What is the difference between a logistic curve and something that overshoots? 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. 
 A: 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. 
A: 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.
