Directed migration can be expected to lead to habitat selection as predicted by the ... (3) We treat migration as part of the environment to which natural selection ...
Evolutionary Ecology, 1992, 6, 360-382
Evolution in heterogeneous environments" effects of migration on habitat specialization J O E L S. B R O W N * and N O E L B. P A V L O V I C Department of Biological Sciences, University of Illinois, Box 4348, Chicago, IL 60680, USA
Summary Richard Levins introduced fitness sets as a tool for investigating evolution within heterogeneous environments. Evolutionary game theory permits a synthesis and generalization of this approach by considering the evolutionary response of organisms to any scale of habitat heterogeneity. As scales of heterogeneity increase from fine to coarse, the evolutionary stable strategy (ESS) switches from a single generalist species to several species that become increasingly specialized on distinct habitats. Depending upon the organisms' ecology, the switch from one to two species may occur at high migration rates (relatively fine-grained environment), or may only occur at very low migration rates (coarse-grained environment). At the ESS, the evolutionary context of a species is the entire landscape, while its ecological context may be a single habitat. Evolution towards the ESS can be represented with adaptive landscapes. In the absence of frequencydependence, shifting from a single strategy ESS to a two strategy ESS poses the problem of evolving across valleys in the adaptive surface to occupy new peaks (hence, Sewell Wright's shifting balance theory). Frequency-dependent processes facilitate evolution across valleys. If a system with a two strategy ESS is constrained to possess a single strategy, the population may actually evolve a strategy that minimizes fitness. Because the population now rests at the bottom of a valley, evolution by natural selection can drive populations to occupy both peaks. Keywords: habitat selection; migration; ESS; fitness set; frequency-dependent selection; evolution of specialization; landscape ecology; adaptive surface
Introduction Evolutionary ecologists consider mutation, drift, natural selection, and migration as the forces of evolution. Migration differs s o m e w h a t from the other forces in two important ways. First, migration per se does not change gene frequencies; gene frequencies m a y change locally as individuals m o v e about, but globally frequencies remain unchanged. Second, migration may be influenced by heritable traits and, hence, subject to the evolution of these traits. In particular, migration may exist along a continuum from completely directed to completely undirected. U n d e r directed migration, the migrant can select a specific spot or habitat as its destination. Directed migration can be expected to lead to habitat selection as predicted by the theory of density-dependent habitat selection (Fretwell and Lucas, 1970; Rosenzweig, 1981). When heritable traits control the rate and direction of migration, natural selection favours a distribution of individuals a m o n g habitats that equalizes the average (ideal free distribution - Fretwell and Lucas, 1970) or marginal (ideal despotic distribution) fitnesses a m o n g habitats. U n d e r undirected migration, the migrant cannot influence its destination. In one scenario, heritable traits control the rate but not the direction of migration. This may lead to the evolution * To whom correspndence should be addressed. 0269-7653
9 1992 Chapman & Hall
Migration and habitat specialization
361
of migration rates as an adaptation for bet-hedging in temporally and spatially variable environments (Venable and Brown, 1988), for avoiding over-crowding (Comins et al., 1980; Levin et al., 1984), or for avoiding sib-sib competition (Hamilton and May, 1977). In a second scenario, heritable traits control neither the rate nor the direction of migration. In this case, the environment and the organism define the migration rate among habitats or populations. This kind of migration is the evolutionary force of migration and migration-selection models of population and quantitative genetics (see Slatkin, 1985). The amount of migration among populations in a heterogeneous environment influences the degree to which populations can adapt to their local circumstances. In this way, natural selection can be viewed as thwarted by migration (Pease et al., 1989). However, when viewed as a property of the environment rather than as a force of evolution, migration becomes part of the circumstances to which evolution by natural selection responds. Towards this latter view, Levins, (1962, 1968) introduced fitness sets as a tool for investigating the evolution of habitat specialization within heterogeneous environments. Levins (1968), in considering adaptations to spatially variable environments, made the distinction between 'finegrained' and 'coarse-grained' spatial scales. Fine-grained refers to a scale of habitat variability such that the organism experiences each habitat in proportion to its occurrence in the environment. Coarse-grained refers to a scale of variability at which the organism may only experience a single habitat (see Morris, 1992). These two scales can also be viewed from the perspective of migration. Under fine grained heterogeneity, movement among habitats is frequent and undirected with respect to habitat; migration is passive and unavoidable. Under coarse-grained heterogeneity, movement among habitats need not occur, and when it does, it is directed; migration is intentional and avoidable. When there is an evolutionary trade-off between an organism's performance in two habitats, a fine-grained environment tends to select for a single generalis~t'species whose strategy represents a compromise between performance in the two habitats, and a coarse-grained environment tends to select for two specialist species that restrict their activities to their preferred habitats (Rosenzweig, 1987; Brown, 1990). Here, we use game theory and evolutionarily stable strategies (ESS, Maynard Smith and Price, 1973) to model the evolution of habitat specialization in response to migration in a