Q17RP - Fundamentals Of Differential Equations And Boundary Value Problems

Question

The direction field for the equation\[\frac{{{\bf{dp}}}}{{{\bf{dt}}}}{\bf{ = }}\frac{{{\bf{p(2p - 1)(p - 3)}}}}{{\bf{9}}}\],where p is the population (in thousands) at time t of a certain species, is plotted in Figure 1.17.

(a) If the initial population is 2500, what can you say about the limiting population as \[{\bf{t}} \to \infty \]?

(b) If the initial population is 4000, will it ever decrease to 3500?

(c) If p(1)= 0.3, what is \[\mathop {{\bf{lim}}\;}\limits_{{\bf{t}} \to \infty } {\bf{p(t)}}\]?

Step-by-Step Solution

Answer

(a) 500

(b) No

(c) 500 

Step 1(a): Finding the result for the initial population of 2500

Here \[\frac{{{\bf{dp}}}}{{{\bf{dt}}}}{\bf{ = }}\frac{{{\bf{p(2p - 1)(p - 3)}}}}{{\bf{9}}}\]

As \[\frac{{{\bf{dp}}}}{{{\bf{dt}}}}{\bf{ < 0}}\] for p=2.5, the population decreases units it reaches 0.5 where\[\frac{{{\bf{dp}}}}{{{\bf{dt}}}} = {\bf{0}}\].

The population remains stable. Hence the initial population of 2500 as \[{\bf{t}} \to \infty \],\[{\bf{p}} \to 500\].

Hence the limiting population is 500.

Step 2(b): Applying initial condition 4000

No, it is clear from the directional field that for any value of p>3,\[\frac{{{\bf{dp}}}}{{{\bf{dt}}}}{\bf{ > 0}}\].the population increases indefinitely. Since 4>3, the population will not decrease.

Step 3(c): Finding the result for \[{\bf{p}} = {\bf{0}}.{\bf{3}}\]

For\[p = 0.3\],the point p tends to \[p = 0.5\;{\rm{or}}\;p = 0\]. The population tends to decrease or 

Increasesat \[p = 0.3\].Since \[\frac{{{\bf{dp}}}}{{{\bf{dt}}}}{\bf{ = }}\frac{{{\bf{0}}{\bf{.3(2(0}}{\bf{.3) - 1)(0}}{\bf{.3 - 3)}}}}{{\bf{9}}}{\bf{ = 0}}{\bf{.36}}\]is positive.

So,p increases and stabilizes at \[{\bf{p = 0}}{\bf{.5}}\]with a population of 500.

Hence the solution is\[\mathop {{\bf{lim}}\;}\limits_{{\bf{t}} \to \infty } {\bf{p(t)}} = 500\]