# The evolution equations under different year classes of a age-structured model?

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This sounds a bit complicated but I want to grab more feelings on age-structured problems. Less than 2 days to the exam so I appreciate any help.

So suppose we only get 3 year classes in a school at first. No one can repeat. If you are kicked out of school you must restart from year class $$1$$. No transfer-in students. Exactly $$N$$ freshman each year. The portion of being promoted from year class $$i − 1$$ to year class $$i$$ is given by $$s_i$$ (which we can again assume it's constant). Then suddenly at one year $$Y$$, the school changes its scheme. Only at that year the school take in $$N$$ year $$0$$ and year $$1$$ students respectively. Since year $$Y + 1$$, the school only take in $$N$$ year $$0$$ students each year.

What I want to know the most is how we should decide (1) the steady state no. of the students in year classes $$1$$ to $$3$$ before year $$Y$$, (2) the steady state no. of the students in year classes $$0$$ to $$3$$ since year $$Y$$, and (3) the school years needed to achieve the steady state since year $$Y$$.

1. I tried putting forward the Leslie matrix of the old scheme: $$egin{bmatrix} 1 & 0 & 0 s_1 & 0 & 0 0 & s_2 & 0 end{bmatrix}$$ and found the eigenvalues would be $$1$$ and $$0$$ (with multiplicity of $$2$$). $$lambda = 1$$ gives $$u_{1, n} + u_{2, n} + u_{3, n} = u_{1, n}(1 + s_1 + s_1 s_2).$$ Am I on the right track? Should I just plug in $$u_{1, n} = N$$ and then claim the steady state no. to be $$N(1 + s_1 + s_1 s_2)$$? Or do I need to sort of prove that?

2. If my thought process is correct, the Leslie matrix of the new scheme becomes $$egin{bmatrix} 1 & 0 & 0 & 0 s_0 & 0 & 0 & 0 0 & s_1 & 0 & 0 0 & 0 & s_2 & 0 end{bmatrix}$$ Right? Then its eigenvalues are again $$1$$ and $$0$$ (multiplicity $$3$$), and then $$lambda = 1$$ gives (let me denote the intake proportion $$s_{-1} = 1$$) $$u_{0, n} + u_{1, n} + u_{2, n} + u_{3, n} = u_{1, n}(1 + s_0 + s_0 s_1 + s_0 s_1 s_2) = u_{1, n}left(sum_{i=-1}^{2} prod_{j = -1}^i s_j ight).$$ Did I make any mistakes so far?

3. I can sort of write out the year class size since year $$Y$$ explicitly. It should take 3 school years to achieve the steady state again right? Any less tedious and/or more persuasive ways to show the same result?

## Objectives:

Leslie matrix is a discrete, age-structured model of population growth that is very popular in population ecology. It was invented by and named after P. H. Leslie. The Leslie Matrix (also called the Leslie Model) is one of the best known ways to describe the growth of populations (and their projected age distribution), in which a population is closed to migration and where only one sex, usually the female, is considered. This is also used to model the changes in a population of organisms over a period of time. Leslie matrix is generally applied to populations with annual breeding cycle. In a Leslie Model, the population is divided into groups based on age classes (see Fig. 1) . A similar model which replaces age classes with life stage is called a Lefkovitch matrix, whereby individuals can both remain in the same stage class or move on to the next one. At each time step the population is represented by a vector with an element for each age classes where each element indicates the number of individuals currently in that class. The Leslie Matrix is a square matrix with the same number of rows and columns and the population vector as elements. The (i,j) th cell in the matrix indicates how many individuals will be in the age class, i at the next time step for each individual in stage j. At each time step, the population vector is multiplied by the Leslie Matrix to generate the population vector for the following time step. To build a Leslie matrix, some information must be known from the population:

• nx, the number of individual (n) of each age class x.
• sx, the fraction of individuals that survives from age class x to age class x+1.
• fx, fecundity, the per capita average number of female offspring reaching n0, born from mother of the age class x. More precisely it can be viewed as the number of offspring produced at the next age class mx+ 1 weighted by the probability of reaching the next age class. Therefore, fx = sxmx + 1.

The observations that n0 at time t+1 is simply the sum of all offspring born from the previous time step and that the organisms surviving to time t+1 are the organisms at time t surviving at probability sx , we get nx + 1 = sxnx. This then motivates the following matrix representation:

Where &omega is the maximum age attainable in our population.

Where the population vector at time t and L is the Leslie matrix.

The characteristic polynomial of the matrix is given by the Euler-Lotka equation.

The Leslie model is very similar to a discrete-time Markov chain. The main difference is that in a Markov model, one would have
fx + sx = 1 for each x, while the Leslie model may have these sums greater or less than 1.

### How to create a Leslie Matrix:

Population vector

s+1 rows by 1 column , (s+1) *1 . Here s is the maximum age.

### Birth :

Newborns = (Number of age 1 females) times (Fecundity of age 1 females) + (Number of age 2 females) times (Fecundity of age 2 females) + . Note: fecundity here is defined as number of female offspring. Also, the term "newborns" may be flexibly defined (e.g., as eggs, newly hatched fry, fry that survive past yolk sac stage, etc.

### Mortality:

Number at age in next year = (Number at previous age in prior year) times (Survival from previous age to current age)

## Intraspecific competition and components of niche width in age structured populations

A model is presented for intraspecific exploitative competition among age classes in animal populations. The animals live for several time units and grown continuously in size until they die. Recruitment takes place at the end of each time unit. It is strictly synchronized, resulting in cohorts of age classes. Newborn individuals are similar to adults in shape and feeding behavior. The juvenile period lasts from one to several time units. The animals use renewable but limited food resources. The average niche position with respect to food size is a function of animal size. Overlap in resource utilization among the age classes results in exploitative competition.

Besides a special case that can be treated analytically, the population dynamics is studied by numerical simulations. An increase in size independent fecundity rates or a decrease in the density dependence of growth rates tends to stabilize the population dynamics, that is when the population has a high rate of increase. An increase in the number of age classes tends to destabilize the population dynamics. In general, cycles or chaos are less likely to occur in our model of intraspecific competition in age structured populations, where death is assumed to be a continuous process, than is predicted from comparable models that assume a discrete death process.

Furthermore, when characterizing the total variance of resource utilization of a population during one time unit, we define besides the three classical variance components, within-phenotype, between-phenotype, age structure, a fourth, temporal component.

## Age-structure and transient dynamics in epidemiological systems

Mathematical models of childhood diseases date back to the early twentieth century. In several cases, models that make the simplifying assumption of homogeneous time-dependent transmission rates give good agreement with data in the absence of secular trends in population demography or transmission. The prime example is afforded by the dynamics of measles in industrialized countries in the pre-vaccine era. Accurate description of the transient dynamics following the introduction of routine vaccination has proved more challenging, however. This is true even in the case of measles which has a well-understood natural history and an effective vaccine that confers long-lasting protection against infection. Here, to shed light on the causes of this problem, we demonstrate that, while the dynamics of homogeneous and age-structured models can be qualitatively similar in the absence of vaccination, they diverge subsequent to vaccine roll-out. In particular, we show that immunization induces changes in transmission rates, which in turn reshapes the age distribution of infection prevalence, which effectively modulates the amplitude of seasonality in such systems. To examine this phenomenon empirically, we fit transmission models to measles notification data from London that span the introduction of the vaccine. We find that a simple age-structured model provides a much better fit to the data than does a homogeneous model, especially in the transition period from the pre-vaccine to the vaccine era. Thus, we propose that age structure and heterogeneities in contact rates are critical features needed to accurately capture transient dynamics in the presence of secular trends.

### 1. Introduction

The recurrent epidemics of immunizing infectious diseases, such as measles, mumps and rubella, represent well-documented examples of cyclic population dynamics [1–5], especially before the advent of routine infant immunization. Historically, such diseases mainly affected children owing to their extreme contagiousness and the long-lasting immunity elicited by infection. These characteristics, along with the direct mode of transmission of these diseases, mean that the epidemiological dynamics of many childhood diseases are capably modelled using the susceptible–exposed–infected–recovered (SEIR) model framework [6–9].

