Absolute limits to population growth II – imagine breeding

The need to critique ‘official’ claims about how fast populations of wildlife grow is clear. In the case of horses and asses, I have described how we learned this truism the hard way.

I have also established the biological limits for horses in preparation for this post.

Now a few simple ‘back-of-the-envelope’ calculations are all that is required to detect ridiculous claims about population growth (each explanation below has a bracketed letter after it which corresponds to the calculation with the same letter in the text boxes).

Say, for example, you read a claim that a population of feral horses, or burros (asses), grows at 30% per year. I use 30% because this is the largest published estimate that I could find [1]* A 30% increase in population size each year can be represented as a proportion of the current population (a) or as the population multiplier (b).

Text box 1Annual changes in population size

Now, imagine you have a closed population of 1000 horses (c).

If the population is growing by 30% each year, then when next year’s breeding season is complete it will be 1300 horses (d).

And before last year’s breeding season it was 769 horses (e), the year before that 592 horses (f), and so on.

Population size and structure

We can use these values to estimate the maximum possible size of our breeding population. Not all horses can breed. Some, for example, are foals from last year that will only become sexually mature this breeding season as 1-year-olds. Thus, they cannot contribute to this year’s population growth and we need to remove them from our breeding population.

If we assume, for the moment, that no horses die during this year (an improbability we will address later), there must be at least 231 foals in our population (g). Thus, our potential breeding population this year is just 769 horses – the same as last year’s total population (e).

There must also be at least 178 2-year-olds in our population (h). For the moment, however, we will also assume that all 2-year-olds are sexually mature and capable of breeding. I will question the plausibility of this critical assumption later in this and other posts.

Text box 2In unmanaged populations, the number of adult males and females, called the sex ratio, will be approximately equal.** Thus, only 385 of our 1000 horses are breeding mares 2 years of age and older (i).


For a population of 1000 horses to continue to grow at 30% per year, our 385 mares must produce at least 300 foals next breeding season (j). To achieve this feat, 78% of mares, or 4 of every 5, will have to foal next year (k).

In a population growing at 30% per year, 2-year-olds must be at least 46% of the breeding mare population (l). Thus, for 78% of all mares to breed we require that at least 93 of our 178 2-year-olds to also foal this year (m) or 52% of 2-year-olds.

… and death

It is clear that a 30% increase in population size each year places extreme demands on breeding mares, especially 2-year-olds breeding for the first time. This simple calculation has helped us identify what information would be required to corroborate claims of extreme population growth – evidence that large proportions of mares, especially 2-year-old mares, are successfully foaling.

Nevertheless, so far the values of mare reproduction necessary for a population to grow by 30% in a single year appear possible. We have not yet, however, factored in death. Horses live, therefore horses die.

I will factor death into our calculations in my next post.


* The estimate of 30% population growth from Wolfe (1980) is from one population of the 18 surveyed. In his article, Michael Wolfe goes to great lengths to warn the reader not to believe the estimates of population growth reported. He begins on the first page by saying “It should be emphasized that the estimates of the demographic parameters given herein are tentative at best and that a primary objective of this paper is to point out the shortcomings of the existing data… the results of the analyses thereof should be interpreted with due caution.” He is also points out that the populations may not be closed. Thus, estimates of population growth may be modified by emigration and immigration.

Although Michael Wolfe then concludes that: “the implied rates of increase based on results of consecutive aerial censuses should hardly be accepted at face value” and  “… annual rates of increase for feral horse populations in the range 20-30%, as suggested by some authors…, are highly questionable“, and advised against using this and the other values of population growth in his article because “Given the questionable reliability of regression measures developed from time series and the problems inherent in the population estimates from which they were computed, the implied rates of increase are suspect”, his extraordinary estimates have been quoted many times since as evidence in support of high growth rates, most recently in [2].

This is one of many examples where values of extreme population growth have been misquoted and misrepresented in subsequent scientific literature. Like Chinese whispers, the values are quoted without their context, caveats and qualifiers until they become a simple truth. I will address the issue of Chinese whispers in the use and interpretation of published science in a future post.

** The tendency for sexually reproducing populations to have a similar number of males and females, although a small number of one sex may monopolise breeding, has fascinated scientists for a long time and is still the subject of study and debate – surely also worth of a future post. In the Perissodactyla, like horses, asses and rhinoceros, some stallions (jacks, or bulls) dominate to prevent many, apparently reproductively superfluous males, from breeding. Nevertheless, each year more males are produced, and in equal number with females, than will breed.


1. Wolfe, M. L. 1980. Feral horse demography: a preliminary report. Journal of Range Management, 33, 354-360.

2. Palmer, G. H. et al. 2013. Using Science to Improve the BLM Wild Horse and Burro Program: A Way Forward: The National Academies Press.


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