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Rueffler, C. (2006, April 27). Traits traded off. Retrieved from
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Chapt
er
2
The Evolution of Simple Life-Histories:
Steps Towards a Classification
Abstract
We present a classification approach for a class of simple life-history models. The considered model class is characterized by the assumptions that individuals can occur in two states, discrete time population dynamics, density-dependent population growth and two evolving traits that are coupled by a trade-off. Individual models differ in the choice of traits that are considered to evolve and in the way population regulation is incorporated. We classify models according to curvature properties of the fitness landscape and to their potential to support an optimization criterion. The first classification allows us to infer whether trait combinations that are characterized by a zero fitness gradient correspond to minima or maxima of the fitness landscape and how this property depends on the curvature of the trade-off. The second classification distinguishes scenarios where evolutionary change is frequency independent from scenarios that involve frequency dependence and therefore potentially allow for the coexistence of different types. Given certain symmetry assumptions we can extend the classification in the latter case by splitting selection into a density-dependent and into a frequency-dependent component. The classification is derived from a fitness measure that is sign equivalent to invasion fitness but algebraically simpler. We apply our approach to several simple life-history models and demonstrate how our classification allows for an analytical analysis. Finally, we draw few general conclusions from out classification and discuss several possible extensions.
This chapter is an unpublished manuscript by
Introduction
Evolutionary change is guided by patterns of selection and constraints. Whether a mutation is advantageous or not depends on the fitness landscape, and constraints limit the availability of advantageous mutations. In case of two-dimensional trait spaces the fitness landscape can be visualized as a three-dimensional graph or as a two-dimensional contour plot (Levins, 1962). Evolutionary change driven by selection can only occur in an upward direction on such a fitness landscape. Constraints to evolutionary change emerge for various reasons. Pleiotropy couples a change in one trait to a change in another trait and can thereby render certain parts of the trait-space unattainable. Regions of the trait space are also no-go areas of the evolutionary dynamics when they correspond to non-viable organisms. Either mechanism can result in trade-offs where we observe that an improvement in one function is bought at the expense of a deterioration in another function. Are there features of selection and constraints that allow us to group different eco-evolutionary models into classes with a common set of properties? Such a classification would be interesting for several reasons. It would allow one to identify conditions that favour certain evolutionary outcomes and might help to explain why some life-history patterns are common while others are rare or absent. Are there specific life-cycles and ecological patterns that are prone to typical evolutionary dynamics? Finally, such a classification might facilitate the analysis of mathematical models of evolution.
In a previous paper we attempted a classification, where possible evolutionary outcomes were categorized according to curvature properties of the fitness landscape and the trade-off curve without specifying how the different curvature combinations correspond to specific eco-evolutionary models (chapter 1). In this paper we take a different approach and start by delineating a specific class of models. We then ask how the evolutionary dynamics depends on assumptions on the population ecology and evolutionary constraints. More specifically, we limit ourselves to life-cycles that can be described with two states and where any set of two demographic parameters is allowed to change mutationally while all others are assumed to be fixed. These two traits are coupled by a trade-off. Density dependence can act on any set of demographic parameters where different parameters can be susceptible to specific subgroups of the population.
The Modeling Framework
This section starts with a description of the envisaged life-cycles and the dynamics of monomorphic and polymorphic populations. We then deal with the ecology of a population by explaining how population density feeds back to population growth. In a next step we introduce mutant types that deviate in some demographic parameters from the resident types. Invasion fitness will be derived as a means to determine the long term fate of mutants. After introducing a sign equivalent fitness proxy which is algebraically simpler then invasion fitness proper, we will briefly describe how evolutionary dynamics can be derived. Finally, we establish a link between the population dynamics and the evolutionary dynamics by explaining the concept of evolutionary feedback environment and its implications for optimization.
The Life Cycle
We restrict ourselves to life-histories can be described with two discrete states in a discrete time framework (fig. 1). Population census takes place just before reproduction and after a potential transition from one state to another. Individuals in state j produce fij offspring of type i. After a potential reproductive event
individuals make a transition from state j at time t to state i at time t + 1 with probability tij. The population projection matrix A is then a two-by-two matrix
with components
aij = tij+ fij. (1)
These matrix components give the total amount of individuals in state i at time t + 1 that descend from individuals in state j at time t. The population dynamics of a population is then given by
Nt+1= ANt, (2)
where Nt is the vector of densities in the two states at time t. This setting
includes age- and stage-structured models but also two-patch and two-sex models. Individuals can occur in either one birth state (e.g. immature or small) or in two birth states (e.g. birth in either of two different locations or as different sexes). This paper is concerned with the evolution of matrix components tij and fij.
We therefore have to extend the model such that it can deal with coexisting phenotypes. We restrict ourselves to the case where different phenotypes deviate from each other in only two traits x1, x2∈ {t11, f11, t12, f12, t21, f21, t22, f22} while
–
1
–
–
2
–
t
21+
f
21t
12+
f
12t
11+
f
11t
22+
f
22Figure 1: Life-cycle with two states. The parameters tijindicate the transition probability of individuals in state j at time t to state i at time t + 1. The terms fij indicate the number of surviving offspring that enter state i and are born to an individual in state j.
that within the two-dimensional trait space phenotypes are confined to a one-dimensional manifold to which we will refer as trade-off curve x2(x1) (fig. 2).
The rationale behind this assumption is as follows. The dominant eigenvalue of a positive matrix, hence population growth, is an increasing function of all matrix components, therefore selection acts to increase each of the evolving traits. We assume that a constraint exists that sets a limit to the possible increase of traits. Once this limit is reached an increase in one trait can only be bought at the expense of a decrease in another trait. We therefore make the simplifying assumption that the evolutionary dynamics are confined to the set of trait combinations (x1, x2)
that constitute the trade-off relationship. We parametrize the trade-off curve x2(x1) with a coefficient θ that lies between zero and one (fig. 2). Hence, any
phenotype is uniquely determined by the trade-off coefficient θ corresponding to the trait values x(θ) = (x1(θ), x2(θ)). A community consisting of n types will be
denoted with Θ = (θ1, . . . , θn). To denote the population dynamics of the ith type
from a community Θ we rewrite equation (1) as Ni
t+1= A(θi)Nti. For numerical
calculations we use the following trade-off parameterization:
x(θ) = (x1(θ), x2(θ)) = (x1max(1 − θ)1/z, x2maxθ1/z), (3)
where x1max and x2max are positive constants. The parameter z determines the
curvature of the trade-off such that z < 1 correspond to a convex (or strong) trade-off (d2x
2/dx21> 0) while z > 1 corresponds to a concave (or weak) trade-off
(d2x
2/dx21< 0). For z = 1 the trade-off is linear.
