Answer:
a) Equilibrium point : [ 947, 53 ]
b) N = 947 is stable equilibrium, N = 53 is unstable equilibrium
c) N0, the population will not go extinct
Step-by-step explanation:
a)
Given that;
r = 2, k = 1000, H = 100
dN/dT = R(1 - N/k)N - H
so we substitute
dN/dt = 2( 1 - N/1000)N - 100
now for equilibrium solution, dN/dt = 0
so
2( 1 - N/1000)N - 100 = 0
((1000 - N)/1000)N = 50
N^2 - 1000N + 50000 = 0
N = 1000 ± √(-1000)² - 4(1)(50000)) / 2(1)
N = 947.213 OR 52.786
approximately
N = 947 OR 53
Therefore Equilibrium point : [ 947, 53 ]
b)
g(N) = 2( 1 - N/1000)N - 100
= 2N - N²/500 - 100
g'(N) = 2 - N/250
SO AT 947
g'(N) = g'(947) = 2 - 947/250 = -1.788 which is less than (<) 0
so N = 947 is stable equilibrium
now AT 53
g'(N) = g"(53) = 2 - 53/250 = 1.788 which is greater than (>) 0
so N = 53 is unstable equilibrium
The capacity k=1000
If the population is less than 53 then the population will become extinct but since the capacity is equal to 1000 then the population will not go extinct.
is calculated from the sample, and used as an estimate for the population variance. Thus, it can be said that it represents both 362 and 5530 customers, and the correct option is D.