I'm new to the forum and looking for some help. So I'm in a combinatorics class, and a lot of the material is over my head, so I'm looking to see if any of you guys would be able to help. I'll list the problem, and also post what I have figured out so far.
2A) Population growth with variable birth rate. For the model pn = pn-1 + pn-2/(n-2), how the solutions grow as n -> infinity depending on given p1 = a and p2 = b? Is it always linear? If yes, calculate the tngent (the rate of growth) in terms of the initial a, b. B) What happens if the birth rate is /(n - 2)^2 instead of 1/(n-2)? WIll this population still enojoy unlimited growth? Justify numerically and try to find online related formuals from (advanced ) calculus. Trees may grow this way.
I have the solution to part a if you would like me to post it, but I can't really figure out part b. I have found some articles online modeling tree growth, and from looking at their plots, I have learned this function does not enjoy unlimited growth. I have written a program to evaluate the recurrence relation given for part B, but the function converges at a different number for every new p1 and p2 value. I thought this function would converge at the same point for any new values of p1 and p2, and because it doesnt I am kind of lost. My teacher gave me a hint to "Look at sinh(pi)/(pi)". Please help. And if I need to post the solution to 1, I will.