3. Overshoot and Collapse
In the last model, the population found its steady state in a smooth manner; the rate of change decreased as the carrying capacity was approached. But this is not always the case; sometimes the population's growth rate does not decline fast enough as the carrying capacity is reached, leading to overshoot, which is usually followed by a collapse. One way of interpreting this is that the population does not process information about its condition fast enough to change its growth rate. For instance, if humans do not communicate the nature of the problem of population growth and do not communicate ways to solve the problem, we will likely overshoot our carrying capacity and a catastrophic population collapse is likely to follow.
Yeast growing in grape juice provide a good example of overshoot and collapse behavior. The yeast go after the sugar in the juice and in the process of metabolizing that sugar, they produce alcohol and carbon dioxide. As they consume the sugar, they also reproduce and their reproductive rate is dependent on the availability of food. Within a few days, at room temperature, the yeast population soars and the juice is bubbling from the release of so much CO2. But, the alcohol is a pollutant as far as the yeast are concerned and as their population rises, so does the level of alcohol. If there is enough sugar in the juice, the yeast will eventually produce so much alcohol that they start to die off rapidly and as the sugar reservoir is depleted, their reproductive rates plummet, leading to a total collapse of the population. So, in turning the juice into wine, the gluttonous, know-no-restraint yeast do themselves in.
This kind of a system is a bit more complex than the previous ones since we need to install another reservoir, representing the resources or some other kind of limiting quantity. Here, we will simply call this new reservoir Resources, and imagine that it represents things like energy and mineral resources, which can be thought of as finite resources (at least to begin with; we can explore the concept of recycling in a bit). The structure of this system, and the equations and graphs that define it are shown in the figure below; note that once again, it includes a graph in the Death Pct converter, but in this case, the Death Pct is a function of the amount of resources per capita.