Density Dependence
tra it 2 trait 1 s s sθ = 0.5 s s sθ = 1 sθ = 0 1/3 1/2 1 2 4
Figure 2: Trade-off curves illustrating the relationship between two traits that are traded-off. The number next to each curve is the parameter z determining the curvature (strength) of the trade-off. The trade-off curve is parameterized in the coefficient θ that lies between zero and one. Filled circles half way on the trade-off curve correspond to θ = 0.5.
explicitly but only implicitly via negative density dependence, such that the growth rate of any specific type i depends both on its own phenotype and on the density of all phenotypes present in the population. Hence, the population projection matrix becomes time dependent: Ni
t+1 = At(θi)Nti (Caswell, 2001). Any of the matrix
components tij or fij can be affected by population density and throughout this
paper any trait that is a function of population density will be marked with a tilde, e.g. ˜f12. In the context of age-structured populations Charlesworth (1994)
coined the term “critical age-group” for the subgroup of the population that affects density-dependent demographic traits. Here we assume that different traits can be affected by different subgroups or “critical state-groups”. We restrict ourselves to functions of population regulation such that the population dynamics settle on a unique nontrivial stable equilibrium ˆNi = ( ˆN1i, ˆN2i) for all i ∈ {1, . . . , n}. Note that at least all fij > 1 have to be density regulated for a stable population
dynamical equilibrium to exist. At equilibrium the time index t of the population projection matrix A can be dropped.
How is density dependence manifested at the level of the trait? Let us consider the case where population density affects one of those demographic parameters x that characterize the different types, hence, for the ith type we have x(θi). In this
paper we restrict ourselves to functional forms of density dependence that allow for “separability”. By this we mean that the realized trait ˜xt(θi) at time t can be
split into two factors. One is dependent on the trait value of the focal type i while the other factor, Dx, captures the effect of all other individuals in the community
on the specific trait: ˜
xt(θi) = x(θi)Dx(θ1, N1t, . . . , θ n, Nn
An example for a function of density Dx that allows for separability and that
we will use throughout this paper it the Beverton-Holt type function with the densities of different states as arguments:
˜ xt(θi) = x(θi)Dx= x(θi)/(1 + c1 n X i=1 N1ti + c2 n X i=1 N2ti ). (5)
The weighting factors c1, c2∈ R ≥ 0 are assumed to be constant and identical for
all types i, however, they are allowed to differ for different demographic parameters ˜
x, indicating that specific traits can be affected by different critical state-groups. In a large region of parameter space the Beverton-Holt function leads to stable population dynamical equilibria.
Invasion Fitness
The fate of a rare mutant θ0 occurring in a specific resident community is given by
its invasion fitness, that is, its long term average growth rate in an environment that is determined by the resident community (Metz et al., 1992; Rand et al., 1994). We assume that mutations are rare and of small effect. The first assumption is made to assure that a resident community has settled on its attractor before a new mutant arises. This means that a resident population is completely described by the vector Θ = (θ1, . . . , θn) because these traits determine the unique
non-trivial population dynamical equilibria Nˆi. Therefore, for a rare mutant θ0
equation (4) can with some abuse of notation be written as ˜
x(θ0) = x(θ0)Dx(Θ) (6)
and its population projection matrix can be written as A(θ0, Θ). The second assumption assures that a mutant that has the ability to invade a resident type but can not be invaded by the resident when common itself, goes to fixation (Metz et al., 1996a; Geritz et al., 1998, 2002).
In the present model invasion fitness is given by the dominant eigenvalue λd(θ0, Θ)
of the projection matrix A(θ0, Θ). In the following paragraph we introduce a sign equivalent fitness proxy w for invasion fitness. This fitness proxy is algebraically simpler than the dominant eigenvalue λdand it will be a fundamental tool in this
paper.
The characteristic polynomial of a mutant’s population projection matrix A(θ0, Θ) equals
P (λ, θ0, Θ) = λ2− traceA(θ0, Θ)λ + detA(θ0, Θ).
P (λ, θ0, Θ). Since λ2 > 0, P (λ, θ0, Θ) is a parabola in λ opening upward. Therefore, if P (1, θ0, Θ) < 0, then λd > 1. If, however, P (1, θ0, Θ) > 0, we need
dP (1, θ0, Θ)/dλ = 2 − traceA(θ0, Θ) < 0 for λ
d> 1. In this case both eigenvalues
are larger than one. Hence, λd> 1 if
traceA(θ0, Θ) − detA(θ0, Θ) = ˜a11+ ˜a22− ˜a11˜a22+ ˜a12˜a21> 0
or
traceA(θ0, Θ) = ˜a11+ ˜a22> 2,
and λd< 1 if and only if
traceA(θ0, Θ) − detA(θ0, Θ) < 0 and traceA(θ0, Θ) < 2.
We mark all matrix components with a tilde as long as we do not specify which terms are density regulated. Note, that for θ0∈ Θ we have λd(θ0, Θ) = 1 while the
subordinate eigenvalue is less than one. Hence, for any resident type at population dynamical equilibrium we find traceA(θ, Θ) < 2 and for any mutant type θ0 that differs but slightly from the resident type such that traceA(θ0, Θ) < 2 is still fulfilled, we find
sign[λd(θ0, Θ)−1] = sign[traceA(θ0, Θ)−detA(θ0, Θ)−1] = sign[−P (1, θ0, Θ)]. (7)
Therefore 1 − P (1, θ0, Θ) = ˜a11+ ˜a22− ˜a11˜a22+ ˜a12a˜21 can be used as a fitness
proxy. We denote this fitness proxy as w(θ0, Θ) and, to simplify matters, we will refer to it as invasion fitness in the remainder of this paper though it is only sign equivalent to invasion fitness proper. Note that this fitness proxy describes the direction of evolutionary change but not its speed (Dieckmann and Law, 1996; Durinx and Metz, 2005). The idea to exploit the characteristic polynomial
evaluated at λ = 1 for invasion considerations has been introduced by
Taylor and Bulmer (1980) and can also be found in Courteau and Lessard (2000). Let us briefly note some interesting properties of w. Firstly, it equals R0 in
age-structured models with t22= 0. Secondly, in models where f12is the only fecundity
term, for instance, in age-structured models where reproduction only takes place in the second year, the condition traceA(θ0, Θ) < 2 is fulfilled automatically. Thirdly, for small mutational steps ˜a11, ˜a22 < 1. To see this, we rewrite traceA(θ, Θ) −
det(θ, Θ) = 1 until we get 0 = (1 − ˜a11)(1 − ˜a22) − ˜a12˜a21. For this equality to hold
either ˜a11, ˜a22> 1 or ˜a11, ˜a22< 1. Since the first case violates traceA(θ0, Θ) < 2,
we have proven the second case.
