Verhulst model

Fourteen years ago, in a post on Lotka-Volterra equations, I have briefly mentioned Verhulst's population dynamics model. But since then, I still haven't written a dedicated post about this model. As in the near future I would like to explore its stochastic modification, let us explore the deterministic version in this post.

Malthusian growth model

In the end of XVIII century Thomas Malthus wrote "An Essay on the Principle of Population" in which he outlined simple exponential population growth model,

\begin{equation} P(t) = P_0 e^{\alpha t}. \end{equation}

In the above \( P_0 \) is the initial population size, \( \alpha \) stands for the population growth rate, and \( t \) is time. It is obvious that given \( P_0 > 0 \), the population eventually explodes (i.e., \( \lim_{t\rightarrow\infty} P \rightarrow \infty \)). Given this outcome Thomas Malthus (1766-1834) predicted that humanity would eventually need more resources than our planet can offer.

Before improving the model let us understand how does the exponential arises. It is well known that simple exponential functions are solutions of simple ordinary equations of the following form,

\begin{equation} \frac{d P}{d t} = \alpha P . \end{equation}

Having written down this differential equation, provides us hints into the underlying dynamics of the model. It simply states that each individual reproduces with rate \( \alpha \). While this model is very simplistic, at the time it was as influential to the population dynamics community as Newton's laws to the physics community.

In the 1960s Malthusian model was tested against world population data, and it was found that it doesn't fit the data well. However, the improved model was not much more optimistic - it predicted human population explosion by 2026 [1]. Though, our quick comparison against the most recent data have shown that it was also wrong (well, and also, in 2026 we already know that population did not explode).

Verhulst's correction

Almost half a century later, Pierre Verhulst (1804-1849) suggested that resource limitations could prevent population explosion. Namely, individuals within population would gradually stop reproducing as limitations would become more obvious as population grows. His model can be cast into the following ordinary differential equation,

\begin{equation} \frac{d P}{d t} = \alpha P \textcolor{blue}{\left( 1 - \frac{P}{K} \right)}. \end{equation}

In the above the blue term could be referred to as the Verhulst's correction. \( K \) is an additional system capacity parameter, which determines the population in the long time limit, \( P\left(\infty\right) \).

Inspired by Verhulst's idea, we have applied a similar correction when trying to adjust the doomsday model to the population data.

Equilibria point analysis

Equilibria points of the model can be determined by setting the derivative to zero,

\begin{equation} 0 = \alpha P \left( 1 - \frac{P}{K} \right). \end{equation}

There are two possible solutions, \( P_s = 0 \) and \( P_s = K \).

We can determine the stability of the equilibria points by taking derivative of the right hand side,

\begin{equation} \frac{d}{d P} \left[ \alpha P \left( 1 - \frac{P}{K} \right) \right] = \alpha - \frac{2 \alpha P}{K} . \end{equation}

The derivative is positive for \( P_s = 0 \), i.e., \( \alpha > 0 \). This means that this particular equilibrium point in unstable.

With \( P_s = K \) the derivative is negative, i.e., \( -\alpha < 0 \). This means that this particular equilibrium point is stable.

Interactive app

Interactive app below shows multiple trajectories with different initial conditions, \( P(0) \). Observe that changing \( \alpha \) changes only the time scale of the trajectories (and the model by extension), while changing \( K \) only influences the equilibrium point towards which (almost) all trajectories converge.

Note that, for the two gray curves the initial condition coincides with equilibria points. Thus, by definition, curves remain flat (they retain constant value).

References

• H. von Foerster, P. M. Mora, L. W. Amiot. Doomsday: Friday, 13 November, A.D. 2026. Science 132: 1291-1295 (1960).