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WHAT IS ECOLOGICAL STABILITY? In 2019 I posed this question informally to colleagues, using Twitter, a professional workshop that I lead, and a conference. Respondents on Twitter consisted mostly of ecological scientists, but the workshop included paleontologists, biologists, physicists, applied mathematicians, and an array of social scientists, including sociologists, anthropologists, economists, archaeologists, political scientists, historians and others. And this happened…

Previous posts in this series

Environmental Variation

The discussion to this point has treated the intrinsic demographic parameters, r, R and K as constants. However, it cannot be overstated that under the current conditions of climate change, and the interactions of multiple anthropogenic drivers of population change, great importance must be placed on understanding the potential environmental impacts on those parameters and population dynamics. The impact of direct environmental perturbations was considered briefly in an earlier post, but in reality the relationship can be, and usually is, more complicated. Our collection of environmental data is expanding at a rapid pace, enabling monitoring of variables such as air temperature, precipitation, etc. at scales ranging from meters to the entire planetary surface. Similarly, we are reconstructing environmental histories in ever more detail, ranging from sub-decadal to multimillenial timescales. Those data are revolutionizing views of the relationship between stability and the environment, but they bring their own challenges. The study and literature of that relationship are vast and growing though, and the discussions in this post and the next will therefore be limited to two important topics; the treatment of time in population dynamics, and the effect of nonlinear relationships between population dynamics and environmentally impactful factors. The topics capture two fundamentally important features of environmental variation: the inconstancy of the environment, and organismal responses to the variation.

Expectations and Averages

Imagine a population growing in a randomly varying, but stationary environment, v. We’ll call it a “noisy” environment. The environment varies from one interval of time to another, but the mean environment is constant. We refer to the mean environment as the expected environment, E(v), and its value is independent of time, i.e. E(v) is expected to have the same value no matter when the population exists. Let us assume that the population growth rate, λ, is at any given time a simple function of the value of the environment, say larger values of v increase birth rates and lower values increase death rates. We use λ to signify that we model growth rate here as independent of populations size. This model was introduced in an earlier post (Eq. 2). Then the long-term value of λ is also a simple expectation or function of the mean environment. We will express this as


Eq. 1: (expected or long-term population growth rate) = (function of the mean environment)

E(r) = f(E(v))

A crucial question is this: Is the observed population size, X(t) , equal to the expected size given E(v) and λ? Returning to one of our earlier and simplest models, the expected size of the population after elapsed time T , given an initial population size X(0) and a deterministic, average growth rate of \langle\lambda\rangle, is

Eq. 2: (deterministic population size) = (initial population size) x (the product of growth rate, multiplied by itself T times)

X_T = X_0\langle\lambda\rangle^T

Mathematical digression
X(0) is the initial population size, and \langle\lambda\rangle is the factor by which the population increases during each interval of time. Therefore, after the first interval, population size is
X_1 = X_0 \langle\lambda\rangle
because \langle\lambda\rangle is an average growth rate.
\langle\lambda\rangle = \frac{1}{N}\sum\frac{X_n}{X_{n-1}}
where N is the number of observations in your population size time series. After the second interval,
X_2 = X_1 \langle\lambda\rangle = \left ( X_0 \langle\lambda\rangle\right )\langle\lambda\rangle = X_0\langle\lambda\rangle^2
Thus, after T intervals,
X_T  = X_0 \langle\lambda\rangle^T

Assume realistically, however, that the environment is a randomly varying one, though, and this environmental stochasticity means that there is no single value of v, but instead a distribution of values. Say that we characterize this distribution as a normal one, a mean and variance. Then, by Eq. 1, the population growth rate λ is also a distribution of values. If we assume here that Eq. 1 is a simple linear function, then r is also distributed normally, with mean \langle\lambda\rangle. A population living in that environment will have a value of r, during each interval of its history, drawn from the distribution. In that case, population size after time T is now given as

Eq. 3: (deterministic population size) = (initial population size) x (the product of observed growth rates)

X_T = X_0\prod_{t=1}^{T}\lambda_t

Surprisingly, actual population size given a varying environment is always smaller than that predicted by the mean environment (Fig. 1)!

The difference between deterministic, or expected growth rate given an environmental mean (red line), versus actual growth trajectories based on variation about the same mean (using Eqs. 4 and 5).
Fig. 1: The difference between deterministic, or expected growth rate given an environmental mean (red line), versus actual growth trajectories based on variation about the same mean (using Eqs. 4 and 5).

The mean growth rate predicts the deterministic population size in the absence of environmental variation, but the series of observed λ(t) predicts otherwise. The environment, v, is a random variable with a mean and variance, and we therefore treat population growth as a resulting random variable. The reason for the discrepancy is subtle — growth rate is now a function of environmental variation from one interval of time to the next — but the implications are important. The mean or expected growth rate, \langle\lambda\rangle, is derived from the environmental mean, and is hence an arithmetic mean,

Eq. 4: (average growth rate) = function of (averageD environmentAL VARIATION)

\langle\lambda\rangle=f[(1/T) \sum v(t)]

(because there are T observations), treating population growth as an additive process. Population growth, however, is in actuality a multiplicative process, where future population size is a multiple of initial population size and a series of randomly varying growth rates (Eq. 3) (Lewontin and Cohen, 1969). The true mean growth rate is therefore the geometric mean of the observed growth rates,

Eq. 5: (average growth rate) = (geometric average of observed growth rates)

\bar\lambda_G = \left (\prod \lambda_t\right )^{1/T}

which are themselves functions of the varying environment during a given interval of time, and not the mean environment over time (use the same logic as in the box above). You should be able to convince yourself that the geometric mean is always equal to or less than the arithmetic mean. In this case the geometric mean reflects the fact that population growth is not based on the average environment, but instead on the population’s history, that in turn reflects the manner in which the environment varies. This distinction is an important one to remember whenever dealing with historical, path-dependent processes, such as paleo-population time series, and there are many ways of saying it, e.g. the performance of an individual over time does not equal the average performance of the group, because the system is non-ergodic, i.e., time matters. Or, the observed value of a random variable is not the same as averaging the expected value over time. For our purposes here, the following is the important implication in a world of changing and increasingly variable environments.

Population size in a varying environment is not equal to expected population size in the mean environment.

Vocabulary
Arithmetic mean — Sum of a set of numbers divided by the size of the set.
Environmental stochasticity — The impact of nearly continuous perturbations on individual birth and death rates.
Ergodic — Property of a dynamic system if the behaviour of the system during a sufficiently long interval of time is typical of the system’s behaviour during other intervals of similar duration.
Geometric mean — The n^{\mathtt{th}} root of the product of a set of numbers, where n is the size of the set.
Stationary — A series is stationary if distributional parameters, such as the mean and variance, do not change over time.

References
Lewontin, R. C. and Cohen, D. (1969). On population growth in a randomly varying environment. Proceedings of the National Academy of sciences, 62(4):1056–1060.