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Ecology : Introduction
Modelling Population Growth
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:
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.
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.
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:
Despite this models can be used to make predictions. If the prediction made by the model does not fit with the might be happening to the population.
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