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:

Note when N << K, N/K is zero.

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.

Some of this behavior is seen in the classic predator-prey case study
of Lynxs and Snowshow hares. Its an excellent example of __
density dependent lag time __ in some system:

Although these systems is in quasi-equilibrium when averaged over long timescales. On short time scales the relative species densities are rapidly changing. Each peak or valley in this cyclical behavior is a point of unstable equilibrium and a small pertubation could drive the system completely out of control. An additional complication is time lags or system response. In real life, this is the limiting factor for accurate modeling.

Non-linear dynamics is very difficult to accurately model because of the degree of non-linear response of the system is unknown and the relevant stress parameters are poorly understood.

Qualitatively we can represent some of this difficulty by consideration of a three parameter system. In the below case, our cows are happiest when there in the center of this parameter space, in a nice meadow. But if they wander too far off any axis direction, they will be in less favorable conditions.

Following is a collection of related notes that may help you further understand this. The key point with respect to this class is the emphasis on the non-linear behavior. That was incorporated into the predator-prey exercise that your groups are working on and, as some of you have noted, simple calculus doesn't work to model that system.