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:
1. Welcome Back Video
2. Introduction

THE SIMPLEST CONCEPT OF ECOLOGICAL STABILITY IS A POPULATION OF STATIC, UNCHANGING SIZE. For example, if we are considering the population size of a species, X, then X in static equilibrium has the same, unchanging value over time. Or, population size might be variable but over time the net change of population size is zero, in which case the system could be in dynamic equilibrium. I will refer to the entire population as a system, by which I mean simply a collection of potentially interacting objects. In this case the interacting objects are conspecific members of the same population.

The basic questions of stability to be answered for any ecological or related (e.g. socio-ecological) systems are:

1. Is the system static or dynamic?
2. If the system is dynamic, can we formulate a dynamical law that predicts the next state of the system given the current state?
3. Given a dynamical law, can we determine the driver or drivers of the system’s dynamism? Is the driver intrinsic? That is, if the system was a closed system, would it remain dynamic, or, how would the dynamics change?

Note that in closing a system by isolating a population, one can include anything in “the box” to the exclusion of other potential drivers, e.g. a population and the fishing pressure on it, excluding climate change. Such an approach is a primary motivation and guide to building scientific models.

LET US BEGIN WITH A VERY SIMPLE MODEL OF POPULATION GROWTH, the logistic model. Here we describe the growth of a population as regulated by two factors: the intrinsic, or population-specific rate of increase of the species, e.g. measured as average birth rate minus average death rate; and the carrying capacity of the environment, i.e. the maximum population size of the species in this place and time based on limiting factors, such as biological resources and the abiotic environment. In mathematical terms, we write this as

EQ. 1: (POPULATION GROWTH RATE) = (INTRINSIC RATE OF INCREASE) x (POPULATION SIZE) x (GROWTH REGULATION)

$\frac{dX}{dt} = rX\left( 1-\frac{X}{K}\right )$

where dX/dt is the population’s growth rate (change in X per unit of time), r is the intrinsic rate of increase of the species, and K is the population’s carrying capacity.

This differential equation expresses a logistic growth rate, where growth is exponentially fast at small population size, but slows toward zero as the carrying capacity is approached. The first product on the right hand side (RHS) of Eq. 1, rX, would if solved for X result in exponential or Malthusian growth,

EQ. 2: (POPULATION SIZE AFTER TIME t) = (INITIAL POPULATION SIZE) x (EXPONENTIAL REPRODUCTION DURING TIME INTERVAL t)

$X_t = X_0 e^{rt}$

where $X_0$ is the initial population size (Fig. 1). The growth rate is the Malthusian growth rate (r) (Malthus, 1798) multiplied by the number of individuals already in the population, and thus accelerates over time, yielding an exponential relationship.

Any population with a positive growth rate will increase exponentially if left unchecked.

The growth rate rX makes sense, as it can be interpreted as the number of offspring that an individual will produce per unit of time, multiplied by the number of individuals in the population. But why does that yield an exponentially increasing population? The answer is given below, but first a brief Malthusian digression.

(From Wikipedia) Thomas Robert Malthus FRS (February 1766 – December 1834) was an English cleric, scholar and influential economist in the fields of political economy and demography. In his 1798 book An Essay on the Principle of Population, Malthus observed that an increase in a nation’s food production improved the well-being of the populace, but the improvement was temporary because it led to population growth, which in turn restored the original per capita production level.

Deriving the Malthusian Equation
Let the Malthusian growth rate be
$\frac{dX}{dt} = rX$
Here the growth of X is a function of X, that is, the rate of population growth depends on the size of the population. In order to calculate the size of the population at any given time, however, we first re-arrange the equation to make it a function of time elapsed, dt.
$\frac{dX}{X} = r \, dt$
Integration to solve for X is then straightforward.
$\int \frac{1}{X}dX = \int r \, dt$
The integral of 1/X is the logarithm of X.
$\Rightarrow \ln{X} = rt + C_1$
$\Rightarrow e^{\ln{X}} = e^{rt + C_1}$
$\Rightarrow X = e^{rt} \cdot e^{C_1}$
$\Rightarrow X = C_{2}e^{rt}$
C 1 and C 2 are simply constants. And of course, e, or Euler’s number, raised to any positive number yields a relationship where the output increases faster as the input increases.

Vocabulary
Carrying capacity — The maximum population size that can be supported by the environment.
Closed system — A system where the passage of specified materials, objects or information into or out of the system is prohibited or controlled. The system is isolated from the rest of the Universe, behaving as if nothing else existed.
Dynamics — The study of how systems change or develop over time.
Dynamic system — A system that varies or changes over time. Dynamic systems can be in equilibrium, if net change is zero.
Equilibrium — A condition where the state of a system either does not change, or experiences no net change over time.
Law A scientific law is a statement, based on multiple observations or experiments, that describes a natural phenomenon. Laws do not have absolute certainty and can be refuted by future observations.
Population A system of conspecific individuals occupying overlapping space at the same time.
System A collection of potentially interacting objects.

References
Malthus, T. R. (1798). An essay on the principle of population. Library of Economics and Liberty.