The direction of evolutionary change is given by the fitness gradient, the first derivative of invasion fitness with respect to the mutant trait. For the time being we limit ourselves to resident communities that consist of single type θ. Points θ∗ in trait space where the fitness gradient equals zero, that is
are of special interest and were named “evolutionarily singular points” by Metz et al. (1996a) and Geritz et al. (1998). Singular points can be classified according to two properties: convergence stability and invadability (Metz et al. 1996a; Geritz et al. 1998; chapter 1). Singular points that are both convergence stable and uninvadable are final stops of evolution and we refer to them as ”continuously stable strategies” or CSSs (Eshel, 1983). Singular points that are convergence stable but invadable by nearby mutants are particularly interesting. Directional selection drives the mean trait value of a population towards such points and once the mean population trait value has reached the singular point, selection turns disruptive and favors an increase in phenotypic variance (chapter 6). It case of clonal organisms this increase can be realized by the emergence of two independent lineages and it is this scenario that earned such points the name ”evolutionary branching points” (Metz et al., 1996a; Geritz et al., 1998). Singular points that lack convergence stability are evolutionarily repelling. When such singular points are invadable we refer to them as evolutionary repellors and when they are immune to invasion by nearby mutants we refer to them as ”Garden of Eden-points” (Nowak, 1990). In the latter case any perturbation results in directional selection away from the singular point and no natural population is ever expected to occupy a Garden of Eden-point.
Metz (unpublished) proved that, given that the trait space is connected, global uninvadability of a singular trait θ∗ is given when w(θ0, θ∗) ≤ 1 for all possible θ0, that is, the condition traceA(θ0, θ∗) < 2 becomes superfluous.
Feedback Environment
A considerable part of this paper will be concerned with finding conditions that allow us to derive evolutionary dynamics from an optimization principle (Christiansen, 1988; Mylius and Diekmann, 1995; Metz et al., 1996b). By this we mean a function from the trait values to the real numbers such that a CSS corresponds to maximum of this function while the minima correspond to evolutionary repellors. In order to get to grips with this problem we take a slightly different perspective where we consider invasion fitness as a function of the mutant’s trait and of an input I from the environment (Heino et al. 1997; Heino et al. 1998; Diekmann et al. 2003; Mesz´ena et al. 2006; chapter 3). The m-dimensional vector I characterizes the state of the feedback environment, that is, those aspects of the environment that are determined by the resident population and simultaneously feed back to affect the fitness of individuals in the population. Each Ii ∈ I channels effects of population density and composition
(Diekmann et al., 2003; Mesz´ena et al., 2006). On this time scale the trait values of the interacting individuals are assumed to be fixed and it is sufficient to collect the densities of the different types in the feedback vector. On an evolutionary time scale the trait values of the interacting types can change and in order to achieve independence between individuals on this time scale, the feedback environment has not only to account for the densities of the conspecifics but also for any direct effect through their traits.
The dimension of I is of great interest because it imposes an upper limit to the number of species that can possibly coexist (Diekmann et al., 2003; Mesz´ena et al., 2006). To see this consider two coexisting types θ1 and θ2. At population
dynamical equilibrium both w(θ1, I(θ
1, θ2)) = 1 and w(θ2, I(θ1, θ2)) = 1. When
dim(I) = 2 these two equalities constitute a system of two equations in two unknowns which can have a robust solution. If, however, dim(I) = 1, then we have a system of two equations in one unknown and no generic solution exists. This proves that in one-dimensional feedback environments robust coexistence is impossible (Metz et al., 1996b; Mesz´ena et al., 2006). If, additionally, invasion fitness w is a monotone decreasing (increasing) function in I, then I is an optimization (pessimization) criterion and the evolutionary dynamics can be predicted by maximizing (minimizing) I (Metz et al., 1996b).
Under the assumption of separability (cf. eq. [4]) the interaction variables Ii∈ I
can be equated with the different functions of density Dxij. In case all transition
rates are density dependent and all functions of density are different, dim(I) can become as high as eight. However, for some special cases dim(I) can be lower. For instance, if we assume that the functions of density Dxij only depend on the
population at time t via the densities of the different types but not directly via their trait values θi, then the maximum dimension of I decreases to two. This
assumption is realized in the Beverton-Holt type function (eq. 5) where Dx is a
decreasing function of the weighted sum of the densities in the two states. Then Pn i=1Nˆ i 1and Pn i=1Nˆ i
2 are the arguments of the functions of density Dxand it is
sufficient to consider I = (Pn i=1Nˆ i 1, Pn i=1Nˆ i
2) as input from the environment in
order to achieve independence between individuals. This holds true independent of the number of types present in the community and of the number of traits that are affected by density dependence and the maximum dimension of I becomes two.
Results
Invasion Boundaries
Invasion boundaries (IBs) are manifolds in trait space consisting of all trait combinations that are selectively neutral in a given resident community Θ as long as they are sufficiently rare. For our model class IBs are implicitly defined by
w((x, y), Θ) = 1. (9)
Hence, IBs are curves given by all trait combinations (x, y) that have an invasion fitness equal to one. IBs divide the trait space into two regions (chapter 1). Trait combinations (x, y) that lie above such a curve are able to invade since w((x, y), Θ) > 1 and trait combinations that lie below an IB are characterized by w((x, y), Θ) < 1 and are therefore not able to invade. An IB necessarily intersects with the trade-off curve at all resident trait values θi∈ Θ. At a singular trait value θ∗ (cf. eq. [8]) an IB is tangent to the trade-off curve. Whether or not a singular point θ∗ is invadable by nearby mutants can be deduced from the
configuration of the trade-off curve and the IB in the neighborhood of θ∗. When,
except for the point of tangency θ∗, the IB lies below the trade-off curve, then all trait values θ0 in the neighborhood of θ∗ have w(θ0, θ∗) > 1 and are therefore able to invade; θ∗ corresponds to a minimum of the fitness landscape, hence, to either a repellor or branching point. If the opposite patterns holds true, that is, if the IB, except for the point of tangency, lies above the trade-off curve, then θ∗ is uninvadable by all nearby mutants. In this case the singular point has to be either a CSS or a Garden of Eden-point. From this brief treatment follows that the relative curvature of IBs and trade-off curves are an important determinant of the of the evolutionary dynamics (chapter 1; de Mazancourt and Dieckmann 2004; Bowers et al. 2005). Whenever the IB is more curved than the trade-off, a singular point will be uninvadable and this case is more likely for convex IBs than for concave ones and vice versa.
