Population Growth: Let P(t) be the number of rabbits in the rabbit population. In the simplest case we can assume the number of rabbits born at any moment of time is proportional to the number of rabbits at this moment of time. Mathematically we can write this as a differential equation:
Here b is the birth rate, i.e. births per time unit per rabbit. In the model above we ignore deaths and assume resources are unlimited.
A. Solve the equation above.
B. Find the number of rabbits in 72 months, if initially there were P(0) = 24 rabbits and after 6 months P(6) = 75 rabbits.
C. Take into account deaths. Assume the death rate per unit of time per rabbit is constant. Solve the equation and analyze the solution at t approaching infinity when: a) the birth rate exceeds the death rate, b) if the death rate exceeds the birth rate.
D. Modify the differential equation above taking into account the fact that resources are limited. In other words, the number of rabbits cannot exceed K, P(t) is less than or equal to K always.
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(b) Write the differential rate equation for the number of
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