species equilibria are locally stable. The outcome of the competition between two species depends on the initial abundance—whichever species has a large initial occupancy will drive out the competitor. This is the situation called bistability. In the case without migration, two species are completely segregated in different habitats and have no contact. The migration between habitats could cause coexistence in the sense that all species are present in all habitats (Levin, 1974). In the following, we study the effect of larval migration at an intermediate level on the conditions for coexistence. Here we concentrate on three solvable cases that assume different types of competitive ability in order to examine how the competitive outcome depends on the magnitude and pattern of migration. 3. One species is inferior in both habitats If one species is inferior to the other in both habitats, it is eliminated from each habitat in the absence of larval migration between habitats. We might conjecture that the inferior species declines monotonically from any abundance to extinction even if larval migration occurs. However, we can disprove this by a counter-example illustrated in Fig. 1. Here, species 1 is inferior to species 2 in both habitats (b11 =d11 ob21 =d21 and b12 =d12 ob22 =d22 ), but the total fraction of species 1 increases in the initial phase (Fig. 1). This is possible if habitat 1 is more favorable for reproduction of both species than habitat 2 (bs1 4bs2 for s ¼ 1; 2) and if the initial fraction of inferior species 1 is higher in favorable habitat 1 but lower in unfavorable habitat 2. In the end, species 1 starts to decrease and gradually approaches zero after a long time. The dynamics of the fraction of inferior species show unexpected behavior if habitats are coupled by larval migration. To examine the final state, we consider the local stability of the equilibrium ð0; 0Þ: The dynamics near the origin are given by x1 ðt þ 1Þ ¼ ð1 d11 Þx1 þ d21
ð1 m21 Þb11 x1 þ m12 b12 x2 ð1 m21 Þb21 þ m12 b22
þ ½higher order terms ;
ð4aÞ
ARTICLE IN PRESS S. Muko, Y. Iwasa / Theoretical Population Biology 64 (2003) 359–368
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analysis, we further assume the symmetric conditions for mortality and reproductive rate as
x1+x2
1.2 1.0 x1
0.8 0.6 0.4
x2
0.2 0 0
20
40
60
80
100
time t Fig. 1. A numerical example of the trajectory of species 1 when species 1 has no advantage in either habitat. The horizontal axis is the time step. The bold curve indicates the total fraction of species 1 in the whole system. The broken curve is x1 ; the fraction of species 1 in habitat 1, and the solid curve is x2 ; that in habitat 2. Parameters are d11 ¼ d21 ¼ d12 ¼ d22 ¼ 0:1; b11 ¼ 2:1; b21 ¼ 2:0; b12 ¼ 1:1; b22 ¼ 1:0; m12 ¼ m21 ¼ 0:4; x1 ð0Þ ¼ 0:9; and x2 ð0Þ ¼ 0:1:
ð1 m12 Þb12 x2 þ m21 b11 x1 x2 ðt þ 1Þ ¼ ð1 d12 Þx2 þ d22 ð1 m12 Þb22 þ m21 b21 þ ½higher order terms :
d11 ¼ d22 d;
d12 ¼ d21 ¼ pd;
b11 ¼ b22 b;
b12 ¼ b21 ¼ qb;
ð5Þ
where d is the mortality of species 1 in habitat 1 and also that of species 2 in habitat 2. The mortality of species 1 in habitat 2 and that of species 2 in habitat 1 are the same but are different from d; denoted by pd: Parameter implies a factor for ‘‘cross’’-habitat mortality and 0opo1=d holds because 04pdo1: In a similar way, b is the reproductive rate of species 1 in habitat 1 and that of species 2 in habitat 2; qb is the reproductive rate of species 1 in habitat 2 and that of species 2 in habitat 1. Parameter q is a factor for the ‘‘cross’’-habitat reproductive rate. The abundance of species 1 in habitat 1 and habitat 2, denoted by x1 and x2 ; respectively, follows the dynamics given as x1 ðt þ 1Þ ¼ ð1 dÞx1 þ dfx1 þ pð1 x1 Þg x1 þ aqx2 ; fx1 þ qð1 x1 Þg þ afqx2 þ ð1 x2 Þg
ð6aÞ
ð4bÞ
The matrix of the linearized dynamics, M; is a positive matrix (see Appendix A). According to Perron–Frobenius’s Theorem (e.g. Karlin and Taylor, 1975), both eigenvalues are real, and the larger one is positive when the corresponding eigenvector is also positive. The local stability of the dynamics around ð0; 0Þ is equivalent to the condition for the larger (or dominant) eigenvalue of the matrix being less than 1. Since this is rewritten in terms of the condition for a quadratic equation of characteristic equation for the matrix, the stability condition can be rewritten as TrðMÞ 2o0; and 1 TrðMÞ þ DetðMÞ40: From the assumptions, d21 b11 od11 b21 ; d22 b12 od12 b22 ; D1 40; and D2 40; and the calculation in Appendix A, it can be shown that both of the inequalities for stability hold. With TrðMÞ40; we can conclude that both eigenvalues are positive and less than 1. Then the origin ð0; 0Þ is stable. This implies that if species 1 cannot be maintained in the absence of migration, it cannot invade the population dominated by the competitor at any values of larval migration (m12 and m21 ).