While the curvature of the trade-off is determined by the morphological, physiological and genetic constraints of the organism under study, the qualitative curvature properties of the IBs depend on the traits that are considered evolvable, or, more precisely, on the exact position of the two traits x and y in the fitness function w. Whether or not IBs are linear, convex or concave does not depend on the ecology, that is, on properties of the density regulation. However, it is the ecology that determines whether a singular point is convergence stable (CSS or branching point) or evolutionary repelling (evolutionary repellor or Garden of Eden-point).
To derive the shape of invasion boundaries we have to solve equation (9) for y. Since the matrix component aij is a linear function of the contributing parameters
tij and fij, it is sufficient to solve for the matrix component aij that depends on
Trade-Off Within One Matrix Component
When evolution occurs in the two traits tij and fij of the same matrix component
aij IBs are linear. This follows from the fact that aij is the sum of the two traits
and that w is linear in aij. As an example we consider ˜t11 and ˜f11:
˜ f11= 1 + ˜ a12˜a21 1 − ˜a22 − ˜t11. (10)
Trade-Off Between Diagonal and Off-Diagonal Components
When the evolution affects both a diagonal component aii and an off-diagonal
component aij IBs are linear again. To see this we rearrange equation (9) to
˜
a12˜a21= (1 − ˜a11)(1 − ˜a22). (11)
The linearity follows from this equation whenever mutations affect components on both the right- and the left-hand side. For instance, if x = (f12, t22), then
˜ f12= (1 − ˜t11− ˜f11)(1 − ˜t22− ˜f22) ˜ t21+ ˜f21 − ˜t12.
Hence, f12 can be written as a linear function of t22.
Trade-Off Between Diagonal Components
If evolution occurs in components that affect the diagonal components ˜a11 and
˜
a22, then invasion boundaries are concave. To see this, we rearrange equation (9)
such that ˜ a22= 1 − ˜ a12a˜21 1 − ˜a11 . (12)
The second derivative of equation (12) with respect to ˜a11 is
d2˜a 22 d˜a211 = − 2˜a12a˜21 (1 − ˜a11)3 . (13)
In the previous section we proved that ˜a11, ˜a22 < 1 for any population at
In case that neither a11 nor a22are density dependent, traits can be rescaled such
that invasion boundaries become linear. If we take the logarithm of equation (11) we get
ln ˜a12+ ln ˜a21= ln(1 − a11) + ln(1 − a22). (14)
From this we see that ln(1 − a22) is a linear function of ln(1 − a11).
Trade-Off Between Off-Diagonal Components
If evolution occurs in traits that affect the off-diagonal components of the projection matrix A, then IBs are convex. This can be seen from equation (11) where these entries occur in the product on the left-hand side. Hence, the traits of the invasion boundary are inversely related to each other and lie on a convex curve. From equation (14) we see that if neither a12nor a21are density dependent,
then invasion boundaries are linear on a logarithmic scale.
Optimization
Only under some rather restrictive conditions can the course of evolution be predicted by seeking the extrema of an optimization criterion. In the section on the feedback environment we gave necessary and sufficient conditions for one specific criterion: if (i) dim(I) = 1 and (ii) w is a monotone decreasing (increasing) function of I, then I is maximized (minimized). Here we prove that for the considered class of models condition (ii) follows from condition (i). Since dim(I) = 1 implies that not two types can coexist, we only need to consider monomorphic resident populations. For this proof we assume that all functions of density are monotonic functions of I that change in the same direction with increasing I. Invasion fitness becomes
w(θ0, θ) = (f
12Df12+ t12Dt12)(f21Df21+ t21Dt21) + f11Df11+ t11Dt11 − − − − −..+ f22Df22+ t22Dt22− (f11Df11+ t11Dt11)(f22Df22+ t22Dt22).
To check for the monotonicity condition (ii) we have to take the derivative of w with respect to I. After some simplification we get
dw dI = (1 − ˜a11)(f22 dDf22 dI + t22 dDt22 dI ) + (1 − ˜a22)(f11 dDf11 dI + t11 dDt11 dI ) + Q,
with Q = d(f12Df12+ t12Dt12)(f21Df21+ t21Dt21)/dI. Since all derivatives have
the same sign and because ˜a11 < 1, ˜a22 < 1, the whole expression is negative
when dDxij/dI < 0 and positive when dDxij/dI > 0 for all functions of density.
Table 1: Combinations of evolving traits (columns) and density regulated traits (rows) that support and optimization criterion where i, j ∈ {1, 2} with i 6= j. The arguments I for the functions of density are assumed to differ because otherwise I always acts as an optimization criterion (see section on optimization for details). Cells give the corresponding optimization criterion. In some cases the shown expression is only an optimization criterion when the corresponding condition is met: (∗) aij consists of a single trait (either fij or tij), (∗∗) aji consists of a single trait. If the condition is not met, then dim(I) > 1 and optimization is not possible. evolving traits regulated traits Z Z Z Z Z Z Z t0
ij& fij0 a0ii& a0ij a0ii& a0ji a011& a022 a012& a021
all dim(I) > 1 dim(I) > 1 dim(I) > 1 dim(I) > 1 a 0 12a021.∗,∗∗ see table 2 ˜ aij, ˜a11& ˜a22 if t 0 ij an d f 0 ij no t den sit y regu lated : t 0 ij + f 0 ij
dim(I) > 1 dim(I) > 1 dim(I) > 1 a012a021.∗ ˜
aii& ˜aij dim(I) > 1 dim(I) > 1 dim(I) > 1 a012a021.∗ ˜
aii& ˜aji dim(I) > 1 dim(I) > 1 dim(I) > 1 a012a 0 21. ∗∗ ˜ ajj & ˜aij a0 ij 1 − a0ii –∗ a0 ji 1 − a0ii dim(I) > 1 a 0 12a 0 21. ∗ ˜ ajj & ˜aji a0ij 1 − a0 ii a0ji 1 − a0 ii –∗∗ dim(I) > 1 a0 12a021.∗∗ ˜
a12& ˜a21 dim(I) > 1 dim(I) > 1 a 0
11+ a022− dim(I) > 1, a011a022 see table 2 ˜
a11& ˜a22 dim(I) > 1 dim(I) > 1 dim(I) > 1,see table 2 a012a021
be an optimization criterion. Note, that this proof holds also if only a subset of parameters is density dependent. For the special case that all functions of density Dxij(I) are identical such that they can be factored out from the fitness function,
Dxij is a pessimization criterion.
that once I is established as an optimization criterion a corresponding criterion ψ exists. However, no general recipe exists to find it. For the following patterns of trade-offs and density regulation we were able to find an explicit optimization criterion ψ (cf. table 1).
