Fishery/Harvest Model: Dimensionless Form and Solution Attempt

[fk378]

Homework Statement

dN/dt= rN(1-N/K) - HN/(A+N)

where H and A >0

Show that the biological system can be rewritten in dimensionless form as

dx/dq= x(1-x) - hx/(a+x)

The Attempt at a Solution

I have no idea how to do this...any suggestions?

 

[Mark44]

Presumably N represents the number of fish at time t. What do r, K, H, and A represent?

For your second diff. eqn. what are x, q, h, and a? I don't think there's any way that we can help you if we don't know the connection between the first set of variables and those in the second DE.

Also, in the first DE, is it HN/A + N or is it HN/(A + N)? These are different.
In the second DE, is in hx/a + x or is it hx/(a + x)?

 

[fk378]

Sorry, just fixed it.

H is the rate of harvest
A is the rate of reproduction?
r is 1/Time?

How do I figure out what they represent?

 

[Mark44]

H might be a rate (the fraction of fish harvested--i.e., caught, a percentage), but I'm guessing that A is the number of new fish, and is not a rate. It wouldn't make any sense to add a rate to a number.

What about x, q, h, and a?

Surely the problem explains what these symbols represent.

 

[fk378]

The only other information I have is that K is the carrying capacity.

x,q,h,a ...I think I am supposed to figure that out on my own. There is no information given in the problem. We are supposed to do some substitutions, like x=something and with that, get to the second DE. That is where I'm lost though.

 

[HallsofIvy]

Since you are asked to put this in "dimensionless form" it would be a good idea to list the dimensions of the various terms! Cerrtainly you cannot do this problem without knowing what "H", "A", "r", and "k" mean!