Logistic Differential Equations and Population Modelling

Posted on by Dinh Quan Tran

So today I want to discuss a little bit about a special type of differential equation – the logistic differential equation, which holds particularly great applications in population modelling. The normal form of the logistic differential equation can be expressed as follows:

f'(x) = r(1- \frac{f(x)}{K})f(x)

And in the simplified case K = 1, then

f' = rf(1-f)

Background of the equation:

The logistic function was introduced by the Belgian mathematician Pierre François Verhulst (1804-1849) in a series of three papers between 1838 and 1847 under the guidance of his tutor at the University of Ghent, Adolphe Quetelet. The function was originally intended to model population growth in the face of obvious limitations in the model currently being used at that time, the exponential growth model. Verhulst devised the function as early as the mid-1930s, but only presented an expanded analysis as well as gave the name of the function in 1944.

The original image of Verhulst’s logistic curve in comparison to a logarithmic curve, or exponential curve in modern language

Mathematical Solution for the Equation:

We start with the equation:

\frac{df}{dx} = r(1- \frac{f(x)}{K})f(x)

which after variables separation turns into

\frac{df}{f(1-\frac{f}{K})} = rdx

and, after simplifying the left side,

\frac{df}{f} + \frac{df}{K-f} = rdx

Integrating both sides yield

ln(f) - ln(K-f) = Ce^{rx}

Replacing x = 0 yield the value of the constant of integration C = \frac{f(0)}{k-f(0)}, and further changes yield the final result

f(x) = \frac{Kf(0)e^{rx}}{K + f(0)(e^{rx}-1)}

Application in Population Modelling

As mentioned above, the exponential model was previously used for population modelling, but it fell out of favor with its long-term inaccuracy (which can easily be observed, as for example if the population of the world has been increasing exponentially than the Earth would have been overpopulated a long time ago). With the introduction of this model, Verhulst for the first time, and correctly so, accounts for the carrying capacity K, which is the maximum population an ecosystem can hold.

The graph can be broken down into 3 phases. The first and second phase is when the population is still small and hence 1- \frac{f(x)}{K} is very close to 1, yielding the rate of f' = rf typical of exponential growth models. The first phase is the “initial” phase in which the population is still very small and increases slowly even with exponential growth. In reality, this is the phase where the population has just started in a new ecosystem and has to adapt to it, which extensively limits its growth.

In the second “booming” phase, the population has completely adapted and has grown enough in numbers to be able to exploit the resources of the ecosystem to its advantage. This period is marked with rapid growth in the population as the resources are still more than enough to fuel the boom. The rate, therefore, increases as the newborns will grow up and add to the reproductive capacity of the population together with already existing individuals.

Of course, resources are not infinite, and the population will soon go to the third phase, the “saturation” phase. It is worth noticing that the factor 1- \frac{f(x)}{K} increases in importance as the population grows and eventually would be enough to start decreasing the growth rate of the population – which ideally is after the rate have reached \frac{rf^{2}}{4} at population \frac{K}{2}. In reality, this is when the resources are no longer enough to support every single individual. Competition will arise in the population, and only the ones with enough nutrition will survive and maintain their reproductive capability, explaining the decrease in the rate.

It can also be pointed out that the three-step model discussed above is only applicable when K is ideally fixed. However, K will never be so, and therefore we can add almost an infinite number of phases according to the changes in the K-value. Catastrophes such as volcano eruptions, storms and tsunamis can destroy the habitat and lower the K-value, sometimes below the existing population and resulting in a negative growth rate. Even without them, the waste from the population itself could be enough to degrade the ecosystem and result in the same phenomenon. On the other hand, positive factors such as technological development can raise the K-value either gradually (for example, soil improvement) or instantly (new agricultural products with better yields). This would more likely to keep the “booming phase” lasting longer, or, sometimes in the case of instant K-value rise, a two-slope development corresponding to two different booming phases.

This concludes my discussion of the logistic equation and arguably its most important application. I just feel that the topic is fascinating enough to deserve a separate blog post; and until my next one, math on!

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