Early attempts to explain the determinants of these dynamics were initially focused on models that were not explicitly age-structured (e.g. [2,10–13]). We call these homogeneous models since ignoring age structure is equivalent to assuming homogeneous mixing, such that individuals from all age classes contact each other at the same rate. These models were able to reproduce the qualitative dynamics of some diseases by capturing key mechanisms: seasonal variation in contacts and susceptible depletion. However, following the classic work of Schenzle [14], Bolker & Grenfell [15] argued that quantitatively capturing pre-vaccine biennial cycles of measles required the explicit consideration of age-stratified pattern of contacts. The importance of age structure was subsequently called into question by Earn et al. [16], who demonstrated that the key ingredient necessary for a homogeneous model to successfully explain measles epidemics was the use of an appropriate seasonal forcing function (mimicking schools opening and closing), rather than age structure per se (see also [17]). These authors further demonstrated that, via a linear change of variables, a single bifurcation diagram may be constructed to summarize measles dynamics in response to changes in per capita birth rates or trends in vaccine uptake.

Models have been less successful at recreating dynamics during transition from the pre-vaccine to vaccine era of disease transmission. Under the assumption that the susceptible population is replenished by births and that a vaccine confers perfect protection against infection, theoretically, the vaccine era dynamics should be similar to the pre-vaccine era but with birth rates reduced to reflect the vaccine coverage reducing entry into the susceptible population, as predicted by Earn et al. [16]. However, a mathematical transmission–vaccination model that can capture key features of the observed transition from the regular pre-vaccine era measles epidemics to the more irregular vaccine era disease dynamics has remained elusive. While a homogeneous model with appropriately discounted susceptible influx rate can adequately reproduce the large decline in incidence after the introduction of vaccination, the transient dynamics accompanying the decline have been difficult to capture, in particular, features such as the changing periodicity [18].

In this paper, we compared the dynamics of a homogeneous model and an age-structured model of measles during the transition from the pre-vaccine to vaccine era. In the age-structured model, school-aged children were assumed to have high, seasonally varying contact rates (due to school-term forcing), while adult contact rates were lower and constant throughout the year. As in a homogeneous model, in the age-structured model, vaccination has the obvious effect of decreasing the fraction of the population susceptible to measles and hence a reduced mean transmission rate. However, age-structured contact rates lead to an additional effect of vaccination: reduced effective amplitude of seasonal forcing. This is due to the shift in transmission from primarily children to older age groups in which contact rates are less seasonal.

To illustrate the dynamical impact of age structure, we compared goodness of fit of a homogeneous model to that of a model with age structure, using historical measles data from London. In order to quantify the performance of models in explaining transient dynamics, we compared model fits for the pre-vaccine (1945–1968), pre-vaccine and early vaccine (1945–1978) and pre-vaccine to modern vaccine era (1945–1990). Our aim in this study was to examine the hypothesis that models that can capture the dynamic feedback between susceptible recruitment rates and the shifting age distribution of prevalence, together with the concomitant impact on the effective amplitude of seasonality will better explain the data. As a result, our models were deliberately simple and deterministic. We found that the age-structured model provided a better explanation of the data than the homogeneous model in both the pre-vaccine and the vaccination era.

Previous studies have recognized that age structure is an important component of the response of measles dynamics to vaccination [19–25]. In this paper, we emphasize that age structure is particularly relevant when there are secular trends in transmission, including the transition period soon after the start of routine immunization. We also provided empirical support for this claim by showing that age structure substantially improves the fit of a minimally complex model of measles vaccination to data. More broadly, our findings imply that age structure and heterogeneities in contact rates should be accounted for to capture transient dynamics associated with trends in the transmission of immunizing infectious diseases.

### 2. A transmission–vaccination model with an arbitrary number of age classes

We considered a standard SEIR model, with an additional V component for individuals vaccinated at birth [9,26–28]. Each compartment was further divided into M age classes (M = 1 for the homogeneous model). For each age class i (i = 1 to M), we set Ni to be the total population of age class i and assume that this remains constant for all time t. Thus, Vi(t) + Si(t) + Ei(t) + Ii(t) + Ri(t) = Ni for all t and we further assume that N = ∑ i = 1 M N i = 1 . The model is illustrated in figure 1 and the model equations are given in (2.1). The parameters of this model are described in table 1 and mathematical properties of this model are discussed in Magpantay [29].

Figure 1. Schematic diagram of the age-structured compartmental model of vaccination used in §2–4.

## SIAM Journal on Applied Mathematics

Various classes of antiretroviral drugs are used to treat HIV infection, and they target different stages of the viral life cycle. Age‐structured models can be employed to study the impact of these drugs on viral dynamics. We consider two models with age‐of‐infection and combination therapies involving reverse transcriptase, protease, and entry/fusion inhibitors. The reproductive number \$\$ is obtained, and a detailed stability analysis is provided for each model. Interestingly, we find in the age‐structured model a different functional dependence of \$\$ on \$epsilon_\$, the efficacy of a reverse transcriptase inhibitor, than that found previously in nonage‐structured models, which has significant implications in predicting the effects of drug therapy. The influence of drug therapy on the within‐host viral fitness and the possible development of drug‐resistant strains are also discussed. Numerical simulations are performed to study the dynamical behavior of solutions of the models, and the effects of different combinations of antiretroviral drugs on viral dynamics are compared.

## Examples of Other Choices of Weights

### SIMPLE ARITHMETIC MEAN

We can compare the selection differentials obtained from equations 4 and 5 for the weighted and unweighted means simply by plugging in for all age classes. Then , and . Inserting this in the expression for still gives when there is no selection, whereas the value of conditioned on the age-specific phenotypes in a large population is (6) which is no longer zero. Hence we will on average find a nonzero selection differential for the arithmetic mean phenotype even if there is no selection at all, in agreement with the transient fluctuations in the differentials for the mean phenotypes shown in Figure 1, only due to differences in mean vital rates and mean phenotypes among age classes. For a general projection matrix the term in equation 6 should be replaced by .

### MEAN PHENOTYPE OF NEWBORNS

Another interesting choice of weights is and for . Then and , that is, is the mean phenotype of newborns. Assuming a large population and inserting this in equation 5 then yields (7)

As in equation 6 the age effect is partly generated by differences in the produced by viability selection on different stages in the life cycle, and will also be affected by differences in the caused by drift or other nonselective mechanisms. Selection within years occurs only as fecundity selection by the fecundity terms of equation 4 with only v1 different from zero. Viability selection within age classes does not appear in equation 4 but may produce differences in the and thus affect .

### MEAN ADULT PHENOTYPE

Defining the adult class as all individuals of age larger than y by choosing for and zero for so that is the mean phenotype of adults in a large population yields (8) where now is the mean phenotype of adults. Notice that the sum in the numerator is taken over the adults as well as the last age class before the adult stage. Hence is strongly affected by the differences in the among adult age classes as well the class before the adult stage.

## Results

The modelled time-varying associations between laying date synchrony and (a) survival and (b) recruitment are shown in Fig. 1. The survival associations are largely linear, and although mean survival varies among years, there is consistent directional selection for early breeding via this component of fitness. The recruitment associations are generally nonlinear and hump shaped, with substantial variation in mean recruitment evident, and the strength and direction of selection via this component of fitness vary from weakly positive in a few years to negative in others. Taken together, the survival and recruitment functions indicate that the population should evolve towards an earlier laying date, with an optimum value of the interval laying date and half-fall date that is at least approximately 7–10 days larger than the current value. Once this optimum is reached, it appears that (stochastic) stasis will be maintained by antagonistic selection on recruitment and survival, that is later laying will generally favour recruitment but earlier laying increases survival.

We used the model to track the expected change in population density and laying date synchrony. The expected mean trajectory of laying date synchrony – given as the change in the mean breeding value () – and the change in breeding pair density are shown in Fig. 2. As expected, the model predicts that the interval between half-fall date and egg-laying date will increase by just over one week (Fig. 2a), although the predicted rate of change is slow on average, a change of approximately 1 day is expected in the first 20 years. Although environmental stochasticity introduces a degree of uncertainty into this prediction, projections from independent simulations are largely consistent. Earlier laying is predicted to increase the survival of established individuals and the recruitment of offspring, and concomitant with the change in laying date synchrony, the model predicts an increase from about 440 to 480 breeding pairs on average (Fig. 2b).