1. Population regulation is such that all functions of density can be factored out from those terms that contain the two evolving parameters. This means that all Dxij that affect an evolving trait multiplicatively are identical and
have the same argument. Then the sum of the terms that contain the
evolving traits is the optimization criterion ψ. Two different scenarios can lead to this case: (i) Both tij and fij of the same matrix component aij
are evolving. (ii) The two evolving traits affect both a diagonal and an off-diagonal components of the projection matrix A and both matrix components consist of only a single trait, that is, of either a transition ratio or a fecundity term.
2. The two evolving traits occur in a single product in the fitness function w. This is can only be the case when the evolving traits affect the off-diagonal components of the projection matrix A and when additionally each off-diagonal component consists only of a single term, that is, when a012a021∈ {f0
12t021, f120 f210 , t012f210 , t012t021}. Then ψ = a012a021.
3. In the fitness function w none of the evolving characters occurs in a product with a function Dxij. This is the case when both diagonal components
a11 and a22 are evolving while density dependence only acts on the
off-diagonal components a12and a21, or vice versa. Then ψ = a11+ a22− a11a22
or ψ = a12a21, respectively. In these cases the corresponding re-scalings
mentioned in the section on invasion boundaries allow us to rephrase these optimization criteria such that invasion boundaries become linear.
4. When the evolving traits affect both an diagonal and an off-diagonal component of the fitness function w an optimization criterion can be found in several other cases. To see this we note that
Higher-Dimensional Feedback Environments
In general, whenever dim(I) > 1 an optimization criterion does not exist. If the interaction of the evolving population with the feedback environment allows for a rare type advantage, coexistence of different types becomes possible. In feedback environments with dim(I) > 1 in addition to CSSs and evolutionary repellors, evolutionary branching points and Garden of Eden-points become possible and we refer selection under this condition as frequency dependent.
The wider array of possible dynamics makes it more difficult to achieve a classification. We can do so only for models that are characterized by a high degree of symmetry, for example as when the two states correspond to two habitats of equal size and quality and a trade-off exists between the same measure of performance in each habitat. In this case we can understand the selective forces that determine the direction of evolutionary change by splitting invasion fitness into a density-dependent and a frequency-dependent component.
• Density-Dependent Component (DDC) Invasion fitness in a two-dimensional feedback environment is given by w(θ0, (I1, I2)). We can
calculate invasion fitness as it would result from a homogenous feedback environment where I1 = I2. Without loss of generality we choose ¯I =
(I1 + I2)/2 as reference environment, hence, we consider the function
w(θ0, ( ¯I, ¯I)). This function can account for density dependence but not for frequency dependence and we therefore refer to it as the density-dependent component of fitness. In the previous section we have proven that when population regulation is mediated via a single variable then this variable is maximized in the course of evolution. Hence, evolution in the reference environment I = ( ¯I, ¯I) would maximize (I1+ I2)/2. As mentioned in the
previous section, under this condition a function ψ from the evolving traits to the real numbers exists such that evolution in the reference environment would maximize ψ. In some cases we might be able to write out ψ explicitly. • Frequency-Dependent Component (F DC) We define the frequency-dependent component of fitness as the difference between invasion fitness proper and its density-dependent component. The frequency-dependent component can be visualized by its effect on invasion boundaries. Let us consider the case of a two-dimensional feedback environment I = (I1, I2).
An invasion boundary in the reference environment is defined implicitly by w((x, y), ( ¯I, ¯I)) = 1 (cf. eq. [9]). Any deviation of the invasion boundary in the reference environment from the real invasion boundary, implicitly defined by w((x, y), (I1, I2)) = 1, is the result of frequency-dependent selection.
the frequency-dependent component by F DC(θ0, θ)). We are interested in the difference of each of these components between a mutant and a resident: ∆DDC := DDC(θ0, θ) − DDC(θ, θ) and ∆F DC := F DC(θ0, θ) − F DC(θ, θ). A
mutant benefits from the DDC when ∆DDC > 0 and it benefits from the F DC when ∆F DC > 0. These two effects determine the direction of evolutionary change and the properties of evolutionary singular points. Whether a mutant benefits from the DDC depends on whether the mutation corresponds to an increase in the optimization criterion ψ. Whether a mutant benefits from the F DC depends on the relative difference between the two interaction variable I1
and I2. Whenever a mutation directs effort away from demographic parameters
that suffer strongly from density dependence towards a demographic parameters that suffer relatively less from density dependence, the mutation benefits from the differential impact of the resident population on the different environmental components and ∆F DC > 0.
In the following we describe a set of rather restrictive conditions that allows us to derive analytical results. We assume that (i) a θ∗exists such that I1(θ∗) = I2(θ∗),
(ii) dIi(θ∗)/dθ has opposite signs for i = 1 and i = 2, and (iii) the optimization
criterion ¯I has a local extremum at θ∗. The first condition means that a resident population with trait value θ∗ affects both interaction variables equally. The second condition means that any deviation from θ∗ alters the two interaction variables in opposite directions. The third condition in combination with the first one means that θ∗ is a singular point (cf. eq. 8). Under these conditions, which amount to a model with a highly symmetric structure, we have a good understanding of the selective forces driving the evolutionary dynamics in the neighborhood of θ∗ (table 2). In the example section we analyze a model where
the above assumptions are fulfilled.
