Individual fitness and models of demographic and environmental stochasticity

We will describe simple cases only, neglecting age and size structure. Usually, we will refer to populations with discrete dynamics in which reproduction is annual and synchronized. We assume that populations either are semelparous (annual plants, univoltine insects ) or have overlapping generations, but consist of organisms that start breeding one year after birth and in which the survival of adults (i.e. individuals with age $\geq$ 1 year) and the fertility are independent of age. For the sake of simplicity, we assume that, if the population reproduce sexually, the sex ratio is constant, so that it is sufficient to consider the dynamics of females only. We will use as the baseline model for our consideration the following

$$N_{t+1} = \lambda _t N_t$$

where $N_t$ represents the number of adult females in year $t$ and $\lambda_t$ is the finite rate of demographic growth during year $t$.

Note that the finite rate of population growth can vary from year to year, which is why this rate is indicated as $\lambda_t$ instead of $\lambda$. The drivers of these variations may be different: density dependence (i.e. the fact that the rate may vary with $N_t$, namely $\lambda_t =\Lambda(N_t)$), demographic and/or environmental stochasticity, interactions with other populations, etc. Remember that for semelparous populations the finite growth rate is

$$\lambda_t= \sigma_{Y,t} f_t$$

where $\sigma_{Y,t}$ is the survival from juvenile to reproductive adult, while $f$ is fertility. As for populations with overlapping generations the growth rate is instead given by

$$\lambda_t = \sigma_{Y,t} f_t + \sigma_{A,t}$$

with $\sigma_{A,t}$ indicating the adult survival from year to year.

To discuss stochasticity from a quantitative viewpoint, one must first consider that the rate of demographic growth $\lambda_t$ is actually the average of the individual contributions of each adult female to the reproductive output in year $t$ and to the survival from year $t$ to year $t + 1$. The contribution of the $i$-th female to the change of population abundance is termed individual fitness. Rather than resorting to convoluted theoretical definitions, it is more appropriate to understand the concept of individual fitness through a simplistic example. One of many great tit females, say the $i$-th, belonging to the considered population produces 5 eggs in year $t$. 4 of these eggs hatch, but only 2 of the 4 birds are female and only one of them survives until the first birthday. Also the $i$-th female succeeds in surviving until the $t + 1$-th breeding season. Therefore, it turns out that

   fitness$$(i, t) = 1+1 = 2.$$

If we denote by $w_{i, t}$ the fitness of the $i$-th female in year $t$, we can then write that

$${N_{t + 1}} = \sum\limits_{i = 1}^{{N_t}} {{w_{i,t}}},$$

which implies that

$$\lambda _t=\frac{1}{N_t}\sum\limits_{i = 1}^{{N_t}} {w_{i,t}}$$

is nothing but the average fitness of the $t$-th season. There is therefore a strong link between the finite rate of population growth and individual fitnesses. It is also instructive to break down each individual fitness into the sum of two terms. The first term is the expected value of the fitness of the $i$-th female (say $\overline {w}$), while the second term $\delta$ is the deviation from the mean. It is almost always reasonable to assume that, while the expected value of the fitness depends on season $t$, the deviation from the average value depends only on the characteristics of each individual.

Figure 8: Annual change in the distribution of individual fitnesses in two species of passerine birds: grey bars report the figures for the song sparrow (Melospiza melodia) at Mandarte Island, Canada, and white ones those of the great tit (Parus maior) in Wytham Wood, England (after Lande and Saether, 2003). The dashed lines display the average fitness of the population in each year $t$ (in the main text indicated as ${\overline {w_t}}$), while the solid line is the mean value of the average fitnesses across years (in the main text indicated as $\overline {\overline w}$). The number of sampled females from which statistics are derived is denoted by $N_t$.