Next, we applied the modified ASPE to the model to better understand the component drivers of change at both the phenotypic and genotypic levels. Figure 3 shows how the expected female contributions to annual change from each term change over time for (a) the mean breeding value, , and (b) the mean phenotype, Δ[ x(t)+g(t) ], summed over age classes. The lines show the mean contribution calculated from 250 simulations, and the points show a subsample of annual contributions from a single representative simulation. Rather than separating the two demographic process terms, we chose to summarize their combined effect in Fig. 3.

In general, the different contributions to the change in the mean breeding value are very small (Fig. 3a), reflecting weak selection on the laying date synchrony near its optimum value and its low heritability in the model (). The survival selection component (purple line, squares/crosses) is consistently negative and exhibits relatively little variation among years. In contrast, the recruitment selection component (blue line, ‘+’ symbols) tends to be negative in the early phase of evolution and then positive, with much larger fluctuations overall. Selection acts antagonistically on recruitment and survival once stochastic, evolutionary stasis is reached. The mean inheritance contributions (yellow line, filled triangles) are initially positive – recruits have more positive breeding values than their mothers – and then decay to zero as stasis is reached. There is no ‘growth’ contribution associated with the mean breeding value, as this is invariant over an individual's lifetime. The aggregate demographic effect (red line, circles) is negligible, reflecting the fact that the mean trait value does not change much with age. The male inheritance term (not shown) is the only nonzero contribution to changes in the mean breeding value from males. This term is always exactly equal in magnitude, but opposite in sign, to that of females.

The annual contributions of different processes to changes in the mean phenotype (Fig. 3b) are larger than their breeding value counterparts, because they include the shared, stochastic component of annual variation. However, although they exhibit larger fluctuations, the temporal change in the recruitment and survival contributions is identical to those associated with the mean breeding value the realized laying date synchrony is an additive function of breeding value and so any change in the latter is reflected in the phenotype. The aggregate demographic contributions to phenotypic changes are also very similar to the breeding value complements these are very small, again reflecting the limited age structuring of vital rate and mean trait differences. The two largest terms are those due to inheritance and plasticity, both of which exhibit relatively large annual fluctuations. These terms are consistently nonzero (on average) even after evolutionary stasis is reached. The inheritance term is generally positive, such that new recruits tend to have later laying dates than established females, while the plasticity term is generally negative, indicating that individuals start egg-laying earlier as they grow older. However, the magnitude of these average effects is modest relative to the scale of their annual fluctuations.

Finally, we examined the age-specific component of the two selection terms associated with the mean breeding value (Fig. 4, lines show the mean contribution calculated from 250 simulations). Initially, there is selection for early laying date synchrony via recruitment differences (left panel). The largest negative contributions were from new recruits (age = 1), tending towards zero in older age classes. As stasis is reached, this pattern becomes hump shaped, such that these terms are near zero in new recruits and older individuals, and positive in intermediate age classes. This pattern reflects the opposing effects of an age-dependent shift in mean and the decreasing demographic weights attached to older individuals. The age-specific contributions due to survival selection are always negative. The age-pattern is monotonic, such that the largest negative contributions are always from new recruits, reflecting the decreasing demographic weights attached to individuals as they age.

## Abstract

We analyze the stochastic demography and evolution of a density-dependent age- (or stage-) structured population in a fluctuating environment. A positive linear combination of age classes (e.g., weighted by body mass) is assumed to act as the single variable of population size, N, exerting density dependence on age-specific vital rates through an increasing function of population size. The environment fluctuates in a stationary distribution with no autocorrelation. We show by analysis and simulation of age structure, under assumptions often met by vertebrate populations, that the stochastic dynamics of population size can be accurately approximated by a univariate model governed by three key demographic parameters: the intrinsic rate of increase and carrying capacity in the average environment, r 0 and K, and the environmental variance in population growth rate, σ e 2 . Allowing these parameters to be genetically variable and to evolve, but assuming that a fourth parameter, θ, measuring the nonlinearity of density dependence, remains constant, the expected evolution maximizes E [ N θ ] = [ 1 − σ e 2 / ( 2 r 0 ) ] K θ . This shows that the magnitude of environmental stochasticity governs the classical trade-off between selection for higher r 0 versus higher K. However, selection also acts to decrease σ e 2 , so the simple life-history trade-off between r- and K-selection may be obscured by additional trade-offs between them and σ e 2 . Under the classical logistic model of population growth with linear density dependence ( θ = 1 ), life-history evolution in a fluctuating environment tends to maximize the average population size.

Classical population genetic models assuming a constant population size or density in a constant environment (1 ⇓ ⇓ ⇓ –5) show that evolution maximizes the mean fitness of individuals in the population. The ecological meaning of this result is that when population dynamics is density-dependent, but selection is not, evolution in a constant environment maximizes the intrinsic rate of population increase, r 0 , the per capita population growth rate at low population density (ref. 6, chap. 1).

Density-dependent selection happens when the relative fitness of genotypes differs in response to change in population density. MacArthur (5) showed that density-dependent selection in a constant environment maximizes the equilibrium population size or carrying capacity, K. This result for density-dependent selection in a constant environment was extended to an age-structured population by Charlesworth (7) for the case where all density dependence in the life history is exerted by a single age or stage class (e.g., adults), showing that evolution maximizes the size of that class.

Contrasting conclusions for density-independent and density-dependent selection in a constant environment led to the conceptual theory of r- and K-selection, proposing a constraint or trade-off between these variables within populations, such that K-selection prevails in stable populations in nearly constant environments, while r-selection prevails in highly variable populations in strongly fluctuating environments (8 ⇓ ⇓ ⇓ ⇓ –13). The dichotomous classification of species being either r- or K-selected lost popularity when more cognizance was taken that r and K are emergent properties of age-structured populations, with complex trade-offs between survival and fecundity among age classes. Furthermore, in every fluctuating population r- or K-selection prevails, respectively, when population density is low or high, corresponding on average to an increasing or decreasing population (7, 14).

Initial analytical investigations of evolution in fluctuating environments for populations without age structure were made by Dempster (15) for single-locus haploid or asexual inheritance, and by Haldane and Jayakar (16) for a single diallelic locus in an obligately sexual, randomly mating diploid population. They modeled random environmental fluctuations with a stationary distribution of fitnesses for each genotype, implicitly assuming density-independent selection, and found a simple condition that determines the final evolutionary outcome. For a haploid or asexual population, the genotype with the highest geometric mean fitness eventually becomes fixed. For a diallelic locus in a diploid randomly mating population, if a homozygote has the highest geometric mean fitness it eventually becomes fixed, but if the heterozygote has the highest mean fitness a genetic polymorphism is maintained by fluctuating selection (neglecting fixation by genetic drift in a finite population). These conclusions are especially interesting because the genotype with the highest geometric mean fitness may not be that with the highest arithmetic mean fitness, if the latter occasionally experiences very low fitness thus, genotypes that are well-buffered against catastrophic loss of fitness in extreme environments may have a decisive long-term evolutionary advantage. For polygenic characters, fluctuating density-independent selection generally does not act to maintain genetic polymorphism except possibly at a single locus (17).

A common misinterpretation of these classical results on fluctuating selection still pervades the evolutionary and ecological literature, which is that the geometric mean fitness of a genotype in a fluctuating environment measures the selection on it. This interpretation is erroneous because fitness is supposed to predict short-term changes in gene frequencies over time spans on the order of one generation, whereas the geometric mean fitness only predicts the final outcome of selection. For density-independent selection in a fluctuating environment, the correct measure of the expected relative fitness of a genotype is its Malthusian fitness (the intrinsic rate of increase in the average environment) minus the covariance of its fitness with that of the population (17 ⇓ –19). A similar definition of expected mean fitness of a genotype also applies with density-dependent selection, with the Malthusian fitness conditioned on the current population density (20, 21).

In a constant environment, assuming constant genotypic fitnesses and approximate linkage equilibrium among loci, the log of mean fitness in a population, ln w ¯ = r , as a function of gene frequencies, describes an adaptive topography that governs both short-term and long-term evolution of gene frequencies by natural selection (3, 4, 22, 23). A similar adaptive topography has been derived for the evolution of mean phenotypes in quantitative characters (24, 25). The gradient of the adaptive topography (or selection gradient) at any given point, ∇ r , is a vector with elements that are the partial derivatives of the log mean fitness in the population with respect to gene frequencies or mean phenotypes. The gradient dynamics imply that evolution in a constant environment is a hill-climbing process, continually increasing the mean fitness of the population until a local maximum in mean fitness, an adaptive peak, is achieved.

