A rabbit population satisfies the logistic equation dy/dt=2*10^-7y(10^6-y) where t is the time measured in months. The population is suddenly reduced to 40%of its steady state size by myxomatosis.
a) If the myxo' then has no effect how large is the population 8 months later?
b)How long will it take for the population to build up again to 90% of its steady state size?
To find the population at a given time, we need to solve the logistic equation dy/dt = 2 * 10^-7y(10^6 - y). We can do this by separating variables and integrating:
∫(y / (y(10^6 - y))) dy = ∫(2 * 10^-7) dt.
Simplifying the left-hand side integral, we get:
∫(1 / (10^6 - y)) - (1 / y) dy = 2 * 10^-7t + C.
Integrating both sides, we have:
ln|10^6 - y| - ln|y| = 2 * 10^-7t + C.
Applying logarithmic properties, we get:
ln|10^6 - y / y| = 2 * 10^-7t + C.
Exponentiating both sides, we have:
|10^6 - y / y| = e^(2 * 10^-7t + C).
To find specific solutions, we need additional information for the initial conditions. However, we can answer the questions based on broader observations.
a) If myxomatosis has no effect, it means the population remains unaffected by the disease. Therefore, the population will continue to grow until it reaches its steady state size. In this case, the steady state size is the y-value that satisfies dy/dt = 0.
Setting dy/dt = 0, we have:
2 * 10^-7y(10^6 - y) = 0.
This equation has solutions y = 0 and y = 10^6.
Since the population starts at 40% of its steady state size, the initial population is 0.4 * 10^6 rabbits. After 8 months, the population will reach its steady state size of 10^6 rabbits.
b) To find how long it takes for the population to build up to 90% of its steady state size, we need to find the time when the population is 0.9 * 10^6 rabbits.
Setting y = 0.9 * 10^6 in the solution obtained earlier, we have:
|10^6 - 0.9 * 10^6 / 0.9 * 10^6| = e^(2 * 10^-7t + C).
Simplifying, we get:
|0.1 / 0.9| = e^(2 * 10^-7t + C).
Taking the natural logarithm, we have:
ln(0.1 / 0.9) = 2 * 10^-7t + C.
Rearranging and solving for t, we get:
t = (ln(0.1 / 0.9) - C) / (2 * 10^-7).
Since C is the constant of integration, we need additional information (initial conditions) to determine its value and find the exact time it takes for the population to reach 90% of its steady state size.