Traits traded off Rueffler, Claus

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Citation

Rueffler, C. (2006, April 27). Traits traded off. Retrieved from

https://hdl.handle.net/1887/4374

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thesis in the Institutional Repository of the University

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Chapt

er

5

Evolutionary Predictions

Should be Based on Mechanistic Models

Abstract

In the recent literature a series of theoretical studies appeared that analyzed the evolution of habitat specialization using either the logistic equation or the Ricker equation. In these studies evolutionary change is assumed to affect habitat-specific intrinsic growth rates r or carrying capacities K directly. However, since these population level parameters vary due to mutations in underlying physiological or morphological traits, such a simplification is only justified when the map from mechanistic traits to population level parameters belongs to a few particularly simple types. Here we show that the evolutionary dynamics of habitat-specific carrying capacities differs strongly when evolutionary change is modeled either directly in these traits or in underlying mechanistic traits contributing to the carrying capacities. These differences are caused by the fact that curvature properties of trade-offs at the level of mechanistic traits are generally not conserved when mapped to population level parameters. Furthermore, we point out that in all mechanistic derivations of the logistic- and the Ricker-equation known to us, traits that contribute to r also contribute to K. Consequently, models that assume evolutionary change in r while K stays constant lack a mechanistic interpretation.

This chapter is adapted with minor changes from:

Claus Rueffler, Martijn Egas, Johan A.J. Metz. in revision.

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Introduction

Long term evolution by mutation and selection is largely driven by invasion of novel genotypes into resident populations. Invasion is a population dynamical process and, when evolution is studied by means of mathematical models, therefore has to be inferred from a population dynamical model. In the recent literature a number of theoretical studies have appeared that analyze the evolution of habitat specialization using either the Ricker equation (Wilson and Yoshimura, 1994; Egas et al., 2004) or the logistic equation (Parvinen and Egas, 2004; de Mazancourt and Dieckmann, 2004) to describe habitat-specific population growth. In these models habitat specialization is subject to a trade-off, such that a high value in the carrying capacity or the intrinsic growth rate in one habitat is bought at the expense of a low carrying capacity or growth rate in the other habitat. Evolution and coexistence are then studied by assuming variation in the degree of habitat-specialization in terms of either r or K, parameters that have an interpretation only at the level of populations. This is a shortcut because mutations causing evolutionary change occur at the level of the individual and a description of how variation at this level is linked to variation in population level parameters is skipped. The question then becomes under which circumstances this shortcut is permissible. The answer to this question depends on the way individual level traits affect population level parameters, or, more formally, on the mapping that links individual traits to population level parameters. Such a mapping should emerge from a mechanistic model based on our biological knowledge of the system. By a mechanistic model we mean a model that is formulated in terms of individual processes underlying survival, reproduction, foraging, growth, and maintenance. A mechanistic model is necessarily simplified and idealized, but since all parameters have a clear biological interpretation, it can, at least in principle, be parameterized in the field or in laboratory experiments on individuals. This opens the path to strong empirical tests of model predictions.

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Evolutionary Predictions and Mechanistic Models 117

Population Level Parameters Derive From Underlying

Mechanistic Traits

In this section we report two observations on how evolutionary predictions can be flawed when these are derived from models that assume evolutionary change in population level parameters. We first illustrate our point using the logistic equation as a specific example and subsequently formulate it in more general terms. The logistic equation is given by

1 N dN dt = r 1 −N K , (1)

where N denotes population density, r the intrinsic rate of increase and K the carrying capacity. The logistic equation can be motivated in at least two different ways. In the first case it is assumed that the current per capita growth rate is proportional to the unused potential of the environment, hence,

1 N dN dt = α(K − N ) = αK 1 − N K . (2)

Here the carrying capacity is a given constant while r results from K and the positive proportionality constant α: r = αK. In the second case it is assumed that some maximum per capita growth rate linearly decreases with increasing population density, hence,

1 N dN dt = r − αN = r 1 − α rN . (3)

In this scenario r is given while K is a resulting constant: K = r/α.