In a fluctuating environment with a stationary distribution of environmental states and no autocorrelation, Lande (17, 18) showed that the long-run growth rate of the population, r − σ e 2 / 2 , provides an unchanging adaptive topography governing the expected evolution. From the theory of stochastic demography, the long-run growth rate of a density-independent population gives the expected (or asymptotic) rate of increase of ln N , where r is the expected growth rate in the average environment and σ e 2 is the environmental variance in growth rate (26 ⇓ ⇓ ⇓ –30).

Density-dependent selection in a fluctuating environment presents the additional complication that stochastic evolution is coupled with the stochastic dynamics of population size. Lande et al. (20) and Engen et al. (21) analyzed density-dependent selection in a fluctuating environment for populations without age structure, respectively both for asexual inheritance and for quantitative (polygenic) characters in an obligately sexual population. Neglecting genetic drift and demographic stochasticity due to small population size, they showed that evolution tends to maximize the expected density dependence averaged over the distribution of population size, E[g(N)]. The density-dependence function g ( N ) may often be nonlinear, as suggested by evidence from a variety of species (31, 32). For the θ-logistic model of density dependence, g ( N ) = N θ (33), evolution maximizes E [ N θ ] = [ 1 − σ e 2 / ( 2 r ) ] K θ . This simple result generalizes to stochastic environments MacArthur’s (5) finding that density-dependent selection in a constant environment maximizes the equilibrium population size, K. In a stochastic environment, genetic trade-offs can constrain adaptive evolution toward increasing r and K and decreasing σ e 2 . This incorporates the trade-offs most often considered in the literature on r- and K-selection, by including the environmental variance in population growth rate.

Explicit age or stage structure is essential for any theory of selection, inheritance, and evolution of life histories. Here we extend the recent results on the evolution of stochastic demography and life-history trade-offs (20, 21) by assuming that all density dependence in the life history is exerted through a single positive linear combination of age classes, defined as the population size or density. The age-specific vital rates are all affected by the same possibly nonlinear function of population density but may differ in their sensitivity to population density and environmental stochasticity. We obtain parallel results for asexual inheritance and polygenic inheritance of quantitative traits.

When scientists want to explain some aspect of nature, they tend to make observations of the natural world or collect experimental data, and then extract regularities or patterns from these observations and data, possibly using some form of statistical analysis. Characterizing these regularities or patterns can help scientists to generate new hypotheses, but statistical correlations on their own do not constitute understanding. Rather, it is when a mechanistic explanation of the regularities or patterns is developed from underlying principles, while relying on as few assumptions as possible, that a theory is born. A scientific theory thus provides a unifying framework that can explain a large class of empirical data. A scientific theory is also capable of making predictions that can be tested experimentally. Moreover, a theory can be refined in the light of new experimental data, and then be used to make new predictions, which can also be tested: over time this cycle of prediction, testing and refinement should result in a more robust and quantitative theory. Thus, the union of empirical and quantitative theoretical work should be a hallmark of any scientific discipline.

Theory has long been celebrated in the physical sciences, but the situation is very different in the life sciences. As Conrad Hal Waddington wrote in 1968, in the preface of Towards a Theoretical Biology: ‘Theoretical Physics is a well-recognized discipline, and there are Departments and Professorships devoted to the subject in many Universities. In strong contrast to this situation, Theoretical Biology can hardly be said to exist as yet as an academic discipline. There is even little agreement as to what topics it should deal with or in what manner it should proceed’.

Yet theory plays a paramount role in biology. The best known example of a theory in biology is, of course, the theory of evolution by natural selection. Charles Darwin may have been a globe-trotting hands-on naturalist and geologist, but his outstanding contribution to science was theoretical. Drawing on fieldwork, fossil records and the breeding records of domestic animals and plants, he observed that variations readily arose and that much of this variability was heritable. After reading Malthus' essay on the repercussions of an exponential growth in population, Darwin reasoned that a struggle for existence must have selected for the variants that were most adapted to their local environment. As different populations adapted to different environments, he argued that these variations accumulated over time, eventually forming diverse species. Despite the success of his theory, Darwin never formalized it in mathematical terms. Rather, he wrote: ‘I have deeply regretted that I did not proceed far enough at least to understand something of the great leading principles of mathematics for men thus endowed seem to have an extra sense’ (May, 2004). Although a theory does not have to be formulated as a mathematical model to be useful, the development of such a model is a hallmark of a maturing theory. The role of theory and mathematical models in the life sciences is the focus of this editorial.

The best known example of a theory in biology is, of course, the theory of evolution by natural selection.

By the end of the 1960s, when Waddington was bemoaning the lowly status of theoretical biology, the field had in fact witnessed major breakthroughs. Early population geneticists such as Pearson, Fisher, Wright and Haldane had developed the formulation that Darwin was unable to construct, providing a mathematical foundation for the theory of evolution by natural selection. In the process, they also generated a number of major advances in statistics. The modern evolutionary synthesis had reconciled the gradualist Darwinian view of natural selection with a Mendelian understanding of genetics, unifying observations from naturalists, experimental geneticists and paleontologists. A crucial contribution from theory came in 1943 when Luria and Delbrück used mathematical reasoning and experiments to conclude that mutations arose in the absence of selection, rather than in response to selection. And in 1953 the structure of DNA was determined with the help of a theoretical physicist and the building of physical models (which were the forerunners of today's computer simulations). Elsewhere, the simple and elegant Lotka–Volterra models of competition and prey-predation had jump-started theoretical ecology, Kermack–McKendrick theory had laid a foundation for mathematical epidemiology, and Burnet had developed the clonal selection theory that lies at the heart of our understanding of the adaptive immune system. In neuroscience it is difficult to overstate the importance of the Hodgkin–Huxley model of action potentials or Rall's use of cable theory to provide a framework for understanding the complex, dendritic structures of neurons.

During the past half-century, theory has continued to advance in diverse areas of biology. Within evolution and ecology, for example, evolutionary game theory provided a framework for thinking about the evolution of strategic behavior, while kin selection and multi-level selection theory helped to explain cooperation and altruism. Life history theory offered a systematic way to think about the evolution of senescence, developmental plasticity and reproductive schedules, among other things, while optimal foraging theory introduced economic reasoning into the study of animal foraging. Other examples include kinetic proofreading in biochemistry, the Hopfield model of neural networks, and the use of bifurcation theory and phase-plane analyses in neuroscience.

Increased computational power has also allowed biologists to study the structure and function of proteins, and to simulate complex biological processes such as morphogenesis, chemotaxis, the cell division cycle, metabolism and, in some cases, the workings of the entire cell. And over the past decade new experimental tools and techniques have generated such a staggering amount of data that we are, in the words of Sydney Brenner, ‘thirsting for some theoretical framework with which to understand it’ (Brenner, 2012). This is true in genetics and genomics, immunology, microbiology, neuroscience and many other areas. New theoretical and computational models are therefore needed to make sense of this abundance of data.

Yet, despite this rich history, the divide between theoretical and empirical biologists seems to persist, even in areas with a long history of both types of work, such as ecology and evolutionary biology (Haller, 2014). One reason for this is that the complexity of real biological systems often requires relatively sophisticated mathematics, which means that many theoretical papers do not resonate with empirical biologists. This complexity has many sources: the number of interacting parts in even the simplest living cell presents a formidable challenge for a theoretical biologist, as does the heterogeneity that is intrinsic to biological systems. Moreover, interactions among these parts can span a large range of time scales (from picoseconds for electron transfer in photochemical reactions, to billions of years for evolution) and length scales (from molecules to cells, from organisms to ecosystems).

Yet, despite this rich history, the divide between theoretical and empirical biologists seems to persist, even in areas with a long history of both types of work, such as ecology and evolutionary biology.

As a result, theoretical biologists often need to make a trade-off between abstraction and realism (or between the qualitative and the quantitative) when building mathematical models. The appropriate level of abstraction will depend on the question of interest. For example, simplifying assumptions can be made to develop a highly abstract model that reveals general features shared by many systems and thus improves our understanding of some aspect of biology. However, such a model is unlikely to produce quantitative predictions for any particular system. On the other hand, a highly detailed model that contains many equations and parameters is unlikely to improve intuitive understanding of a system or process. However, if the various parameters in the model can be measured to a credible level, then these models should be able to make quantitative predictions about a given system or process. Part of the challenge in model building is to choose the right level of abstraction despite the complexity of biological processes. In other words, we need to work out what aspects of this biological complexity we can ignore and still gain critical insights about a biological phenomenon.

