The following autonomous system of differential equations de

The following autonomous system of differential equations describes two populations in interaction:

x\' = 3x - x2 + 1/2xy y\' = 1/5xy - y

What does this mean for the long-term behavior of the population y?

Solution

given that x\' = 3x - x2 + 1/2xy

y\' = 1/5xy - y

we can do the double derivative for this,then

with respect to x, x\"=3-2x+1/2y

=-2x+1/2y+3

with respect to y dy/dx=3x/y-x^2/y+1/2x

y\'=1/5xy-y

with x dy/dx=1/5y-y

with y y\"=1/5x-1