Help with change of variables in ODE

[jim_s]

I'm auditing a math course, some 25+ years after having received a BS in math...

I have the differential equation:

dN/dt = rN(1-N/K-a/(1+bN)), N(0)=N0>0

I am given the change of variables:

Tau=rt
and x(Tau)=bN(t)
with M=bK, M>1

to arrive at:

dx/dTau = x(1-x/M-a/(1+x)), x(0)=x0>0

My simple-minded side says to just start plugging in the substitutions algebraically, but some long-dormant vague memory says that I need to employ the chain rule or such to get from dN/dt to dx/dTau. I'm looking for a brief demo of how I'd make this transition. (or to have someone tell me I'm making it more complicated than it needs to be

Thanks for any assistance!

 

[Deleted member 4993]

I'm auditing a math course, some 25+ years after having received a BS in math...

I have the differential equation:

dN/dt = rN(1-N/K-a/(1+bN)), N(0)=N0>0

I am given the change of variables:

Tau=rt
and x(Tau)=bN(t)
with M=bK, M>1

to arrive at:

dx/dTau = x(1-x/M-a/(1+x)), x(0)=x0>0

My simple-minded side says to just start plugging in the substitutions algebraically, but some long-dormant vague memory says that I need to employ the chain rule or such to get from dN/dt to dx/dTau. I'm looking for a brief demo of how I'd make this transition. (or to have someone tell me I'm making it more complicated than it needs to be

Thanks for any assistance!

Please define explicitly which parameters are constant, and which are variables.

And yes - you would have to use chain rule here.

 

[jim_s]

Thanks for the reply! Sorry for the delayed response - I thought I'd subscribed to the thread with email notification, but never got an email and just happened to check the forum now. :-(

a, b, K and r are constants in the first equation. t (time) is the independent variable. N is the dependent variable

In the second equation, a and M are constants (M=bK via the substitution). Tau is the independent variable. x is the dependent variable.

FWIW, this is a modification of the Lotke-Volterra predator/prey model. r is the reproduction rate of the prey. K is the carrying capacity (max sustainable population) for the prey. a is an 'attack rate' for the predator (how frequently the predator attacks prey) and b appears to be an 'escape rate' for the prey (seems to be an inverse of a 'success' rate for the predator - larger b leads to smaller dN for a given set of param)

Glad to hear I wasn't over-complicating things thinking that I needed to use the chain rule!