Recursion formula for a given population for each interval,n, at a time, and k is a constant called the Malthusian factor:
a_n=ka_n-1(1-a_n-1)
- Assuming n is measured in weeks, calculate the insect population for weeks 2, 3, and 4 if the initial population is 112, and the Malthusian factor for this population is -9.5
-Assuming n is measured in years, and the Malthusian factor for a species of insects is -1, what is the population in year 9 if the population in year 10 is 200?
Anyone know what's up?
Sure. It's all clearly explained. For the first one,
a_1 = 112
a_2 = -9.5*112(1-112) = 118104
and so on.
Similarly for the other example.
Sure! Let's solve these step by step.
1. Assuming n is measured in weeks, calculate the insect population for weeks 2, 3, and 4 if the initial population is 112, and the Malthusian factor for this population is -9.5:
To calculate the population for each week, we'll use the given recursion formula: a_n = k * a_n-1 * (1 - a_n-1)
Given:
- Initial population (week 1): a_1 = 112
- Malthusian factor: k = -9.5
Week 2:
a_2 = k * a_1 * (1 - a_1)
= -9.5 * 112 * (1 - 112)
≈ -9.5 * 112 * (-111)
≈ 118,584
Week 3:
a_3 = k * a_2 * (1 - a_2)
= -9.5 * 118,584 * (1 - 118,584)
≈ -9.5 * 118,584 * (-118,583)
≈ 133,703,674,680
Week 4:
a_4 = k * a_3 * (1 - a_3)
= -9.5 * 133,703,674,680 * (1 - 133,703,674,680)
≈ -9.5 * 133,703,674,680 * (-133,703,674,679)
≈ 178,121,982,203,463,972,600
So, the insect population for weeks 2, 3, and 4, starting with an initial population of 112 and a Malthusian factor of -9.5, is approximately:
- Week 2: 118,584
- Week 3: 133,703,674,680
- Week 4: 178,121,982,203,463,972,600
2. Assuming n is measured in years, and the Malthusian factor for a species of insects is -1, what is the population in year 9 if the population in year 10 is 200?
Given:
- Year 10 population: a_10 = 200
- Malthusian factor: k = -1
We need to calculate the population for year 9 (a_9) using the recursion formula: a_n = k * a_n-1 * (1 - a_n-1)
Year 10:
a_10 = 200 (given)
Year 9:
a_9 = k * a_10 * (1 - a_10)
= -1 * 200 * (1 - 200)
= -1 * 200 * (-199)
= 39,800
So, the population in year 9, assuming a Malthusian factor of -1, is 39,800, given that the population in year 10 is 200.
To calculate the population for each interval using the given recursion formula, you need to substitute the values into the formula and recursively apply it.
For the first question, where n is measured in weeks, and the Malthusian factor is -9.5:
a_1 = 112 (initial population)
k = -9.5
To calculate the population for week 2 (a_2), you substitute the values into the formula:
a_2 = -9.5 * a_1 * (1 - a_1)
= -9.5 * 112 * (1 - 112)
= -9.5 * 112 * (-111)
= 112980
So, the population for week 2 is 112980.
To calculate the population for week 3 (a_3), you substitute the values again:
a_3 = -9.5 * a_2 * (1 - a_2)
= -9.5 * 112980 * (1 - 112980)
= -9.5 * 112980 * (-112979)
= 120586782740
So, the population for week 3 is 120586782740.
To calculate the population for week 4 (a_4), you substitute the values again:
a_4 = -9.5 * a_3 * (1 - a_3)
= -9.5 * 120586782740 * (1 - 120586782740)
= -9.5 * 120586782740 * (-120586782739)
= 13842482139768931662930
So, the population for week 4 is 1.3842482139768932e+22.
For the second question, where n is measured in years and the Malthusian factor is -1:
a_10 = 200 (population in year 10)
k = -1
To calculate the population in year 9 (a_9), you need to work backward using the formula:
a_9 = (a_10 / (k * (1 - a_10)))
Substituting the values:
a_9 = (200 / (-1 * (1 - 200)))
= (200 / (-1 * (-199)))
= (200 / 199)
So, the population in year 9 is approximately 1.0050251256281406.
Please note that because the given recursion formula is nonlinear, the values can quickly become very large or approach infinity, depending on the initial population and Malthusian factor.