This is the final principle of population dynamics, at least from my (narrow = numerical) point of view. It is also the most difficult to deal with from the animal’s standpoint. With plants, it is really fairly easy to break down the inorganic elements required into their components, and to find out which one is limiting the population. Thinking in this manner, the chemist Justin von Liebig proposed (Liebig 1840), what Blackman later called the law of the minimum (Blackman 1905). What this law means is that, of all the elements needed by the plant, the only one that limits its growth is the one in the minimum. I note that there is some disagreement amongst botanists over this law, but it is difficult to argue with it from a feedback standpoint (see later). There are some constraints on the law: For instance, it requires that the limiting factor be independent of other factors in the environment. It also states that the limiting factor only becomes dominant as the population gets close to equilibrium. This concept has far reaching consequences for it says that, of all the elements required by plant growth, only one acts to limit growth at any particular time and place. This greatly simplifies the equations for projecting plant growth.

With animals, there are some difficulties, brought to light by the fact that this law is rarely thought of importance to animals. Animals eat plants or other animals and, therefore, obtain all the nutrients they need in this manner. But they sometimes use many species as food, and are also attacked by multiple species of viruses, bacteria, parasites, predators, etc. They also need water, and nesting places or territories. The law of the minimum implies that only one of these factors can regulate a population in the vicinity of equilibrium.

Other people have come to the same conclusions from different points of view. For instance, Paine (1980, 1992) and Michalski and Arditi (1995) found that food webs simplified as the system evolved towards equilibrium, when food web architecture became dominated by a few strong interactions. The strong interactions were often associated with predators at the top of the web, the so-called keystone predators. Thus, the food web simplified, in terms of the number of interactions, as it evolved towards equilibrium. Berryman (1993) came to a similar conclusion on the basis of feedback architecture. It is fairly easy to show that, in a multi-feedback system, only a single negative feedback loop will regulate a variable (e.g., the density of a population) as it approaches equilibrium.

The law of the minimum is a sweeping principle since it states that, near equilibrium, only a single variable is likely to regulate the size of a population. This principle also explains why the dynamics of populations are usually of low order, the oscillations in population size being caused by one or two variables rather than a large number. If many factors were operating we would expect chaotic dynamics, which is almost never seen in nature.

Now although this principle states that a single limiting factor is likely to regulate the abundance of a population at any given time, it does not say that the limiting factor cannot change with time. This is easiest to see, once again, with plant populations. Say nitrogen is limiting the population at a particular time, and more nitrogen is then added to the soil by a farmer. The plant population will grow to a higher level. However, at some point another chemical, say phosphorus, will become in short supply, and then that nutrient will take over as the limiting factor. Thus, we can see shifting limits as the properties of the environment change with time. It is also possible for more dramatic changes to occur, changes in response to shifts in population density rather than in the limiting factors. Under these conditions we can observe two potential equilibrium densities in some populations that are determined by two different limiting factors, i.e., a low density (endemic) and a high density (epidemic) level, that is separated by an unstable threshold (cooperative equilibrium). This occurs with some of the aggressive bark beetles which can overcome resistant hosts by coordinated mass attack (Berryman et al. 1984). Random variations in the size of the population can cause it to cross the unstable equilibrium and thereby unleash the alternative behavior. This is called bi-stable dynamics.

Up to this point we have discussed populations that do not vary far from equilibrium. However, we know that some cyclic populations can vary tremendously from their steady states. These populations can be affected by different factors as they go through their cycle. In the case of bi-stable dynamics, the high density equilibrium is often cyclic. Bark beetles, for instance, kill the trees they have attacked and thereby diminish the food for future populations.

I hope I have given you an understandable view of the law of the minimum. It is a complicated principle but it gives rise to simplified mathematics, for it states that a single factor, or a group of interdependent factors (for instance the number of species used as a food by an animal), will determine the equilibrium population.

Berryman, A. A. 1993. Food web connectance and feedback dominance, or does everything really depend on everything else? Oikos 68: 183-185.

Berryman, A. A., N. C. Stenseth and D. J. Wollkind. 1984. Metastability of forest ecosystems infested by bark beetles. Res. Popul. Ecol. 26: 13-29.

Blackman, F. F. 1905. Optima and limiting factors. Ann. Bot. 19: 281-295.

Liebig, J. 1840. Chemistry and its application to agriculture and physiology. Taylor and Walton.

Michalski, J. and R. Arditi. 1995. Food web structure at equilibrium and far from it: is it the same? Proceedings of the Royal Society of London B, 259: 217-222.

Paine, R. T. 1980. Food webs: linkage, interaction strength and community infrastructure. J. Anim. Ecol. 49: 667-685.

Paine, R. T. 1992. Food-web analysis through field measurement of per-capita interaction strength. Nature 355: 73-75.