Lab 8

This lab is a study in nonlinear systems of ODE's. The first system reflects the interaction between two competitive species. The second reflects a predator-prey situation.

CANNIBALS

OR

What shall I eat today or should I ask, Who?

Prologue: Two tribes of cannibals live on an isolated island. The population of tribe $x$, the more remote tribe, at time $t$ is $x(t).$ The population of tribe $y$, the more remote tribe, at time $t$ is $y(t).$ These populations are governed by ODE's that contain terms for each of the following:

Natural Growth: + $ax$ or + $ay$

Overcrowding of their own region: - $bx^{2}$ or - $by^{2}$

Cooperation, eating $+\ cxy$

Harvesting, being eaten $-\ dxy$

The First Course: Natures own, $x$ and $y$ are in thousands and $t$ in years. The equations: NG OC Coop. Harv.
MATH

Graph the system and find equilibrium point(s) and create a story line.

The Second Course: Enter the Ecologist, you know the person who knows ''How it should be'', This person declares, ''too many people for one island, cannibalism is a bad diet, and vegetarians are better for the ecology''. The remote tribe ($x)$ is too hard to get to so he/she convinces the other tribe to become vegetarians and to quit dining out on the other tribe. Therefore the harvesting term in $\frac{dx}{dt}$ becomes $0$ as well as the cooperation term of $\frac{dy}{dt}$ while all the other terms remain unchanged.

Graph the system and find equilibrium point(s) and create a new story line.

Final Course: Shortly, very shortly, the village shaman (tribe $y$ ) runs a projection of this situation on his new laptop computer and plans a feast with the ecologist as the guest of honor. Should he/she accept?

Adapted from an article in The College Mathematics Journal by J M McDill and Bjorn Felsager.

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