Can someone help me solve this? Suppose a population (P) of yeast is changing ac

Question

Can someone help me solve this?

Suppose a population (P) of yeast is changing according to the differential equation: dP/dt = 3P(20-P) uoP - 3P 2 rightarrow uo-up What are the equilibrium values for the autonomous differential equation? 3P = 0 70-P rightarrow P = 70 Which of the equilibrium values are stable? Which are unstable? 0 = stable, 70 = unstable Construct a phase line. Sketch solution curves below. (Be sure to label your axes!) Suppose you start with a population of 10. What will eventually happen to the population? increase rightarrow 300

Explanation / Answer

dP/dt= 3P(20-P)

dp/P(20-P) =3dt

[1/p -1/(P-20)] dp =3dt

integrating both sides

ln |P|- ln |P-20|=3t+C

ln|P/(P-20)|=3t+C

for equilibrium

dP/dt= 3P(20-P)=0

p=0 and 20

population cannot be zero

so P=20

so zero is unstable and 20 is stable

ln|P/(P-20)|=3t+C

|P/(P-20)|=e3t+c

P= 20(e3t+c)/(1-e3t+c)

for t=0

P=10

so we can see that with time it will increase

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