A common application of Chaos Theory is population growth. The simplest equation to explain Chaos Theory is the following applied mathematical equation:
\[x_{n+1} = \lambda x_n(1 - x_n)\]where \(\lambda\) is the growth factor, and \((1-x_n)\) is the limiting factors. \(n \in \mathbb{N}\) such that \(x_n\) is the current population. Thus, \(x_0\) is the initial population size, and \(x_{n + 1}\) is the population size of the next generation.1
The simple model equation is commonly used for bifurcation plots, showing the predictability of where a population is stable versus not stable. In bifurcation plots,2 there is a point that if the population becomes unstable, the population diminishes. Thus, populations of species commonly stay within the stable regions of the bifurcation plot. However, external factors can cause a population to become unstable, for example, bringing in an overabundance of predator species. Although predicting the amount of a species may be difficult, bifurcation plots allow biologists, ecologists, and applied mathematicians to predict the point at which a population can become unstable.
An example of population growth is the number of bunnies in a population. Many external factors can effect the population of the bunnies in an ecosystem including predators, weather, food, water, and shelter. If one or more of the factors listed increases or decreases, the population of bunnies will increase or decrease. Since all of the factors intertwine, the population of bunnies can be seen as a nonlinear equation as their population increases and decreases over time. Thus, the bunny population is hard to predict in the long-term future. This scenario can be explained using nonlinear ordinary differential equations and applied mathematics.
The link below is a Geogebra Applet for exploring population growth and Chaos Thoery, specifically using foxes and rabbits (as explained in the example above).
This applet follows Applied Mathematics and Ordinary Differential Equations, but also applies to Chaos Theory. The applet addresses initial conditions for specific variables and how the number of foxes and rabbits change based on one another's population. For example, if there are too many foxes, the rabbit population will decrease. The way this applies to Chaos Theory is an ecosystem relies on many small variations, but one tiny difference could cause a completely unpredictable final outcome.