Since under the assumptions above, ∆F DC equals zero at θ∗, invadability of θ∗ purely depends on ∆DDC. If ∆DDC has a local minimum at θ∗, then θ∗ is invadable by θ0 while the opposite holds true if ∆DDC has a local maximum at θ∗ (left and right column in table 2). Whether ∆DDC has a local minimum or maximum at θ∗ is determined by the curvature of the trade-off curve and the invasion boundary at θ∗. For θ∗ to be invadable by nearby mutants the invasion boundary has more concave than the trade-off curve such that the invasion boundary is a tangent to the singular point below the trade-off curve (chapter 1). The opposite pattern has to hold true for θ∗ to be uninvadable by nearby mutants. This paragraph can be read equivalently by replacing ∆DDC with the optimization criterion ψ.
Table 2: Classification of evolutionarily singular traits θ∗ with I1 = I2. The given signs of the frequency-dependent and density-dependent component have to hold for all mutants θ0and residents θ from within some neighborhood B = (θ∗− r, θ∗+ r), with r ∈ R > 0 and θ < θ0< θ∗ or θ∗< θ0< θ. ∆DDC > 0 ∆DDC < 0 ∆DDC + ∆F DC > 0 Branching Point ∆DDC + ∆F DC < 0 ∆F DC > 0 CSS Repellor ∆DDC + ∆F DC > 0 CSS ∆DDC + ∆F DC < 0 ∆F DC < 0 Garden of Eden Repellor
are positive, that is, when DDC and F DC act in the same direction, or when the two summands have opposite signs but with the positive summand overruling the negative one (table 2). From the preceding paragraph follows that a convergence stable singular point is a CSS when ∆DDC > 0 and an evolutionary branching point when ∆DDC < 0. An analogue distinction can be made for evolutionarily singular points that lack convergence stability, that is, when either both summands are negative or when the negative summand is larger in absolute value than the positive one. When ∆DDC is positive such an evolutionarily repelling singular point is uninvadable, that is, a Garden of Eden-point while the singular point corresponds to an evolutionary repellor when ∆DDC is negative (table 2).
Examples
Spatially-Structured Population with Juvenile Dispersal
We start with this model, because it easily allows us to assume a series of symmetry conditions that permit a more detailed qualitative analysis. Assume an iteroparous population which occupies two different habitats. Newborns disperse and settle in one of the two habitats where they stay for the rest of their life. Mutational change occurs in the habitat specific adult survival probabilities t11 and t22, which are
assumed to be traded off. We distinguish two scenarios of population regulation. (i) Adult fertility depends on one common resource (e.g., freely floating plankton) and therefore the offspring number decreases with increasing total population size N1+ N2. (ii) Adult fertility depends on a local resource (e.g., space within each
habitat) and therefore the offspring number decreases with increasing population density Ni in each habitat. Invasion fitness is then given by
w(θ0, θ) = ˜f11+ t11(θ0) + ˜f22+ t22(θ0) − ( ˜f11+ t11(θ0))( ˜f22+ t22(θ0)) + ˜f12f˜21. (16)
To be able to apply the results from the previous section we assume that adults have equal fecundity in both patches (f11+ f21= f22+ f12) and that juveniles are
equally likely to settle in either patch, hence: f11= f12= f21= f22. Furthermore,
we assume that the trade-off is symmetric, that is, t11max= t22max(cf. eq. [5]) and
that all juveniles are equally susceptible to crowding: Df11 = Df12 = Df21 = Df22.
From these symmetries follows that the habitat generalist with θ∗ = 0.5 is a singular point.
First we consider the case where all fecundities decrease with total population size. We know from the section on optimization that evolution maximizes total population size. Since evolution affects the diagonal components of A, invasion boundaries in this model are concave and therefore the bifurcation from a repelling generalist to an evolutionarily stable generalist occurs for a concave trade-off (z > 1). The exact bifurcation points can only be calculated numerically (see fig. 3a). These calculations confirm our qualitative predictions. The pitchfork bifurcation of singular points that is revealed by the numerical analysis is generic for symmetric systems like this one. However, with our method we are not able to predict whether the bifurcation is supercritical or subcritical.
Next we analyze the case where fecundities are decreasing functions of local densities: Df11( ˆN1), Df21( ˆN1), Df22( ˆN2), Df12( ˆN2). This model corresponds to the
cell given by the 1st row and the 4th column of table 1 where selection is frequency dependent. From the previous section we can conclude that a threshold z∗ > 1
exists such that for all combinations of mutants and residents with θ ≶ θ0 ≶ 0.5
(a)
z
θ
(b)
z
θ
Figure 3: Bifurcation of singular points for the example of a spatially structured population with bifurcation parameter z. Trade-off between t11and t22where t11 is decreasing in θ while t22is increasing in θ. Solid black lines: CSS; solid gray lines: evolutionary branching point; hatched lines: evolutionary repellor. (a) All fecundities decrease in N1+ N2 (Dfij = 1/(1 + N1+ N2)
for i, j ∈ {1, 2}), (b) ˜f11and ˜f12are decreasing functions of N1 (Df11 = 1/(1 + N1) = Df12)
while ˜f22and ˜f21are decreasing functions of N2(Df22= 1/(1 + N2) = Df21). Other parameter
values: t11max= 0.7 = t22max, f11= f22= f21= f12= 10.
z > z∗ the generalist θ∗ is a CSS. If z is slightly smaller than z∗, then ∆DDC becomes negative, however, ∆DDC + ∆F DC stays positive and θ∗ turns into an evolutionary branching point. When z becomes small enough such that the ∆DDC overrules the positive ∆F DC the singular trait-value θ∗turns into an evolutionary repellor (table 2). Figure 3b shows a numerically calculated bifurcation diagram of singular points that confirms our qualitative predictions concerning the generalist.
Age-Structured Life Cycle
Consider an age-structured population where fecundity of yearlings is given by f11.
These individuals survive with probability t21to the second year. Once this age is
reached, individuals produce each year f12offspring and survive with probability
t22 to the next breeding season. First we consider the case where mutational
change occurs in f11 and t21 which are traded off: individuals that invest a lot in
reproduction when they are young suffer from a decreased survival to adulthood. For a population dynamical equilibrium to exist reproduction has to be density dependent. In the first scenario we assume that individuals in both age-groups rely on a common resource for the production of offspring and that therefore both f11and f12decrease with I = c1N1+ c2N2 (c1, c2∈ R ≥ 0). In a second scenario
we assume that each age-group makes use of a different resource and that therefore reproduction decreases with the density in the corresponding age-group such that I = ( ˆN1, ˆN2). For both cases invasion fitness is given by
For the first scenario it follows from the section on optimization that I is an optimization criterion. When both age-groups are equally susceptible to competition, that is, when Df11 = Df12 we can deduce from the same section
that f11− f11t22+ t21f12 in another (numerically simpler) optimization criterion.
