Roff D.A. Modeling Evolution: An Introduction to Numerical Methods

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selection pressures could be evaluated from total population size. Caswell et al.

(2004) explored this problem and produced a general theorem on density-depen-

dent sensitivity in matrix population models. The effective equilibrium density,

, is not the census number but rather a weighted value of each stage, the weights

being a function of the contribution to density-dependence and the effect of the

stage on l (¼ the dominant eigenvalue of the density-dependence matrix). At

equilibrium l ¼ 1. The effect of variation in some parameter y on l is measured

by its elasticity, which is deﬁned as the proportional change in l resulting from an

inﬁnitesimal proportional change in y. For detailed discussion of elasticity, see

Grant (1997), Grant and Benton (2000, 2003), Caswell (2002), and Van Tienderen

(2000). The elasticity of l to y is proportional to the elasticity of N

to y



@ N

ð1:9Þ

Any change that increases l will increase N

but not necessarily the total census

population. The sensitivity of the invasion exponent to a change in the parameter

y, is given by the elasticity of l to y



;

ð1:10Þ

from which it is evident that the invasion of a mutant will increase the effective

equilibrium density and the ESS (Evolutionarily Stable Strategy, a strategy that

cannot be invaded by another mutant) will maximize the effective equilibrium

density.

As noted earlier, for a homogeneous population at stable equilibrium r equals

zero and the characteristic equation reduces to equation (1.2) and ignoring the

density-dependent effect we have the net reproduction rate, R

(see equation [1.3]).

This parameter is one of the most widely used operational metrics of ﬁtness (e.g.,

Roff [1992]; Stearns [1992]; Charnov [1993]; see Chapter 2) but its use implies a

particular deﬁnition of the biological scenario, which is often not overtly acknowl-

edged. In order for R

to be an appropriate deﬁnition of ﬁtness either the density-

dependence is selectively neutral or the density-dependence is neutral with respect to the trait

under study (Roff 1992, p. 39). Determination of the optimal life history using r may

give a different answer to that obtained using R

(Roff 1992, pp. 183–184; Stearns

1992, pp. 31–33): Both answers cannot be right and the correct one (if either is

correct) depends upon the population dynamical assumptions. If the population is

assumed to be at equilibrium and the above assumption(s) of density-dependence

hold, then R

is appropriate. On the other hand, if the population is in a growing

phase and again the above assumption(s) of density-dependence hold, then r is

appropriate. If density-dependence is not selectively neutral, then neither metric

is appropriate and the analysis must take the selective effects of the density-

dependence into account (Mylius and Diekmann 1995; Benton and Grant 2000;

Brommer 2000).

8 MODELING EVOLUTION

1.2.5 Constant environment, variable population dynamics

Even in a constant environment a population may still show ﬂuctuations as a

result of the deterministic properties of the population model. A general and

much used example of this is the Ricker function (see Chapter 3):

tþ1

¼ lN

MN

ð1:11Þ

where N

is the population size at time t, l is the ﬁnite rate of increase at low

population numbers, and M is a parameter that could be the mortality of juveniles

resulting from competition or cannibalism by the parents. Depending on the

value of l, the population is either stable (1  l  2), oscillates with a period of

(where n is a positive integer, the value of n depending on the value of l, with

< l <e

2.6924

) or displays chaotic ﬂuctuations (l > e

2.6924

What we would like to know is whether a mutant can invade such a population,

which is generally termed the resident population. To ﬁnd this out we consider

the situation at the beginning of the process when the mutant is so rare that it

cannot have a signiﬁcant effect on the dynamics of the system. If under these

circumstances the mutant can increase in frequency, then we presume that it will

increase to ﬁxation in the population. Note that this assumption presupposes no

frequency-dependence. Nor does it suppose that there is necessarily a unique

parameter set that is resistant to invasion by all other mutants (see below and

Chapter 3 for further discussions). We can write the trace for the resident popula-

tion as

R;t

¼ l

M

R;t

R;t1

¼ l

M

R;t2

R;t

¼ N

R;0

i¼0

M

R;i

ð1:12Þ

where the subscript R designates the parameters of the resident population.

