Problem Set 9:

Leslie Matrices

Scenario

You are given the task of establishing a viable population of caribou (Rangifer tarandus) in northern Idaho. You released 80 caribou into a large, unoccupied area of ideal habitat. Assume a 50:50 tertiary sex ratio, birth-pulse fertility, and annual post-birth sampling. Age-specific female survival and fecundity rates are shown below, as well as the population structure (n_x) for the initial introduction (i.e., t = 0).  Note: since your n_x column is not the normal sample of a population, but instead the numbers in each age class you are going to introduce, you must use this fomula to calculate your lx column: l_x+1 = S_x*l_x (ask your TA if you have any questions about this).

age (x)	nx	Sx	mx	Fx	lx	lx mx	x lx mx			Cx
0	10	0.5	0	0.1	1.00	.	.	.	.	.
1	2	0.8	0.2	.	0.50	.	.	.	.	.
2	8	0.9	0.8	.	0.40	.	.	.	.	.
3	5	0.9	1.0	.	0.36	.	.	.	.	.
4	14	0.9	1.0	.	.	.	.	.	.	.
5	0	0.9	0.9	.	.	.	.	.	.	.
6	1	0.0	0.3	.	.	.	.	.	.	.

1. Complete the extended life table. Calculate the expected finite rate of change and the stable-age distribution for this population.

2. Use the Leslie-matrix method to predict population size and age structure for the first 2 years. Do these calculations by hand and show your work.

3. Use program LESLIE to project population size and age structure for 10 years (use the initial age structure provided in the life table). Remember to input F_x and not m_x when using program LESLIE. Note: if n_x = 0 then enter 0.1; if F_x = 0 then enter 0.0001.

4. Calculate the finite rate of change for t = 1, 2, 3, 4, and 5 (hint: use the output from program Leslie). Compare these values to the finite rate of change calculated in question #1. If these rates differ, explain the reason for the difference.

5. Construct a table that compares the stable age distribution (SAD) calculated in question #1 to the age structure observed in the population at year 0 and year 10 (hint: look at the Leslie -matrix projections). Did the population at time 0 have a SAD? Did the population reach a SAD by year 10?

6. Modify survival rates of calves and yearlings to reflect slightly lower survival (i.e., set S₀ = 0.4 and S₁ = 0.7). Remember to calculate new F_x values for ages 0 and 1. Use the same initial age structure (from life table) and project population growth for 10 years with program Leslie. Did the small decrement in survival of calves and yearlings affect the population projections? (hint: compare differences in population size, age structure, and growth rates).

7. Use the same life table as used in question #1, but now use the following population vector: n_x = 0, 0, 5, 5, 5, 5, 1 for ages 0, 1, 2, 3, 4, 5, 6, respectively. Use program LESLIE to project the population growth for 10 years. Plot the 2 projections (i.e., total population size, starting with N₀) on 1 graph. Did the difference in initial age distribution, with all else being equal, affect population growth? Explain your answer.

Turn in any output(s) from program Leslie that you used to answer the questions.

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Revised: 29 October 2001