A cell population is growth in a Petri dish in a laboratory for scienti c purpos

Question

A cell population is growth in a Petri dish in a laboratory for scienti c purposes. Cells grow according to a logistic equation, with growth rate r = 1.5 day1 and carrying capacity k = 5 × 105 cells. However, every day the biologists working in the lab take h cells away from the Petri dish to perform experiments with them. (a) Write down a di erence equation for the cell population, considering 1 time-step to be equal to 1 day. Find the equilibria and determine their stability analytically. (b) Determine the equilibria and their stability graphically (with cobweb- bing) you should get the same results as before. (c) Interpret your results: is a stable equilibrium possible? Should there be any limitation on the number of collected cells per day, h?

Explanation / Answer

given growth rate r = 1.5 /day;

maximum capacity = 5*10^5 cells;

given

dN/dt = rN -h;

solving above we get;

dN/(rN-h) = dt; or ln(rN-h)=rt+c;

rN-h=ke^(rt)

N = (h+ke^(rt))/r;

t=0;

k= 1.5N0-h;

N = (h+(1.5N0-h)e(rt))/1.5

now given N0=5E5;

we have equilibrium when

rN-h =0;

h=1.5N;

or h=7.5E10 cells daily;

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