4. Symmetric competition and migration To study the dynamics in more detail, let us concentrate on the case in which the migration rate from one habitat to another is the same as that in the opposite direction ðm21 ¼ m12 ¼ mÞ: For simplicity of
x2 ðt þ 1Þ ¼ ð1 pdÞx2 þ dfpx2 þ ð1 x2 Þg ax1 þ qx2 ; afx1 þ qð1 x1 Þg þ fqx2 þ ð1 x2 Þg
ð6bÞ
where a ¼ m=ð1 mÞð0oap1Þ; which indicates the ratio of the immigration rate to the emigration rate. a ¼ 1 holds when larvae are completely mixed, and a becomes small as migration between the habitats decreases. These are the two-dimensional discrete time dynamics of x1 and x2 : 4.1. The equilibria We denote the abundance of adults at a coexistence equilibrium by symbols with a hat ðxˆ i ; s ¼ 1; 2; i ¼ 1; 2Þ: By setting xi ðt þ 1Þ ¼ xi ðtÞ ¼ xˆ i and using the calculation in Appendix B, we obtain three types of equilibria. There are always two trivial equilibria—one with species 1 only, ðxˆ 1 ; xˆ 2 Þ ¼ ð1; 1Þ; and the other with species 2 only, ðxˆ 1 ; xˆ 2 Þ ¼ ð0; 0Þ: In addition to these single-species equilibria, the system has one equilibrium indicating the coexistence of two species, ðxˆ 1 ; xˆ 2 Þ ¼ ðx ; 1 x Þ with qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðp qÞ 2apq þ ðp qÞ2 þ 4a2 pq : ð7Þ x ¼ 2fp q þ að1 pqÞg If migration rate is a special value, i.e., a ¼ a ¼ ðp qÞ=ð1 pqÞ; there is a line of equilibria, i. e., x1 ¼ x2 :
ARTICLE IN PRESS S. Muko, Y. Iwasa / Theoretical Population Biology 64 (2003) 359–368
4.2. Invasibility condition
q
ð8Þ
If Eq. (8) is violated, one of the eigenvalues is greater than 1, and the species can invade the habitats occupied by the competitor. Hence, the sufficient invasibility condition is ðp qÞfðp qÞ þ að1 pqÞgo0:
ð9Þ
Similarly, the condition for the successful invasion of species 2 into a system dominated by species 1 can be derived from Eq. (9). Two equilibria with a single species, ð1; 1Þ and ð0; 0Þ; are stable or unstable simultaneously. According to numerical analysis, the coexistence equilibrium is stable when both single-species equilibria are unstable, and the coexistence equilibrium is unstable when both single-species equilibria are stable.