heterogeneous environment. We investigate the ESS of an environment containing two habitats that exchange individuals via passive migration. The question of interest - how does this inevitable rate of passive migration influence the number and positions of strategies in the ESS? We achieve the following: (1) We generalize Levins' (1962) concept of fitness sets in variable environments to conditions of density- and frequency-dependent selection, and to conditions that can vary along the environmental continuum from fine- to coarse-grained. (2) We provide insights into the conditions and scale under which spatial heterogeneity promotes or inhibits species diversity. (3) We treat migration as part of the environment to which natural selection responds. (4) We highlight the importance of considering different spatial scales for an organism's ecological and evolutionary context. (5) We compare our results with models of the evolution of habitat specialization under directed migration (Brew, 1982; Rosenzweig, 1987; Matsuda and Namba, 1989; Brown, 1990). The model
Consider an environment subdivided into two habitats A and B. A number of competing species
362
B r o w n and Pavlovic
live in these habitats. The different species are identified by their value for a heritable phenotype or strategy, u, that can take on any value between 0 and 1; 0 ~< u ~< 1. Let u = (ul . . . . ,un) be the vector of strategies among the n different species where ui is the strategy of species i = 1 , . . . , n . Let N A = (NA 1. . . . . NA n) and Na = (NB 1,. 9 .,NB") be the vector of species population sizes in habitats A and B, respectively, where NA i gives the population size of species i in habitat A. Let the fitness of an individual within habitat A or B, F g or Fa, be influenced by its strategy, u, the strategies of others, u, and the densities of each species within habitat A or B, N A or NB, respectively: FA(U,U,NA) and FB(u,u,NB). We assume that individuals in habitat B (or A) do not directly influence the fitness of those in habitat A (or B). Let m A be the p e r capita migration rate of individuals from habitat A to habitat B, and let mB be the p e r capita migration rate of individuals from habitat B to A. These migration rates are passive in that we assume that m A and m B are independent of strategies (=species), u, and species population sizes, N A and NB. Let p be the probability that an individual with strategy u finds itself in habitat A; p is influenced by u, u, NA, and Na. We assume that species are identical in all respects save for the values of their strategies. This assumption allows us to use a fitness generating function (so called because the expected per capita growth rate of an individual using strategy ui can be found by replacing u in G(u,u,Ng,Na) with ui; see Vincent and Brown, 1988) to characterize the fitness of all individuals regardless of species. The fitness generating function is given by: G(u,U,NA,N8) = p ( u , U , N A , N B ) F A ( U , U , N A , N 8 ) + [1 -- p(u,U,NA,N8)]FB(u,U,NA,N8)
(1)
and the changes in the population size of the ith species within the two habitats are given by (these equations follow Holt, 1985): dNAi/dt = NAiFA(Ui,U,NA) -- mANA i + m a N a i
(2a)
dNBi/dt = NBiFB(Ui,U,NB) -- mBNB i -I- mANA i
(2b)
We assume that the population dynamics given by Equations 2a and 2b lead to stable equilibria, NA* and NB*, at which one or more species have non-zero equilibrium population sizes. Before using the above three equations to characterize the ESS's of this model, we discuss the fitness set defined by FA(U,U,NA,NB) and FB(u,U,NA,Na) , and the distribution of individuals among habitats given by p ( u , u , N g , N a ) . The fitness set
Holding values for the species' strategies, u, and species' abundances in habitats A and B, N A and NB, fixed, the fitness set gives the combinations of FA and Fa that result from all of the evolutionarily feasible values for u. Suppose that u represents a trade-off between fitness in the two habitats by assuming that FA increases with u and FB declines with u (OFA/OU > 0 and OFa/Ou < 0), and suppose that the fitness set is convex (to the origin): there are diminishing returns to fitness in habitat A from increasing u, and diminishing returns to fitness in habitat B from decreasing u (02FA/OU2< 0 and OZFB/OU2 < 0). We assume that individuals compete with each other for limiting resources within each habitat; so, fitness within a habitat declines with an increase of individuals within that habitat: OFA/ONA ~ < 0 and OFB/ONB ~ < 0 for all i = 1 , . . . , n . These above assumptions define the fitness set in the state space of FA and FB (Fig. 1). The strategy u = 0 gives the upper-most point of the fitness set (minimum value for FA and maximum value for FB). As u increases from 0 to 1, one moves down the fitness set to the lower-most point given by u = 1 (maximum value for FA and minimum value for FB). Increasing values for population sizes in habitat A or habitat B causes the fitness set to shift downwards or leftwards,
Migration and habitat specialization
0.015
363 I
I
l
a
0.005 Z m
-0.005 1.1.1 Z Ira U.
C
r
-0.01 5
-0.025 -0.025
I
I
-0.015
-0.005
0.005
0.015
FITNESS IN A Figure 1. The shape and behaviour of density- and frequency-dependent fitness sets. The fitness set gives all combinations of fitness in habitat A and fitness in habitat B as determined by all evolutionarily feasible values of the strategy, u. By assumption, moving down the fitness set represents ever higher values for u, and, by assumption, the fitness set always remains convex. Moving from fitness set 'a' to 'b' represents the effect of increasing total population size while holding the frequency of different strategies constant. Moving from fitness set 'a' to 'c' represents the effect of increasing the frequency of habitat B specialists (the fitness set rotates in a counter-clockwise direction). Increasing the frequency of habitat A specialists would cause a clockwise rotation of the fitness set.
respectively. Increasing the frequency of habitat A or habitat B specialists causes the fitness set to rotate in a clockwise or counter-clockwise direction, respectively (Fig. 1).