We can thus write

$${w_{i,t}} = \overline {{w_t}} + {\delta _i}$$

where both $\overline {{w_t}}$ and $\delta_i$ are stochastic variables. In particular, $\overline {{w_t}}$ is the average of $w_{i, t}$ taken with respect to females that make up the population at time $t$ while, by definition of $\delta$, we have that E$\left[\delta _i\right]=0$, i.e. the expected value of the deviations vanishes. Basically, the individual fitness in a breeding season is the sum of a random variable that reflects the effect of environmental stochasticity ( $\overline {{w_t}}$) and one that accounts for demographic stochasticity ($\delta_i$). Fig. 8 provides an idea of ​​the variation of individual fitnesses - and particularly the diversification of the value of $\overline{w_t}$ and the values ​​of $\delta_i$ - in two populations of birds: the song sparrow (Melospiza melodia) and the great tit (Parus maior). The contributions of the two different sources of stochasticity to the finite growth rate can be calculated as

$${\lambda _t} = \dfrac{1}{N_t}\sum\limits_{i = 1}^{{N_t}} {w_{i,t}... ...overline {w_t}} + \frac{{\sum\limits_{i = 1}^{{N_t}} {{\delta _i}} }}{{{N_t}}}.$$ (3.1)

In conclusion, the random variable $\lambda_t$ - remember that it is the average fitness in year $t$ - is then the sum of two random variables, one depending on the year (environmental stochasticity) and the other depending on the variability between individuals (demographic stochasticity). Therefore it is reasonable to assume that the two random variables are independent.

The simplest statistical properties of $\lambda_t$ can be obtained by assuming that environmental stochasticity is a process without a trend, i.e. it is stationary. In other words, we assume that the mean and variance of the process $\overline{w_t}$ are independent of time, namely

$$\left[\overline{w_t}\right] = = \overline{\overline w}$$

$$\left[\overline {{w_t}}\right] = = \sigma_e^2 .$$

If we denote by $\sigma_d^2$ the demographic variance - i.e. Var$\left[\delta _i\right]$ - and keep in mind that $\overline {{w_t}}$ and $\delta_i$ can be supposed to be independent random variables (because $\overline{w_t}$ depends on the variability of environmental conditions, while $\delta_i$ depends on the variability of individuals), we obtain from eq. 6

$$\left[\lambda_t\right] = \overline{\overline w}$$

$$\left[\lambda_t\right] = \sigma_\lambda ^2 = \left[\overline {w_t}\right]+\frac{1}{N_t^2}\sum\limits_{i = 1}^{{N_t}} {\text{Var}\left[\delta _i\right]} = \sigma_e^2 + \frac{\sigma _d^2}{N_t}.$$

Tab. 1 shows that the demographic variance is in the order of $0.1$-$1$ for many populations, while the environmental variance is usually one order of magnitude smaller. However, demographic stochasticity influences the variance of the growth rate through the factor $\frac {1}{N_t}$ and therefore it is practically immaterial for large populations, namely whenever

$$N \gg \frac{\sigma _d^2}{\sigma _e^2} .$$

In an empirical way, for a given population we can define a critical population number

$$N_c = 10 \frac{\sigma _d^2}{\sigma _e^2}$$

below which demographic stochasticity cannot be neglected. The critical number $N_c$ is often in the order of hundreds, but can indicatively vary between $10$ and $1000$, as one can check by calculating $N_c$ for the populations listed in Tab. 1.

Table 1: Demographic variance ( $\hat {\sigma }_d^2$) and environmental variance ( $\hat {\sigma }_e^2$) in populations with different ages at first reproduction ($\alpha$). Bibliographic references and data after Table 1.2 in Lande and Saether (2003).

Species	Location	$\alpha$	$\hat {\sigma }_d^2$	$\hat {\sigma }_e^2$
Barn swallow (Hirundo rustica)	Denmark	1	0.18	0.024
White-throated dipper (Cinclus cinclus)	Southern Norway	1	0.27	0.21
Great tit (Parus maior)	Wytham Wood, U.K.	1	0.57	0.079
Pied flycatcher (Ficedula hypoleuca)	Hoge Veluwe, The Netherlands	1	0.33	0.036
Song sparrow (Melospiza melodia)	Mandarte Island, Canada	1	0.66	0.41
Soay sheep (Ovis aries)	Hirta Island, U.K.	1	0.28	0.045
Brown bear (Ursus arctos)	Southern Sweden	4	0.16	0.003
Brown bear (Ursus arctos)	Northern Sweden	5	0.18	0.000