Chapter 7.P2, Problem 4P

Aconstant-yield model, applied to species $x$, assumes that

$dx/dt$ isreduced by a positive constant $H$, the yield rate. For

the situation described by Eqs. (i), the modified equations are

$\begin{array}{c}\frac{dx}{dt}=x(1-x-y)-H\\ \frac{dy}{dt}=y(0.75-y-0.5x).\end{array}$(vi)

(a) For $H=0$, the $x$-nullclines are the lines $x=0$ and $x+y=1$. For H > 0, show that the x-nullcline is a hyperbola whose asymptotes are $x=0$ and $x+y=1$.

(b) How do the critical points move as $H$ increases fromzero?

(c) For a certain value of $H$, denoted by $H_c$, the asymptotically stable node originally at $(0.5,0.5)$ coincides with thesaddle point originally at $(0,0.75)$. Determine the value of $H_c$. Also determine the values of $x$ and $y$ where the two critical points coincide.

(d) Where are the critical points for $H>H_c$.? Classify themas to type.

(e) What happens to species $x$ for $H>H_c$? What happens tospecies $y$?

(f) Draw a direction field and/or phase portrait for $H=H_c$. and for values of $H$ slightly less than, and slightlygreater than, $H_c$.