Distribution of individuals among habitats From Equation 1, the fitness of an individual using strategy u in a biotic environment defined by strategies u, and population sizes N A and NB is influenced by the expected exposure of strategy u to each of the habitats, p(u,U,NA,NB). The fitness of the strategy in each habitat, FA(U,U,NA ) and Fa(u,u,Na), and the migration rates among habitats, mA and ma, determine the strategy's exposure to each habitat. For a fixed biotic environment of u, NA, NB, an arbitrary strategy u will converge on an exposure to habitats such that the strategy has the same per capita growth rate in each habitat (see Holt, 1985). (If the strategy-u has a higher per capita growth rate in habitat A
Brown and Pavlovic
364
than B then its relative frequency in habitat A will increase, and vice-versa if the strategy has a higher per capita growth rate in habitat B.) We rearrange Equations 2a and 2b and substitute q for N A i / N B i t o obtain:
( 1 . 1 (dNAil-\NA'/\/dt
= FA(Ui,U,NA) -- mA + mB/q
(3a)
We equilibrate the per capita growth rates by setting Equation 3a equal to Equation 3b. The resulting expression gives the following quadratic equation for q: m A q 2 + [Fa(u,U,NB)
-
FA(U,U,NA)
+
mA
--
mB]q - mB = 0
with solution: (FA q =
--
FB + mB -- mA) + N/[(FB -- FA + mA -- mB)z + 4mAmB] 2mA
(4)
Equation 4 determines the frequency, p, with which a strategy u will eventually experience habitats A and B for any fixed biotic environment:
p(u,u,NA,NB) = q/(q+ 1)
(5)
In determining how FA, FB, mA, and mB influence p it will be useful to subdivide migration rates into three components. The first two components, fA and fB, determine the relative rates of migration between habitats. The third component, m, represents the overall magnitude of migration, independent of relative rates. We incorporate these components by letting mA = fAro and mB = fBm. These relationships can be substituted into Equation 4. We will determine the properties ofp indirectly by analysing how FA, fA, and m influence q. Because p is monotonically related to q, p will share all of the salient properties of q. First consider how q depends on FA. From Equation 4: Oq
=
~/[(FB
-- F A + m A --
ma) 2 +
4mAmB]
-- ( F B - - F A + m A -- rnB) >
0
(6)
aFA Increasing fitness in habitat A increases a strategy's relative exposure to habitat A, OplOFA> O. A similar analysis shows that increasing fitness in habitat B decreases a strategy's exposure to habitat A, Op/OFB < 0. When FA = FB, it can be shown that p = mB/(mA + mB) which is the expected distribution of individuals across habitats in response to the balance of passive migration rates. When FA > FB, p > mB/(mA + roB) and the distribution of individuals with strategy u becomes biased towards habitat A. By assumption, increasing u increases FA, lowers FB, and, hence, increases exposure to habitat A, i.e. cgp/Ou> 0. A change in strategy directly changes fitnesses in each habitat and indirectly changes relative exposure to the different habitats. Even under passive migration, the strategy, u, gives the organism evolutionary control over habitat selection. To determine the effect of the overall intensity of migration on a strategy's exposure to the two habitats, we differentiate q with respect to m (this changes both mA and mB without changing relative migration rates between habitats). We find that Oq/Om has the same sign as (FB - Fa):
Migration and habitat specialization
365
f(A)
j/
f(B)
P
F(A) > F(B) U
" ~ m
Figure 2. A summary of the effects of changing migration rate, m, the evolutionary strategy, u, the relative rates of migration from habitat A to B, f(A), and the relative rate of migration from habitat B to A, f(B), on the proportion of a strategy's exposure to habitat A, p. A '+' and '-' indicate positive and negative effects, respectively. The effect of migration rate on p depends upon which habitat offers the higher fitness; i.e. F(A) > F(B) or F(B) > F(A).
when FA > F8 then aplam < 0, and when FB > FA then aplam > 0. As m goes to infinity, the distribution among habitats of individuals with strategy u approaches that expected from a balance of the passive migration rates; p approaches fB/(fA + fB). Increasing overall migration rates diminishes the bias of a strategy's exposure towards the habitat with higher fitness. T o determine the effect of migration from a habitat on a strategy's exposure to that habitat, we differentiate q with respect to fB (this changes both mB and relative migration rates between habitats). From this derivative, we can show that increasing migration into habitat A from B increases q, aq/afB > 0 (Fig. 2).
Characterizing the ESS Strategy u~ will be the ESS for species 1 if the fitness generating function, given by Equation 1, takes on a global maximum at u = ui when ul is c o m m o n (i.e. NA i = Ns i = 0 for i > 1) and species 1 is at its equilibrium population size of NA 1. and NB 1. (Vincent and Brown, 1984). At the ESS, any individual will suffer a loss in expected fitness by unilaterally changing its strategy from u~. T h e first and second order necessary conditions for maximizing G with respect to u at u~ are given by OG(Ul,UI,NAI*,NBI*)/au = 0 and 02G(Ul,Ul,NA 1., NBI*)/Ou2 < O, respectively. U n d e r our assumption of equilibrial population dynamics, ecological stability requires that Equations 2a and 2b equal zero when evaluated at Ul, NA 1. and Na ~*.