Extrema of this optimization criterion are characterized by dw(f110 , f11) df0 11 = 1 − t22+ t21 df12 df11 = 0. (18)
Note, that we have written fitness as a function of f11instead of θ. Since evolution
affects both diagonal and off-diagonal components, IBs are linear and we can conclude that extrema are maxima and therefore correspond to a CSS when the trade-off is concave while they are minima, corresponding to a repellor, when the trade-off is convex. Figure 4a shows a numerically calculated bifurcation diagram showing the features we predicted a priori. It is worth noticing that in this case expected density-independent life time reproductive output is also a optimization criterion (see Mylius and Diekmann (1995)). R0 is given by
R0(θ0, θ) = ˜f11(θ0) +
t21f˜12(θ0)
1 − t22
, (19)
and from equation (15) follows that sign[w(θ0, θ) − 1] = sign[R0(θ0, θ) − 1]. If
Df11 = Df12, then the function of density can be factored out and it follows the
result of R0-maximization.
If f11 and f12 are decreasing functions of N1 and N2, respectively, then the
model corresponds to the cell given by the 3rd row and 2nd column in table 1 and selection is frequency dependent. This is a case where we cannot provide qualitative predictions on analytical grounds because we cannot sensibly assume the necessary symmetry conditions. We therefore rely on a numerical exploration (fig. 4b-c). The first case (fig. 4b) shows the bifurcation of singular points when the same parameters as in the previous case are used (fig. 4a) and we see that the change in population regulation affects results only quantitatively. For figure 4c we assumed that two-year old individuals die after reproduction (t22= 0) and
that fecundity in the second year is lower than in the first year. For this set of parameters we find a bifurcation pattern that shows evolutionary branching for moderately strong trade-offs.
As a last example of an age-structured model we assume the same life-history but a different trade-off. Individuals that increase their chance to survive to adulthood t21 suffer from a decrease in future fecundity f12. From the last column of table
1 follows that t21f12 acts as an optimization criterion for all possible scenarios of
(c)
z
θ
(a)
z
θ
(b)
z
θ
Figure 4: Bifurcation of singular points for the example of a age-structured population with bifurcation parameter z. Trade-off between f11and t12where f11is decreasing in θ while t12is increasing in θ. (a) Fecundities decrease in N1+ N2(Df11= 1/(1 + N1+ N2) = Df12), (b) & (c)
˜
f11is decreasing in N1 while ˜f12is decreasing in N2 (Df11= 1/(1 + N1), Df12= 1/(1 + N2)).
Other parameter values: (a-c) f11max = 5, t21max = 0.8, (a) & (b) f12 = 10, t22 = 0.5, (c) f12= 3, t22= 0.
For the trade-off parameterization given by equation (5) it is easy to prove that θ∗= 0.5 is a unique maximum of the optimization criterion and therefore a CSS for all z > 0.
Size-Structured Life-Cycle
Assume that individuals can be categorized as either small or large with only the latter capable of reproduction. In this model we assume a trade-off between survival of mature individuals t22 and their reproductive output f12. The
model therefore addresses the question whether selection favours a single large reproductive event (semelparity, t22 = 0) or a compromise between reproduction
and survival that results in several reproductive events (iteroparity, t22 > 0).
Invasion fitness is given by
(a)
z
θ
(b)
z
θ
Figure 5: Bifurcation of singular points for the example of a size-structured population with bifurcation parameter z. Trade-off between f12and t22where f12is decreasing in θ while t22is increasing in θ. (a) Fecundity decreases with total population size (Df12= 1/(1 + N1+ N2)), (b)
newborn survival decreases with density of small individuals (Df12= 1/(1 + N1)) and survival of
large individuals decreases with density in this size class (Dt22 = 1/(1 + N2). Other parameter
values: f12max= 10, t22max= 0.8, t11= 0.5, t21= 0.5.
In this case IBs are linear. To analyze these models further we make use of equation (15) from which follows that we can use v(θ0, θ) := t21f12(θ0)/(1 − t22(θ0)) + t11as
a sign-equivalent fitness measure. We investigate three alternative scenarios with respect to population regulation where either only f120 , or f120 and t11 or f120 and
t022carry a tilde to indicate that they decrease with increasing population size.
In the first case only fecundity is a decreasing function of total population density ( ˜f21= f21Df12(N1+ N2)). This model was analyzed by Takada (1995). From the
section on optimization follows that ˆN1+ ˆN2 acts as an optimization criterion.
From the previous paragraph we see that t21f12(θ0)/(1 − t22(θ0)) is an alternative
optimization criterion. The fact that the feedback-environment is one-dimensional and that invasion boundaries are linear, allows us to conclude that intermediate singular points correspond to evolutionary repellors in the case of strong trade-offs and to CSSs in the case of weak trade-offs (see fig. 5a).
In the second version of this model we assume that fecundity of mature individuals and survival of small individuals are density-regulated according to ˜f12 =
f12Df12(N2) and ˜t11= t11Dt11(N1). This case corresponds to the cell given by the
3rd row and 3rd column in table 1. As in the previous paragraph we can conclude that t21f12(θ0)/(1−t22(θ0)) is maximized in the course of evolution. Hence, the fact
that density dependence affects two independent parameters has no influence on the existence of an optimization criterion and the bifurcation diagram is identical to the one of the previous version (fig. 5a).
In a last version of this model we assume that both fecundity and survival of mature individuals are density dependent according to ˜f12 = f12Df12(N1) and
˜
the 3rd column in table 1. In this case selection is frequency dependent and because of the inherent asymmetry we can only predict that the invadability property is lost when z < 1. Figure 5b shows a numerically calculated bifurcation diagram of singular points for this case. We see that the altered form of population regulation has only a quantitative effect on the evolutionary dynamics.
