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\noindent{\bf Math 3226
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Laboratory  3A}\\
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{\bf Predator-Prey Models and Hunting\footnote{This laboratory is based on a group project in {\em ``Fundamentals of Differential Equations''} by R. Kent Nagle and Edward B. Saff.}}
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\noindent{\em Consider the classical predator-prey model
\begin{eqnarray*}
\frac{dx}{dt}&=&A x - B x y\\
\frac{dy}{dt}&=&-C y +D x y.
\end{eqnarray*}
Such a system typically has a periodic solution, i.e., there is some time constant $T$ so that 
$x(t+T)=x(t)$ and $y(t+T)=y(t)$ for all $t$. Because of this periodic behavior, it is useful to consider the average populations $\overline{x}$ and $\overline{y}$, defined by 
\begin{eqnarray*}
\overline{x}&=&\frac{1}{T}\int_0^T x(t)\, dt\\
\overline{y}&=&\frac{1}{T}\int_0^T y(t)\, dt.
\end{eqnarray*}}

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\item 
Show that $\overline{x}=C/D$ and that $\overline{y}=A/B$. {\em Hint: } Use the first equation above and the fact that $x(0)=x(T)$ to show that 
\[\int_0^T (A-B y(t))\, dt=\int_0^T \frac{x'(t)}{x(t)}\, dt=0.\]

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\item Assume that a percentage of both species is hunted; more precisely: Assume that the prey is  hunted reducing its rate of change  by a constant $\epsilon$ times the prey population, while the predators are  hunted reducing their rate of change  by a constant $\delta$ times the predator population.  
Write down a system of first-order differential equations  describing this new predator-prey model with hunting.

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\item Assume $\epsilon<A$. What effect does this model of hunting have on the average prey and predator populations? 

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\item Assume that only the predator is hunted. What effect does this model of hunting have on the average prey and predator populations? 

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\item Assume that only the prey is hunted. What effect does this model of hunting have on the average prey and predator populations? 

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\item In a rural community, foxes prey mainly on rabbits, but occasionally include a chicken in their diet. The farmers decide to put a stop to the chicken killing by hunting the foxes. What do you predict will happen?
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\flushright\tiny\copyright\  H. Knaust. \today .
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