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Math Help - modelling population growth/decay

  1. #1
    Junior Member
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    modelling population growth/decay

    I have another question I would like to ask about, this is the question as follows


    ************************************************** *****************

    The simplest useful model for a shery comes from the logistic model for population growth,
    together with a harvest h which is proportional to the current population P, that is,
    h = EP; (1)
    where the constant E is called the eort. E measures the fraction of the population harvested,
    so that 0 <= E <= 1. This gives the model

    \frac{dP}{dt}=kP(1-\frac{P}{a})-EP

    where P(t) is the number of these sh at time t years and k (the natural growth rate) and a (the
    carrying capacity) are constants for a particular sh population. In what follows take k = 1 and
    a = 4, for simplicity.
    (a) Determine the equilibrium solution(s) for a given eort E.
    (b) Plot the phase plane dP
    dt versus P for general E, 0 E 1, and determine for which values
    of P the population is increasing or decreasing with time.
    (c) Make a qualitative sketch of P versus t for various initial population sizes P(0).
    (d) Hence determine which equilibrium population size(s) are stable and which are unstable.
    (e) The harvest h corresponding to the stable population size is called the sustainable yield
    from the shery. Using equation (1), and your answer to part (d), nd the sustainable yield
    as a function of the eort, plot it, and show it has a maximum at E = \frac{1}{2}; the harvest at
    this eort is called the maximum sustainable yield (MSY). What is the MSY and the stable
    population size at this eort?


    ************************************************** ************

    So I was able to do Question a to d, although I'm not certain if I did it correctly or not. How does one do question e?

    for question a), i let \frac{dP}{dt}=0 because it is an equilibrium solution, thus EP=P(1-\frac{P}{4}) so  E=1-\frac{P^2}{4}


    for question b), I use the equation \frac{dP}{dt}=P-\frac{P^2}{4}-EP where E =0 and 1 seperately
    thus I have two inverse parabolic graph with one of them touch the origin and the other passed through P when P = 0 and 4, I then shade the area between these graphs



    for C) I made a graph of P vs t, where when P(0) and P(4) is a straight line, because of the shape of the graph in question b), I assume that P(4) is stable while P(0) is unstable, this is also the answer for question 4

    modelling population growth/decay-1239613_433161736801610_893594901_n.jpg


    How would you do question e)?

    Thank Junks
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  2. #2
    MHF Contributor
    Joined
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    Re: modelling population growth/decay

    Hey junkwisch.

    Hint: You need to take your answers and evaluate P(t) given your initial conditions. If the function only depends on P and not t and the derivative is 0, then it means that the yield won't change over time. What kind of function is one that does not change over time?

    You should also think about what initial conditions allow you to go to the stable position and which ones do not.
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