Suppose a population grows according to the logistic equation but is subject to a constant total harvest rate of H. If N(t) is the populatio

Question

Suppose a population grows according to the logistic equation but is subject to a constant total harvest rate of H. If N(t) is the population size at time t, the population dynamics are dN dt = r 1 − N K N − H. Different values of H will result in different equilibrium population sizes, and if H is large enough we might expect extinction.(a) Suppose r = 2, K = 1000, and H = 100. Find all equilibria. (Round your answers to the nearest integer. Enter your answers as a comma-separated list.) N hat =(b) Determine whether each of the equilibria found in part is locally stable or unstable. (Round your answers to the nearest whole number. Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) stable N hat = unstable N hat =Is the population predicted to go extinct?YesNo

in progress 0
Mít Mít 5 years 2021-09-05T04:05:12+00:00 1 Answers 43 views 0

Answers ( )

  1. taiduc
    0
    2021-09-05T04:06:33+00:00
    Reply

    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.

Leave an answer

Browse

Giải phương trình 1 ẩn: x + 2 - 2(x + 1) = -x . Hỏi x = ? ( )