Taking logs gives

lnN

R;t

 lnN

R;0

¼ tlnl

 M

i¼0

R;i

ð1:13Þ

Taking limits gives

lnl

 M

i¼0

R;i

¼ lim

t!1

EðlnN

R;t

 lnN

R;0

Þð1:14Þ

which is the dominant Lyapunov exponent, given the symbol s by Ferriere and

Gatto (1995). Because a mutant will be in insigniﬁcant numbers in the initial

invasion, the trace of population numbers is given by the trace of population

numbers of the resident population, that is,

i¼0

R;i

. Thus, the invasion (Lyapu-

nov) exponent of a mutant, s

, is given by

¼ lnl

 M

i¼0

R;i

ð1:15Þ

OVERVIEW 9

and the condition for the mutant to invade is

lnl

ð1:16Þ

In the above example, it is possible to derive an exact expression for the invasion

(Lyapunov) exponent: This will frequently not be the case and numerical methods

will have to be employed (see Chapter 3). Nothing in the above theory precludes

the existence of a polymorphism, and indeed the origin of the theory for temporal

variation, discussed later, was initiated by the presence of dimorphism for dor-

mancy in plants (Cohen 1966).

1.2.6 Temporally stochastic environments

Environments are rarely if ever temporally stable and such variation is likely to be

reﬂected in variation in vital rates. In general, a population growth rate converges

to a ﬁxed quantity, which Tuljapurka (1982) labeled a to distinguish it from the

Malthusian parameter. In a constant environment a is equivalent to the Malthu-

sian parameter. Population size at some time t can be represented by

¼ l

t1

¼ l

t2

¼ N

i¼0

ð1:17Þ

Taking logs gives

lnN

¼ lnN

i¼0

ln½l

ð1:18Þ

As noted earlier, under relatively unrestricted conditions – namely, (a) demo-

graphic weak ergodicity, (b) the random process generating vital rates is stationary

and ergodic, and (c) the logarithmic moment of vital rates is bounded (Tuljapurkar

1989; see Tuljapurkar [1990] for a deﬁnition of demographic weak ergodicity) –

the value of N(t) becomes independent of the initial condition, N

, and the long-

run growth rate and hence the ﬁtness of a particular life history is given by Cohen

(1966), Tuljapurkar and Orzack (1980), and Caswell (2001):

lnl ¼ lim

t!1

EðlnN

 lnN

Þð1:19Þ

Fitness is measured by the geometric mean of the ﬁnite rate of increase. The

geometric mean rate of increase, r



, is a function of the arithmetic mean ﬁnite

rate of increase,



, and its variance, s

. Using a Taylor series expansion an approx-

imate formula is (Lewontin and Cohen 1969)



¼ EðlnlÞln l



2 l

2

ð1:20Þ

10 MODELING EVOLUTION

The important point is that increases in the variance in the rate of increase

decrease ﬁtness and thus selection will favor strategies that both increase the

arithmetic rate of increase and decrease it variance. One such manner in which

the latter can be achieved is by producing variation in offspring phenotypes. This

concept appears to have been put forward at least three times since 1966. It is

implicit in Cohen’s analysis (1966) of the optimal germination rate in a randomly

varying environment, was explicitly advanced verbally by den Boer (1968), who

referred to it by the term “spreading the risk,” and ﬁnally discussed by Gillespie

(1974, 1977) in the context of variation in offspring number. Slatkin (1974), in

reviewing Gillespie’s work, labeled the phenomenon as “bet-hedging,” a term

that has stuck. The forgoing arguments apply to populations of inﬁnite size, but

we might expect from the analysis of Demetrius and Ziehe (2007) that this ﬁtness

measure may break down at low population sizes. Indeed, for a particular scenario

in which there is a common and a rare environment (King and Masel 2007) showed

that bet-hedging would not be favored when

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ



s þ 1







ð1:21Þ

where N is the population size,s is the selective advantage associated with switch-

ing in the rare environment, and y is the rate of encountering the rare environ-

ment.

