Yellowstone Wolf Lab

Year	Time (years)	Wolf Population	Births	Deaths	Elk Killed	Available Data		Births/Population	Deaths/Population	Elk Killed/Population
1995	1	21	9	2	43	GYA	t(0)=14	0.4285714286	0.09523809524	2.047619048
1996	2	37	14	9	124	GYA		0.3783783784	0.2432432432	3.351351351
1997	3	80	49	16	234	GYA		0.6125	0.2	2.925
1998	4	83	36	10	197	GYA		0.4337349398	0.1204819277	2.373493976
1999	5	72	38	14	276	GYA		0.5277777778	0.1944444444	3.833333333
2000	6	119	71	14	281	Both		0.5966386555	0.1176470588	2.361344538
2001	7	132	43	6	311	Both		0.3257575758	0.04545454545	2.356060606
2002	8	148	65	4	291	Both		0.4391891892	0.02702702703	1.966216216
2003	9	174	59	16	313	GYA		0.3390804598	0.09195402299	1.798850575
2004	10	169	59	20	241	Both		0.349112426	0.1183431953	1.426035503
2005	11	118	22	25	244	Both		0.186440678	0.2118644068	2.06779661
2006	12	136	60	10	219	Both		0.4411764706	0.07352941176	1.610294118
2007	13	171	64	6	272	Both		0.3742690058	0.0350877193	1.590643275
2008	14	124	22	17	463	Both		0.1774193548	0.1370967742	3.733870968
2009	15	96	23	10	302	Both		0.2395833333	0.1041666667	3.145833333
2010	16	97	38	9	211	Both		0.3917525773	0.09278350515	2.175257732
2011	17	98	34	10	267	Both		0.3469387755	0.1020408163	2.724489796
2012	18	83	20	15	159	Both		0.2409638554	0.1807228916	1.915662651
2013	19	95	41	7	193	Both		0.4315789474	0.07368421053	2.031578947
2014	20	104	41	5	148	Both		0.3942307692	0.04807692308	1.423076923
							Average	0.3827547299	0.1156443443	2.342890475
							Variance	0.01381157161	0.003830577555	0.5263891894

This is the data we got from http://www.nps.gov/yell/learn/nature/wolfreports.htm which is the Yellowstone National Park website. They had yearly reports for wolf population, including births, deaths, and elk killed. (Population is total population, not change in population) We got the coefficient for W by averaging the number of deaths/population per year. This number signifies the natural decay of wolves. Since we had the change in population for that year, wolf population, and the natural death rate, we could solve for the coefficient for E (which is elk killed, not total elk population) for each year, then took the average to get a result of .07789.

dW/dt = -.116W + .07789E 

Since we don't have any information about the total elk population, we only have one equation for this predator-prey system of equations model. This equation is unsolvable because of the reason stated before. There is not enough historical data recorded on Yellowstone elk.


The following graph shows the number of wolf births and deaths per year in the Greater Yellowstone Area. In all but one of the years, births outnumbered deaths. However, due to emigration, wolf population in the area does not reflect continuous growth as we would expect looking at this graph.

We are allowed to use the mean as an estimator for our coefficient because our histogram of deaths per population is relatively close to normal. If the distribution was not close to normal, then the mean would not be an effective estimator for the wolf death rate per year.
 







This is a bar graph of the total number of wolves in the Greater Yellowstone Area. This population includes pups older than 6 months to adult wolves. These population numbers were taken in December of the year listed.



 We found this project very interesting the way that the wolves were relocated from Canada and reintroduced into their natural habitat. We appreciated the thoroughness of the annual reports published by the National Park Service, which allowed us to gather an abundant amount of data. We do wish, however, that a total population of elk had been included in the reports so that we would have been able to determine the percentage of the whole population of elks that the wolves were killing each year. Without this information, we were only able to come up with one equation in our "system" of equations.