So how can we increase interactions and collaborations between theoretical biologists and empirical biologists for the benefit of the discipline as a whole? First, universities and institutions should ensure that biology students are taught more about theoretical and mathematical techniques, including ideas from physics that have already been successfully applied to biological questions (such as statistical mechanics and nonlinear dynamics). Laboratory work could also be extended to include exercises that involve computer simulations. These changes would help biologists to better communicate with theorists and, more importantly, to incorporate quantitative thinking into their own work. There are signs that this is starting to happen: the sixth edition of Molecular Biology of the Cell, for example, includes examples where ordinary differential equations are used to model gene regulation and to explain switch-like and oscillatory dynamics. It would be good to see more mathematics in biology textbooks.

Second, theoretical biologists could do more to increase the chances that their papers will resonate with empirical biologists. The primary audience for some theory papers will be other theorists, and like all papers aimed at a specialist readership, these papers will be a challenging read for non-specialists. However, the potential impact of most theoretical papers—especially modeling papers—could be increased by following a few simple guidelines. The first thing to do is to clearly state the goal of the modeling: is the aim to organize data emerging from high-throughput experiments, to test a particular hypothesis, to uncover the basic mechanisms driving some phenomenon, to evaluate the feasibility of an intuitive argument, to make specific predictions, or something else? How does the model or theory relate to and differ from previous models, and what are its advantages and disadvantages? What assumptions have been made, and what are the justifications for these? How were the parameters in the model chosen?

Theoretical biologists could do more to increase the chances that their papers will resonate with empirical biologists.

Mathematical papers can be made more accessible by giving step-by-step derivations for equations, and intuitive explanations for how these equations reflect the biological process under investigation, even if this involves covering material that may already be familiar to other theoretical biologists. Schematic diagrams can also help. Finally, it is important to relate the conclusions back to biology. This includes clearly stating which conclusions are not surprising (in the sense that they are straightforward derivatives of the empirical results used to constrain a model), which insights are novel, and which predictions are worthy of empirical tests. Theoretical biologists can also benefit from wet lab experience to help them appreciate what doing an experiment involves.

Third, empirical biologists could make their work more accessible and valuable to theorists. For example, all the relevant datasets should be included in papers. Moreover, where possible, time-course data should be collected, rather than just ‘end-point’ data, as this will allow dynamical processes to be studied. And when the experimental measurements in a paper differ from previous measurements in a significant way, it would help theorists (and others) to build on the work if the authors discussed possible reasons for these differences. Taken together the recommendations outlined above should lead to improved collaboration between theoretical and empirical biologists.

eLife welcomes theoretical and modeling papers in all areas of biology, especially papers that report new biological insights, make substantial predictions that can be tested, or help to resolve contradictory empirical findings. Papers that report new theories or algorithms that have the potential to solve important biological problems are also welcome. Papers can also be as long (or as short) as necessary. Across the life sciences we aim to publish papers that are insightful and change the way that other researchers think about their subject (Malhotra and Marder, 2015). Theory and modeling are no exception.

## Principles of Evolution, Ecology and Behavior

### Chapter 1. Introduction [00:00:00]

Professor Stephen Stearns: Okay, today we’re going to talk about life history evolution, and life history evolution deals with some big questions. It’s explained why organisms are small or large, why they mature early or late, why they have few or many offspring, and why they have a short or a long life.

Basically what life history evolution does is it analyzes the evolution of all of the components of fitness, all the different things that combine to result in lifetime reproductive success, and in so doing it visualizes the design of the organism as an evolutionary solution to an ecological problem. So it’s fundamentally about the interface between evolution and ecology, and it is one of the places where scientists confronted the problem of how do we explain phenotypic evolution rather than genetic evolution? So this is really about the design of the large-scale features of organisms, and it brings us to ask questions about ourselves as well, of course. Why is it that we have a lifespan of about eighty years? Why is it that we’re about three kilos when we’re born, etcetera? Okay, so we fit into this matrix of questions.

Now here are a few world records. Biggest baby is a blue whale, twelve tons. And the interesting thing about it is that it will grow to be sixty tons in the next six months. So it’s really pumping it in. And by the way, a mother blue whale, and most whale mothers, actually have muscles in their breasts so that they actively pump the milk into their offspring. Baby isn’t just sucking. Baby is attached to a fire hose. Okay? [Laughter]

And look at what happens to the mom. She goes to warm tropical waters to give birth, has her baby, and then she nourishes that child until it is independent, without eating herself. Imagine how big she is, because he turns into something sixty tons. And you can imagine how cranky she is before she swims back to Antarctica to get lunch. Okay?

Then if you ask yourself, for a given body weight what is the biggest thing? It’s not the blue whale baby, it’s the babies–the twin babies of a bat are the largest weight of any offspring in mammals, and she actually flies with them. And in the kiwi, it has a 400 gram egg. If you take a radiograph, if you put a kiwi into an x-ray machine, take a radiograph of it, you are to imagine an egg that’s occupying about two-thirds of the body cavity of the kiwi, it’s got a giant egg.

The fewest offspring per lifetime of anything that’s out there bearing a significant risk–and this is actually less than humans–is the Mexican dung beetle, that only has four to five babies per lifetime, which is pretty remarkable when you think about how risky you would think life would be for a Mexican dung beetle. How can it get away with only having four or five babies, if some of them are likely to die? But, in fact, it has such good parental care that it’s around, and it’s doing just fine, thank you, with only four to five babies.

And the most offspring per reproductive event is orchids. Orchids produce typically billions of seeds and they are extremely tiny and the only reason that they can hatch is that they have a fungal midwife that helps them. Orchid hatching is dependent upon fungi. So the mother doesn’t have to put the nutrients into the seed. So she makes billions of tiny seeds. And in bivalves and codfish, they can get up to hundreds of millions of eggs per reproductive attempt. So you can see that just by comparing some numbers and looking broadly–and this is a typical thing that happens in comparative biology, it’s one of the neat things about it–if you look across the Tree of Life and you see how different things live their life histories, you’ll immediately start to ask questions.

You guys have all been generating wonderful questions this week. You look at that stuff and you say, “Well why are things big and small? Why do they have few babies or many babies? What has caused the evolution of all of this diversity?”

So here is the largest whale and the smallest dolphin. So this is the whale radiation. You can see that since the ancestor, there’s been considerable change in body size. Here’s Pipistrellus, flying with babies. Here’s a dung beetle, and it’s going to lay its egg into that pile of dung. That’s why it’s going to have an extremely well protected baby. Not too many things are going to come along and eat baby. [Laughter] And here’s a kiwi with its egg. Okay? So diversity.

### Chapter 2. Life History and the History of Ideas [00:04:53]

So in the history of ideas, life history theory and the rest of evolutionary and behavioral ecology fit about here. Darwin showed us that natural selection and descent with modification from ancestors can explain a lot, but then genetics remained a problem until 1900. Then we had the genetical reaction to that issue, which is the neo-Darwinian synthesis that basically says Darwin works with genetics. And this concentration on genetics then, in its own turn, elicited a reaction. So this is a reaction to that. And what’s the role of phenotypes in evolution is the reaction to the neo-Darwinian synthesis.

So the phenotypic reaction, it’s been going on for about forty years. It has a selectionist part–that is, how are phenotypes designed for reproductive success–and it has a developmental part: what are the restrictions on the expression of genetic variation? So the phenotypes are actually both being designed by natural selection for reproductive success and, in the process of their production, they are themselves editing genetic variation.

So life history evolution is the part that explains the design of phenotypes for reproductive success, and it concentrates on size at birth, how fast things grow, age and size at maturity, reproductive investment, and mortality rates and lifespan. So part of life history evolution is why do we grow old and die?

And after a lot of discussion, it was possible–this is after about twenty years of discussion–to make this simple statement: What causes life histories to evolve? They result from the interaction of extrinsic and intrinsic factors.

So the extrinsic factors are things that are influencing the age-specific rates of mortality and reproduction, and that’s where ecology comes in. It’s not just ecology, there’s a lot of phylogenetic effects on this stuff, but the point is that if you look at whatever is affecting changes in mortality and reproduction, in age and size of the organism, you will be able to explain a great deal of what you see in the life history.

But that’s not enough. There’s interaction between that and factors that are intrinsic to the organism, and the intrinsic factors are conceptualized as tradeoffs among traits. The idea here is there’s no free lunch. If you change one thing in evolution, a byproduct of that change will be a change in another trait. So even though you are gaining fitness through changes in one trait, almost inevitably, whatever you change is going to cause a decrease in fitness in some other trait, and this forces compromises.

So the intrinsic factors then can be looked into, and we find phylogenetic effects, developmental effects, genetic effects, physiological effects all sorts of things. Tradeoffs in a evolutionary situation are often conceptualized as being strictly energetic. If I take calories away from my growth in order to reproduce and make more babies, then I won’t be so big next year and I can’t have so many babies next year. That would be kind of a standard physiological story about a tradeoff. But they can also occur in many other ways. So that would be a physiological story.

But certainly there are developmental and genetic influences on tradeoffs as well. So there’s a lot of biology that’s hiding behind these simple summary statements, on this slide. In the rest of the lecture I’m just going to show you how to explain age and size at maturity, reproductive investment, and aging and death. So, not too much.

### Chapter 3. Age and Size at Maturity [00:08:56]

This is kind of a standard statement out of life history theory, and this generic statement could be applied to clutch size and lifespan and a lot of other things. But let’s just look at age and size at maturity. They will be optimal when the positive difference between the benefits and the costs–so the difference between the benefits and the costs–is maximized. And we can conceive of that as either being maximized just at a stable equilibrium point–that’s kind of a simple statement, that’s a theoretical statement so that would be, okay, everybody in this species, they ought to mature at just one age and size, which is a little unrealistic. Or we can use that kind of analysis to predict a stable equilibrium reaction norm. So here we’re beginning to use this idea that we got of a reaction norm.

And that one summarizes pretty easily. You’re going to–whatever problem you’re faced with, you’re going to mature at the age and size where the payoff in fitness is going to be greatest. The problem analytically is to decide what you have to bring in to the mix in order to successfully make that prediction. You want to keep it as simple as possible, because it can get very complex, but you want to keep it realistic enough to actually be successful. So it’s a balancing act.

Now with–I’m going to show you one way to do this. If we make four general assumptions, we can predict age and size at maturity. Here they are. The first one is that if you’re older when you first reproduce, your offspring are going to have better survival rates, they will be of higher quality so one reason to wait is that you get higher quality offspring. Another reason to wait is that because you’ve been growing for longer, you’ve taken longer to grow before you start to reproduce, you can have more of them, because you’re bigger especially important in plants and in fish.

However, these advantages of delaying maturity are counter-balanced by the advantages of having a shorter generation time, and you can only get a shorter generation time if you mature earlier. Let me just illustrate the advantage of a shorter generation time. I give you a hundred bucks and I tell you you can invest it in a bank that’s going to give you compound interest once a day, on the one hand, or once a year, on the other hand. You all know the advantages of compound interest you get interest on your interest. Right? A shorter generation time is the bank that gives you interest earlier you get grandchildren quicker. Okay?

So that is basically the elements that you need to put into a quantitative tradeoff. Delaying, you can get higher quality, or more offspring doing it quicker, you’re going to get a shorter generation time and a quicker payoff. Now in a population that’s at evolutionary equilibrium, these advantages and disadvantages should have come into balance. So let’s see how that might work.

Here’s a simple example. This is using data from the Western Fence Lizard, and what you’re looking at here, this plot here, where you see these curves going up and down, that’s a fitness profile. So we have some kind of trait along the y-axis in this case it’s age of maturity–along the x-axis. Along the y-axis we have relative fitness so this is the rate at which a population of organisms with that age at maturity would grow, given what we know about the physiology and mortality rates of fence lizards.

And if we just put in one of those assumptions, which is that the bigger they are the more babies they have – so their fecundity grows linearly with size – their optimal age at maturity is just about twelve months. If we put in that if they get higher quality offspring as they delay maturity, given the assumptions in the model, we predict actually that they ought to be maturing at about six months. Their observed age at maturity is ten months.

That indicates that this effect is probably important and perhaps accurately modeled. This number tells us that well perhaps we don’t really understand what makes for a good baby lizard. Okay? And you can see that interestingly the age at maturity is pretty strongly peaked the fitness profile has a peak that’s pretty close to one value. That means there’s pretty strong selection operating on this. It’s not flat.

Now if you repeat that kind of thing–and by the way, there’s a bunch of math behind that I’m just waving my hands and covering up that black box. If you repeat that for a bunch of fish species that are growing in different kinds of conditions–these are haplochromine cichlids in Lake Victoria the painted greenling lives in Seattle these roaches are living in Greece–and these are all cases in which very good population biology has been done for long periods of time in the field. So we know growth rates and mortality rates, and we have some estimate of tradeoffs. Then that kind of thinking says this is the predicted age at maturity and this is the observed age of maturity, and the correlation is .93.

So it looks like that way of thinking is capturing something that is not a bad reflection of what’s going on in Nature. This sort of result doesn’t mean you’ve got the right answer. You can have the right answer for the wrong reason, because this is just descriptive work, this is not a manipulative experimental study. We’ll see such an experimental study later on.

However, that’s not the whole story. I now want to extend that to the case when growth rates vary, and I want to introduce you to the idea that age and size at maturity can have a reaction norm. And the way I want to do that is by dealing with some incredibly blockheaded strategies. Okay? So here we have rapid growth. So this is an organism that is born down here, and it’s well fed and it grows rapidly. So it gains weight well, reaches a large size. And this is an organism that grows slowly it’s under food restriction, down here.

Now let’s take the blue strategy–this is a very, very simple one–and what it says is I’m always going to mature at the same weight. If that organism is growing rapidly, it matures at a pretty early age, but if it’s growing slowly and it adheres to this rule, it has to wait a long time until it matures, and its problem here is that it might die before it matures. So that strategy has the cost of mortality.

On the other hand, if it’s always the same age when it matures, under good circumstances, it’s doing okay, but under poor circumstances it’s much smaller, and therefore it can have fewer babies. And so the problem here is fecundity it’s going to not have as many babies if it does that. And so just intuitively you might think that there is some kind of intermediate compromise so that when it is not being fed as much, it changes both its age at maturity and its size at maturity.

And, in fact, this kind of thing can be calculated. This is an optimal reaction norm for age and size at maturity. They don’t all look like this. Okay? This is a common one, but there are conditions under which you can make this thing bend. You can actually sometimes get them so that they go up like this, under very special circumstances.

It depends on a bunch of stuff. I don’t want to trouble you with the complexities. I just want you to take home the message that you can predict what the plastic flexible response should be if evolution has come to equilibrium. And for this one, basically what this graph is telling you is this–this is the reaction norm here, these are growth curves here so this is good conditions, this is poor conditions–and what this picture is telling is that when life is good, you should mature when you are young and big, and when life is bad, you should mature when you’re old and small. Okay? That’s the English take-home message, out of that picture.

Well when Nile perch were introduced to Lake Victoria, there hadn’t been any Nile perch in there before, and they went bananas and ate their way around the lake–and in the process, by the way, they probably drove about 200 haplochromine species to extinction–but while they going through their initial population burst and they had a lot of food, they were about six feet long. This is the business end of a Nile perch. You can see it’s a big fish.

After they had expanded in the lake, which occurred between 1976 and 1979, they ate down the population of their prey, there wasn’t as much food and they didn’t grow as well, and they slid down this reaction norm, and now instead of being six feet long, the Nile perch in Lake Victoria are about that big. They still form a fishery and people are still making money on selling Nile perch fillets, but they’re much smaller. And that was a predictable thing. Okay? And this will happen whenever population densities change.

Back in the 1930s and 1940s, there was a huge sardine fishery off the coast of California. John Steinbeck wrote novels about it, short stories. There’s a book called Cannery Row that talks about the Monterey Bay sardine canneries. In the 1950s that fishery collapsed, not because of over-fishing, but because of changes in the oceanic conditions where the baby fish were growing up. At the time that it collapsed, there were sardines that had been born under better conditions and started to grow, and then all the competition went away nobody else came along because all the baby sardines were getting killed by bad conditions out in the ocean.

Just before the fishery folded and there were no longer enough sardines to catch, the fishermen in Monterey were catching female sardines that were one meter long. So they had gone in the other direction, they’d gone up the reaction norm. These things are predictable as population density changes.

I’d like to give you one more example, and it has to do with the issue of whether or not the mammals died out because of bad weather, or over-hunting. Dan Fisher, who’s a paleontologist at the University of Michigan, has recovered a lot of mammoth bones from a Native American mammoth slaughterhouse that was outside of Ann Arbor. They used to kill the mammoths and then store them under ice, in a lake, over the winter, so that the other predators wouldn’t get the meat, and there are a lot of mammoth bones very close to Ann Arbor. And you can ask yourself–when you look at a mammoth bone, you can tell how big the mammoth was and whether or not it is mature, because the bones of all mammals undergo a change when they reach maturity.

Now if it was bad weather, then they would’ve been growing slowly, and they should’ve been small and older when they matured, according to the reaction norm. If it was hunting, then just like the California sardine, when the population density drops, each individual has more to eat, and they should have been big and young when they matured. Do you think they were old and small, or big and young? How many for old and small bad weather? A few. How many for young and large hunting? Most people believe the over-kill hypothesis. Yes, they were young and large, and some of them had arrow points embedded in their ribs. So you can use that for various things.

This is what that model tells us about human females. These are some pretty theoretical growth curves for human females under poor conditions and under good conditions. We actually have data on how female age and size at maturity has changed. There are measurements on women working in industrial squalor in North England, in the nineteenth century, and there are good records measured on Hutterite colonies in North America in the twentieth century.

The nineteenth century women were poorly nourished. The twentieth century women were well nourished. They moved right up a reaction norm. They got younger and bigger when they matured and it was about four year’s difference. So they went–there are various measures of when a woman is–physiological measures–but they kind of all move together. So it’s about a four-year advance, earlier maturity in the twentieth century.

And this other line here illustrates another point that I want you to take away from this. If modern medicine were to keep juvenile mortality rates as low as it currently does, then it would cause a further shift in age at maturity in humans, and that shift is represented here. This probably would take somewhere around 5 or 10,000 years to occur. This is the evolutionary genetic response this is the immediate developmental response to better nutrition and this is the evolutionary genetic response to a drop in juvenile mortality rates. The whole reaction norm evolves it will move up and down. It’s embodying an evolutionary set of rules of thumb, contingent decisions–if I’m well nourished, do this if I’m poorly nourished do that–and those things evolve.

### Chapter 4. Size and Number of Babies [00:23:38]

Okay, now the second major life history trait is once you’ve matured how many babies should you have, and how big should they be? You want to be an orchid with billions of tiny ones, or you want to be a kiwi with one big one? Well the ideas on this go back to David Lack. David Lack was the man who more or less created the idea of Darwin’s finches in the Galapagos.

Darwin’s finches, as a concept, emerged in the middle twentieth century. They were never called Darwin’s finches before David Lack went to the Galapagos, studied them, came back and wrote a book called Darwin’s Finches. It was 120 years after Darwin had been there. And he went on to become head of the Edward Gray Institute at Oxford, which is an ornithological institute and one of the best places in the world to go if you’re interested in bird biology and you’re not working with Rick Prum at Yale.

So what David said basically was this. If nestling survival decreases as clutch size increases, then an intermediate number of eggs produces the most fledglings. The idea behind that was this. If you make too many babies, you won’t be able to feed them. There are only so many hours in the day. You might be able to work as hard as possible and not bring off a clutch of say ten babies, but you could do quite well with five.

Now I’m going to show you that he was wrong on the details, but he got the main point, which is that fitness is often maximized at intermediate reproductive investments, particularly in organisms that reproduce more than once per lifetime. You don’t do it all now, you hold some back, and you actually do better if you spread it out.

So if we then take Lack’s idea, and we make a simple model out of it–basically what he was saying is this. As clutch size goes up, well that just means that eggs go up, but if survival goes down, as eggs go up–this is the per egg survival probability basically this is saying that if you only laid one egg, you’d have very good survival, and if you lay ten eggs they all die.

You can turn that into an equation for how many fledglings do you get for a given number of eggs? Well it’s going to be 1 minus a constant, times the number of eggs you lay, which means, if you multiply that out, that you’ve got a quadratic term here in eggs and that is what leads to the parabola, it’s this quadratic term that means that as clutch size goes up, the number of babies that you get out of it has a parabolic form with an intermediate optimum.

And you then just do the standard basic calculus thing of taking the first derivative, setting it equal to zero. It tells you that this point right here is going to be at 1/2C, in this equation, and if C is 0.1, this optimal number of eggs will be 5, and the number of fledglings that you get out of it will be 2.5. Of course, you never get 2.5, but that’s just because the model’s continuous and the eggs are discontinuous.

Well if this is the case, if birds are laying the optimal clutch, then a larger or a smaller clutch should have lower fitness. Basically all we’re saying is that if we were able to take a bird, and she wants to do this, but we give her either fewer eggs or we give her more eggs, then she should have lower fitness. This should be the best, which she naturally does, and we perturb that, she should have less fitness.

This was done on kestrels in the Netherlands by Dutch ecologists, in a rather remarkable study. A kestrel is a sparrow hawk, and these animals live, these birds live for several years, and the Dutch ecologists actually followed them long enough to count the grandchildren they went three generations.

So, this is the setup. They reduced the size of 28 clutches, enlarged the size of 20 clutches, and in 54 clutches they took the eggs out and put them back again those were the controls. And if you just look at this, it looks like these birds should be laying more eggs, because if you look over at the enlarged clutches, they’ve been able to change the brood size up by 2.5. They’ve been able to get more fledglings out–they’ve gotten nearly two more fledglings out of the enlarged clutches–and the reproductive value of that clutch, which is how many grandchildren do I get out of that clutch, is higher.

So it looks like these birds are blockheaded, they should be laying more eggs. But that’s only looking at what happens that season. While they were examining these birds, one of them, Serge Daan is a good physiologist, and so he did the experiment with doubly labeled water. He wanted to find out how hard the birds would work, and they were coming into nest boxes.

So mommy and daddy kestrel fly into a nest box with food for baby evil Dutch ecologist, sitting in back of the nest box, takes food away from baby. Baby cries. Baby gets hungry, mommy and daddy work harder. Evil Dutch ecologist takes away food. Mommy and daddy work even harder. How hard do mommy and daddy work? Mommy and daddy work about eight hours that day–daylight’s about sixteen hours a day in the summer in North Holland–and they hit a rate of physiological output which is nearly four times basal metabolic rate, which is what Lance Armstrong puts out on the Alpe d’Huez in the middle of the Tour de France.

So the Dutch ecologists basically forced these birds to work as hard as a peak human athlete would, and then they quit after eight hours, because they didn’t want to die. And then the Dutch ecologists gave the babies their food. Just so you don’t have nightmares about that. Okay?

So that introduces parental survival. If you increase the clutch size, the parents died the next winter at a higher rate, because they worked harder. Okay? And if you add all of that up, the residual reproductive value of the rest of their lifetime the number of grandchildren they would get out of the rest of their lifetime was highest for the reduced broods, intermediate for the control broods, and strikingly lower for the enlarged broods, because of this effect. If you die before the next year, you get zero babies next year.

So if you look at their total reproductive value, which is the value they got this year, plus the value they got in the rest of their life, it’s highest for the control group, and if their clutches were enlarged, they had one grandchild less, and if their clutches were reduced, they had a half a grandchild less. Which has an interesting take-home message. These Dutch kestrels know what’s best for them. They lay the right number of eggs. That’s the control group.

So the take-home points basically are that what’s going on here is that clutch size is trading off with an important fitness component, but it’s not fledgling survival, it’s parental survival. In this case–it’s different in other species–but in this case the reason that they don’t lay more eggs is that they themselves are more likely to die not that their offspring are more likely to die. And these kestrels are optimizing their reproductive investment with a clutch that’s of intermediate size. They could lay more eggs but they don’t. They know how many to lay.

### Chapter 5. Lifespan and Aging [00:31:49]

Okay, so that’s just one example of clutch size analysis. It’s a big literature, there’s a lot of experiments on this. Now let’s go to lifespan. So I’m taking you through the major life history traits from birth to reproduction to death. In Fragment of an Agon, T.S. Eliot wrote, “Birth, reproduction and death. That’s all the facts, when you come to brass tacks, birth and reproduction and death.” He wrote that in the 1930s I think. I didn’t realize that T.S. Eliot was a behavioral ecologist evidently he was.