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proportionally. (ii) K can change independently of r only through a change in α, a parameter inversely related to K. Hence, any change in α affects K in a non-linear way. A consequence of the first observation is that genetic variation cannot be assumed in r while K is kept constant.The consequences of the second observation are more complex and we devote the remainder of this paper to it. To formulate the difficulties that can arise from a non-linear relationship between the level of mechanistic traits and population level traits let us consider two trait spaces. The n-dimensional trait space X consists of morphological, physiological or behavioral traits to which we will refer as mechanistic traits. The m-dimensional space Y consists of population level parameters like carrying capacity K or intrinsic growth rate r. These parameters are a function of the mechanistic traits in X. The map F that links mechanistic traits to population level parameters is therefore of the form

F : X → Y : (x1, . . . , xn) 7→ (f1(x1, . . . , xn), . . . , fm(x1, . . . , xn)). (4)

Let us assume for simplicity that mutational change occurs only in x1 and x2

and that all other traits xi ∈ {x3, . . . , xn} are fixed. Let us furthermore assume

that x1 and x2are traded off, that is, a functional relationship x2(x1) exists with

dx2/dx1 < 0. Often the evolutionary dynamics of x1 and x2 will depend on the

curvature of the trade-off, that is, on the second derivative d2_x

2/dx21. A positive

second derivative implies a convex relationship while a negative second derivative implies a concave relationship. The function F maps x2(x1) onto a manifold

in Y -space. Crucially, properties of the curvature of x2(x1) are generally not

inherited from X to Y when F is a non-linear function. In this case the shortcut of assuming a trade-off directly at the level of population level parameters is invalid. Instead, we have to take the more cumbersome but accurate route where the relationship between population level parameters is derived from the corresponding characteristics of the underlying relationship between mechanistic traits while taking account of the mapping F . Through the analysis of two examples we show that the predictions based on these different methods can vary drastically, in as much that a resource generalist in one case is a stable evolutionary endpoint while in the other case it is an evolutionary repellor.

Two Examples

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to compare the evolutionary dynamics along a trade-off curve when evolutionary change is implemented directly between carrying capacities with a version where evolutionary change is implemented in underlying mechanistic traits. Before we conduct this comparison, we show how the curvature of a trade-off between two mechanistic traits can change when translated into a trade-off between population level parameters.

Continuous Time: Fast Migration Between Habitats

Assume that individuals migrate at a high rate between habitats and that travel time is negligible. This means that the environment is fine-grained. Under the assumption that birth and death rates are small compared to the migration rates, we can calculate the probability for an individual to be in habitat one, p1, or to be

in habitat two, p2= 1 − p1 (see app. A). Furthermore, we assume that birth and

death processes are determined instantaneously, that is, whether an individual dies or gives birth at a certain moment in time, purely depends on its current habitat. If population growth within each habitat can be described by the logistic equation (eq. 1), then the population dynamics is given by

dN dt = N p1r1 1 − p1N K1 + p2r2 1 − p2N K2 , (5)

with ri and Ki the habitat-specific intrinsic growth rates and carrying capacities,

respectively. The dynamics has a single, nontrivial, stable equilibrium at ˆN = (p1r1+ p2r2)K1K2/(p21r1K2+ p22r2K1).

In order to give the parameters riand Kia mechanistic interpretation, we present

one possible derivation of equation (5) based on individual processes in appendix A. This derivation is inspired by Royama’s derivation of the Ricker equation (1992, pp. 144, see following section). We assume that individuals die at some minimum rate when they can exploit a competition neighborhood of size a on their own. The size of this neighborhood is determined by a variety of biological properties like the conversion efficiency of food into energy or the efficiency with which resources are gathered. The minimum rate increases incrementally by l, a positive constant, with each additional conspecific that enters the competition neighborhood of the focal individual. For a single habitat this mechanism yields

dN dt = N (b − d) 1 − la b − dN , (6)

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these descriptions into equation (5), we get the following two-habitat version: dN dt = N p1(b1− d1) 1 − l1a1 b1− d1 p1N + p2(b2− d2) 1 − l2a2 b2− d2 p2N . (7)

Discrete Time: Juvenile Dispersal

Here we assume that generations are discrete and non-overlapping. At birth individuals are randomly distributed over two habitats with probabilities p1 and

p2 = 1 − p1. These probabilities are proportional to the relative size of the

two habitats and independent of the habitat of birth. The latter is the case either because of very effective dispersal or due to a fine-grained environment. After birth individuals stay within their habitat until death. In case of a fine-grained environment this implies that organisms are sessile. If we assume that the population dynamics within each habitat can be described by the Ricker equation (Ricker, 1952), then population growth is given by:

Nt+1= Nt p1exp r1 1 − p1Nt K1 + p2exp r2 1 −p2Nt K2 . (8) Only in case of symmetric parameter values r1 = r = r2, K1 = K = K2 and

p1= 0.5 = p2can we calculate the equilibrium population size analytically as ˆN =

2K. The population dynamics of this case are well understood. The equilibrium ˆ

N = 2K is stable for r < 2, larger value of r lead to cycles and eventually chaotic dynamics (May and Oster, 1976). Here we restrict ourselves to parameter values that produce stable equilibria.

