##
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.