Imagine two species of fish - trout and bass - in the same body of water competing for the same limited resources. Let [math]x(t)[/math] represent the number of trout, and [math]y(t)[/math] represent the number of bass.[br][br]In the absence of competition, the population of each each species would increase at a rate proportional to its size. In this dynamic figure, the initial trout growth rate is set at 5% and the initial bass growth rate is set at 4%. You can change these to other reasonable rates and explore what happens.[br][br]The model for this scenario is [math]x'(t)=.05x[/math] and [math]y'(t)=.04y[/math].[br][br]However, the two species are in competition. Each will [i]decrease [/i]at a rate proportional to the frequency with which the two species interact, which in turn is proportional to [math]xy[/math], the product of the two populations. In the interactive figure, the model is initially set so that the trout population decreases at a rate of [math]0.0001xy[/math] and the bass population decreases at a rate of [math]0.0002xy[/math].[br][br]The revised model taking into account this interaction rate is [br][math]x'(t)=.05x-.0001xy[/math] and [math]y'(t)=.04y-.0002xy[/math][br][br]Explore this model by setting parameters and then dragging the blue point in the plane that represents the initial population of trout and bass.

[i]Developed for use with Thomas' Calculus, published by Pearson.[/i]