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.
: 3.14