1
(a)
0
1
p
1
p
q cross-habitat reproductive rate
Let us now concentrate on the condition for the successful invasion of species 1 into a system dominated by species 2. This is the same as the condition for the corresponding terminal equilibrium ðxˆ 1 ; xˆ 2 Þ ¼ ð0; 0Þ to become unstable. According to the calculation in Appendix C, the Jacobi matrix for the dynamics linearized around ð0; 0Þ; M; is a positive matrix. The conditions for the local stability of ð0; 0Þ are TrðMÞ 2o0 and 1 TrðMÞ þ DetðMÞ40: If both inequalities hold, both eigenvalues of matrix M are less than 1 in magnitude. Thus, the equilibrium ð0; 0Þ is stable and species 1 cannot invade the habitats occupied by species 2. After some arithmetic, as shown in Appendix C, the stability condition of the equilibrium ð0; 0Þ is given by ðp qÞfðp qÞ þ að1 pqÞg40:
363
(b)
1
0 q
4.3. Effects of larval migration rate 1
Fig. 2 illustrates the parameter regions in which the coexistence equilibrium is stable on a ðp; qÞ-plane. Figs. 2a–c are three cases with different values of the relative migration rate a: The open regions are the parameters for which the coexistence equilibrium ðx ; 1 x Þ is stable, while the shaded regions are those in which the coexistence equilibrium is unstable. The regions stable for the coexistence equilibrium increase as a becomes small, implying that the coexistence of two species is more likely to occur when the migration of larvae between the two habitats is weak than when such migration is strong. When a ¼ 1; larvae produced in different habitats are completely mixed in a common larval pool (Muko and Iwasa, 2000) (Fig. 2a). The stability condition given by Eq. (3) becomes ð1 pÞðp qÞo0:
ð10Þ
In this region, the coexistence equilibrium is stable irrespective of the relative migration rate a: In the region
(c)
0
1 cross-habitat mortality
p
Fig. 2. Phase plane of the dynamics for three different relative migration rates a: Open regions indicate that the coexistence equilibrium is stable. Shaded regions indicate that the coexistence equilibrium is unstable. The horizontal axis is the cross-habitat mortality rate p; and the vertical axis is the cross-habitat reproductive rate q: The solid curve indicates the boundaries of Eq. (9), which has an intersection point with horizontal axis at ð0; aÞ: (a) a ¼ 1 (complete mixing case). (b) a ¼ 0:5: (c) a ¼ 0:17:
in which Eq. (10) does not hold, the stability of the coexistence equilibrium depends on the value of a: Let a ¼ ðp qÞ=ð1 pqÞ: If a4a ; the coexistence
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equilibrium is unstable. Fig. 3a illustrates the dynamics of Eq. (6) for this case. The point ðx1 ; x2 Þ would move and converge either to ð1; 1Þ or to ð0; 0Þ; implying that species 1 wins in the system and species 2 disappears, or species 2 wins and drives out species 1, respectively. The competitive outcome depends on the initial condition. In 1.0
0.5
(a)
0
0.5
1.0
contrast, Fig. 3c shows the trajectories of Eq. (1) for aoa ; in which the coexistence equilibrium is stable. Starting from any initial condition, the system converges to ðx ; 1 x Þ; which is globally stable. In between these two cases, we have a ¼ a ; in which the dynamics would converge to a line of equilibria. Fig. 3b illustrates such a case. Fig. 3 suggests that the line of x1 þ x2 ¼ 1 might be a trajectory of the dynamics for all the parameters. In fact, from Eqs. (6a) and (6b), we can show that x1 ðt þ 1Þ þ x2 ðt þ 1Þ ¼ 1 holds if x1 ðtÞ þ x2 ðtÞ ¼ 1: The internal equilibrium ðx ; 1 x Þ with Eq. (7) is on this line. When the internal equilibrium is unstable (a saddle), the line x1 þ x2 ¼ 1 is a separatrix. The system converges to ð1; 1Þ if the initial condition is x1 þ x2 41; while it converges to ð0; 0Þ if the initial condition is x1 þ x2 o1: Biologically, this implies that the species with the greater total abundance will win in the end. These results illustrate that the coexistence of species is more likely to occur when the rate of larval migration between habitats is low than when it is high.
