Differential Equations | Part IA, 2001

Carnivorous hunters of population hh prey on vegetarians of population pp. 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

p˙=p(1pa)pha,h˙=h8(pb1),\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 aa and bb are positive constants, and h(t)h(t) and p(t)p(t) are non-negative functions of time, tt. By giving an interpretation of each term explain briefly how these equations model the system described.

Consider these equations for a=1a=1. In the two cases 0<b<1/20<b<1 / 2 and b>1b>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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