## Predator-Prey Interaction

Lynx and snowshoe hare. © Tom and Pat Leeson

It should come as no surprise that predators influence the numbers of their prey. What may require a bit more reflection is that prey, in turn, affect the number of predators because, when prey become scarce, predators may die of starvation or fail to reproduce. This can lead to cyclical patterns of predator and prey abundance, where prey increase in number and then, with abundant food, predator number increases until the predators begin to suppress prey numbers and then decrease as well. As long as predator and prey numbers don't drop to zero, this cycle can repeat indefinitely. Interestingly though, there are situations where predators are absent, such as on islands or in other isolated areas where they either never became introduced or where they have died out, and yet prey continue to oscillate in number. What causes prey number to cycle in the absence of predators? Generally the answer is that without predators to suppress their number, prey outstrip available food resources, nesting sites, or some other limited resource and thus begin to suppress their further growth through competition. Watch these short video lectures for a very nice overview of predator-prey interactions.

Part 1: How to eat another organism

Part 2: How to avoid being eaten

One of the classic studies of predator-prey interactions is the 90-year data set of snowshoe hare and lynx pelts purchased by the Hudson's Bay Company of Canada. While this is an indirect measure of predation, the assumption is that there is a direct relationship between the number of pelts collected and the numer of hare and lynx in the wild. As you can see, there does appear to be cycling over time in both hare and lynx number, but it's not as clean as in the simple mathematical models. Life rarely is!

Here's a simple predator-prey model. Let's say the number of wolves is represented by w, the number of rabbits is represented by r. The reproductive rate of rabbits is k1 and the reproductive rate of wolves is k2. The mortality rate for wolves is k3. Download the excel spreadsheet to play around with different numbers for each of these parameters and watch what happens to the graphs. Watch the video to see more about how the equations were derived. The following linked equations model changes in predator and prey number:

Change in number of rabbits: Δr=(k1*r)-(k2*r*w)

Change in number of wolves: Δw=(k2*r*w)-(k3*w)

The Lotka-Volterra model of predator and prey interactions is a classic one, but adds another variable to the 3 constants in the above model. The new variable is a predator-prey encounter rate. In the Lotka-Volterra model, it's easy to give it values that drive predator or prey below zero, which makes no sense, or to drive prey to such small numbers that predators should go extinct. However, with the right values, we can get stable oscillation for hundreds of generations.

Lotka-Volterra Predator-Prey.xls