1.0
species 1 in habitat 2 x2
5. Unidirectional migration
0.5
(b)
0
0.5
1.0
1.0
Since larvae are carried away by the water current, the migration of the larvae between habitats may be strongly asymmetric. In an extreme case, the water current is unidirectional during the critical period during which the larval migration is determined. The larvae produced in habitat 1 are transported to habitat 2 and become mixed with the larvae produced in habitat 2—but there is almost no successful migration of larvae from habitat 2 into habitat 1 ðm21 ¼ m; m12 ¼ 0Þ: The dynamics of the abundance of species 1 in two habitats are given by x1 ðt þ 1Þ ¼ ð1 d11 Þx1 þ fd11 x1 þ d21 ð1 x1 Þg b11 x1 ; b11 x1 þ b21 ð1 x1 Þ
0.5
ð11aÞ
x2 ðt þ 1Þ ¼ ð1 d12 Þx2 þ fd12 x2 þ d22 ð1 x2 Þg mb11 x1 þ b12 x2 : mfb11 x1 þ b21 ð1 x1 Þg þ fb12 x2 þ b22 ð1 x2 Þg ð11bÞ (c)
0
0.5 species 1 in habitat 1 x1
1.0
Fig. 3. Trajectories of the dynamics of Eq. (3). The horizontal axis is x1 ; the fraction of species 1 in habitat 1, and the vertical axis is x2 ; that in habitat 2. (a) a ¼ 0:67: The coexistence equilibrium ðxˆ 1 ; xˆ 2 Þ ¼ ð0:47; 0:53Þ is unstable. Two single-species equilibria, ð0; 0Þ and ð1; 1Þ; are stable. The broken line indicates the separatrix that is the boundary between two regions with different competitive outcomes. (b) a ¼ a ¼ 0:5: There is a line of equilibria which is stable. (c) a ¼ 0:25: The coexistence equilibrium ð0:60; 0:40Þ is stable. Other parameters are d ¼ 0:2; p ¼ 0:8; and q ¼ 0:5:
Without loss of generality, we assume b11 =d11 ob21 =d21 ; implying species 1 becomes extinct in habitat 1 (otherwise we can switch the names of the two species). After the convergence of x1 ðtÞ-0; as t-N; we have the equation for species 1 in habitat 2 as follows: x2 ðt þ 1Þ ¼ ð1 d12 Þx2 þ fd12 x2 þ d22 ð1 x2 Þg b12 x2 : mb21 þ fb12 x2 þ b22 ð1 x2 Þg
ð12Þ
ARTICLE IN PRESS S. Muko, Y. Iwasa / Theoretical Population Biology 64 (2003) 359–368
This equation may have a positive equilibrium. If d12 þ d22 b12 =ðmb21 þ b22 Þ40; x2 increases when it is small, and the species 1 can increase and be maintained in the system. The equilibrium fraction of species 1 in habitat 2 is xˆ 2 ¼ 1 m
d12 b21 : d22 b12 d12 b22
ð13Þ
For this to satisfy 0oxˆ 2 o1; we have the following two conditions: b12 b22 4 ; d12 d22 b b b21 : mo 12 22 d12 d22 d22
ð14aÞ
ð14bÞ
According to the calculation in Appendix D, we can prove that the derivative of the right-hand side of Eq. (12) estimated at x2 ¼ xˆ 2 satisfies 0odx2 ðt þ 1Þ= dx2 o1: From these results, we can conclude that when Eq. (14) holds, the origin is unstable and there is the equilibrium ð0; xˆ 2 Þ with Eq. (13), which is locally stable. If instead either one of the two inequalities in Eq. (14) is violated, there is no equilibrium with positive abundance of species 1 in habitat 2, and ð0; 0Þ is locally stable. Eq. (14a) implies that species 1, which is defeated in habitat 1, can win in habitat 2 without migration ðm ¼ 0Þ: Under Eq. (14a) and b11 =d11 ob21 =d21 ; two species can exist if there is no migration. Eq. (14b) indicates that, for the two species to coexist with migration, the migration rate should not be very high when there is larval migration. Hence, we can again conclude that the coexistence becomes more difficult for large larval migration.
