Predation and Parasitism

Lotka-Volterra Equations

Competition involved two species, each of which negatively affected each other. Each species reduced the carrying capacity of the environment for the other.

Predation or parasitism, however, is an interaction where one species benefits (the predator or parasite) and the other is harmed (the prey or host). This system is often modeled by using exponential growth, not the logistic equation which we studied previously. Exponential growth means that the rate of increase (or decrease) of the population of each species depends on how many of that species there are. In the Lotka-Volterra model, we add a term to the rate of change of population for the predator. This term depends upon the number of prey. So, the more prey there are, the more positive the rate of change of predator population. The rate of change of prey has a term subtracted. This term depends upon the number of predators. So, the more predators there are, the more negative the rate of change of prey population becomes. Even these simple equations are difficult to solve but when we do so, we find the following, very interesting cyclic change in predator and prey populations. Notice that this Mathematica simulation done by your instructor shows that the cyclic changes in predator and prey numbers do not follow a nice sinusoidal curve. The changes are cyclic but are not a sin function as shown in many textbooks! (The drawing to the right is a more whimsical version.)

Another way to show this is with what is called a phase plane plot. Here we plot predator numbers on the vertical axis and prey numbers on the horizontal axis:

Notice on either type of plot how the peaks in predator population followthe peaks in prey population. The predator eats the prey and so population of predator follows closely after peaks in the prey population.

Problems with Lotka-Volterra

There are some serious problems with the Lotka-Volterra model. One of these is the following: A realistic addition to the model might be the inclusion of a condition that there be a minimum sustainable population. This is really important since we do need at least two individuals to reproduce. In practice, we probably need more for a sustainable population! If we make this assumption, then the phase plane plots don't show closed paths for cyclic changes. The paths spiral outward! On a conventional plot, this would mean that the oscillations would become larger and larger. Sooner or later, one of the populations, predator or prey, will go to zero and that species will become extinct. If the predator goes extinct first, then the prey lives happily ever after. If the prey goes extinct first, then the predator will soon become extinct also since it has nothing to eat. In either case, the relationship between the predator and prey (or parasite and host) ends. In the following example, the minimum sustainable population size is set at 10:

Another serious problem with Lotka-Volterra is that, in using exponential growth, we have neglected the carrying capacity built in to the logistic growth model. This means that if the predator and prey are competing for some resource (space, for example), Lotka-Volterra methods might not give an accurate answer. Even if only one species is resource limited, Lotka-Volterra will not give accurate results.

Some solutions to problems with Lotka-Volterra

The model which leads to the results to follow puts in a carrying capacity for the prey (not the predator). This is a realistic scenario since the prey may be eating some plant material which runs out at some point. Also, this model incorporates a more realistic "encounter function" which describes the interaction between the predator and prey. The results given here are only shown as a plot of predator and prey numbers vs. time. The phase plane plot is omitted for lack of time. However, note that in this model, the results do not show sustained oscillations - predator and prey numbers approach an equilibrium point. Thus, the trajectory on a phase plane plot will spiral inwards (not outwards as shown in the last example) towards a stable equilibrium point. But, how fast the oscillations die down depends upon the various parameters in the equations. In real life, the time for the oscillations to die down could be very long. By the time they were supposed to die down, something could get them going again.

The Lynx and the Snowshoe Hare

Between 1845 and 1935, the Hudson Bay Company of Canada kept records of the number of lynx (a cat) and snowshoe hare (a rabbit) pelts which were sold by them. Trappers would bring in pelts and the Hudson Bay Company would buy them and resell them to furriers. Of course, the lynx is a predator on the hare and so we expect to find cyclic changes in the populations. The Hudson Bay Company records do show a remarkable, long lived cyclic behavior. But, in the period between 1875 and 1905, the cyclic change goes in the wrong direction. Plotted on a phase plane, the cycle goes clockwise, instead of counterclockwise, as we would expect. This indicates that the hare is the predator of the lynx, a very unlikely possiblity. Many people have thought about this anomaly - One possible explanation is that the trappers are also a predator. Or that the plant material which the hares eat is a third species which itself is the "prey" for the hare. So, the lynx eats the hare and the hare eats the plants. Click on the picture below to go to a link which has an article on recent population changes in the lynx and showshoe hare populations: