The nonlinear system x9 a y 1 y x 2 x y9 y 1 y x 2 y b arises in a model for the growth

Chapter 11, Problem 21

(choose chapter or problem)

The nonlinear system

\(\begin{aligned}

&x^{\prime}=\alpha \frac{y}{1+y} x-x \\

&y^{\prime}=-\frac{y}{1+y} x-y+\beta

\end{aligned}\)

arises in a model for the growth of microorganisms in a chemostat, a simple laboratory device in which a nutrient from a supply source flows into a growth chamber. In the system, x denotes the concentration of the microorganisms in the growth chamber, y denotes the concentration of nutrients, and \(\alpha>1\) and \(\beta>0\) are constants that can be adjusted by the experimenter. Find conditions on \(\alpha\) and \(\beta\) that ensure that the system has a single critical point \((\hat{x}, \hat{y})\) in the first quadrant, and investigate the stability of this critical point.