With age structure, the equivalent measure of the long-term population growth

rate in relation to the arithmetic average is (Orzack and Tuljapurkar 1989)

a  lnl 

ð1:22Þ

where lnl is the dominant eigenvalue of the average Leslie matrix, S is a column

vector of the sensitivities of l to a ﬂuctuation in the matrix elements (i.e.,

¼ @lnl=@x

, where x

is the ij element), S

is its transpose, and V is a variance–

covariance matrix of the elements (x

Equation (1.22) can be illustrated with a simple two-age class model described by

Tuljapurkar (1989). Population change is described by the equation

tþ1

¼ A

ð1:23Þ

where

1;t

2;t



and A

S 0

ð1:24Þ

Fecundity at age i equals m

and survival from age class 1 to age class 2 equals S.

Uncorrelated temporal variability is described by the parameter x which follows a

gamma distribution with probability density function:

PðxÞ¼

ðv  1Þ!

v1

vx

ð1:25Þ

OVERVIEW 11

The parameter x measures the variance, with the variance increasing as n ap-

proaches zero and x approaching 1 as n approaches inﬁnity. If the parameters are

ﬁxed at their average values the ratio m

converges to a stable value, say R*.

The growth rate of the population is then given by

r ¼ ln l



¼ ln







ð1:26Þ

The long-run average growth rate of the population with temporal variability, a,is

approximately

a  r 

2xl





ð1:27Þ

where C ¼ 2 fm

x=½ðx  1Þlg. As in the case of equation (1.20) the average

growth rate is diminished by variability in the vital rates. Thus it is insufﬁcient

to determine the most ﬁt life history using the growth rate from the averaged

values of the life history.

While the fate of a gene or mutant can be determined by the geometric mean or

long-run growth rate, and thus ﬁtness can be so deﬁned for the sake of modeling,

Lande (2007, p. 183) has shown that “these measures fail to describe the expected

short-term dynamics of gene frequencies or mean phenotypes, by which expected

selection coefﬁcients and expected relative ﬁtnesses should be deﬁned.” The

expected relative ﬁtness of an individual is the Malthusian ﬁtness of the genotype

or phenotype in the average environment minus the covariance of its growth

rate with that of the population. A consequence of this is that the expected

relative ﬁtness is frequency-dependent (Land 2007). This result is important in

correctly deﬁning ﬁtness but, as noted earlier, this does not change the utility

of the geometric mean or long-run growth rate as a metric by which to calculate

the optimal combination of trait values.

1.2.7 Temporally variable, density-dependent environments

From the following discussions the most appropriate measure of ﬁtness is the

invasion exponent. Given the complexity of the interactions it is likely that

analytical solutions will not be typically available and one will have to resort to

simulation analysis. Benton and Grant (2000) investigated the reliability of alter-

nate measures of ﬁtness for models in which there was both density-dependence

and temporally uncorrelated variation. Four models of density-dependence were

investigated: Beverton and Holt-type, Ricker-type, Usher-type with gradual onset

of density-dependence, and Usher-type with sudden onset of density-dependence.

Beverton and Holt-type models produce a stable equilibrium, whereas the Usher-

type with sudden onset of density-dependence generally produces chaotic behav-

ior. The dynamical behavior of the other two depends on parameter values,

though Benton and Grant (2000, p. 773) state that “the vast majority of

other combinations of density-dependence ...resulted in equilibrium dynamics.”

Given the predicted differences between models with equilibrium versus

12 MODELING EVOLUTION

nonequilibrium dynamics it is unfortunate that the analysis did not divide the

results both according to the four-model types and the two-dynamical behaviors.

Benton and Grant (2000) considered the following “surrogate” measure of ﬁtness:

r, R

and a estimated both with and without density-dependence effects and the

average (both arithmetic and geometric) population size, K.

First, Benton and Grant simulated constant environments and found, as expected,

that for the chaotic models none of the ﬁtness criteria performed well. On the other

hand, the DI R

and K performed well for the Beverton–Holt model, which does not

exhibit chaotic behavior. In a stochastic environment the best predictor of the

invasion exponent was K, although it has to be remembered that the density-depen-

dence in the models was a direct function of total population size. The general

message from the analyses is that if the population is expected to show variable

dynamics, either due to environmental ﬂuctuation or intrinsic population dynami-

cal properties, and density-dependence is not a consequence of a response to total

population number. the only viable measure of ﬁtness is the invasion exponent.