6. Discussion The analyses in the present paper suggest that coexistence results when one species is superior to the competitor in one habitat but inferior in the other, and that two species are more likely to coexist when the larvae produced from different habitats are mixed at a smaller rate. We can compare the conditions for the coexistence of two species between the case without migration ðm12 ¼ m21 ¼ 0Þ and the case with migration ðm12 40; m21 40Þ: When habitats are isolated, the competitive outcome in each habitat concludes with the elimination of one species by the other. Coexistence is possible only when two species are completely segregated in different habitats. This situation implies the species coexistence at the regional scale. In contrast, if larvae produced in different habitats are mixed partially or completely, the system shows complex
365
behaviors. If the coexistence equilibrium is stable, the two species are observed in each local habitat. However, if the coexistence equilibrium is unstable while the two single-species equilibria are locally stable, the outcome of the competition depends on the initial abundance. The whole system is dominated by either one of the two species depending on the initial abundance. In symmetric competition and symmetric migration of the two species, the regions where the system is bistable become wider as the migration rate increases while the regions of coexistence become narrower (see Fig. 2). Bistability is a characteristic behavior of the complete mixed lottery model with two heterogeneous habitats, because the corresponding model with a sufficiently low rate of larval migration does not show bistability. Following Muko and Iwasa (2000), the mechanism of bistability can be explained in the symmetric case with two habitats as follows. When the inequality in Eq. (10) is violated, each species has a shorter life in the habitat where it can enjoy greater lifetime reproductive success. In such a situation, when species 1 dominates, it holds that the sites in habitat 2 are more effectively than those in habitat 1, resulting in a low average reproductive success of rare species 2 which cannot do well in habitat 1. Similarly, when species 2 dominates the system, species 1 cannot increase because most of the vacant spaces are created in habitat 2. In either case, the lifetime reproductive success of the resident species is higher than the invader species, and bistability occurs. This condition is met when reproductive rate bsi varies between sites and between species in such a way to reverse the order of longevity (Proposition 5 in Muko and Iwasa, 2000). The bistability is hence caused by larval settlement to an unproductive habitat that is not suitable for the species. This effect disappears when the rate of migration between the two habitats is at a low level. In symmetric competition, each species is adapted to its respective habitat and can do well there even if its initial density is low. Sufficiently low migration can weaken the interspecific competition at the regional scale and thus promote the species coexistence. The effect of dispersal on the condition of species coexistence has been studied intensively. Levin (1974) reported that limited migration between two heterogeneous habitats tends to work to maintain species diversity. In his model, the habitats were initially homogeneous but became heterogeneous because one species happened to dominate a part of habitats and the other species occupied the other. The dispersal of individuals from the occupied area to the unoccupied one prevented the competitive exclusion of one species. The difference in dispersal abilities between the competitors has been emphasized as another mechanism which promotes species coexistence, especially when there is hierarchical competition among species (e.g., Levin, 1974; Shmida and Ellner, 1984; Comins and Noble,
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1985; Nee and May, 1992; Tilman, 1994; Shigesada and Kawasaki, 1997). An inferior competitor can coexist with a superior one if it has a higher colonization rate and can preempt the bare space. However, these theoretical studies focused on the role of transient spatial heterogeneity created by biological activities within homogeneous habitats. There have been relatively fewer studies on permanent spatial variation in habitat quality (e.g., Chesson, 1985, 2000; Roughgarden and Iwasa, 1986; Holt, 1997; Muko and Iwasa, 2000), although its importance has been documented empirically in many different taxa, especially in marine organisms (e.g., Stoner, 1992; Hunt and Scheibling, 1998). Recently, Chesson (2000) examined spatial variation in the environment in a general way by classifying the mechanisms promoting coexistence into spatial storage effect and competitive nonlinearlity. He argued that the limited dispersal between heterogeneous habitats substantially broadens the range of opportunities for coexistence compared with the cases in which there is complete mixing of individuals within the spatiotemporal variable habitats. The present paper is a realistic case employing Chesson’s model. We modeled the lottery competition in a system consisting of spatially heterogeneous but temporally homogeneous habitats, and showed that sufficiently low migration promotes species coexistence. The retention of larvae in the local pool, which has been reported recently by empirical studies, might work to maintain the species diversity in heterogeneous environments. Although many theoretical studies on spatially structured habitats have often assumed symmetric migration, the condition of coexistence may be sensitive to the directionality of migration between habitats. Some theoretical studies have addressed permanent spatial heterogeneity in metapopulation dynamics by including a favorable patch (source population) and an unfavorable one (sink population) connected by migration. The sink population can be maintained by continuous immigration from the favorable source patch (Pulliam, 1988). In the present paper, an asymmetric migration case shows that the species which is adapted to an upstream habitat does not experience competitive exclusion. However, the species, which is adapted to a downstream habitat, but not to an upstream one, may be excluded if the migration is very intense. The effect of a low rate of migration between habitats on the maintenance of species diversity has an important implication. In addition to climatic change, land use change, and environmental pollution, one of the most threatening anthropogenetic processes is the invasion of habitats by exogenous species, which subsequently spread in the new habitat, driving the endemic fauna and flora to extinction (Mooney and Drake, 1989). This is caused by an enhanced rate of migration of organisms
by human activities. As suggested by the simple model studied in the present paper, a larger opportunity of migration between regions and between continents often results in a decline in total biodiversity. This demonstrates the importance of the theoretical study of multiple species systems in heterogeneous environments.
