A predator-prey model: whales and krill
Imagine two predator and prey species - whales and krill - in the same body of water competing for the same limited resources. Let represent the number of whales, and represent the number of krill.
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.
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.
The model for this scenario is and .
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 . Initially, the model in the interactive figure is set so that the whale population increases at a rate of . 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 .
The revised model taking into account this interaction rate is
and
Explore this model by setting parameters then dragging the point in the plane that represents the initial population of whales and krill.
Developed for use with Thomas' Calculus, published by Pearson.