366
Brown and Pavlovic
From Equation 1:
op OFB Op FA + ova Ou p -Ou FB + ~Ou [1 - p ] Ou
OG Ou
02p
02G
0//2
-
0/1 2
Op Of A ( F A -- F a ) + 2
(7)
Of B
0// 03//
Oq//
02FB
02FA + p
Oq//2
+ (I-p)
- -
0//2
(8)
Setting Equation 7 equal to zero and rearranging yields:
(aFa/au) (aFAlau)
--
p
(1 - p)
+
(aplOu)(Fa -- FB)
(9)
(1 - p)(aFAlau)
where the functions FA, FB, p and their derivatives are evaluated at u = ul, u = u 1, N A = N A 1., and NB 1.. The left side of Equation 9 is the magnitude of the slope of the fitness set. The right side has two components. The first is that of Levins' (1968) analysis where p and ( l - p ) would be the frequencies of the two habitats. The second component emerges from the effect of a species' strategy on its exposure to the two habitats. In the absence of the second term, Equation 9 reduces to Levins' solution for spatial variability. T h e value for ul will be higher (or lower) than Levins' value when F A ~" FB (or Fa > F A ) . Increased specialization on a habitat results in greater selection for that habitat and the species evolves even greater specialization on the habitat in which it has the higher fitness. T o see this consider the second term on the right in Equation 9. Because Op/Ou > 0, (1 - p) > 0, and OFA/ Ou > 0, the sign of this term is determined by (FA -- FB). If FA > FB then the right side of Equation 9 is more positive than it otherwise would be in the absence of this second term. Hence, the magnitude of the slope of the fitness set at ul is greater, and ul shifts down the fitness set to greater specialization on habitat A. Conversely, if Fa > FA then Equation 9 is smaller than it would be in the absence of the second term. The magnitude of the slope of the fitness set at u~ is less than otherwise, and uj shifts up the fitness set towards greater specialization on habitat B. Consider the extreme cases of m = 0 (no passive migration) and m -* co (complete mixing of populations between habitats). As m ~ ~ , p = fB/OrA-k-fB), Op/Ou = 0, and OZp/OU2= O. Substituting these into Equation 9 shows that ul is the point where the fitness set's slope is p/ (1 - p ) = fa/fA, and substituting the equalities into Equation 8 shows that the second order necessary condition, 02G/Ou2 < 0, is satisfied. The single species with strategy u~ appears to be an ESS. The case of rn ~ oo corresponds to a fine-grained environment, and the resulting ESS has a single species with a strategy that represents a compromise between the conflicting selective pressures of the two habitats (Levins, 1968). As m ~ 0, F A = FB = 0 (this can be shown by considering Equations 3a and 3b when m = 0). Substituting this into Equation 9 shows that u~ is the point where the slope of the fitness set equals p/(1 - p). Equation 9 can be rearranged to solve for p = - (OFB/Ou)/[OFA/Ou -- OFB/aU] and p can be differentiated to obtain an expression for Op/Ou. These expressions for FA = FB, p, and Op/Ou can be substituted into Equation 8 to show that the second order necessary conditions for u~ to be an ESS are not met, 02G/Ou2 > 0. When m = 0, which correspohds to a coarse-grained environment, a single strategy is not the ESS. Rather, the ESS at m = 0, contains two species with strategies u~ = 1 and u2 = 0. Species 1 is the extreme specialist on habitat A. Species 1 cannot be invaded in habitat A by any strategy u < 1. Species 2 is the extreme specialist on habitat B. Species 2 cannot be invaded in habitat B by any strategy u > 0.
Migration and habitat specialization
367
Logistic growth The logistic growth model for a single species and the Lotka-Volterra competition model for multi-species systems have been used as evolutionary models by considering heritable traits that influence parameters such as carrying capacity and the competition coefficients (Lawlor and Maynard Smith, 1976; Roughgarden, 1976; Slatkin, 1980; Taper and Case, 1985, 1992; Brown and Vincent, 1987). Here, we assume that a species growth rate within a habitat is logistic, and we assume that an individual's strategy, u, influences its carrying capacity, K(u). With these assumptions, the fitness of an individual using strategy u in habitats A and B, respectively can be written as:
Ka(u) -
2 NAj ]=1
FA(U,u,NA) = rA
(
)
(lOa)
)
(10b)
KA(U)
KB(U) -FB(U,U,NB) = ra
Z NBj j=l
( KB(U)
These expressions for FA and FB can be substituted into Equations 1, 2a, and 2b to define an evolutionary game. To solve for the ESSs of this model, we use the following functional forms for Kg(u) and Ka(u), and the following parameter values: KA(U) = 100u KB(U) = 5 0 ( l - u ) rA = rB = 0.025 0~ FB. Habitat A supports a source population and habitat B holds a sink population (c.f. Pulliam, 1988). Because habitat A offered higher fitness than habitat B, the theory predicts that the ESS
Brown and Pavlovic
368
LOGISTIC 1~
1.0
O0
r.JO 14/
0.9
-
0.8
-
0.7
-
0.6
-
0.5
m
0.4 0.3 0.2 0.1
m
0.0 -0.1 0.001
I
I
I
0.010
0.1 O0
1.000
MIGRATION
RATE
(M)
Figure 3. The effect of migration rate (m) on the ESS of the logistic model. Below a migration rate of 0.012 the ESS contains two strategies that represent the extreme habitat specialists. Above rn = 0.012, the ESS contains a single strategy. Increasing m results in a decline in the value of this ESS. The relative rates of migration from habitat A to B and from B to A were set to unity; i.e. fA = fa = 1.
should be shifted towards a greater degree of specialization on habitat A than Levins' (1968) solution (u = 0.42). With increasing migration rates the ESS converges on Levins' solution. In comparison to Levins' solution, individuals using the ESS benefit twice: first, their higher valued strategy increases their exposure to the source habitat A, and their greater p e r f o r m a n c e in habitat A allows them to capitalize on this increased exposure. As m increases, the ESS declined from ul = 0.68 to ul = 0.42 (Fig. 3). In this example, the relative rates of migration between habitats influenced the ESS. For illustration, we examined the case where m = 0.2. We then varied fB from 0 to 1 with fA = 1 -fB. Increasing f~ increases the migration rate from habitat B to A, and decreases the migration rate from habitat A to B. Regardless of the value for fB, the ESS always contained a single strategy. As fB increased from uni-directional migration from A to B (fB = 0) to uni-directional migration from habitat B to A (fB = 1), the ESS increased from the extreme habitat-B specialist (u~ = 0) to the e x t r e m e habitat-A specialist (u~ = 1). T h e ESS tended to specialize on the habitat with high immigration rates and low emigration rates (Fig. 4).
Migration and habitat specialization.
369
LOGISTIC 1,0
"
0.9 0.8 0.7
=
0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0
!
I
I
I
I
0.2
0.4
0.6
0.8
1.0
PROPORTION OF MIGRATION INTO A Figure 4. The effect of changing relative migration rates on the ESS of the logistic model. The overall migration rate was set to m = 0.2 which results in a single strategy ESS. The x-axis represents an increase in the relative migration rate from habitat B to A, fB, and a decline in the relative migration rate from A to B by constrainingfA: fA = 1 -- )ca. IncreasingfB causes the value of the ESS to increase. Natural selection favours a strategy that specializes on the habitat with high immigration and low emigration rates.
