Models of Elk and Wolves

Predator/prey models have been around for a long time. I am sure that some quite sophisticated ones exist today to describe reasonably complex ecosystems. My research on existing models is only beginning.  I’ll do some simple scoping work on my own to help figure out what to look for.

My first model will address the relationship of wolf killing of elk (not necessarily the same as wolf eating of elk) and the relative populations. My first hypothesis is that the number of elk killed by the wolves in a given ecosystem equals the amount of new elk production each year; i.e. the total surviving calves accounting for reproduction rate (births) and all causes of death of both the calves and the rest of the herd except for the losses to wolves. Or,

kW = rE

Where:

W = number of wolves

E = number of elk

k = number of elk killed each year by each wolf

r = average surviving net overall reproduction of the elk, as a percentage of the total elk population, not accounting for those killed by wolves

There are many assumptions in this simple model which I will not address at this time. But I will note that elk kill by other predators and natural deaths can be included in r.

Now let’s consider some numbers that are at least close for the ecosystem of Yellowstone National Park. The number of wolves we are told they think they want to have is about 100. I haven’t heard how many elk they’d like to have but historical counts range from just under 20,000 to the last count in 2013 (they didn’t report one for 2014) of somewhat under 4,000.

Numbers in the literature for k range from a low of 11 (as used in the elk “reintroduction EIS) to a high of about 35, with a frequently claimed average of 21.6. The number varies depending on prey concentration and other factors. But I’ll use 20 as a start.

r also varies. The estimates are confounded by the makeup and quantity of predators included in the estimate. But about .20%, or .2, seems to be an average number for a healthy elk herd.

So, substituting in these numbers:

20 W = .2 E, or

E = 100 W

So if Yellowstone wants 100 wolves (W = 100) then E = 10,000. Oops. That means they need more than twice as many elk as currently remain to satisfy the 100 wolves they currently estimate. That means the remaining elk are going to be killed at a rate far above what they can match by reproduction. The elk herd is on its way to extinction.

Or, if Yellowstone has about 4,000 elk (E = 4,000), the stable number of wolves W = 40. Oops. They say there are currently about 100 wolves. So the current wolves are going to have to change their diet, leave, or starve.

Clearly 100 wolves and 4,000 elk are out of balance. Worse, as the number of elk declines because there are too many wolves killing them, the number of sustainable wolves also declines. From the current situation the only stable result for this simple model is E = 0 and W = 0. Double oops.

You might argue that the 20 elk killed per wolf is too high. But then so is the 20% recruitment not including the other predators;  in Yellowstone most notably grizzly bear.  I have found models relating the Elk kills by wolves to the ratio of elk to wolves. It appears to max out at over thirty Elk per wolf per year when the elk to wolf ratio is above about 60; i.e. about 6,000 wolves in Yellowstone with about 100 wolves. It reduces at lower elk to wolf ratios. Although it should go through zero kills when there are no more elk (elk/wolf ratio of 0) there does not appear to be good data on the relationship at the lower level. That is what would help predict a “stable” population.

Wolves also reproduce. Our simple model doesn’t account for that. So whatever the number of wolves, unless they are killing each other, taken by disease, or starving at a rate that matches their reproduction, then their number will increase. We have witnessed just how fast the wolves can increase over the last twenty years, where about 50 multiplied to nearly 2,000 in about ten years. It has been proposed that they reproduce faster when prey are abundant and slower when prey are harder to kill (and thus the kills per wolf go down).

One can move on to more complex models that bring into play second order and non-linear effects. For example the number of elk kills per wolf probably may depend on other factors such as competing predators (bear and cougar primarily) and weather. The number of elk depends on available forage; in particular that available for the winter.

One of the major assumptions of this model is an isolated population. Both wolves and elk move. Elk can move to escape their predators and wolves can move to another area once they have depleted the elk in their present area. They seem to be doing that in Idaho; e.g. after depleting the Lolo, Sawtooth, and Selway regions the wolves seem to have moved on to adjacent regions.

Finally I’ll note here that the classic predator relationship is expected to vary over time, as for example with jackrabbits and coyote. The prey goes through a sine like up and down population over time, with the predator population following but usually at reduced amplitude.  Or you can look at the other way: the prey follows the predator. Both perspectives are right. It is like chickens and eggs: neither comes first. They vary together.

In conclusion this very simple model provides some numerical insights to start with.  Things aren’t stabilized yet in Yellowstone. The final result is likely to be even fewer elk and wolves; possibly the wildlife desert that Lewis and Clark described when they traversed this area.