Return-Path: <greiner%eccx.dnet@esu36.cfsat.honeywell.com>
Newsgroups: bit.listserv.scuba-l
Date:         Sat, 7 Nov 1992 07:13:34 -0700
From:         "(Everything you know... is wrong!)"
              <greiner%eccx.dnet@ESU36.CFSAT.HONEYWELL.COM>
Subject:      Tiger sharks in Hawaii

[stuff deleted]
                                                            ust so I am not
 misunderstood, I can understand a few sharks in the
>water and the sighting of them. These problems this year in Hawaii are
>a little different. Something has changed in the environment there.
>Do we have any one out there doing their Masters or PHD in this
>type of area?

>Tim Chase Computer Operator Specialist  PSU


Well since it is Saturday and I am forced to work and I am bored stiff
as you people could probably tell from my previous mail messages (feeling cauky)
I will attempt to field this question:

I think they need someone who studies marine life and a mathmatician who like
differential equations to set up a problem of two or more species
would look something like this:

dp/dt= ap where "a" is a constant > 0, and "p" is people in millions
ds/dt= -bs where "b" is a constant > 0 ,and "s" is sharks in thousands

Now if we model for the effect of two populations on one another, Clearly the
 presence
of Sharks is bad for the people.

dp/dt= ap - (effect of sharks on people)

on the other hand - sharks do better with people around

ds/dt= -br + (effect of people on sharks)

Now lets assume the effect of one popluation on the other will be porprotional
to the number of encounters (meaning when a shark meets and eats a person)
The number of encounters is likely to be porpotional to the products of the
populations because the more there are of either population the more encounters
there will be. So lets model this

dp/dt= ap - cps        and
ds/dt= -bs + kps
                       where c,k are positive contants
to look at the system lets look at your specifice example where a=b=c=k=1

dp/dt= p-ps
ds/dt= -s+ps

Before we continue we need to figure a way to handle the three variables
sharks,people populations and Time "t"  from a biological point of view it
would be of more help to know many of each population exist simultaneously than
to know each population  at particular times, so for a moment we will think of
"s" as function of "p", and both "s" and "p" as a function of time "t"

s= F(p) and p=G(t)  ==-> s=F(G(t)) so use the chain rule

ds/dt=F'(G(t))*G'(t)= (ds/dp)*(dp*dt)

so we will show

ds/dp= (ds/dt)/(dp/dt)

therefore we get the following equation

ds/dp= (-s+ps)/(p-ps)

which did seem to explain things graphically (sorry I could not send the graph
 over mail)
 but what it looks like is this (connect the dots type of graph "slope field"):
if you have a standard cartesion plane in coordinate section I then
( ah what the hell - I will attempt to draw it

s   3 ||||/___
h     |||/__\\\\\
a     ||||___\\\\\
r   2 |||____\\\\\\
k     ||/___\\\\\\\\
s     |||/  \\\\\\\\\
    1 |||/ \ ||||||||
      ||.\_/ ////////
      | \------------------
      |------------------------- people
          1       2        3
any way it sort of looks like that where you have many slope curves (connect the
dots in the direction they are pointing)
with a center at point (1,1) so if you increase sharks then more encounters will
happen and more people get eaten for a while then it will die offf for a while
then it will increase again - the same happens when you add more people.
 rotating counter
clockwise.

I hope this helps???????????????
Hans