Populations that can be modeled by the modified logistic equation

$\displaystyle \frac{{{d}{P}}}{{{\left.{d}{t}\right.}}}={P}{\left({b}{P}-{a}\right)}$

can either trend toward extinction or exhibit unbounded growth in finite time, depending on the initial population size. If $\displaystyle {b}={0.005}$ and $\displaystyle {a}={0.5}$, use phase portrait analysis to determine which of the two limiting behaviors will be exhibited by populations with the indicated initial sizes.



There is also a constant equilibrium solution for the population. Find this solution (note that the solution often is not a whole number, and hence unrealistic for population modeling).

$\displaystyle {P}{\left({t}\right)}=$ Preview Question 6 Part 2 of 4  

Solve the modified logistic equation using the values of $\displaystyle {a}$ and $\displaystyle {b}$ given above, and an initial population of $\displaystyle {P}{\left({0}\right)}={243}$.

$\displaystyle {P}{\left({t}\right)}=$ Preview Question 6 Part 3 of 4  

Find the time $\displaystyle {T}$ such that $\displaystyle {P}{\left({t}\right)}\rightarrow\infty$ as $\displaystyle {t}\rightarrow{T}$.

$\displaystyle {T}=$ Preview Question 6 Part 4 of 4