I have a problem understanding an ODE and using it to find something particular. Consider the following :
ODE : dC/dt= S-r*C
S: synthesis rate
r : death rate
Co: initial population
the analystical solution is simply C(t) =S/r -(S/r-Co)*exp(-r*t)
My questions are : i) How can we calculate the population that has died at time t
ii) How can we calculate the population that has synthesized at time t
My problem in understanding is relevant the rates S and r and how these can be applied to evaluate the exact quantity of population that has followed death and generation.
So If I want to find how many have died (or synthesized) at time t (given r or S) is it simply the multiplication with C(t) or with C(t-1)?
I would appreciate your help with this problem. Thanks a lot.
Thank you for your responses,
Concerning the first repply,
I am sorry but I do not get your point. There are two parallel processes taking place. Death and generation with rates r and S respectively.
Lets say at time 0 these rates are 0 and for all different times t remain constant at values greater than zero. If I would substract Co from C(t) what do I find ?
For the second repply again I can get your point.
The analytical solution displays the population at any time t given the rates S and r. These rates are positive values and zero only at time zero. I dont understand....Where to plug the "relevant value of t" and what how to find that?
I am wondering....considering the previous model.
Is it true that ??:
C(prev.time) + S(current time) - r*[ C(prev.time) + S(current time)]
where C : population
S : newly synthesised polulation (assume that is independt of population size(const.))
r : death rate of population (assume constant)
and if also the population that dies at any time t is labeled with D we must have :
D(current time)=r*[ C(prev.time) + S(current time)] ??
Please anyone who can take a look at it and let me know an opinion about it.