However, the result in a model with chaotic population dynamics may also depend

upon the mode of inheritance (compare Scenario 3 of Chapter 3 with Scenario 5

in Chapter 4). In populations showing more or less stable equilibria the density-

independent R

appeared to be a reasonable measure, which is reassuring, given the

considerable number of analyses based on this ﬁtness measure.

1.2.8 Spatially variable environments

Starting with Levene (1953) there has been a considerable number of population

and quantitative genetic analyses of the conditions required for the maintenance

of genetic variation (reviewed in Roff [1997]). So far as I am aware, these analyses

have assumed nonoverlapping generations (i.e., no age structure). The solution to

deﬁning ﬁtness when the environment is spatially variable and there is a stable-

age distribution was enunciated independently by Houston and McNamara (1992)

and Kawecki and Stearns (1993). The critical realization in deriving the solution

was that ﬁtness must be measured over the entire environment simultaneously

and not patch by patch. Thus, if we take r as the appropriate ﬁtness measure

(meaning that we assume an equilibrium population) the measure that selection

will maximize is the rate of growth of the population as a whole

PðhÞ

Vðx; hÞe

r

Pop

dx ¼ 1 ð1:28Þ

where r

Pop

is the rate of growth of the entire population (as opposed to the rates of

growth within each patch), P(h) is the probability of patch of type h occurring,

and V(x, h) is the value of l(x)m(x) for patch of type h. One would expect that in a

spatially variable world a reaction norm would evolve to modify the life history

patterns in response to the habitat parameters, the evolutionary change obviously

being dependent on the presence and predictability of cues that indicate habitat

type. Nevertheless, the maximization of ﬁtness within each patch is subject to the

constraint imposed by equation (1.28).

OVERVIEW 13

For density-dependent populations in which equilibrium is attained and for

which density-dependence is assumed to be selectively neutral the appropriate

criterion is the net reproduction rate, R, and the ﬁtness criterion becomes

Pop

PðhÞ

Vðx; hÞdx ð1:29Þ

meaning that selection will favor the life history that maximizes R for the popula-

tion s as a whole (Charlesworth 1994). If density-dependence is not selectively

neutral, then equation (1.29) must include those effects.

1.2.9 Social environment

In the environments so far discussed, the relationship between individuals is of no

consequence because social interactions are absent. In this book I shall not explic-

itly consider the social environment, although it can be accommodated within the

various analytical frameworks. When survival or reproduction depends upon

interactions between individuals that might be related it is necessary to take

into account the increment of ﬁtness accruing to the individual by virtue of

such interactions. Two relatively well-studied social phenomena are altruism

(Koenig 1988; Dugatkin and Reeve 1994, 1998; Thorne 1997; Ratnieks and Wen-

seleers 2008) and “helpers-at-the-nest” behavior (Koenig et al. 1991; Bshary and

Bergmueller 2008; Carranza et al. 2008).

The overall ﬁtness, inclusive of interactions among relatives, was termed inclu-

sive ﬁtness by Hamilton (1964), though, because of the obscurity of Hamilton’s

deﬁnition, it was, at least initially, frequently interpreted incorrectly (Grafen

1982). Operationally, inclusive ﬁtness can be deﬁned, or replaced by, Hamilton’s

rule, which states that organisms are selected to perform actions for which



b  c > 0 ð1:30Þ

where r* is relatedness, and b, c refer to the effects of an allele on offspring

production: bearers of this allele behave in such a manner that each has c fewer

offspring, and the bearer’s sib has b more offspring (Grafen 1984). Queller (1996)

noted that it is phenotypes that interact not genotypes and suggested replacing r*

with Cov(G

, P

)/Cov(G

, P

), where G

is the genetic value of the “actor” or focal

individual, P

is its phenotypic value, and P

is the phenotypic value of the average

phenotype. For other formulations of the relatedness coefﬁcient see Pepper

(2000). Taylor et al. (2006) expanded Hamilton’s rule to a class-structured model,

while Gardner et al. (2007) provide a multilocus version of the rule. Oli (2002)

provides a method of estimating inclusive ﬁtness in an age-structured population

using a Leslie matrix formulation. For other modiﬁcations of Hamilton’s rule that

have been advanced to account for such things as nonadditivity of ﬁtnesses see

Fletcher and Zwick (2006).

