Imagine two predator and prey species - whales and krill - in the same body of water competing for the same limited resources. Let [math]x(t)[/math] represent the number of whales, and [math]y(t)[/math] represent the number of krill.[br][br]Without the presence of krill - a food source for whales - the number of whales would decrease proportionally to its size. In this dynamic figure, the initial whale decay rate is set at 1%, meaning without krill to eat, the whales would slowly die off, losing 1% of its populations in each time period.[br][br]Likewise, without the presence of whales, the number of krill would increase proportionally to its size. In this dynamic figure, the initial krill growth rate is set at 3%, meaning without whales to eat them, the krill population would increase 3% in each time period.[br][br]The model for this scenario is [math]x'(t)=-.01x[/math] and [math]y'(t)=.03y[/math].[br][br]In the presence of krill, the whale population would increase with a rate proportional to the frequency of whale/krill interactions. The frequency of interaction, in turn, is proportional to the product [math]xy[/math]. Initially, the model in the interactive figure is set so that the whale population increases at a rate of [math].00001xy[/math]. Along the same lines, the population of krill would decrease as a result of the frequency of whale/krill interactions. This initial rate of increase is set to [math].00002xy[/math].[br][br]The revised model taking into account this interaction rate is [br][math]x'(t)=-.01x+.00001xy[/math] and [math]y'(t)=.03y-.00002xy[/math][br][br]Explore this model by setting parameters then dragging the point in the plane that represents the initial population of whales and krill.
[i]Developed for use with Thomas' Calculus, published by Pearson.[/i]