(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

As long as N isn't too small, the growth of cancerous tumors can be modeled by the Gompertz equation,

dN/dt=-rN*ln(N/K),

where N(t) is proportional to the number of cells in the tumor, and r, K > 0 are parameters.

(a) Sketch the graph f(N)=-rn*ln(N/k) verses N, find the critical points, and determine whether each is asymptotically stable or unstable.

(b) For the initial condition N(0)=N_{0}(N_{0}>0), solve the Gompertz equation for N(t). Does this agree with your results from (a) in the limit as t -->∞?

(c) Sketch N(t) vs. t for initial conditions (i) 0 < N_{0}< K, (ii) N_{0}=K, and (iii) N_{0}> K.

2. Relevant equations

3. The attempt at a solution

(a)

Below is my sketch. I figured, since r,K,N>0 (can't have negative size of tumor), this is just a natural log function upside down and stretched out a little bit.

I'm not sure how to tell if it's asymptotically stable. I know that on the graph of N(t) versus t, it will start out with a positive slope; decrease in slope until the slope is zero at N(t)=K; then the slope will gradually become more negative.

(b)

dN/dt=-rN*ln(N/K)

==> dN / (N*ln(N/K) = -rdt

==> d/dN (ln(ln(N/K))) = -rdt

==> ln(ln(N/K) = -rt + C

==> ln(N/K) = e^{-rt+C}= ©e^{-rt}(©=e^{c}. I'm going to merge my constant shizzle while I solve)

==>N/K = e^{©ert}

==>N=N(t)= Ke^{©ert}= ®e(-rt) (®=ke^{©})

....... This is where I'm stuck. I've always been told that I can merge constants as I go along; but doing so in this problem screws everything up because I need K to stay with me!

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Tumor Growth word problem (due tomorrow, bro!)

**Physics Forums | Science Articles, Homework Help, Discussion**