Question

• The population of deer is currently 2,000 individuals. The population can be described by a logistic model with K = 5,000 and r = ½. If 600 deer are hunted this season, will the population of deer go up, or down, or stay at 2,000? Is this harvest rate sustainable? What is the maximum sustainable yield? Explain.

Answer 1

Growth rate for logistic growth us given by the equation:

(dN/dt) = rN (K-N)÷K

Where, N is the initial population, K is the carrying capacity and r is the intrinsic rate of natural increase.

After hunting 600 deers, the population size becomes 1400. That is 2000 - 600.

Now, to calculate growth rate in this situation, we should consider 1400 as the initial population.

Growth rate : (r=0.5, K=5000 are given in question, and N=1400)

(dN/dt) = rN (K-N)÷K

= (0.5)*(1400)*(5000-1400)÷5000

= 504 deers

Therefore new population of deer = initial population + growth rate = 1400 + 504 = 1904 deers.

Since the new population is less than the reference population (by 96) of 2000, the harvest rate of 600 is not sustainable.

Maximum sustainable Yield:

Maximum sustainable yield is the hunting rate that will not change the population of deer from reference population at the end of season. To make it more clear, maximum sustainable yield is the hunting rate that would decrease the initial population size of deer to a value that will increase over a season's growth to reach 2000.

That is, the sum of new initial population after hunting and population growth rate after hunting should be equal to 2000.

N' + (dN'/dt) = 2000, where N' is the new initial population after hunting and (dN'/dt) is the new growth rate after hunting.

Also, (dN'/dt) = rN' (K-N')÷K -->discussed earlier

Hence, 2000 = N' + 0.5N' (5000-N')÷5000

On calculating, we get :

N'^2 - 1.5 N + 2000 = 0 ; is of the form :

aX^2 + bX +C = 0

Where, N'^2 is N' squared and X^2 is X squared.

On solving the quadratic equation, we get two values for N'.

N' = 13521 and N' = 1479

Since 13521 is much higher than 2000 and on hunting the value of population of deers will only be reduced, 13521 can be ruled out.

Hence, we take N' = 1479 deers.

Since the new population size after maximum sustainable hunting is 1479, maximum sustainable hunting = Reference population - Initial population after sustainable hunting.

Maximum sustainable yield = 2000 - N'

= 2000 -1479

= 521 deers.