Develop an age-structured population model for elk, where you divide the population to two cohorts: juvenile females and adult females. Juveniles do not reproduce; they reach reproductive maturity in 1 year. The probability of juveniles to reach adulthood is 50%. Adult females have on average 1 calf. a: Compute a long term proportion between (juveniles/adults) in this population. b: Will the population survive in the long term? c: How many calves would a female elk have to have every year for population to survive?
Assumptions:
At the start, the number of juveniles and adults are not equal
Each adult has 1 calf per year ???? (your problem is not clear on that)
Major flaw: When do adults no longer reproduce or die?
current juvenile females --- x
current adult females ----- y
after 1 year:
juves = y , half of surviving became adults + each of the adults had a calf (juvenile)
= y
adults = y + x/2 <----- current adults + half of juves
= (x + 2y)/2
ratio is y : (x + 2y)/2
= 2y : x+2y
after year 2:
juves = (x + 2y)/2 <--- this will always be the previous adults
adults = (x + 2y)/2 + y/2
= (x + 3y)/2
ratio = x+2y : x + 3y
after year 3:
juves = (x+3y)/2
adults = (x+3y)/2 + (x+2y)/4
= (3x + 8y)/4
continue this to see if you can see a pattern
This would be nice to see develop in some kind of computer simulation
I might work on it a bit
Made up this rinky-dink program in an ancient computer language, (which I still like for stuff like this)
10 input j,a
20 for n = 1 to 10
30 newj = a : newa = a+j/2
40 j = newj : a = newa
60 print n-1,j,a,j/a
70 next n
here are the results for 5 different inputs of j (juveniles) and a (adults)
? 16,32
0 32 40 0.8
1 40 56 0.714286
2 56 76 0.736842
3 76 104 0.730769
4 104 142 0.732394
5 142 194 0.731959
6 194 265 0.732075
7 265 362 0.732044
8 362 494.5 0.732053
9 494.5 675.5 0.73205
? 100,20
0 20 70 0.285714
1 70 80 0.875
2 80 115 0.695652
3 115 155 0.741935
4 155 212.5 0.729412
5 212.5 290 0.732759
6 290 396.25 0.731861
7 396.25 541.25 0.732102
8 541.25 739.375 0.732037
9 739.375 1010 0.732054
? 20,100
0 100 110 0.909091
1 110 160 0.6875
2 160 215 0.744186
3 215 295 0.728814
4 295 402.5 0.732919
5 402.5 550 0.731818
6 550 751.25 0.732113
7 751.25 1026.25 0.732034
8 1026.25 1401.875 0.732055
9 1401.875 1915 0.73205
the ratio seems to approach 1:0.73205
no matter how many juveniles and adults we start with
Spent a lot of time answering this question.
No response form the poster, not even sure if he/she even looked at it.
Frustrating, will remember that name for future.
Hey i'm sorry for response from my side. just now i have seen. i'm very happy with your answer and thanks for spending lot of time to solve my question. tq so much
To develop an age-structured population model for elk, we need to consider the cohorts of juvenile females and adult females separately. Let's assume that the initial population consists of 100 individuals, with 50 juvenile females and 50 adult females.
a) To compute the long-term proportion between juveniles and adults in this population, we need to understand the transition rates between different age groups.
Given that juveniles reach reproductive maturity in 1 year and the probability of juveniles reaching adulthood is 50%, it means that 50% of the juvenile females will transition to the adult female cohort each year.
After the first year, there would be 25 juvenile females and 75 adult females (50% * 50 juvenile females + 50 adult females). From this point forward, the number of juvenile females will remain constant because they transition to the adult cohort at a rate of 50%.
In the long term, the proportion of juveniles to adults will converge to a stable ratio. Since 25 juvenile females remain and there are 75 adult females, the long-term proportion between juveniles and adults would be 1:3 (25:75).
b) To determine if the population will survive in the long term, we need to examine the overall reproductive output and mortality rates.
Given that adult females, on average, have 1 calf, the population will sustain itself as long as the net reproductive rate is greater than 1. The net reproductive rate (R0) is the average number of female offspring produced per female over their lifetime.
In this scenario, the elk population will survive in the long term since each adult female produces, on average 1 calf, which is equal to the net reproductive rate (R0 = 1).
c) Now let's consider how many calves a female elk would need to have every year for the population to survive.
To maintain a stable population, the net reproductive rate (R0) needs to be equal to or greater than 1. Since each adult female, on average, has 1 calf, this is sufficient to maintain a long-term stable population.
If the net reproductive rate were less than 1, it would indicate that the population is declining over time. In that case, the average number of calves per female would need to be increased to ensure population survival.
In summary, based on the given parameters, the long-term proportion between juvenile and adult females in this population would be 1:3. The population will survive in the long term since each adult female has, on average, 1 calf, which is sufficient to maintain a stable population. The number of calves a female elk would need to have every year for the population to survive depends on the net reproductive rate, which in this case is already met with an average of 1 calf per adult female.