In order to get a mechanistic interpretation of equation (8) we follow Royama (1992, pp. 144). Royama assumes that the maximum offspring number an individual can produce, R0, decreases exponentially with the number of

conspecifics n within a competition neighborhood of size a. Hence the realized number of offspring is given by Rn = R0kn. The sensitivity parameter k is a

positive constant smaller than one. For the case that individuals are Poisson distributed, Royama shows that

Nt+1= Ntexp log R0 1 − a(1 − k) log R0 Nt . (9)

This is the Ricker equation where r = log R0 and K = log R0/(a(1 − k)). By

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Evolutionary Predictions and Mechanistic Models 121 Nt+1= Nt " p1exp log R01 1 − a1_{log R}(1 − k1) 01 p1Nt +p2exp log R02 1 −a2_{log R}(1 − k2) 02 p2Nt # (10)

The Mapping of Trade-Offs

According to our mechanistic derivations we find:

Ki = bi− di aili ri = bi− di logistic equation (11) Ki = log R0i ai(1 − ki)

ri = log R0i Ricker equation. (12)

These equations represent the mapping F from the level of mechanistic traits to the level of population level parameters (see eq. [4]).

Here we assume that mutational change occurs for a1 and a2. We choose these

traits because they can, at least in principle, be measured at the level of the individual and because they influence the carrying capacities in the logistic and the Ricker equation in the same way, which facilitates a comparison between these models. We assume that a1and a2are coupled by a trade-off, which can be written

as a function a2(a1) with da2/da1 < 0. All other traits are assumed to be fixed

parameters. From now on we refer to a trade-off between a1and a2 as a trade-off

in a, and similarly, to a trade-off between K1 and K2 as a trade-off in K. How

does the function a2(a1) map onto a function K2(K1)? This is determined by the

way the trade-off a2(a1) mediates the trade-off in K and we can formalize this

dependency as

K2(a2(a1(K1))). (13)

Next we introduce specific trade-off parameterizations for a2(a1) and K2(K1).

This allows us to visualize how a trade-off in a maps onto a trade-off in K, or, vice versa, what trade-off in a is implicitly assumed when a specific trade-off in K is chosen. Our parameterizations generalize the type of trade-off function used by Egas et al. (2004) and Parvinen and Egas (2004).

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Here a1max, a1var, a2max, a2var, K1min, K1var, K2min, K2var are positive constants

determining the range of possible parameter values while the positive parameter z determines the curvature of the trade-off curve.

Note the following difference between the two trade-off parameterizations. In equation (14) θ = 0 corresponds to a low value of a1 and to a high value of a2.

In equation (15) the opposite holds true, θ = 0 corresponds to a high value of K1 and to a low value of K2. Since population growth is decreasing in ai (eq. 7

and equation 10) but increasing in Ki (eq. 5 and eq. 8) this means that in both

cases θ = 0 corresponds to a specialist for habitat one while θ = 1 corresponds to a specialist for habitat two. Similarly, z < 1 corresponds to a concave trade-off in a and to a convex trade-trade-off in K. The opposite pattern holds for z > 1. Here we will use the terms “strong trade-off” and “weak trade-off”. The word strong refers to a concave trade-off in a and to a convex trade-off in K while the word weak refers to a convex trade-off in a and to a concave one in K. This terminology is motivated by the following observation. In case of symmetric values for aimax, aivar, Kimin, and Kivarall phenotypes that lie on a linear trade-off (z = 1)

have exactly the same value for a and K, respectively, when averaged over both habitats. On a convex trade-off a generalist has a lower competition neighborhood size (eq. 14) and a lower carrying capacity (eq. 15) than a specialist when averaged over both habitats. Such a generalist is superior over the specialists in terms of its average competition neighborhood size but inferior in terms of its average carrying capacity. The opposite pattern holds for concave trade-offs.

