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Math Help - non linear differential eqns population models

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    Post non linear differential eqns population models

    2 animal populations are in competition. Population sizes measured in thousands are given by X and Y. They are modelled by the diff eqns:-

    dx/dt = px -qxy, dy/dt = ry -sxy

    p, q , r & s are constants. p does not = r

    a) if there are no species Y ie (y = 0) find an eqn for x at time t if
    \[x(0)= x_{0}\] where x > 0

    b) determine the equilibrium points for the differential eqns above.

    Many thanks
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    Quote Originally Posted by acu04385 View Post
    2 animal populations are in competition. Population sizes measured in thousands are given by X and Y. They are modelled by the diff eqns:-

    dx/dt = px -qxy, dy/dt = ry -sxy

    p, q , r & s are constants. p does not = r

    a) if there are no species Y ie (y = 0) find an eqn for x at time t if
    \[x(0)= x_{0}\] where x > 0

    b) determine the equilibrium points for the differential eqns above.

    Many thanks
    \frac{dx}{dt} = px - qxy

    \frac{dy}{dt} = ry - sxy



    So \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}

    \frac{dy}{dx} = \frac{ry - sxy}{px - qxy}

    \frac{dy}{dx} = \frac{y(r - sx)}{x(p - qy)}

    \left(\frac{p - qy}{y}\right)\frac{dy}{dx} = \frac{r - sx}{x}

    \left(\frac{p}{y} - q\right)\frac{dy}{dx} = \frac{r}{x} - s

    \int{\left(\frac{p}{y} - q\right)\frac{dy}{dx}\,dx} = \int{\left(\frac{r}{x} - s\right)\,dx}

    \int{\left(\frac{p}{y} - q\right)\,dy} = r\ln{|x|} - sx + C_1

    p\ln{|y|} - qx + C_2 = r\ln{|x|} - sx + C_1

    p\ln{|y|} - qx = r\ln{|x|} - sx + C, where C = C_1 - C_2.


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