In the previous section, we considered a very general model of population regulation. In that model, the per capita birth and death rates were linear functions of population size. Here we will consider a different model of population regulation, but one which has exactly the same features as the more general model. This model is the model of logistic population growth. This model was first proposed by Pierre François Verhulst in 1838 as a model of population growth. The model takes its name from the shape of the relationship between population size over time – it has a “logistic” shape. The specific mathematical function used for logistic growth is ${dN \over dt} = r N (1 - {N \over K})$

where r is the intrinsic rate of population increase and K is the carrying capacity. The intrinsic rate of increase is interpreted as the per capita population growth rate when the population is very small. The parameter K is called the “carrying capacity” because it is the stable equilibrium of the population if r>0, and is interpreted as the number of individuals that the environment in which this population resides can support.

Let’s first analyze this model in the typical way to evaluate where the equilibria may lie, and when those equilibria are stable. Equilibria: To find where the equilibria lie, set the equation equal to zero and solve for N. Remember, we do this because an equilibrium is defined as any population size having ${dN \over dt} = 0$. When we do this we find two: N*=0 and N*=K. Stability: Now evaluate when each is stable or unstable. Remember, to do this, we take the partial derivative of dN/dt with respect to N and evaluate this function at each equilibrium. The derivative of the logistic equation with respect to N is ${\partial ({dN \over dt}) \over \partial N} = {\partial \over \partial N} (N r (1 - {N \over K}))$ $= {\partial \over \partial N} (r N - {r \over K} N^2)$ $= r - 2 {r \over K} N$

First, evaluate the stability of N*=0. Substituting N*=0 into this last equation gives ${\partial ({dN \over dt}) \over \partial N} = r$. Remember, that the equilibrium is stable if this is negative. Thus, N*=0 is a stable equilibrium if r<0.

However, N*=0 being a stable equilibrium, is bad for the population, because it means that the population will go extinct when population size is small. But this is what r<0 means: the population will decline when rare. N*=0 being an unstable equilibrium is a good thing for the population, because this means that the population will increase when rare. This is true when r>0, which is also the criterion for N*=0 to be an unstable equilibrium. Now, evaluate the stability of N*=K. Substituting into the above equation gives ${\partial ({dN \over dt}) \over \partial N} = -r$. Thus, N*=K is a stable equilibrium if r>0, and is unstable if r<0.

##### Compared to the General Model of Population Regulation

The logistic model is typically used as the basic model of population regulation, because of the intuitive interpretations that can be placed on r and K. However, this intuition comes at a price. Our general model of population regulation was based on the relationship between per capita birth and death rates. Where are the birth and death rates in this model? The typical way the logistic model is considered is as a modification of exponential population growth. Remember that the exponential population growth model is ${dN \over dt} = r N$

Compare this equation to the logistic equation above. The logistic equation is simply exponential growth multiplied by (1-(N/K)). This second term simply decreases population growth rate as N increases. The population has the same per capita growth rate as a comparable exponentially growing population when N=0, and population growth rate is zero at N=K because (1-(N/K)) = (1-(K/K)) = (1-1) = 0 (this is why the parameter K is the equilibrium). Let’s now return to the question of where are the birth and death rates in this model. Specifically, compare the logistic model to the model of population regulation based on per capita birth and death rates. First, manipulate the logistic equation to put it in a comparable form: ${dN \over dt} = N r (1 - {N \over K})$ $= r N - {r \over K} N^2$

Then compare it to the birth-death model of population regulation: ${dN \over dt} = (b_{0} - d_{0}) N + (\beta - \delta) N^2$

When written next to one another, one can see the analogy that r=(b0-d0) and -(r/K)=(β-δ). In other words, the intrinsic rate of increase has exactly the interpretation that corresponds to its interpretation in the more mechanistic model, namely that it is the per capita growth rate of the population when it is very small (remember that b0 and d0 are the intercepts of the per capita birth and death rate functions where N=0). –r/K is more difficult to imbue with an intuitive interpretation. However, this term is the measure of how per capita birth and death rates change relative to one another. In this model, it is simply specified.

As discussed in the first section of this primer, a critical issue to continually be aware of is the relationship between total and per capita population growth rates. For the logistic equation, total growth rate changes as a humped-shaped function of N that crosses the N axis at zero and K (remember that these are the equilibrium population sizes) and has its maximum value at K/2. (As an exercise, calculate the total population growth rate at this maximum value.  Also, explain whether each of these equilibria are stable or unstable based on the features of only this figure.) In contrast, the per capita growth rate of the population declines linearly. The line intercepts the ${dN \over Ndt}$ axis at ${dN \over Ndt} = r$, it has a slope of –r/K, and it intercepts the N axis at N=K. Note that in contrast to total growth rate, the per capita growth rate has only one value where it is zero. This is why you find the location of the equilibria using total population growth rate. (As an exercise for yourself, make sure you know exactly how to plot these functions of the logistic equation but doing the algebra and then plotting the function. Memorizing what the graph looks like and what are the important points and features is absolutely the wrong way to learn this.)