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)=N0 (N0>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 < N0 < K, (ii) N0=K, and (iii) N0 > 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 (©=ec. 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!