Q5E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

The logistic equation for the population (in thousands) of a certain species is given by $\frac{\mathbf{dp}}{\mathbf{dt}}\mathbf{=}\mathbf{3}\mathbf{p}\mathbf{-}\mathbf{2}{\mathbf{p}}^{\mathbf{2}}$ .

⦁    Sketch the direction field by using either a computer software package or the method of isoclines.

⦁    If the initial population is 3000 [that is, p(0) = 3], what can you say about the limiting population?

⦁    If $\mathbf{p}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{8}$ , what is ${\mathbf{lim}}_{\mathbf{t}\to \mathbf{+}\infty }\mathbf{p}\mathbf{\left(}\mathbf{t}\mathbf{\right)}$ ?

⦁    Can a population of 2000 ever decline to 800?

Step-by-Step Solution

Answer

⦁    The Sketch is drawn for the direction field

⦁    The limiting population is  $\frac{3}{2}$

⦁    The limiting population is  $\frac{3}{2}$

⦁    No

1(a): Drawing the Sketch for the direction field of the given equation

Hence, the Sketch is drawn for the direction field.

3(b): Applying the initial condition p ( 0 ) = 3

Hence, the limiting population is  .

4(c): Applying the initial condition p ( 0 ) = 0 . 8 in the solution

$$\begin{gathered} \frac{\mathbf{3}}{\mathbf{2}\mathbf{-}\mathbf{2}{\mathbf{c}}_{\mathbf{2}}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{8} \\ {\mathbf{c}}_{\mathbf{2}}\mathbf{=}\mathbf{1}\mathbf{-}\frac{\mathbf{3}}{\mathbf{1}\mathbf{.}\mathbf{6}} \\ {\mathbf{c}}_{\mathbf{2}}\mathbf{=}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{875} \\ \mathbf{Now}\mathbf{,}\mathbf{p}\mathbf{=}\frac{\mathbf{3}{\mathbf{e}}^{\mathbf{3}\mathbf{t}}}{\mathbf{2}{\mathbf{e}}^{\mathbf{3}\mathbf{t}}\mathbf{+}\mathbf{1}\mathbf{.}\mathbf{75}} \\ {\mathbf{lim}}_{t\to \infty }\mathbf{p}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\frac{\mathbf{3}}{\mathbf{2}} \end{gathered}$$

Hence, the limiting population is $\frac{\mathbf{3}}{\mathbf{2}}$ .

5(d): Analyzing the graph and the different initial conditions

From the above two parts (b), (c) and the graph, 

the limiting value of population approaches 1.5 (i.e., 1500) as t tends to infinity.

Hence, the population of 2000 can never decline to 800.