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
3. Malthusian Populations

We were introduced to the logistic equation (Eq. 1) in the previous post,

EQ. 1: (POPULATION GROWTH RATE) = (INTRINSIC RATE OF INCREASE) x (POPULATION SIZE) x (GROWTH REGULATION) $\frac{dX}{dt} = rX\left( 1-\frac{X}{K}\right )$

and then explored Malthusian populations, where populations grow according to the first term of the function, rX. Examining Eq. 1, however, we note that the second term on the RHS includes K, the carrying capacity of the environment. K is a bit of an odd parameter because it is never something that we can determine a priori, but instead must be deduced or inferred from observations (preferably long-term observations). It is basically our assessment of the number of individuals of the species that can be accommodated in the environment, whatever that assessment might be based on. Knowing, however, that all resources on the planet are finite, we infer that all species must have finite carrying capacities. In that second term, we note that as X increases and approaches K, the entire second term approaches zero. K puts the brakes on growth.

The sizes of all populations are ultimately limited on a finite planet.

The logistic model captures this intuitive fact.

Figure 1 shows our population from the previous post, but this time with a carrying capacity set at 100 individuals. The population grows from a small initial size and asymptotes toward K. The growth rate is initially very rapid, as the population expands to “fill the environment”, but slows down as fewer additional individuals can be accommodated. There are two equilibria for this population, i.e. points at which, once attained, X will remain. The first is trivial, being zero, or an extinct population, and the other is K. “Zero” is an unstable equilibrium, because the population will always increase away from zero if X > 0, i.e. if reproducing individuals are added to the environment. Logistic growth. Population size is limited by carrying capacity, growing to that limit and remaining there. X(0) = 1, r = 1 and K = 100. That is, the environment can sustain up to 100 individuals of the species.

Parameter r

Like K, r is also an interesting parameter. What do we mean when we say “intrinsic rate of increase”? r is the instantaneous rate of change of population size, and can be estimated over an interval of time as the per capita number of births minus the per capita number of deaths. It is not a population growth rate, the actual population growth rate, dX/dt, being a function of population size (X), r, and, in the case of limitation by carrying capacity, K. The proper way to interpret r, and other such modifiers of growth rate (dX/dt), is as the net reproduction per individual in the population which, being the number of individuals produced per individual per unit of time, yields units of 1/time . Population growth rate is a function of this individual productivity and standing population size, or the number of individuals produced, in total, per unit of time. Contrast what happens when r is positive versus when it is negative. If r > 0, then dX/dt is positive and the population increases. If r < 0 then population growth rate is negative, and the population shrinks.

Deriving r is instructive, and helps us to understand this widely-estimated and applied parameter. We begin with a simple, density-independent model (Lewontin and Cohen, 1969), i.e. a population whose growth is not limited by the number of individuals in the environment.

EQ. 2: (FUTURE POPULATION SIZE) = (POPULATION GROWTH RATE) x (CURRENT POPULATION SIZE) $X_{t+1} = \lambda X_{t}$

where λ is the proportional change in population size between times t and t+1, sometimes termed the “deterministic population growth rate”. Re-write population growth, letting growth in a single time interval be a function of the total numbers of births (B) and deaths (D) during the interval: X(t+τ) = X(t) + BD, where τ is the amount of time elapsed. The growth rate during the interval then is given as the increase (decrease) of population size, divided by the duration of the interval.

GROWTH RATE WITHIN A SINGLE INTERVAL IS THE NUMBER OF BIRTHS MINUS THE NUMBER OF DEATHS DIVIDED BY THE DURATION OF THE INTERVAL $\frac{\Delta X}{\tau} = B-D$

If we now express B and D as results of the production and losses of all individuals in the population, then B = bX and D = dX, where b and d are the per capita birth and death rates. It should then be easy to see
that if we shrink the duration of the interval to be infinitesimally small (i.e. τ tends toward zero, the basis of calculus; nice connection to Isaac Newton and the pandemic of his time), we derive the differential formula $\frac{dX}{dt} = bX - dX = (b-d)X = rX$

which if you recall is both the basis of the Malthusian exponential growth formula, and also Eq. 1 without the limitation of carrying capacity!

Finally, it should be noted that the simulation illustrated in Fig. 1 is not a plot of Eq. 1. The formula is for the logistic growth rate, whereas the plot is of population size over time — the integral of Eq. 1. In the next post I will show you how to derive the logistic equation itself, $X(t) = \frac{K}{Ae^{-rt}+1}\, \mathtt{,}$

discuss a discrete and accessible parallel to the continuous logistic function, and give a real-world example of logistic population growth.

Vocabulary
Carrying capacity The maximum population size that can be supported by the environment.
Intrinsic rate of increase Intrinsic birth rate minus intrinsic death rate, or the per capita productivity of an individual per unit of 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.