Figure 1a&b show how a trade-off in a translates into a trade-off in K when the function F is applied. Note that the curvature of the derived trade-off in K can change from convex to concave along a single trade-off curve (see fig. 1b, right panel). In the following we refer to the curvature around the generalist’s trait with θ = 0.5. In figure 1a the trait space encompasses values of ai between 0.1 and 0.2

(aimax = 0.1, aivar = 0.1) while in figure 1b the trait space encompasses values

between 0.1 and 1 (aimax = 1, aivar = 0.9). In the first case all trade-offs in a

corresponding to z < 2 are mapped into a strong trade-off in K. In other words, for trade-offs in a with 1 < z < 2, that is, for moderately weak trade-offs, their

Figure 1: Mapping of five different trade-off curves for the habitat-specific size of competition neighborhoods aionto trade-off in carrying capacities Kithrough function F (a & b), and vice

versa through function F−1(c & d). The left graph in each pair shows trade-offs for five values of the curvature parameter z before the application of the mapping while the right graph shows trade-offs as a result of the mapping. Curves with the same gray scale correspond to the same value of z ∈ {4, 2, 1, 0.5, 0.25} with z decreasing with lighter coloration. Other parameters: b = (0.3, 0.3), d = (0.1, 0.1), p = (0.5, 0.5), (a) amax= (0.2, 0.2), avar= (0.1, 0.1), l = (10, 10), (b)

amax= (1, 1), avar= (0.9, 0.9), l = (2, 2), (c) Kmin= (0.1, 0.1), Kvar= (0.1, 0.1), l = (10, 10),

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weakness is not inherited through F . In the second case, only extremely weak trade-offs in a corresponding to z >≈ 12 are mapped onto weak trade-offs in K. For offs with 1 < z < 12, the weakness is not inherited. Hence, weak trade-offs in a are mapped onto strong trade-trade-offs in K for z-values below some threshold. In appendix B we show that this threshold increases with increasing values of aivar.

In the limit of aivar = 0 the threshold becomes one and the curvature property

is always inherited through F . In the limit of aivar = ∞ the threshold becomes

infinity, hence, any trade-off in a, weak or strong, is mapped onto a strong trade-off in K.

Models assuming a trade-off directly at the level of the carrying capacities make an implicit assumption about the shape of the trade-off in a mechanistic trait. This implicit assumption can be laid bare by applying the inverse function, F−1, to the trade-off in the carrying capacities. The result of this exercise is shown in figure 1c&d.

Evolutionary Dynamics

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Fast Migration Between Habitats

First we consider the case where evolutionary change is assumed directly to affect K1 and K2. Invasion fitness of a mutant type K0 = (K10, K20) into a resident

population with carrying capacities K = (K1, K2) can be derived from equation

(5) as s(K0, K) = p1r1+ p2r2− ˆN (K) p2 1r1 K₁0 + p2 2r2 K₂0 . (16)

For a resident at population dynamical equilibrium s(K, K) = 0, its population neither grows nor shrinks. When a mutation changes the K-values such that the term in brackets decreases, the mutant has s(K0, K) > 0 and is therefore able to invade. Hence, evolution will lead to ever smaller values of the term in brackets or, equivalently, to ever larger values of this term when multiplied with minus one. The latter expression serves as an optimization criterion O:

O(K) = − p 2 1r1 K1 +p 2 2r2 K2 . (17)

Maximizing this optimization criterion is equivalent to maximizing population size ˆ

N (whose formula is given after eq. [5]). Equation (17) has a critical point when dK2/dK1 = −(p21r1K22)/(p22r2K12). Whether such a point is a maximum or a

minimum, depends on the second derivative: d2O dK2 1 = p 2 2r2 K2 2 d2K2 dK2 1 −2p 2 1r1 K3 1 −2p 2 2r2 K3 2 dK2 dK1 2 . (18)

If the trade-off in K is concave (negative second derivative), then the first term and therefore the whole expression is negative. Any critical point of the optimization criterion O is then a maximum and therefore a CSS. If the trade-off is convex (positive second derivative), the sign of equation (18) depends on the relative magnitude of the positive and negative terms. With increasing convexity of the trade-off (increasing value of second derivative) it becomes more likely that the critical points of O are minima. Figure 2a illustrates the location of minima and maxima of the optimization criterion as a function of z for one specific set of parameters. In appendix C we show that whether the generalist with K1 = K2

constitutes a minimum or a maximum of the O depends on the parameters Kimin

and Kivar, i.e., on the range of possible parameter values for Ki. Whenever Kimin

lies below a certain threshold, O has a maximum for all possible values of z. By contrast, when Kivar decreases towards zero a situation is approached where O

has a maximum for z > 1 and a minimum for z < 1. In this limit weak trade-offs select for a generalist while strong trade-offs select for specialists.

