2 Interacting populations 1: competition
Case study: Red and grey squirrels
Populations rarely (if ever) exist in isolation. In reality, the growth rate of a given population depends not only on itself, but also on other populations that it interacts with either directly or indirectly. Such interactions lead to a range of ecological relationships, including competition for resources, predation, mutualism, parasitism and more besides. In the next chapters we will study models that represent a few of these different interactions.
Our first example will be of two species that compete for a common resource. This is sometimes known as the Lotka-Volterra model (though I personally try to avoid using labels named after people – they can be confusing and ingrain bias) or interspecific competition. To guide our thinking we will have a case study in mind, namely red and grey squirrels – well-known competitors in the UK – but we could equally consider microbes competing for substrates, plants competing for water or nutrients or a whole host of species competing with one another for territory or food. As with our previous single population models, we begin by considering what factors might contribute to the birth and death mechanisms that control the rates of changes of the two populations. We will assume:
- There is only a limited supply of food for squirrels to eat. This means that even in the absence of interspecific competition, the environment would have a limited carrying capacity (i.e. a maximum number of squirrels that it could support). The population dynamics for each species on its own can be modelled using a logistic growth equation (that does include intraspecific competition between individuals in the same population).
- The effect of one species on the other is proportional to its population size (since the more there are, the less food, territory and other resources will be available). Therefore there will be a term in each equation causing a reduction in growth that is given by the product of the two species’ densities.
We also assume that the carrying capacity for the two species of squirrel can be different (perhaps because one species can eat a wider variety of food than the other), and that the two species can have different susceptibilities to competition (perhaps because one species is more timid than the other). Without interspecific competition, we could model the two populations using two logistic growth equations:
[latex]\begin{align} \frac{dR}{dt} & =R(a-R) \\ \frac{dG}{dt} & =G(c-G), \end{align}[/latex]
where [latex]R[/latex] and [latex]G[/latex] represent the respective population densities of red and grey squirrels. Note that we have already done a little non-dimensionalising here, and that [latex]a[/latex] and [latex]c[/latex] are thus the respective carrying capacities (since if, for example, [latex]R\gt a[/latex], we have [latex]dR/dt\lt 0[/latex]). To include competition between the species, we add an additional term to eac