A population of a species of animal had the following differential equation:
$${dP\over dt} = rP \left(1-{P\over K}\right)$$
$P$ was the population. $t$ was the time in years. $K$ was the carrying capacity (which is the ecologically sustainable mass of population the place can hold).
After solving the differential equation (by getting the final equation, rearranging the formula for $P(0)$ and putting it back into this final equation, the formula was:
$$P = {KP_0\over P_0 + (K-P_0)e^{-rt}}$$
The carrying capacity ($K$) is between 50 and 60 kT.
$P_0$ is the initial population which in this situation is between 42 KT and 68 KT.
The rate of increase ($r$) is between $-0.05$ and $0.05.$
What parameter is the most important to monitor the population? (I was thinking $r,$ is this right?)
What parameters together leads to the most ecologically sustainable population (I was thinking $r$ and initial population but I'm not too sure)
Production of food needs to be increased and it is acceptable for the population of the animal species to be reduced by 10% over the next 10 years (1% per year). What $r$ value would be put on the population? (I was not very sure of what to do for this)