# Thread: non linear differential eqns population models

1. ## 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

2. Originally Posted by acu04385
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
$\displaystyle \frac{dx}{dt} = px - qxy$

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

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

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

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

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

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

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

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

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

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

Can you go from here?