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(a) (b) (c) (d) Trade-Off in K Trade-Off in a F ast M ig rat ion Juv enile Disp er sa l

z

θ

z

θ

Figure 2: Bifurcation diagram of extrema of optimization criterion as a function of curvature parameter z for model with: (a & b) fast migration between habitats (logistic equation), (c & d) juvenile dispersal (Ricker equation). Ordinate shows value of the specialization coefficient θ. (a & c) Trade-off directly in K. (b & d) Trade-off in a. Solid lines indicate maxima of the optimization criterion and correspond to CSSs while dashed lines indicate minima of the optimization criterion and correspond to evolutionary repellors. Other parameter values: p = (0.5, 0.5), (a & c) Kmin= (0.1, 0.1), Kvar= (0.4, 0.4), r = (0.2, 0.2), (b & d) amax= (0.5, 0.5), avar= (0.4, 0.4),

l = (4, 4), (b) l = (4, 4), b = (0.3, 0.3), d = (0.1, 0.1), (d) k = (0.1, 0.1), logR0= (0.2, 0.2).

Ki can be expressed as functions of four mechanistic traits (see eq. [11]). Only

two of them, the size of the competition neighborhood ai and the sensitivity

to competition li can cause evolutionary change in Ki while not affecting the

growth rate ri. Here we assume that mutations occur for ai while all other traits

are fixed parameters. Invasion fitness of a rare mutant type with competition neighborhood size a0 = (a0₁, a0₂) in a population with a = (a1, a2) can be derived

from equation (16) when we replace the carrying capacities with the expression from the mechanistic derivation (eq. 11):

s(a0, a) = p1r1+ p2r2− ˆN (a)(p21a01l1+ p22a02l2). (19)

As in the previous version, the term in brackets multiplied by minus one is an optimization criterion and it is easy to show that ˆN is an equivalent optimization criterion. Differentiating the term in brackets twice yields −p2

2l2(d2a2/da21),

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It is illuminating to plot these results in terms of fitness sets (Levins, 1962). Figure 3a shows trade-off curves for K plotted on top of a fitness landscape given by the optimization criterion (eq. 17). The contours of the fitness landscape are convex hyperbolas. They can be derived from equation (17) as K2= −(p22r2K1/(p21R1−

cK1), where c is the height of the contour line. Such a fitness landscape favors

generalists and it is only for very strong trade-offs (e.g., z = 1/4) that specialists do better (cf. fig. 2a). Figure 3b shows that fitness contours become linear when the optimization criterion is derived from equation (19). Consequently, for a linear trade-off and symmetric parameter values all phenotypes lie on the same contour line while weak trade-offs favor generalists and strong trade-offs favor specialists (cf. fig. 2b). In figure 3c the same result is shown, however, now the trade-off in K is derived from the trade-off in a via equation (13). We see that the trade-off curve given by z = 1 exactly follows a contour line of the fitness landscape which accounts for the evolutionary neutrality of all traits.

Juvenile Dispersal

Fitness in discrete time models can be expressed more easily as w = exp(s). If w > 1, a mutant is able to invade while a mutant with w < 1 cannot invade. As in the previous section we first perform an evolutionary analysis under the assumption that K1and K2 are traded-off directly. Invasion fitness is given by

w(K0, K) = p1exp r1 1 −p1 ˆ N (K) K₁0 + p2exp r2 1 − p2 ˆ N (K) K₂0 . (20) This equation is monotonically decreasing in ˆN . Mylius and Diekmann (1995) proved that this a sufficient condition for ˆN being an optimization criterion. Unfortunately, we can calculate ˆN analytically only for the symmetric case with p1 = 1/2 = p2, r1 = r = r2 and K1 = K = K2. Under this condition ˆN = 2K.

Hence, the maximization of ˆN has to be done numerically. We find a pitchfork bifurcation of θ-values corresponding to extrema in ˆN (fig. 2c). For small values of z the generalist is an evolutionary repellor. The generalist turns into a CSS when some threshold value of z is passed. For r < 2, i.e., for stable population dynamical equilibria, we prove in appendix C that this threshold always has a value smaller than one.

