# non linear differential eqns population models

• Jun 7th 2010, 11:27 AM
acu04385
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
• Jun 7th 2010, 11:56 PM
Prove It
Quote:

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

$\frac{dx}{dt} = px - qxy$

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

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 $C = C_1 - C_2$.

Can you go from here?