The amount of variation that is present in biological systems makes predicting their growth difficult. However, there are some systems which have been successfully modelled and for which there are equations that permit us to make predictions:

• in genetics Mendel’s laws make predictions about simple inheritance patterns,

• in population genetics the principle of Hardy Weinberg predicts inheritance, 

• in enzymology it is possible to predict the behaviour of enzymes at low temperatures by applying the temperature coefficient, the Q₁₀.

Modelling population growth, the maths

Population growth follows the numbers of individuals in a population through time. So the models try to trace what will happen little by little as time passes by.

A small change in time is given by which is usually reduced to dt.

In population models time may be measured in regular units such as years or even days or it may be measured in units such as generations.

A small change in numbers is given by which is usually reduced to dN

Exponential growth

We know the population growth of species in a newly colonised habitat will start exponentially.

This is an example of positive feedback. The more individuals there are in a population the faster they will breed. The growth curve looks like this (often called the J-shaped curve):

The curve gets steeper and steeper with time.
Even Darwin was aware of this. He calculated that 1 pair of elephants could produce 19 million elephants in 700 years.

Modelling the curve

The exponential growth curve can be modelled by the equation:

r is the rate of increase (the rate of reproduction aka biotic potential) of a species.
For example if a population increases by 4% per year we would write:

Of course the higher N becomes the bigger the increase in the population becomes. This leads to exponential growth.

Real examples of exponential growth

The periodic explosion of pest species populations, such as locusts, are modelled by ecologists to predict their growth.

Pest species show exponential growth because humans provide them with a perfect environment for population growth.

Alien species, which often become pest species, also show this pattern. When a new species is introduced accidentally or deliberately into a new environment it has no natural predators or diseases to keep it under control.

European starling (Sturnus vulgaris), a bird, was introduced into the United States. Between 1890 and 1891, 160 of these birds were released in Central Park New York. By 1942 they had spread as far as California. An estimate population of between 140 and 200 million starlings now exist in North America, making it one of the commonest species of bird on Earth.

The Allee Effect and the living dead

In our example of starlings above, it should be added that there were several attempts to introduce starlings to the US during the 19^th century. In Pennsylvania 1850, in Ohio 1872 and in Oregon in 1889. These all failed. The starlings in Central Park New York were introduced in two waves April 1890 and March 1891. It took them10 years to establish themselves in New York city and it’s suburbs but only 40 more years to cover the rest of the country (7.8 million km²).

This slow start is called the Allee Effect and it seems to be related to population density. For animals showing sexual reproduction they need to find a partner. This is not easy when the population density is low. This gives the growth curve a slower start than exponential growth would predict.

This effect of under population can explain why when species threatened with extinction are pushed below a certain level, they cannot recover and eventually die out. These species on the verge of extinction are called the living dead by conservationists.

The Carrying Capacity

One of Darwin’s important observations was that a population never continues to grow exponentially for ever. There is a resistance from the environment as the food supply nesting sites decrease (i.e. competition increases) and the numbers of predators and pathogens increase. This resistance results from negative feedback.

This leads to the classic s-shaped or sigmoid population curve (below):

This too can be modelled but it needs a component in it that will slow down the population growth as it reaches a certain point, the carrying capacity of the environment (K).

The equation is called the logistic equation:

Models are only approximations

Mathematical models rely on a number of assumptions. These assumptions do not always hold. In the case of population growth models, the following assumptions are made:

1. all the individuals are identical at any one time.

2. limiting factors are constant throughout the period of growth.

3. the rate of population growth is proportional to the degree of unsaturation of the habitat.

4. the age distribution of the population is stable.

5. time lags of the factors affecting the population growth are insignificant.

6. the carrying capacity (K) is constant [which goes with (2)].

Despite this models can be used to make predictions. If the prediction made by the model does not fit with the actual results, the model can be used to identify what might be happening to the population.