More generally, b and c in equation (1.30) are referred to as the beneﬁts and

costs, respectively. A potential problem with using Hamilton’s rule is in opera-

tionally deﬁning these costs and beneﬁts, leading some to attempt to use a more

14 MODELING EVOLUTION

direct deﬁnition of inclusive ﬁtness, which in turn has led to discussion over how

to correctly calculate this quantity. The issue lies in the verbal description given by

Hamilton (1964) that inclusive ﬁtness is the sum of the ﬁtness that would be

obtained in the absence of the social environment (e.g., helpers at the nest) and

the added increment due to the presence of the social environment. The problem

is in calculating the former quantity. Creel (1990) pointed out that a potential

paradox can arise if the social environment is essential for successful reproduc-

tion, as is almost the case for the dwarf mongoose, Helogale parvula. Stripping away

the social environment leaves the reproductive individual with zero ﬁtness, all the

ﬁtness being attributed to the helpers. Thus there should be contest to be helpers

and not reproductives, which is clearly not the case and makes no sense geneti-

cally. Creel’s solution to this paradox was shown by Queller (1996) to be inappro-

priate and that the solution resides in recognizing that Hamilton’s rule applies

strictly only when ﬁtnesses are additive, which in the mongoose case they are not.

The paradox is removed when nonadditive versions of Hamilton’s rule are used

(Queller 1996; Pepper 2000; West et al. 2002).

1.2.10 Frequency-dependence

A reasonably general deﬁnition of frequency-dependent selection is that given by

Ayala and Campbell (1974, p. 116): “The selective value of a genotype is frequency

dependent when its contribution to the following generation relative to alterna-

tive genotypes varies with the frequency of the genotype in the population.”

There are, however, other deﬁnitions, which though similar, can be subtlety

different, or more restrictive in the sense that stable coexistence is required

(Heino et al. 1998). There is no reason why a stable equilibrium frequency of

genotypes should be a requirement of frequency-dependent selection and some

very simple games such as “Rock-Paper-Scissors” which are clearly frequency-

dependent do not have a stable equilibrium (Maynard Smith 1998; see Chapter

6). Most models of frequency-dependent selection assume either competition

between clones or Mendelian inheritance with a ﬁxed generation time. In either

case ﬁtness is deﬁned in terms of the contribution of types (genotype or pheno-

type) to the subsequent generation.

An example of frequency-dependence is the occurrence of two types of males in

several ﬁsh species, particularly salmon: One type of male is territorial whereas

the other is typically smaller, matures earlier, cannot maintain a territory, and

attempts to sneak fertilizations (Gross 1982, 1985; Hutchings and Myers 1988).

The analysis of the equilibrium combination of the two types in the population

has either used R

as the ﬁtness measure (Gross and Charnov 1980) or r (Hutchings

and Myers 1994). A more frequently used approach is that of Game theory, in

which the relative ﬁtness of each type when interacting either with another of its

type or another type is represented by a payoff matrix. The classic example of this

approach is the Hawk-Dove game (Maynard Smith 1982): In this scenario there is a

2  2 payoff matrix indicating the payoff to a hawk when it interacts with either

another hawk or a dove and the payoff to a dove when it interacts with either a

OVERVIEW 15

hawk or a dove. The game is frequency-dependent because although a hawk

interacting with a dove has a higher ﬁtness than the dove, a hawk interacting

with another hawk suffers a decrement in ﬁtness. The equilibrium frequency of

hawks and doves in the population depends upon the relative values in the payoff

matrix and is called an ESS. It is obtained simply by equating the payoff to hawks

with the payoff to doves: at equilibrium the two must be equal. In simple terms an

ESS is one that cannot be invaded by a mutant playing an alternate strategy (see

Hammerstein [1998] for a more formal deﬁnition). Game theoretic models are

discussed in detail in Chapter 6.