Next we analyze the evolutionary dynamics of this model where evolutionary change is directly implemented between the size of the two competition neighborhoods a1 and a2. Invasion fitness is given by

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By numerical calculations we find again a pitchfork bifurcation (fig. 2d). For this case we can prove that the change from a repelling generalist strategy to a generalist which is a CSS takes place for a z-value larger than one (app. C). In appendix C we also show that the bifurcation point moves towards higher z-values with increasing population growth rate r. This means that fast growth favors habitat specialization. Figure 3d-f illustrate these results in terms of fitness sets.

Application to the Recent Literature

In a series of models from the recent literature mutational change is assumed for habitat-specific carrying capacities and/or intrinsic growth rates. We show how the authors’ assumptions can be motivated from a more mechanistic point of view; however, we will not provide any re-analysis.

Wilson and Yoshimura (1994) explore the scope for coexistence of two habitat specialists and a habitat generalist. In their basic model version without environmental fluctuations the carrying capacity for a specialist is ten times higher in the habitat it is adapted to when compared to the habitat it is not adapted to. This situation corresponds to our figure 1b&d, where the two specialists are characterized by K = (1, 0.1) and K = (0.1, 1). The shape of the trade-off is determined by the carrying capacities of the generalist. To cover trade-off relations from strong to weak, Wilson and Yoshimura varied the carrying capacity of the generalist between 0.3 and 0.99. According to our parameterization (see eq. [15]), a generalist that is characterized by K = (0.3, 0.3) lies on a strong trade-off parameterized by z = 0.46. From figure 1d we can see that such a trade-off still corresponds to a weak trade-off in a (the least convex curve in fig. 1d corresponds to z = 1/2). Hence, all trade-offs examined by Wilson and Yoshimura correspond to a weak trade-off in a.

Figure 3: Fitness contour plots for model with: (a-c) fast migration between habitats (logistic equation) and (d-f) juvenile dispersal (Ricker equation). Fitness contours represent values of optimization criteria as functions of K1and K2(a, c, d, f) and a1and a2(b, e). Lighter coloration

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Egas et al. (2004) present a re-analysis of the model of Wilson and Yoshimura where they focus on the evolutionary dynamics along a trade-off curve. In one version of their model they assume that the habitat-specific carrying capacities can vary continuously between zero and 100. A carrying capacity of zero corresponds to an infinitely large competition neighborhood size with aimax = ∞ = aivar.

In appendix B we prove that in this case any trade-off in a is mapped into a strong trade-off in K. Therefore no biological mechanism is known to us, that can produce weak trade-offs in K and we cannot give a biological interpretation for those results in Egas et al. (2004) that are based on the assumption of a weak trade-off in K.

Parvinen and Egas (2004) studied the evolution of habitat specialization in a metapopulation model with two types of habitat and logistic growth within patches. Evolutionary change is assumed in either habitat-specific intrinsic growth rates or carrying capacities. In their model version where carrying capacities are allowed to evolve, the authors assume that the carrying capacity of a habitat specialist is twice as high in the habitat it is adapted to. This situation corresponds to figure 1a&c, where the two specialists are characterized by K = (0.2, 0.1) and K = (0.1, 0.2). In this situation a slightly convex trade-off has to be assumed in a, when a linear trade-off in K is investigated. More importantly, no derivation of the logistic equation known to us provides a mechanism that would allow for varying the intrinsic growth rate while leaving the carrying capacities constant, since the latter is linearly dependent on the former. Hence, we have no biological interpretation of habitat specialization in terms of intrinsic growth rates.

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Discussion

In the face of trade-offs, theory predicts two qualitatively different evolutionary outcomes. Natural selection can lead either towards an intermediate phenotype where the gain from improving one trait is exactly balanced by the loss through the accompanying change in another trait or to an extreme phenotype at the boundary of the trait space. It is known for a long time that the curvature of a trade-off can decide between these two cases (e.g., Levins, 1962; Stearns, 1992; Roff, 2002). In the recent literature a series of papers appeared that studied the evolution of habitat specialization and coexistence by means of mathematical models in populations showing variation in intrinsic growth rate r and carrying capacity K (Wilson and Yoshimura, 1994; Egas et al., 2004; Parvinen and Egas, 2004; de Mazancourt and Dieckmann, 2004). These population level parameters are determined by traits at the individual level. In this paper, we argue that the consequences of mutational change have to be studied directly at the level of mechanistic individual-based traits. Implementing mutational change at the level of population level parameters, and assuming that those are traded off, can lead to results that are either not interpretable at all at the level of mechanistic traits or only under the assumption of rather unrealistic properties of the underlying trade-off at the level of mechanistic traits.

