How Do You Solve a Differential Equation for Wildlife Population Growth?

[Coldie]

Let P(t) represent the number of wolves in a population at time t years, when t>=0. The population P(t) is increasing at a rate directly proportional to 800 - P(t), where the constant of proportionality is k.

a) If P(0) = 500, find P(t) in terms of t and k.
b) If P(2) = 700, find k.
c) Find lim t -> x P(t)

I'm really not sure where to start with this question, and any help would be appreciated. It's an entrance exam our teacher got for us to practice on, so I suspect we may not have covered material on these types of questions yet.

 

[Rogerio]

The diff eq:
dp = k(800-p) dt

rewrite as:
dp/(800-p) = k dt

Then you have to integrate both sides to get something like
p = 800 - (800-p0)e^-kt

 

[M98Ranger]

No problem, let's break this down step by step.

a) To find P(t) in terms of t and k, we need to use the given information that the population is increasing at a rate directly proportional to 800 - P(t). This can be written as the differential equation dP/dt = k(800-P(t)).

Now, we can use separation of variables to solve for P(t).

dP/dt = k(800-P(t))
dP/(800-P(t)) = kdt

Integrating both sides, we get:
∫ dP/(800-P(t)) = ∫ kdt
-ln(800-P(t)) = kt + C

Using the initial condition P(0) = 500, we can solve for C:
-ln(800-500) = 0 + C
C = ln(300)

Substituting this value back into our equation, we get:
-ln(800-P(t)) = kt + ln(300)
ln(800-P(t)) = ln(300) + kt

Now, we can exponentiate both sides to get rid of the natural logarithm:
800-P(t) = 300e^(kt)

Finally, we can solve for P(t):
P(t) = 800 - 300e^(kt)

b) To find k, we can use the given information that P(2) = 700. Substituting this into our equation from part (a), we get:
P(2) = 700 = 800 - 300e^(2k)

Solving for k, we get:
2k = ln(100/3)
k = ln(100/3)/2

c) To find the limit as t approaches infinity, we can use the equation we found in part (a) and take the limit as t approaches infinity:

lim t->∞ P(t) = lim t->∞ 800 - 300e^(kt)

As t approaches infinity, e^(kt) also approaches infinity. Therefore, the limit will be -∞, which means the population will continue to increase without bound.

I hope this helps and good luck on your entrance exam!