1.3 Some general principles of model building

Models are not replicas of nature: If they were they would be just as complicated

and equally hard to understand. The purpose of a model is to extract the essential

elements that deﬁne the problem under study. Having done this we investigate

the impact of the model components and compare the predictions of the model

with nature. Should there be an obvious discrepancy we return to the model and

examine the underlying assumptions: A model is simply the logical outcome of

the assumptions and thus any failure to ﬁt reality is a failure of the assumptions.

Having modiﬁed the model we again compare predictions and observations,

repeating the process until a satisfactory ﬁt is obtained.

In constructing a model the following should be kept very much to the fore:

1. Keep the model as simple as possible and focus upon the problem. Modeling the

mechanism for telling time provides an instructive example of this process.

The modern digital watch is a highly complex affair and seemingly vastly

different from the earliest mechanical clocks. Further, when one looks at the

history of clocks and watches one sees an enormous variety of mechanisms. Yet

under all this complexity and variety, all mechanical or electrical clocks have

ﬁve elements in common that determine how time is monitored: “(1) a source

of energy (spring or battery); (2) an oscillating controller (balance or quartz

crystal); (3) a counting device (escapement or solid state circuit); (4) transmis-

sion (wheelwork or electric current); (5) display (hands or liquid crystal seg-

ments)” (Landes 1983, p. 377). All mechanical or electrical clocks must satisfy

these requirements. Thus to ﬁnd out how a clock works one must strip away

the extraneous details such as the size of the clock, whether it gives the date or

altitude or compass direction and look for these ﬁve preceding elements.

2. Make assumptions explicit. Verbal models are frequently “preferred” because

they seem less conﬁned than a mathematical model but in reality verbal

models are generally full of “hidden” assumptions that may well result in any

conclusions to come crashing down once these assumptions are noted. In this

book I adopt the policy of beginning with a general conceptual model and then

move to a mathematical construct based on the general assumptions. For

example, we might assume that there is a negative relationship between the

size and number of offspring that a female produces. This statement is very

16 MODELING EVOLUTION

general and might be sufﬁcient in some analyses but most cases an analysis will

require a more detailed speciﬁcation such as that the number of offspring is

proportional to the reproductive biomass divided by offspring size.

3. This book is primarily concerned with numerical analysis of models: If an

analytical solution is possible, then it is to be preferred. Such solutions may

be possible only on very simpliﬁed versions of the model and numerical

analysis of more complex scenarios may reveal inadequacies in the simple

analytical solution.

4. While simplicity is desirable it is important to maintain a reasonable level of

realism. In this regard it is important to provide operational deﬁnitions of all

parameters and variables in the model. If a variable cannot be measured, then it

is not useful and an alternate approach should be sought.

5. As much as possible, write the model incrementally and as a series of modules

that can be examined and debugged separately.

To illustrate these points the next section constructs a model of the evolution of

migration in a spatially and temporally heterogeneous environment.

1.4 An introduction to modeling in R and MATLAB

The purpose of this section is twofold: First, it is to outline, by using a simple

example, the process of creating a model to address an evolutionary question, and

second to illustrate the most important R and MATLAB codes used in the remain-

der of the book.

The problem we shall consider is that of the evolution of migration in a hetero-

geneous environment. As used in all the scenarios throughout this book we begin

ﬁrst by outlining a conceptual model and then convert this model into one that

can be programmed.

1.4.1 General assumptions

1. The environment is heterogeneous in time and space.

2. This heterogeneity affects population dynamics by causing variation in the vital

statistics of the population (e.g., fecundity and survival) and the carrying capac-

ity of the environment.

These two assumptions are too general to be programmed as such and must be

converted into a suitable form by addressing the underlying mathematical as-

sumptions, which will necessarily restrict the model to some extent. While we

could pose a mathematical model that included the processes outlined above it

would include factors, such as age structure, that may not be important to the

central issue but could complicate the analysis. Thus to start we begin with a very

simple model and ask if in this case spatial and temporal heterogeneity could

be an important selective agent. This does not prove that such variation is an

OVERVIEW 17