Population dynamics:

Biologists like to talk a lot about r-selected (growth selected) and k-selected (carrying capacity limited) species usually in the context of predator-prey relations.

While this is a useful framework it generally oversimplifies the problem. Most real systems are subject to non-linear or chaotic dynamics, involving unstable oscillations around points of equilibrium. But I digress for now ..

K-selected species are idealized by the equation below where it can be seen that the growth rate, R, becomes zero as N approaches K. This is called negative feedback - the idea being as conditions get "crowded" it is detrimental to growth. The problem with real systems is that estimates of K are very difficult and uncertain:

r-growth is standard exponential growth which leads to population crashes. The functional difference between r and K selected growth is shown below:

In reality, K-selected growth qualitatively looks like this:

In this case we have r-selected growth up to the carrying capacity line but then we have oscillations about that line. These oscillations have points of unstable equilibria (i.e. the smaller scales peaks and valleys). Small perturbations in the system, when species growth is at one of these points, could trigger catastrophic continued r-growth or rapid decay. This is called non-linear dynamics (sometimes called chaotic dynamics).

In non-linear dynamics, small perturbations in some system can cause large changes in overall system evolution.