LOGO-L> Volterra-Lotka model

[MHELHEFNY@FRCU.EUN.EG]

Erich Neuwirth wrote :-

================
ok, 
once again i cannot resist.
we are discussing the volterra-lotka model here,
and in many textbooks it is introduced
with equations like this:

;Hello Everybody
;Here is another example demonstrating Predator-Prey relation.
;This model was formulated by the famous Italian mathematical
;biologist E. Volterra.
;The model solves the two simultaneous D.E. equations:-
;
;             dx/dt = a*x + b*xy + c*x*x   &

;             dy/dt = d*y + e*xy + f*y*y
;
;constants are given in fun1 & fun2 of the initial procedure.

i think one should discuss a little bit more
why these equations make sense,
and we also should talk about george's remark
that it is not so easy to find "working" values
for the parameters.

let us talk about growth processes.
growth processes are defined by

(dx/dt)/x = const

the change per time unit is proportional
to "what's there already"

5% interest rate on an bank account,

5% newborn children (in relation to the current population)
5% increase in the number of bacteria per hour 

....

we can think of many more examples of growth with a fixed 
growth rate.

the rate also can be negative, then the population
eventually will die out.

this model is very simple,
and constant growth over a wide range for population
is probably not a very good model.

so let us try a first modification.

let us assume a petri dish with bacteria,
but the dish has a finite capacity, no more than
E bacteria can live on the disk.

we start with a small number of bacteria
x << E

at first, the growth rate still is const
but as the dish gets fuller and fuller,
the growth rate goes down.

a simple model is:

75% of the original growth rate
when the number of bacteria is 25% of the capacity E,
50% growth rate for a 50% full dish
25% growth rate for a 75% full dish

so x/E is the saturation

and we have the model

(dx/dt)/x=const*(1-x/E)

renaming some constants we can rewrite this equation as

dx/dt=c1*x-c2*x*x

(just take c1=const, c2=const/E)

i think that the first way of expressing the equations
"tells more of the story" than the second way.
and const and E are parameters which still can directly be
connected to what we are modeling.

let us state the fundamental idea once more
the equation

(dx/dt)/x=const*(1-x/E)

for more...models a process which is a modified constant growth
process, and the simplistic growth factor is modified
by a factor depending linearly on the state variable.

now let us change to rabbits and foxes

x1 rabbits
x2 foxes

the model is
if there were no foxes the rabbits would
grow in number as described by

(dx1/dt)/x1=c1

on the other hand, the foxes would starve without rabbits

(dx2/dt)/x2=-c2

(written this way c2 would be positive, and the minus sign 
in the formula acts as a reminder, telling us who the predator is)

of course, things are not that simple.

rabbit growth rate, which would be constant without foxes,
gets reduces the more foxes are around.

(in german we have the saying
"viele fuechse sind des hasen tod", many foxes are rabbit's 
death)

therefore we write the equation

(dx1/dt)/x1=c1*(1-d1*x2)

written this way, d1 also can be interpreted:

it is the percentage by which the growth rate of the
rabbits is lowered by one more fox.

on the other hand, the growth rate of foxes
(which is negative without any rabbits,
is changed by the number of rabbits

(dx2/dt)/x2=-c2*(1-d2*x1)

d2 is positive, and it is the amount of change for the
growth rate of the foxes induced by one rabbit more.

if x2 gets large enough, (1-d2*x1) finally gets negative
and the right hand side of

(dx2/dt)/x2=-c2*(1-d2*x1)

becomes positive, so with enough rabbits around
the foxes don't starve and eventually even grow in number.
so if we write the volterra-lotka-system this way:

(dx1/dt)/x1=c1*(1-d1*x2)
(dx2/dt)/x2=-c2*(1-d2*x1)

we still can interpret the values of the parameters
and find "sensible" values.

modeling the finite capacity effect we discussed
in the (one-variable) bacteria model 
we can extend this model to

(dx1/dt)/x1=c1*(1-d1*x2-e1*x1)
(dx2/dt)/x2=-c2*(1-d2*x1-e2*x2)

and that is equivalent to
;             dx/dt = a*x + b*xy + c*x*x   &
;             dy/dt = d*y + e*xy + f*y*y

it still has the advantage of better understanding for
the parameters.

additionally, looking at

(dx1/dt)/x1=c1*(1-d1*x2)
(dx2/dt)/x2=-c2*(1-d2*x1)

we can easily see that
x2=1/d1
x1=1/d2

produces changes of 0,
therefore is the steady state of the system.

from then on, one can do all the usual experiments.


but i think it is very helpful to introduce
volterra-lotka systems starting from
constant growth rate models with
linear functions as the modifications of
the growth rates.

--
Erich Neuwirth <neuwirth@smc.univie.ac.at>
Computer Supported Didactics Working Group, Univ. Vienna
======================

Hi Erich !

I agree comleetely with your openion;however I thought that sending
both code and such introduction will be too lengthy for a mail;
Though I am not sure that I could write such a simple and elegant
introduction as your's.
However I have posted a short code for the prediction of the USA
population somedays ago (which follows the logistic curve model)

The posted Volterra-Lotka model invites users to do various experiments
such as :

1)Studying the effect of increased fishing (third term of the equations)
2)The effect of the increase population groth rate (first term)
3)Increasing the interaction of both spicies (second term)
4)approachin an oscilatory phase which indicate that equilibrium
 of the system had faild.

Finally let me thank you for your comment.
Best Regards
Mhelhefny
==================== 
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