T h e result of specializing on the habitat with high immigration rates makes sense from the perspective of evolving aptitude on the habitat most likely encountered by an individual or its offspring, but the result seems to leave an unfilled opportunity for individuals that specialize on the habitat with high emigration rates. For our specific example, this is not the case. While a species specializing on the habitat with the high emigration rate will indeed have a very high fitness in that habitat, this high fitness (which cannot exceed the intrinsic growth rate of r = 0.025) does not exceed or c o m p e n s a t e for the loss of individuals through emigration. Interestingly, regardless of the relative migration rates between habitats, habitat A always remained the source habitat: FA >~ 0 >t F B.
Consumer-Resource model Models of consumer-resource interactions (Tilman, 1982) have provided evolutionary models by
370
B r o w n and Pavlovic
considering heritable traits that influence parameters such as resource uptake rates, feeding costs, and growth rates (Abrams, 1986, 1987, 1990). Here, we assume that individuals within each habitat compete exploitatively for resources. We let the strategy of the individual influence its functional response on resources in habitat A and B, bA(U,RA) and b B ( u , R a ) where RA and RB are resource abundances in habitat A and B. With these assumptions, the fitness of an individual using strategy u in habitats A and B, respectively, can be written as: FA(U,U,NA) = H A ( b A [ U , R A ( U , N A ) ] -- CA)
(12a)
F a ( u , u , N a ) = H a ( b a [ u , R B ( u , N B ) ] - ca)
(12b)
where HA and HB give fitness as a function of the net rate of resource consumption, bA and ba are the rates of resource consumption as a function of the individual's strategy (u) and the abundance of resources (RA and RB), and CA and CB give the cost of maintaining the individual in units of resources. Relative to changes in consumer population sizes, we assume that resource abundances within a habitat equilibrate quickly in response to resource harvest. And so, we can write the resource abundances of each habitat as a function of the strategies and numbers of individuals within the habitat: RA(U,NA) and Ra(u,Na). Equations 12a and 12b when substituted into Equations 1, 2a and 2b define an evolutionary game. T o solve for the ESSs of this model, we considered the following simple functional forms and p a r a m e t e r values: HA{bA[U,RA(U,NA)] -- CA} = bA[U,RA(U,NA)] -- CA
(13a)
H B { b B [ u , R B ( u , N a ) ] - cB} = bB[U,Rs(u,NB) ] -- cB
(13b)
bA(U,RA) = hA(U)RA
(13C)
hA(U) = U/1000
(13d)
bB(U,RB) = h a ( u ) g a
(13e)
ha(u) = (0.000001 - u2) ~
(13f)
n
RA(U,NA) = 100/ X h A ( u i ) N g i
(13g)
i=1
RB(U,NB) = CA = CB = 1
50/ Z hB(Ui)NB i i=1
(13h) (13i)
Equations 13a and b assume that fitness increases linearly with net profit from resource harvest. Equations 13c-f define the consumer's functional response in each habitat as a function of its strategy. In this example, the consumer has a linear or Type I functional response. Equations 13g and h describe the equilibrium abundance of resource that would result in each habitat if resource renewal rates were 100 and 50 in habitats A and B, respectively. In this example, habitat A exceeds habitat B in productivity. For convenience, the maintenance costs described by Equation 13i have been scaled to one. Because the environment-wide resource renewal rate equals 150 and because maintenance costs equal 1, the environment at equilibrium can support a total of 150 consumer individuals. If there was no migration, this example would lead to 100 individuals in habitat A and 50 individuals in habitat B. T h e strategy u generates a trade-off between a consumer's functional response in habitat A and its functional response in habitat B. Based on Equations 13d and 13f, the set of points
Migration and habitat specialization
371
CONSUMER 1.1
-
RESOURCE
m
1.0 0.9 0.8 0.7
0.8
U)
0.5
ILl 0.4 0.3 0.2 0.1 0.0 -0.1
0.0
I
I
I
I
!
0.2
0.4
0.6
0.8
1.0
M I G R A T I O N RATE Figure 5. The effect of migration rate (m) on the ESS of the consumer-resource model. Below a migration rate of 0.95 the ESS contains two strategies. As the ESS declines below this level, the two strategies become increasingly specialized on the two habitats, respectively. Above m = 0.95, the ESS contains a single strategy, that, in this example, is somewhat specialized on habitat A. The relative rates of migration from habitat A to B and from B to A were set to unity; i.e. fA = fB = 1.
describing all combinations for the consumer's functional responses is convex. The relationship between the consumer's harvest rate and Equations 13d and 13f are linear, and the relationship between the c o n s u m e r ' s fitness and harvest rate is linear. Because linear transformations of convex functions remain convex, the fitness set relating all evolutionarily feasible combinations of fitnesses in habitats A and B is also convex. Solving numerically for the ESSs, we investigated the effect of the migration rate, m, and the effect of the relative migration rates between habitats, fA and fB, on the ESS. When fA = fB = 1, the switch in the diversity of the ESS from two to one strategy occurs at m = 0.95. A t values of m close to zero the ESS, as expected, contains two strategies ul = 1 and u2 = 0. As migration increases from rn = 0 to m = 0.95, the ESS of species 1 declines steadily from ul = 1 to ul = 0.85, and the ESS of species 2 increases from u2 = 0 to u2 = 0.53. For m > 0.95, species 2 no longer remains in the ESS, and the strategy of species 1 continues to decline and converge on
Brown and Pavlovic
372
CONSUMER - RESOURCE Z
_o
1.0
m
0.9 "~
0.8
0-
0.7
~,~....~....,..,.. ul ~ e l e m~~ 9 9 gwwwl ~ e I w l m m e ~ g q m l e ~
am
%
0.6
~
q w l l IR
0.5
m
m
0.95, the single strategy ESS remained unchanged at the value of Levins' solution. The migration rates at which the ESS switches from one to two species differed greatly between the Logistic and Consumer-Resource models. Hence, the ecology embedded within the model of habitat-specific population dynamics may be more important than the passive migration rate in determining the diversity of strategies within the ESS. For determining the likelihood of sympatric speciation (Feder et al., 1988; Rice and Salt, 1990), an understanding of the ecological interactions determining habitat-specific fitnesses may be as important as measurements of gene flow.