Discussion
In this article we classify a set of simple life-history models with respect to some criteria driving the evolution in two traits that are connected by a trade-off. Our main tools are (i) a sign-equivalent and algebraically simpler expression for invasion fitness, (ii) curvature properties of invasion boundaries, (iii) the dimension of the feedback environment, which allows to distinguish models that support optimization criteria from models that do not, and (iv) the decomposition of invasion fitness into a density-dependent and a frequency-dependent component. The results we present are not primarily motivated by specific biological questions but rather by a desire to understand the mechanisms that govern the evolutionary dynamics in a larger class of models. These mechanisms are formulated in a rather formal way, nevertheless, they can be translated to biological characteristics and we therefore think that this approach can help to understand patterns of biological evolution. The following general conclusions can be drawn: (i) Evolution in different diagonal components of the projection matrix A corresponds to concave IBs. This favors the occurrence of disruptive selection and diversification. (ii) Evolution in different off-diagonal components corresponds to convex IBs. This favors the occurrence of stabilizing selection on intermediate phenotypes. (iii) Evolution of traits that occur in different summands of the fitness function are a prerequisite for evolutionary branching. Such traits correspond to “alternative routes” in the life cycle. (iv) Evolution of traits that occur in a single product in the fitness function makes evolutionary branching impossible. Such traits correspond to “consecutive steps” in the life-cycle.
Open Questions and Extensions
First we discuss two unresolved issues of our classification that occur within the described model family. Then we will discuss several ways in which the model class can be extended by relaxing various assumptions.
selection. However, once one has decided that frequency dependence does act in a specific model, further analysis is only possible when several symmetry assumptions are met. Though moderate deviations from symmetric conditions will only lead to small quantitative changes in the bifurcation pattern of singular points, we have to admit that we lack strong analytical tools for the general case. Developing such tools seems to be a most challenging and rewarding extension. Another unsatisfactory issue is that cases exist where optimization is possible based on a one-dimensional I while no optimization criterion can be found that can be calculated directly from the trait values, that is, without determining population dynamical equilibria. This is for example the case when the two evolving traits affect different diagonal components of the population projection matrix while both are affected by density dependence in the same way. In this case the function of density Dxoccurs both as a linear and quadratic multiplier of the evolving traits
in the fitness function. This constellation renders all recipes as discussed above for optimization based on traits impossible.
In our model class we allow only for the simultaneous evolution of two traits. However, it is possible that one trade-off affects more than two matrix components. This is the case in a model analyzed by Kisdi (2002) who studies the evolution of habitat specific fecundity in a two-patch model. Relaxing this assumption makes the derivation of invasion boundaries more complicated and the existence of optimization criteria will be more restricted.
Another possible route to extend our results is to allow for non-equilibrium attractors. Especially for simple attractors like 2-cycles it might be possible to extend the logic of our approach. The population dynamics for a large class of two-state models has been described by Neubert and Caswell (2000).
The described model structure becomes considerably more complex when we drop the “separability”-assumption. Separability is not given when the effect of a resident type θ on a focal individual with trait θ0depends on the traits of both types as it is the case when competition is mediated by competition coefficients such as α(θ0− θ) = exp[−(θ − θ0)2/2σ] with σ being the width of the competition kernel
(e.g. Roughgarden, 1979; Doebeli and Dieckmann, 2000). The interpretation is that intraspecific interactions are mediated by quantitative traits like body size that determine the outcome of a competition. In this particular case the dimension of the feedback environment becomes potentially infinite. When the feedback is mediated through competition coefficients of the above form the tools developed here do not work anymore. Firstly, the equation for invasion boundaries lose their simple form and in some cases it will even be impossible to find an analytical expression. Secondly, optimization becomes impossible.
two states can be extended to models with more states. The fitness proxy w we used can be derived for models with more than two states in an analogous manner. Result (iii) & (iv) of the above list are obviously not restricted to two-state models but apply for any number of states.
Acknowledgments
Appendix: Spatially-Structured Population with Local
Density Dependence
Given f := f11 = f12 = f21 = f22, Df := Df11 = Df12 = Df21 = Df22 and
t11max = t22max we prove in this appendix that for all combinations of mutant
θ0 and resident θ with θ ≶ θ0 ≶ 0.5 we find ∆F DC > 0. First we note that at population dynamical equilibrium
ˆ
N1= (t11(θ) + ˜f11) ˆN1+ ˜f12Nˆ2 ⇔ 1 = t11(θ) + ˜f11+ ˜f12Nˆ2/ ˆN1
ˆ
N2= (t22(θ) + ˜f22) ˆN2+ ˜f21Nˆ1 ⇔ 1 = t22(θ) + ˜f22+ ˜f21Nˆ1/ ˆN2.
For a resident specialized for habitat type 1 (θ < 0.5 ⇐⇒ t11> t22) it follows
˜
f22+ ˆN1/ ˆN2f˜21> ˜f11+ ˆN2/ ˆN1f˜12.
Given the above symmetries we can rewrite the last inequality as f Df( ˆN2) 1 − ˆ N2 ˆ N1 ! > f Df( ˆN1) 1 − ˆ N1 ˆ N2 ! ,
which can only hold when ˆN1> ˆN2. An analogous reasoning holds for θ > 0.5.
Next we calculate the frequency-dependent component of invasion fitness (eq. [16]) for both a mutant θ0 and a resident θ with respect to the reference environment
¯
I = ( ˆN1+ ˆN2)/2 as determined by the resident θ:
F DC(θ0, θ) = w(θ0, [I1(θ), I2(θ)]) − w(θ0, [ ¯I(θ), ¯I(θ)])
= f (Df( ¯I)t22(θ0) − Df( ˆN1)t22(θ0) + Df( ¯I)t11(θ0) − Df( ˆN2)t11(θ0))
F DC(θ, θ) = w(θ, [I1(θ), I2(θ)]) − w(θ, [ ¯I(θ), ¯I(θ)])
= f (Df( ¯I)t22(θ) − Df( ˆN1)t22(θ) + Df( ¯I)t11(θ) − Df( ˆN2)t11(θ))
The fitness benefit for a mutant compared to that of the resident is given by the difference of the two frequency-dependent components:
∆F DC = F DC(θ0, θ) − F DC(θ, θ)
= f [(t11(θ) − t11(θ0))(Df( ˆN2) − Df( ¯I)) + (t22(θ) − t22(θ0))(Df( ˆN1) − Df( ¯I))].
From the first paragraph follows
θ ≶ 0.5 ⇐⇒ [ ˆN1(θ) ≷ ¯I(θ) ∧ ˆN2(θ) ≶ ¯I(θ)]
and from our trade-off parameterization equation (5) we know θ ≶ θ0⇐⇒ [t11(θ) ≷ t11(θ0) ∧ t22(θ) ≶ t22(θ0)].
