Consider a bioreactor used by a yogurt factory to grow the bacteria needed to make yogurt....

 

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Consider a bioreactor used by a yogurt factory to grow the bacteria needed to make yogurt. As long as no bacteria are being harvested, the growth of the bacteria is governed by the logistic equation dP/dt = kP(M-P), where P is the population in millions and t is the time in days. Recall that M > 0 is the carrying capacity of the reactor, and k is a positive constant. a) Through observation it is found that after a long time, the population in the reactor stabilizes at 50 million bacteria and that when the population of the reactor is 20 million bacteria the population increases at a rate of 12 million per day. From this, find k and M in the governing equation. b) If the colony starts with a population of 10 million bacteria, how long will it take for the population to reach 80% of carrying capacity? c) Assume the reactor allows for continuous harvesting of the bacteria, without shutting down the reactor. Let h be the rate at which the bacteria are harvested, in millions per day. Write down the new differential equation governing the bacteria population. What is the maximum rate of harvesting h that will not cause the population of bacteria to go extinct? (Harvesting at less than this rate will ensure that there is always a stable equilibrium point where P is positive.) Hint: This question is all about phase diagram analysis and doesn't require any solution formulas. Consider a bioreactor used by a yogurt factory to grow the bacteria needed to make yogurt. As long as no bacteria are being harvested, the growth of the bacteria is governed by the logistic equation dP/dt = kP(M-P), where P is the population in millions and t is the time in days. Recall that M > 0 is the carrying capacity of the reactor, and k is a positive constant. a) Through observation it is found that after a long time, the population in the reactor stabilizes at 50 million bacteria and that when the population of the reactor is 20 million bacteria the population increases at a rate of 12 million per day. From this, find k and M in the governing equation. b) If the colony starts with a population of 10 million bacteria, how long will it take for the population to reach 80% of carrying capacity? c) Assume the reactor allows for continuous harvesting of the bacteria, without shutting down the reactor. Let h be the rate at which the bacteria are harvested, in millions per day. Write down the new differential equation governing the bacteria population. What is the maximum rate of harvesting h that will not cause the population of bacteria to go extinct? (Harvesting at less than this rate will ensure that there is always a stable equilibrium point where P is positive.) Hint: This question is all about phase diagram analysis and doesn't require any solution formulas.