Evolutionary dynamics along adaptive landscapes Adaptive landscapes plot the fitness generating function as a function of strategy, and can be used to illustrate the ESS and the selective pressures when strategy frequencies are away from the ESS. When the ESS has a single strategy, the adaptive landscape, at the ESS, has a single peak. Despite the single peak, it is often possible for two species with different strategies to coexist in the community, so long as their strategies are on opposite sides of the ESS. While the coexistence of the two species may be ecologically stable, the two species' strategies are not evolutionarily stable. Each species is under directional selection to converge on a common strategy (Fig. 10a). If each species' strategy evolves in response to this directional selection then each will, in time, achieve the single strategy of the ESS (Fig. 10b); under the assumptions of the model, the two species would no longer be distinguishable. When the ESS contains two strategies, the adaptive landscape, at the ESS, contains two peaks which maximize fitness (Fig. lla). How are peaks crossed in such a landscape, and what is the consequence of constraining all individuals to share the same strategy? If one begins with a single species where all individuals must share a common strategy, the adaptive landscape no longer has two peaks. In fact, the single species will experience directional selection. If the species' strategy responds to this directional selection it can actually evolve a strategy that sits at the bottom of a valley of the adaptive landscape (Fig. 1 lb). If a species, at this point of minimum fitness, responds by evolving a slightly higher strategy then the adaptive landscape changes form drastically and the species resumes experiencing directional selection for a lower strategy value (Fig. lld). The reverse occurs if the species responds to the point of minimum fitness by evolving a slightly lower strategy value (Fig. llc). By evolving to the strategy that minimizes fitness, the species experiences strong disruptive selection to diversify. Should a diversity of strategies emerge either from speciation (competitive speciation provides such a mechanism; Rosenzweig, 1978) or a species invasion, then disruptive selection can drive the strategies of the two species towards their respective peaks via directional selection (Figs l l e and llf). When adaptive landscapes are primarily density-dependent, they remain rigid in response to changes in strategy frequency or number of strategies. Crossing valleys between peaks becomes an evolutionary dilemma for which Wright (1977) proposed the shifting balance theory of evolution. When adaptive landscapes are frequency dependent, crossing valleys may often not be a problem for at least three reasons: (1) as in the present example, a strategy may actually evolve to a point where it occupies the bottom of the valley, (2) as evolution proceeds, the valley may cross under the strategy rather than vice-versa (T.L. Vincent and colleagues, in preparation), and (3) valleys may not exist as such until they are crossed by the appropriate strategy or strategies (Rosenzweig, 1978).
Brown and Pavlovic
378 0.01 0.04
m = 0.6 -0.00
0.03 -0.01 to itl Z
m = 0.6 0,02
-0.02
ul
= 0.855
u. 0.01
-0.03 -0.04
0.00 I
I
I
!
I
20
40
50
80
100
-0.05
-0.01 0
(a) 0.02
I
I
I
50
80
100
0.20
0.01
m = 0.6
0.15 0.00 u) u) uJ Z I,m u.
I
40
(b)
/ /
m = 0.6
I
20
ul
= 0.9
0.10 -0.01 0.05 -0.02 0.00
-0.03 f
-0.04 0
I
I
I
I
20
40
60
80
I 100
-0.05 0
I 20
I 40
I 60
80
100
(c)
(d)
0.04
0.02 m
0.03
m
0.02
u.
0.01
co w
m = 0.6
0.01
ul
= 0.8
0.00
u2
= 0.9
-0.01
u2 = 0 . 9 2
-0.02 0.00
= 0.6
\
I
-0.03
-0.01 0
I 20
I 40
I 60
STRATEGY
I 80
I 100
-0.04 0
(e)
I 20
I 40
I 60
STRATEGY
I 80
I 100
(f)
Figure 11. For the consumer-resource model, some configurations of the adaptive landscape when the ESS contains two strategies, ul = 0.2 and u2 = 0.98 (m = 0.6). Landscape (a) is the configuration at the ESS; the landscape takes on global maxima at the two ESS strategies. Landscape (b) shows where the system will evolve if only a single strategy is permitted. The resulting strategy of u] = 0.855 resides at a minimum of the adaptive landscape. If the single strategy evolves to a slightly smaller value (ul = 0.8), the
Migration and habitat specialization
379
Discussion
We have characterized the ESSs that emerge from a class of models that consider passive migration between two habitats, and an evolutionary strategy that represents a trade-off between habitat-specific fitnesses. The migration rate was a fixed property of the organism's environment and it was independent of the evolutionary strategy. The migration rate influenced both the values and the diversity of strategies in the ESS. The ESSs have implications for a number of theories on the evolution of habitat specialization in heterogeneous environments.