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traits. Because of the involved non-linearity this assumption will often be rather unrealistic and in some cases, as we have shown, a certain curvature cannot be derived from an underlying trade-off at all.

The issue put forward here can be placed in a wider context. Mutations ultimately occur at the level of chromosomes and the mapping that links this variation to higher levels is better described by a cascade of mappings: genotypes are mapped to enzymes and their regulation. During development these characteristics might be mapped to morphometrical properties of the beak of a bird, which determine its handling times for different seeds, which determines energy uptake, which determines offspring production, which determines population growth rate. Our plea for mechanistic models is not new. Here we mention a few references that treat problems related to ours. Matessi and Gatto (1984) illustrate that much confusion about the concept of r and K selection arises from the use of models that involve just these parameters. Apparent contradictions disappear when competition equations are derived from an explicit predator-prey model. Schoener (1986) delivers an extensive philosophical discussion of advantages and disadvantages of a mechanistic modeling approach in ecology. Kuno (1991) noticed that the assumption of independence between r and K leads to population dynamical anomalies in the logistic equation (see also Ginzburg, 1992).

Finally, we want to draw attention to the results of our evolutionary analysis as such. The evolution of habitat specialization in terms of the size of the interaction neighborhood a shows a marked difference in the continuous and the discrete time model. In the continuous time model, where fast migration between habitats is assumed, the generalist is selected for in case of weak trade-offs while specialists are selected for in case of strong trade-offs. This scenario coincides with the intuition of many evolutionary ecologists about the evolution of resource specialization (e.g., Benkman, 1993; Robinson, 2000) and with Levins’ predictions for evolution in an environment stable in time but heterogeneous in space (Levins, 1962). By contrast, in the discrete time model, where juvenile dispersal and no migration is assumed, specialists are also selected for in combination with relatively weak trade-offs. This scenario shows that Levins’ result does not hold generally but can be modified when fitness is a non-linear function of the traits considered evolvable.

Acknowledgments

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Appendix A: Fast Migration Between Habitats

The population dynamics of one consumer species exploiting two different habitats can be described by the following system of coupled differential equations:

dN1 dt = m12N2 A2 A1 − m21N1+ b1N1− d1N1 (A1) dN2 dt = m21N1 A1 A2 − m12N2+ b2N2− d2N2. (A2)

N1 and N2 denote the population density in habitat one and two, respectively.

Individuals migrate at rate mij from habitat j to habitat i. Absolute habitat size

is denoted by Ai. In each habitat individuals reproduce and die at the

habitat-specific rates bi and di, respectively. If we assume that migration rates are high

in comparison to the birth and death rates, we can calculate the equilibrium distribution of the population over the two habitats as ˆN1 = ˆN2A2m12/A1m21.

Combining this with n = A1N1+ A2N2, where n denotes the total population size,

we find that p1= A1Nˆ1/ˆn = m12/(m21+ m12) and p2= A2Nˆ2/ˆn = m21/(m21+

m12).

In order to write the model purely in terms of mechanistic traits, we assume that the habitat-specific death rates increase linearly with the number of competitors and that therefore the realized rate of increase of a consumer with n competitors in its competition neighborhood of size a is given by rn = b0− (d0+ ln). Here

l is a positive constant. It describes the sensitivity to competition such that the sensitivity is increasing with increasing values of l. If we substitute r0 for b0− d0

we can rewrite the realized rate of increase as rn = r0− ln. The expected rate of

increase is then given by E[r] = r0− l X n nP (n) = r0− l¯n = r0− lN a = r0 1 −laN r0 , (A3) i.e., K = r0/la.

Appendix B: Trade-Off Curvature Under Mapping F

Consider a trade-off between a1 and a2 that can be described by the function

a2(a1). This trade-off curve is translated into a curve K2(K1) by the map

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with c1 and c2 positive constants. This map encompasses both mechanistic

derivations (see eq. [11] and [12]). To study the curvature of the trade-off in K, we have to differentiate equation (13):

d2_K 2(a2(a1(K1))) dK2 1 ∝ 2da2 da1 da2 da1 1 a2 − 1 a1 −d 2_a 2 da2 1 . (B2)

This gives us an expression for the shape of the trade-off in terms of the first two derivatives of a2(a1). Further analytical results can be obtained for specific

parameterizations of the trade-off curve. Here we choose equation (14) which can also be written as 1 = a1max− a1 a1var z + a2max− a2 a2var z . (B3)

