Consider a population with the following numbers of individuals over time:
Year 1: 10
Year 2: 25
Year 3: 30
If the population size in year 4 is determined by a logistic model with delayed density dependence from a 1-year time lag and K = 50 individuals, what value of r would lead to the population size to exceed carrying capacity in year 4?

Question

Quizplus · Accepted Answer

We can use the logistic equation with a delayed density dependence term to model the population growth:

Nt = Nt-1 + rNt-1 (K - Nt-1*lag(Nt-1)/K)

where Nt is the population size in year t, Nt-1 is the population size in the previous year, r is the intrinsic rate of increase, lag is the time lag for density dependence, and K is the carrying capacity.

For this problem, we can substitute the given population data to get the following:

Year 1: N1 = 10
Year 2: N2 = 25 = N1 + rN1(K-N1*lag(N1)/K)
Year 3: N3 = 30 = N2 + rN2(K-N2*lag(N2)/K)

To solve for r, we can rearrange the equation for N2 as follows:

r = (N2 - N1) / (N1 * (K - N1*lag(N1)/K))

Substituting the values, we get:

r = (25 - 10) / (10 * (50 - 10*lag(10)/50)) = 0.5

Therefore, the answer is D.

We can also check if the population size in year 4 exceeds the carrying capacity:

N4 = N3 + rN3(K - N3*lag(N3)/K)
    = 30 + 0.5*30(50 - 30*lag(30)/50)
    = 43.125

Since N4 > K, the population size exceeds carrying capacity in year 4.

Consider a Population with the Following Numbers of Individuals Over