Levins' fitness sets in spatially heterogeneous environments Our migration term corresponds to the continuum from coarse- (migration rate of zero) to finegrained (very large migration rate) spatial variability in Levins' (1962, 1968) theory of fitness sets. Like Levins' theory, a low migration rate encourages the evolution of two specialist species on the two habitats, respectively. A high migration rate encourages the evolution of a single generalist species. Our model differs from Levins' theory in that the species' strategy influences its habitat selection; this occurs despite the assumption that the strategy does not influence migration rates. The trade-off between habitat-specific fitnesses induces positive evolutionary feedback. If a species evolves higher fitness in habitat A, then its fitness in habitat B declines, and its relative exposure to habitat A increases. This increased habitat selection for A selects for greater fitness in A. The positive evolutionary feedback provides an additional source of disruptive selection that does not occur in Levins' theory. As a result, the ESS may still contain two specialist species (instead of a single generalist species) even under high migration rates.
Density-dependent habitat selection Our migration term has similarities to the costliness of density-dependent habitat selection in models that allow the organism to choose its habitats (Rosenzweig, 1981). A low migration rate corresponds to cost-free habitat selection in which an organism can select for one habitat without the costs of searching for the one habitat or the costs of avoiding other habitats (see Morris, 1992). Under a very low migration rate, selectivity by an organism for a subset of habitats occurs passively; while under density-dependent habitat selection, it occurs through active choice. In either case, the end result is the same; the ESS tends to contain specialist species behaving selectively (Brew, 1982; Rosenzweig, 1987; Matsuda and Namba, 1989; Brown, 1990). Conversely, under high migration rates or under costly habitat selection, the ESS contains a generalist species behaving opportunistically. Varying the relative migration rates between habitats in our model is analogous to varying the frequency of habitats or varying the orientation and shape of habitat patches in models of density-dependent habitat selection (Morris, 1992). In this case, the ESS may contain a specialist species behaving selectively and a generalist species behaving opportunistically. Of course, in our model, all habitat selection behaviour results from passive choice; the balance between habitat-specific growth rates and migration rates determines an organism's habitat selection.
landscape changes abruptly to one of strong directional selection for a larger value of u (landscape c). If thc single strategy evolves to a slightly larger value (ut = 0.9), the landscape changes abruptly to one with strong directional selection for a smaller value of u (landscape d). If both strategies of ul = 0.8 and t~ = 0.9 are allowed to evolve then they occupy different sides of an evolutionary valley (landscape e) and both are under directional selection to diverge. The partial results of such an evolutionary divergence are shown in landscape (f), Continued divergence results in the ESS of landscape (a).
380
Brown and Pavlovic
Some of the ESSs of our model differed from those expected under density-dependent habitat selection. When habitat selection results from active choice, behaviour tends to be either selective or opportunistic. In response to selective behaviour, natural selection favours the appropriate extreme specialist strategy. When the ESS contains two selective species, both are extreme specialists (Rosenzweig, 1987), and if the ESS has two species at least one is an extreme specialist (Brown, 1990). This is not so for passive habitat selection, over some range of migration rates the ESS may contain two relatively generalist species that overlap broadly in their patterns of habitat use. This may be particularly appropriate for plants that have varying rates of passive migration. Habitat heterogeneity within a field may still promote coexistence and diversification of plant species even though the species appear to overlap considerably in space, and even though individual plants may seem to owe their existence more to chance events rather than habitatspecific events. Gene flow: is migration a force of evolution? The migration rate of our model closely resembles gene flow in population and quantitative genetic models that consider migration as a force of evolution (see Slatkin, 1985 and references therein). In such models, a balance between the force of selection operating at a smaller spatial scale and the force of migration operating at a larger spatial scale determines the resultant distribution of strategies or gene frequencies. Pease et al. (1989) using quantitative genetics modelled how the value of a trait changed spatially. Because of migration, the trait, at any point along the spatial cline, did not possess the value which would maximize fitness at that point. Natural selection may be seen as victimized by migration. Such a viewpoint emerges when selection is modelled at the smaller spatial scale of a single subpopulation and migration is modelled at the larger spatial scale of numerous weakly-coupled subpopulations. Our model presents a different view of natural selection and migration as evolutionary forces in spatially variable environments. Like genetical models of selection and migration, we considered two scales of processes. At the smaller scale (within a habitat) only other individuals within a habitat influenced the success of a given individual. At the larger scale (between habitats) individuals within the other habitat were influential insofar as they or their offspring may at a later time migrate to the other habitat. In our model an extreme strategy always would maximize the success of an individual within a pre-selected habitat. But, the ESS (the strategy or strategies promoted by natural selection) often contained a single or pair of strategies that deviated considerably from either extreme strategy. While the ecological context of the individual might be the habitat within which it resides, the evolutionary context of the individual includes all of the habitats experienced by its ancestors (Holt and Gaines, 1992). By elevating natural selection to the larger spatial scale, the resulting ESSs were context-dependent (context dependencies in genetical models are often referred to as 'group selection'). The context being determined by the different habitats and their associated migration rates. We were able to consider natural selection within the larger context by calculating the expected exposure of a strategy to the two habitat types independent of the present habitat of a strategy or individual; this was p in Equation 5. Migration was not a force of evolution in our model, rather it was part of the context to which natural selection should respond. Taking the approach of this paper in viewing the consequences of migration may lead to subtle but important changes in perspective. Migration may prevent the local specialization of subpopulations to their local peculiarities, migration may maintain the integrity of species over broad geographic distributions, and migration barriers may be a requisite to some kinds of speciation events. But, these outcomes may not result from migration per se; they may be the
Migration and habitat specialization
381
adaptive consequences of natural selection responding to a particular migration regime. In addition to migration, the nature of the fitness set and the ecological interactions within and between habitats are co-equals in defining the context to which natural selection may then respond with greater or lesser degrees of specialization, or greater or lesser species diversity (witness the very different effects of migration on the ESSs that emerged from the examples based on Logistic equations and Consumer-Resource equations). The integrity of species with large and variable ranges may not so much prevent local adaptation but be the result of adaptations to a larger spatial scale than the predominate ecological interactions. For this reason, it perhaps should not surprise ecologists to discover that their species is nowhere perfectly adapted. Furthermore, many important patterns of community structure may be obscured, especially when the ecological context of species coexistence is much smaller than the evolutionary context of the species' adaptations (also see Holt and Gaines, 1992). Many ecological communities may be 'species takers rather than species makers' (Kotler and Brown, 1988).
