2.II.8B

Carnivorous hunters of population $h$ prey on vegetarians of population $p$. In the absence of hunters the prey will increase in number until their population is limited by the availability of food. In the absence of prey the hunters will eventually die out. The equations governing the evolution of the populations are

$\begin{aligned} \dot{p} &=p\left(1-\frac{p}{a}\right)-\frac{p h}{a}, \\ \dot{h} &=\frac{h}{8}\left(\frac{p}{b}-1\right), \end{aligned}$

where $a$ and $b$ are positive constants, and $h(t)$ and $p(t)$ are non-negative functions of time, $t$. By giving an interpretation of each term explain briefly how these equations model the system described.

Consider these equations for $a=1$. In the two cases $0<b<1 / 2$ and $b>1$ determine the location and the stability properties of the critical points of $(*)$. In both of these cases sketch the typical solution trajectories and briefly describe the ultimate fate of hunters and prey.

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