1 Single population models
A first population model
Our aim in this textbook is to model the dynamics of populations over time. By ‘population’ we simply mean some collection of individuals that are subject to the same underlying mechanisms. For example, we may consider the human population of Sheffield, or the tapir population of South America, or the maize crop population of a field in Nigeria. In later chapters we will move to smaller biological scales, considering perhaps less intuitive definitions of populations, such as of cells in the body, or proteins within cells. However, a key message to take from this textbook is that we can consider pretty much any biological populations in the same way from a modelling perspective, and subject to the same fundamental biological mechanisms.
We will model these dynamics using ordinary differential equations, and our focus will be on how the size of a population varies over time. We will not consider where individuals are in space – this would likely require extending our methods to partial differential equation models, and plenty of excellent courses and textbooks can be found to explore such systems. When we talk about the ‘size’ of a population, we might think we mean the number of individuals. However, as we will be using ordinary differential equations we need our variables to be continuous, not discrete. We will therefore instead keep track of a population’s density – the number of individuals within some unit area.
Introducing our mathematical notation, we might call [latex]N(t)[/latex] the density of individuals within our population at time [latex]t[/latex]. We now wish to write down an ordinary differential equation that describes its dynamics – that is what causes the population to increase or decrease. Our first modelling challenge, then, is to decide what mechanisms we should include. What mechanisms would lead a population to change in size? Using some biological intuition for the simplest possible population we can think of, we might expect our model to look something like,
[latex]\begin{align} \frac{dN}{dt} &= \textrm{births} -\textrm{deaths}. \end{align}[/latex]
Now to make any mathematical progress we will need to decide on functional forms for each of these mechanisms. This is another important modelling decision, and will depend on the specific biology. There are three main forms we might use:
- constant rate, [latex]\textrm{births}=b[/latex]. This would be suitable when new individuals appear in the environment at some constant rate, for example due to migration, or production of cells by the body.
- per-capita rate, [latex]\textrm{births}=bN[/latex]. This would be suitable when each individual produces offspring at a certain rate, such that there are more offspring produced when the population is larger.
- density-dependent rate, [latex]\textrm{births}=b(N)N[/latex]. This would be suitable if the per-capita rate is not fixed but instead varies with the population size, for example