Acknowledgments This work was supported in part by Grants-in-Aid from the Japan Society for the Promotion of Science to S.M. and to Y.I. We greatly appreciate Professor Peter Chesson for his very important comments on a previous version of this paper. We also thank the following people for their useful comments: T. Namba, A. Sasaki, N. Shigesada, and T. Takegaki.
Appendix A The matrix of the linearized dynamics is 8 d21 ð1 m21 Þb11 > > < 1 d11 þ D1 M¼ > d22 m21 b11 > : D2
9 d21 m12 b12 > > = D1 ; d22 ð1 m12 Þb12 > > ; 1 d12 þ D2
ðA:1Þ where D1 ¼ ð1 m21 Þb21 þ m12 b22 and D2 ¼ ð1 m12 Þ b22 þ m21 b21 : From the assumptions, d21 b11 od11 b21 ; d22 b12 od12 b22 ; D1 40; and D2 40; we have TrðMÞ 2 1 ½fðd21 b11 d11 b21 Þð1 m21 Þ d11 m12 b22 g ¼ D1 D2 þ fðd22 b12 d12 b22 Þð1 m12 Þ d12 m21 b21 g o0
ðA:2aÞ
1 TrðMÞ þ DetðMÞ 1 ¼ ½ðd21 b11 d11 b21 Þðd22 b12 d12 b22 Þð1 m21 Þ D1 D2 ð1 m12 Þ ðd21 b11 d11 b21 Þð1 m21 Þd12 m21 b21 ðd22 b12 d12 b22 Þð1 m12 Þd11 m12 b22 þ ðd11 b22 d12 b21 d21 b12 d22 b11 Þm12 m21 40:
ðA:2bÞ
With TrðMÞ40; we can conclude that both eigenvalues are positive and less than 1.
Appendix B We denote the abundance of adults and larvae at a coexistence equilibrium by symbols with a hat (xˆ i and Lˆ s ; s ¼ 1; 2; i ¼ 1; 2). By setting xi ðt þ 1Þ ¼ xi ðtÞ ¼ xˆ i ;
ARTICLE IN PRESS S. Muko, Y. Iwasa / Theoretical Population Biology 64 (2003) 359–368
Eq. (6a) and (6b) are rewritten as ðp qÞxˆ 1 ð1 xˆ 1 Þ axˆ 1 ð1 xˆ 2 Þ þ apqð1 xˆ 1 Þxˆ 2 ¼ 0; ðp qÞxˆ 2 ð1 xˆ 2 Þ axˆ 1 ð1 xˆ 2 Þ þ apqð1 xˆ 1 Þxˆ 2 ¼ 0;
367
1 TrðMÞ þ DetðMÞ ðB:1aÞ
¼
ðB:1bÞ
pd d þ qþa
pd þ
qd aq þ 1
apqd ad
40: q þ a aq þ 1
ðC:2bÞ
If p ¼ q; Eqs. (B.1a) and (B.1b) are the same and are equal to xˆ 1 ð1 xˆ 2 Þ ¼ p2 ð1 xˆ 1 Þxˆ 2 ; which corresponds to a curve on the ðx1 ; x2 Þ-plane. Any point on this curve is the equilibrium of Eq. (6) because it satisfies Eq. (B.1). This gives the ‘‘curve of equilibria’’ of the dynamics. If paq; Eq. (B.2) gives xˆ 1 ¼ xˆ 2 ; or xˆ 1 ¼ 1 xˆ 2 : First ˆ Eqs. (B.1a) and we consider the case with xˆ 1 ¼ xˆ 2 ¼ x: (B.1b) are the same and are rewritten as
If both inequalities hold, both eigenvalues of matrix M are less than 1 in magnitude. Then the equilibrium ð0; 0Þ is stable and species 1 cannot invade the population occupied by species 2. In contrast, if Eq. (C.2a) holds, but Eq. (C.2b) is violated, there is one eigenvalue exceeding 1 and the other is less than 1. If Eq. (C.2b) holds but Eq. (C.2a) is violated, both eigenvalues of matrix M exceed 1. In both of these cases, species 1 succeeds in spreading throughout the system occupied by species 2. After some arithmetic, the stability condition Eq. (C.2b) is rewritten as
ˆ xÞfp ˆ xð1 q að1 pqÞg ¼ 0:
ðp qÞfðp qÞ þ að1 pqÞg40:
respectively. Subtracting Eq. (B.1b) from Eq. (B.1a), we obtain ðp qÞðxˆ 1 xˆ 2 Þð1 xˆ 1 xˆ 2 Þ ¼ 0:
ðB:2Þ
ðB:3Þ
Let a ¼ ðp qÞ=ð1 pqÞ: When a ¼ a ; all the points satisfying are equilibria of the dynamics, Eq. (6). Hence, x1 ¼ x2 is the line of equilibria. If aaa ; the dynamics have only two equilibria: the one with species 1 only ðxˆ 1 ; xˆ 2 Þ ¼ ð1; 1Þ; and another with species 2 only ðxˆ 1 ; xˆ 2 Þ ¼ ð0; 0Þ: In the case of xˆ 1 ¼ 1 xˆ 2 ¼ x ; Eqs. (B.1a) and (B.1b) are the same and are rewritten as ax2 ðp qÞx ð1 x Þ þ apqð1 x Þ2 ¼ 0; which can be solved as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðp qÞ 2apq þ ðp qÞ2 þ 4a2 pq x ¼ : ðB:4Þ 2fp q þ að1 pqÞg x satisfies 0ox o1 for p ð0opo1=dÞ; any q; and a ð0oap1Þ:
Since the relative migration rate a is smaller than 1, a should satisfy pq oao1: 1 pq
The Jacobi matrix for the dynamics linearized around ð0; 0Þ is 2 3 pd apqd 1dþ 6 7 qþa qþa 7; ðC:1Þ M¼6 4 ad qd 5 1 pd þ aq þ 1 aq þ 1 which is a positive matrix. The conditions for the local stability of the matrix are pd TrðMÞ 2 ¼ d þ qþa qd þ pd þ o0; ðC:2aÞ aq þ 1
ðC:4Þ
This inequality is satisfied either for 1oqop or for 1opoq: From Eq. (C.4), the left hand side of Eq. (C.2a) can be written as pð1 pqÞ TrðMÞ 2od ð1 þ pÞ þ qð1 pqÞ þ ðp qÞ qð1 pqÞ þ ; qðp qÞ þ ð1 pqÞ ¼
Appendix C
ðC:3Þ
pq ; 1q
o0;
ðC:5Þ
which is negative for the parameter region which satisfies Eq. (C.4). Therefore, the stability condition (C.2a) holds when the stability condition (C.2b) is satisfied. Thus, the stability of the equilibrium ð0; 0Þ is determined by the sign of the left-hand side of Eq. (C.3). If Eq. (C.3) is violated, one of the eigenvalues is greater than 1, and the species can invade the population occupied by the competitor.
Appendix D By setting x2 ðt þ 1Þ ¼ x2 ðtÞ ¼ xˆ 2 ; the dynamics of x2 given by Eq. (12) in the text becomes d12 ¼ fd12 xˆ 2 þ d22 ð1 xˆ 2 Þgb12 =D; where D¼ mb21 þ b12 xˆ 2 þ b22 ð1 xˆ 2 Þ; at equilibrium.
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The derivative of the right-hand side of Eq. (12) at x2 ¼ xˆ 2 is dx2 ðt þ 1Þ 1 dx2 b xˆ 2 ¼ d12 þ ðd12 d22 Þ 12 þ fd12 xˆ 2 þ d22 ð1 xˆ 2 Þg D b12 xˆ 2 ðb12 b22 Þ 1 ; D D xˆ 2 ðD:1Þ ¼ ðd22 b12 þ d12 b22 Þ; o0 D because b12 =d12 4b22 =d22 : Subtracting the numerator of Eq. (D.1) from the denominator of Eq. (D.1), we obtain D ðd22 b12 þ d12 b22 Þxˆ 2 ¼ b12 xˆ 2 þ ðd22 b12 d12 b22 Þxˆ 2 þ
d22 b12 ð1 xˆ 2 Þ; d12
40 because b12 =d12 4b22 =d22 : Hence1odx2 ðt þ 1Þ=dx2 1 when x2 ¼ xˆ 2 : Therefore the derivative of the right-hand side of Eq. (12) estimated at x2 ¼ xˆ 2 satisfies 0odx2 ðt þ 1Þ= dx2 o1: This implies that x2 ðtÞ converges to xˆ 2 without oscillation.
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