Alright, let's analyze the given data and determine the necessary conditions to keep the population growth rate of cheetahs in 2013 the same as it was in 2011 and 2012.
First, we need to calculate the net population change for each of the years 2011 and 2012 based on births, deaths, immigration, and emigration.
Step 1: Calculate the net population change for 2011:
- Deaths in 2011: 2
- Births in 2011: 5
- Immigration in 2011: 6
- Emigration in 2011: 8
Net population change for 2011:
[tex]\[
\text{Net change}_{2011} = (\text{Births}_{2011} + \text{Immigration}_{2011}) - (\text{Deaths}_{2011} + \text{Emigration}_{2011})
\][/tex]
[tex]\[
\text{Net change}_{2011} = (5 + 6) - (2 + 8) = 11 - 10 = 1
\][/tex]
Step 2: Calculate the net population change for 2012:
- Deaths in 2012: 1
- Births in 2012: 4
- Immigration in 2012: 1
- Emigration in 2012: 3
Net population change for 2012:
[tex]\[
\text{Net change}_{2012} = (\text{Births}_{2012} + \text{Immigration}_{2012}) - (\text{Deaths}_{2012} + \text{Emigration}_{2012})
\][/tex]
[tex]\[
\text{Net change}_{2012} = (4 + 1) - (1 + 3) = 5 - 4 = 1
\][/tex]
Step 3: Determine the desired net population change for 2013:
The desired net change in population for 2013 should be the same as the average of the net changes for the previous two years (2011 and 2012).
[tex]\[
\text{Desired net change}_{2013} = \frac{(\text{Net change}_{2011} + \text{Net change}_{2012})}{2}
\][/tex]
[tex]\[
\text{Desired net change}_{2013} = \frac{(1 + 1)}{2} = 1
\][/tex]
Step 4: Evaluate each scenario for 2013:
1. (4 deaths, 7 births, 2 immigration, 6 emigration)
[tex]\[
\text{Net change}_{2013} = (7 + 2) - (4 + 6) = 9 - 10 = -1
\][/tex]
2. (3 deaths, 6 births, 5 immigration, 7 emigration)
[tex]\[
\text{Net change}_{2013} = (6 + 5) - (3 + 7) = 11 - 10 = 1
\][/tex]
3. (5 deaths, 2 births, 8 immigration, 3 emigration)
[tex]\[
\text{Net change}_{2013} = (2 + 8) - (5 + 3) = 10 - 8 = 2
\][/tex]
4. (1 death, 5 births, 3 immigration, 7 emigration)
[tex]\[
\text{Net change}_{2013} = (5 + 3) - (1 + 7) = 8 - 8 = 0
\][/tex]
Step 5: Identify the scenario that matches the desired net change:
From the calculations, the second scenario:
- (3 deaths, 6 births, 5 immigration, 7 emigration)
results in a net change of 1, which matches the desired net change of 1.
So, the scenario that would keep the population growth of cheetahs in 2013 the same as the previous years is:
3 deaths, 6 births, 5 immigration, 7 emigration.