Acknowledgements We thank Brent Danielson, Bill Mitchell, Doug Morris, and Tom Poulson for their interest and discussions throughout the development of these ideas. Marc Mangel and an anonymous reviewer provided valuable comments. This work was supported by National Science Foundation grant No. BSR 9106785.
References Abrams, P. A. (1986) Character displacement and niche shift analyzed using consumer-resource models of competition. Theor. Pop. Biol. 29, 107-60. Abrams, P. A. (1987) Alternate models of character displacement. I. Displacement when there is competition for nutritionally essential resources. Evolution 41,651-61. Abrams, P. A. (1990) Mixed responses to resource densities and their implications for character displacement. Evol. Ecol. 4, 93-102. Brew, J. S. (1982) Niche shift and the minimization of competition. Theor. Pop. Biol. 22, 367-81. Brown, J. S. (1990) Habitat selection as an evolutionary game. Evolution 44, 732-46. Brown, J. S. and Vincent, T. L. (1987) Coevolution as an evolutionary game. Evolution 41, 66--79. Comins, H. N., Hamilton, W. D. and May, R. (1980). Evolutionarily stable dispersal strategies. J. Theor. Biol. 82, 205-230. Feder, J. L., Chilcote, C. A. and Bush, G. L. (1988) Genetic differentiation and sympatric host races of the apple maggot fly, Rhagoletis pomonella. Nature (London) 336, 61-4. Fretwell, S. D. and Lucas Jr., H. L. (1970) On territorial behavior and other factors influencing habitat distribution in birds. I. Theoretical development. Acta Biotheor. 19, 16-36. Hamilton, W. D. and May, R. (1977) Dispersal in stable environments. Nature (London) 269, 578-81. Holt, R. D. (1985) Population dynamics in two-patch environments: Some anomalous consequences of an optimal habitat distribution. Theor. Pop. Biol. 28, 181-208. Holt, R. D. and Gaines, M. S. (1992) Analysis of adaptation in heterogeneous landscapes: implications for the evolution of fundamental niches. Evol. Ecol. 6, 433-47. Kotler, B. P. and Brown, J. S. (1988) Environmental heterogeneity and the coexistence of desert rodents. Ann. Rev. Ecol. Syst. 19, 281-307. Lawlor, L. R. and Maynard Smith, J. (1976) The coevolution and stability of competing species. Am. Nat. 110, 79-99. Levin, S., Cohen, D. and Hastings, A. (1984) Dispersal strategies in patchy environments. Theor. Pop. Biol. 26, 165-91.
382
B r o w n and Pavlovic
Levins, R. (1962) Theory of fitnesses in a heterogeneous environment, I. The fitness set and adaptive function. Am. Nat. 96, 361-78. Levins, R. (1968) Evolution in Changing Environments. Princeton University Press, Princeton, New Jersey. Matsuda, H. and Namba, T. (1989) Co-evolutionarily stable community structure in a patchy environment. J. Theor. Biol. 136, 229-43. Maynard Smith, J. and Price, G. R. (1973) The logic of animal conflict. Nature (London) 246, 15-18. Morris, D. W. (1992) Scales and costs of habitat selection in heterogeneous landscapes. Evol. Ecol. 6, 41232. Pease, C. M., Lande, R. and Bull, L J. (1989) A model of population growth, dispersal, and evolution in a changing environment. Ecology 70, 1657-64. Pulliam, H.R. (1988) Sources, sinks, and population regulation. Am. Nat. 132, 652-61. Rice, W. R. and Salt, G. W. (1990) The evolution of reproductive isolation as a correlated character under sympatric conditions: experimental evidence. Evolution 44, 1140-52. Rosenzweig, M. L. (1978) Competitive speciation. Biol. J. Linn. Soc. 10, 275-89. Rosenzweig, M. L. (1981) A theory of habitat selection. Ecology 62, 327-35. Rosenzweig, M. L. (1987) Habitat selection as a source of biological diversity. Evol. Ecol. 1,315-30. Roughgarden, J. (1976) Resource partitioning among competing species - a coevolutionary approach. Theor. Pop. Biol. 9, 388-424. Slatkin, M. (1980) Ecological character displacement. Ecology 61, 163-77. Slatkin, M. (1985) Gene flow in natural populations. Ann. Rev. Ecol. Syst. 16, 393-430. Taper, M. L. and Case, T. J. (1985) Quantitative genetic models for the coevolution of character displacement. Ecology 66, 355-7l. Taper, M. L. and Case, T. J. (1992) Models of character dispIacement and the theoretical robustness of taxon cycles. Evolution 46, 317-33. Tilman, D. (1982) Resource Competition and Community Structure. Princeton University Press, Princeton, N J, USA. Venable, D. L. and Brown, J. S. (1988) The selective interactions of dispersal, dormancy, and seed size as adaptations for reducing risk in variable environments. Am. Nat. 131,360-84. Vincent, T. L. and Brown, J. S. (1984) Stability in an evolutionary game. Theor. Pop. Biol. 26, 408-27. Vincent, T. L. and Brown, J. S. (1988) The evolution of ESS theory. Ann. Rev. Ecol. Syst. 19, 423--43. Wright, W. (1977) Evolution and Genetics of Populations. Vol. 3, Experimental Results and Evolutionary Deductions. University of Chicago Press, Chicago, IL, USA.