So reproductive lifespan, under this kind of analysis, is a balance between selection that increases the number of reproductive events per life–you live longer, you can reproduce more–and effects that increase the intrinsic sources of mortality with age. And it’s this idea that there’s an evolution of aging or of senescence there’s an evolution of the body falling apart, as a byproduct of something, which is the key feature of this part of life history theory.

So the first kinds of selection pressures are going to lengthen life to give you more reproductive opportunities, but if there are byproducts that are causing intrinsic increases in mortality rate, those will shorten your lifespan. So these things then come into some kind of balance. Any increase in intrinsic mortality rates, or decrease in reproductive rates with age, is called aging or senescence. So now we’re talking about why people fall apart when they get old, and why organisms age and die.

To do that I need to introduce you first to the way selection operates at different ages. Selection is quite age specific in its impact. Any selection pressure that lengthens life is going to be one that decreases the relative contribution to fitness of offspring, and increases that of adults.

So if an adult has survived to some intermediate age, and juvenile mortality in that species is pretty high, then the adult represents a relatively improbable event that’s quite valuable, and if it’s making babies in that environment, each of them has a relatively low chance of surviving to be that big and that old, and therefore there is a certain fitness advantage in investing in the preservation of that adult, because it’s unlikely that you’ll get another one up to that state.

The things that will do this are lower adult mortality rates and higher juvenile mortality rates. So if life is relatively good for adults and pretty risky for juveniles, and infants, then you’re going to get the evolution of a longer lifespan.

But in contrast, if adult mortality rates increase, then organisms should evolve more rapid aging, basically because there really isn’t much point in maintaining a body that’s going to be dead anyway for other reasons. Why should I take away from my reproduction and invest it in say disease resistance, or running away from predators, if I’m not going to be able to avoid them anyway? Then I should make more babies. Okay?

So those are the basic ideas. And I’d like to illustrate a little bit of the math behind this, with a pictorial model. So this is why senescence evolves. I’m going to use the fruit fly Drosophila as the model organism. We’re going to start this thing off, not when it’s an egg, but when it ecloses and is an adult, and we’re going to say that our model has no intrinsic mortality at all. So this one doesn’t age this is our baseline, this is what happens if an organism doesn’t age.

Its risk of dying is 20% per day, and every day it lays ten eggs. Okay? So on the first day it gets ten eggs. On the second day 80% of them are still around, and each of them lays ten eggs, and on the third day 64% of the original are still around (.8 times .8), ten eggs, ta-da ta-da. And this thing is potentially immortal. Okay? So it can just go on pumping out the eggs, if it survives for as long as ever and its probability of survival isn’t changing with age, it’s 80% each time. This one gets 50 progeny. We do that just by using an infinite series. Okay? And the numbers were set up to give you a nice simple output. Okay? The numbers are cooked. So this one gets 50.

Now, what happens if everybody dies between the nineteenth and the twentieth day? That one gets 49.3. That’s all the difference that death at old age makes. And this is in a case where there’s no senescence. Right? This is kind of like light bulbs failing or something like that.

However, now let’s throw in a little life history tradeoff, and it’s a really small one. This genotype here, because it can lay eleven instead of ten eggs, on the first day of life, dies, between the nineteenth and the twentieth day, it leaves 50.3 progeny. It has a .6% fitness advantage. If we introduce this genotype into the populations of the ones that live forever, it will take over. There won’t be any immortal flies anymore. There will be flies that have evolved a shorter lifespan because they had a reproductive advantage early in life and it didn’t take much of one to do it.

As you contemplate your own mortality, I hope you realize that the Drosophila example in fact is non-trivial it’s giving you an important message. This is the strength of selection on further survival in human males in the United States in the year 1960, calculated from real demographic data from the U.S. Census. This is the partial derivative of fitness with respect to further survival. And it’s a very interesting picture.

What it shows you is that as soon as you become a teenager and you have some probability of surviving in that human population, your fitness starts to drop, because as soon as you’ve had a baby, you have some probability of grandchildren. And it shows you that after the age of 46, evolution doesn’t care if you’re there anymore, from the point of view of getting grandchildren. As someone who is out here, I would like to congratulate all of you. There’s a reason I look different from you.

Now this way of looking at aging basically says that aging is a byproduct of selection for reproductive performance, and the reason that it occurs is that there’s an accumulation of a lot of genes, and they have positive or neutral effects on fitness components early in life, and they have negative effects on fitness components late in life. The positive effect is called the antagonistic pleiotropy hypothesis. The idea is that the gene has two effects: good early and bad late. It’s like that one that gave the fly one more baby, on the first day of life, but killed it off at the nineteenth day of life. And neutral effects early and negative effects late is called the mutation accumulation hypothesis.

And these two hypotheses formed sort of the intellectual basis of research on the evolution of aging for quite awhile, and they turn out to be not too productive. It looks like–in fact, most of the cases that have been well investigated, suggest that it’s positive effects early and negative effects late not neutral early and negative late. Okay? But it’s hard to distinguish between these sometimes.

A general take-home point is this: that organisms age is actually the best evidence we have that it’s the replication of genes, not the survival of organisms, that is the object of evolution. So that gives you strong empirical evidence that a gene-centered view of evolution is in fact empirically correct. This is extremely discouraging for the organisms that have consciousness and the ability to analyze a situation. [Laughter]

So, a bit of experimental evidence. By the way, there have been five or six experiments like this. I’m just showing you because this is the one I did. We had two treatments. We had high and low adult mortality. And if you followed the logic so far, then you already know that if you apply high adult mortality, then the organism should age rapidly, and if you apply low adult mortality, they should evolve to age more slowly. So if you make the environment risky, why try to invest in surviving, because somebody’s going to kill you anyway? And in this case it was a Swiss laboratory technician that was doing the killing, but one can imagine that it might have been a lion or something like that.

The result is that after five years, which is about 70 to 110 generations, in these flies, aging evolved as expected. The higher extrinsic mortality rates have produced shorter intrinsic life spans, and the change was about five days. It’s convenient that a day in the life of a Drosophila is about a year in the life of a human. So that gives you some feel, some kind of intuitive feel for what this means.

Basically what that means is that if we had started applying this strength of selection at the time of the Trojan War, we would have produced a response in the human population of about five years by now. Okay? Just to put it back into the human time scale. There’s a paper here. You can go read about that if you want. It gives you an entry into that literature.

### Chapter 6. Summary [00:42:32]

To summarize today’s lecture, all the major life history traits–age and size at maturity number and size of offspring lifespan reproductive investment–are involved in tradeoffs, and that causes them to come to evolutionary equilibrium at intermediate, not at extreme values. They are all under stabilizing selection caused by tradeoffs. Age and size at maturity, number of offspring per birth and per lifetime, and lifespan and aging have all evolved.

I’ll just riff on this for a moment, to tell you how you’ve changed, compared to chimpanzees and bonobos. Humans live a bit longer, about oh twenty years longer. The unique human life history traits that appear to have evolved since we shared ancestors with chimpanzees and bonobos are menopause, which does occur, but rarely, in zoo chimps, and is almost never observed in the wild. The most striking thing though is that we can have babies twice as fast as they can.

The average time in a Neolithic or hunter-gatherer society, between births is two years, in humans, and in chimps it’s five to six. That, despite the fact that human babies are much more helpless and need a lot more parental care when they’re born. So, in fact, humans have somehow managed to almost double the reproductive output of chimpanzees, and it appears that they’ve done it through social interaction.

So family members help raise the kids. Sometimes even partners help raise the kids. Grandmothers help raise the kids. But there’s a lot of help. And so the reason that the inter-birth interval in humans has been shortened dramatically in the last five or six million years is because we have become a much more highly integrated–we have a much better integrated family life.

The evolution of all of these traits can be understood, in general, as an interaction between extrinsic ecological conditions, that determine mortality rates, and conditions inside organisms to cause tradeoffs. So if you’re looking for a general explanatory structure, it is that the environment poses problems, and when you answer that problem with a solution, you are forced to make compromises and we know usually which kind of compromises, and we are now in a position to say if you’re looking in the environment you should look for these kinds of factors.

Okay, next time we’re going to extend this framework into a particular part of life history evolution called sex allocation, and how investment is divided between male and female function, and when it pays to switch sex and to be born as one sex and turn into the other.