In population dynamics, the growth of a population can have exponential growth, or a logistic curve growth up to its carrying capacity, or it can overshoot the carrying capacity and fluctuate before eventually settling down at the carrying capacity.

The logistic curve equation is a modification of the exponential curve equation. How should I modify the logistic curve equation to allow it to overshoot?

And when I have actual noisy time-series data, how could I predict if the population is going to smoothly arrive at the carrying capacity or if it is going to overshoot?

I would also be grateful for any information about how to derive the parameters of the overshooting equation from actual time-series data.

  • $\begingroup$ Overshooting can happen in the discrete time logistic growth model. This book appears to have useful notes starting on page 19 in the section 1.3.1 Logistic difference equation: ethz.ch/content/dam/ethz/special-interest/usys/ibz/… $\endgroup$
    – mkt
    Aug 6, 2017 at 10:49
  • $\begingroup$ If we need to stick to a continuous curve, we could add either a random or a periodic term, but I am not sure if that is useful in your particular model $\endgroup$
    – David
    Mar 12, 2019 at 11:23

2 Answers 2


I know the question is old, but when you take the steepness parameter $k$ to be complex, then the curve oscillates (as opposed to vary randomly or chaotically) when it approach the asymptote. The real part of $k$ controls the actual steepness, and the complex part controls the oscillation frequency.


The logistic equation, most commonly written, is an ordinary differential equation (ODE), meaning it depicts the rate of change instantaneously:

$$\frac{dN(t)}{dt} = rN(t) - \alpha N(t)^2.$$

Since the rate of change occurs instantaneously, as written above, it is impossible to overshoot equilibria, and population sizes asymptotically approach equilibria (e.g., $N^* = \frac{r}{a}$).

There are many ways to modify the logistic equation to make it overshoot like we see in many populations. Two common ways are to either introduce a time lag in the differential equation or make the population change over discrete-time increments. In the form of a delay-differential equation (DDE), the population would change according to

$$\frac{dN(t)}{dt} = rN(t) - \alpha N(t-\tau)^2.$$

Here, $\tau$ represents a lag in time. As the lag increases, so does the propensity of the population to overshoot its equilibria. Discretization of time yields the difference equation:

$$N_{t+1} = \lambda N_t - \alpha N_t^2.$$

Here, the population changes over discrete time steps. With a large enough $\lambda$, the population will tend to overshoot equilibria.

Most real populations have many, many different factors affecting their dynamics. In general, when applying these generalizable, single-species models to real populations, it is difficult to reliably discriminate between them unless one has a lot of data or the population is grown under highly controlled conditions. Ultimately, I think that predicting the behavior of a population will need a much more complex, realistic model if you want realistic results. The logistic model is a heuristic model and doesn't necessarily fit or predict real populations well.


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