We can spell out the trade-off in a as a function a2(a1) by solving equation (B3)

for a2. The first two derivatives of this function can be simplified using equation

(14) and equation(B3) until we get: da2 da1 = −(1 − θ)θ 1 za_2var (1 − θ)1zθa_1var (B4) d2a2 da2 1 = (1 − θ)θ 1 za_2var(z − 1) (1 − θ)2zθ2a2_1var . (B5)

Now we are able to simplify condition (B2) using equation (B4) and (B5): d2_K 2(a2(a1(K1))) dK2 1 ∝ 1−z +2 " θ a1var(1 − θ) 1 z a1max− a1var(1 − θ) 1 z + (1 − θ) a2varθ 1 z a2max− a2varθ 1 z # (B6) This condition depends on the magnitude of z relative to 1 plus two times some complicated expression in brackets. If aivaris small for i ∈ {1, 2}, then the term in

brackets is small as well, and the sign of the second derivative will be determined by the difference between 1 and z. In this case, whether K2(a2(a1(K1))) is a strong or

a weak trade-off is inherited for most values of z from the corresponding property in the underlying trade-off a2(a1). By contrast, if aivar is large for i ∈ {1, 2},

i.e., very similar to aimax, then the term in brackets will be large as well and can

dominate the whole expression. In this case moderately weak trade-offs in a will be mapped into strong trade-offs in K. When aivar approaches infinity, that is,

when Kiminapproaches zero, then K2(K1) will be a convex trade-off for any value

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Appendix C: Results for Evolutionary Analysis

First we present some analytical results for the case of fast migration and mutational change in K. The derivation of these results is very similar to the one in the previous section and will not be repeated here in as much detail. The optimization criterion is given by equation (17) and its curvature by equation (18). The curvature depends on first two derivatives of K2(K1), a function that can be

derived from equation (15). If we evaluate equation (18) at the critical point of O, that is where dK2/dK1 = −p2

1r1K22/p22r2K12, and use the first and second

derivative of K2(K1), a straightforward calculation shows that equation (18) can

be simplified to d2_O dK2 1 ∝ 1 − z − 2 θ K1var(1 − θ) 1 z K1min+ K1var(1 − θ) 1 z + (1 − θ) K2varθ 1 z K2min+ K2varθ 1 z ! . (C1)

When the term in brackets is larger than 0.5, then d2_O/dK2

1 will be negative

for any value of z and the critical point will be a maximum. This is the case when Kimin is sufficiently small for i ∈ {1, 2}. In case of symmetric parameter

values for r, p, Kimin, Kimax, we find a critical point of the optimization criterion

at θ = 0.5. In this case a sufficient condition for d2O/dK12to be negative is given by

Kimin< Kivar0.5

1

z. On the contrary, when K_ivar is very small for i ∈ {1, 2}, then the term in brackets will be close to zero as well. In this case the difference between one and z is dominating over the term in brackets and the qualitative curvature of the optimization criterion will be identical to the qualitative curvature or the trade-off K2(K1). In the limit Kivar = 0, the term in brackets becomes zero and

a strong trade-off in K corresponds to a minimum in O while a weak trade-off in K corresponds to a maximum in O.

For the case of symmetric parameter values (p1= 1/2 = p2, r1= r = r2, K1min =

K2min, K1var = K2var, a1max = a2max, a1var = a2var, R01= R02, k1= k = k2)

we can prove that for the model with juvenile dispersal the bifurcation from a repelling generalist to a generalist that is a CSS occurs for z < 1 when the trade-off is directly in K, and for z > 1 when the trade-trade-off is in a. Under the assumption of symmetry the first derivative of the fitness function (eq. [20] and eq. [21]) equals zero at the generalists trait where K1 = K = K2 and a1 = a = a2, respectively.

At these trait values holds dK2/dK1 = −1 and ˆN = 2K in case of a trade-off

directly in K and da2/da1= −1 and ˆN = 2r/(a(1 − k)) in case of a trade-off in

a. The bifurcation point is given by the z-value where the second derivative of the fitness function (eq. [20] and eq. [21]) equals zero: d2_w(K0_{, K)/dK}02

1= 0 and

d2_w(a0_{, a)/da}02

1= 0, respectively. Under the above mentioned conditions we can

derive that the bifurcation points are characterized by

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and

d2a2/da21= 2r/a, (C3)