1. I really don't understand what this problem is asking for
The population P(t) of fish in a lake satisfies the logistic differential equation
dP/dt = 3P - (P^2)/6000
If P(0) = 4000, what is ? Is the solution increasing or decreasing?
&& how would you find inc/dec of a solution?
To understand what this problem is asking for, we need to find the value of P(t) at a specific time t, given an initial population P(0) of 4000 fish. We also need to determine whether the solution is increasing or decreasing.
To find the solution to the logistic differential equation, we need to solve the equation:
dP/dt = 3P - (P^2)/6000
To do this, we can separate the variables and integrate both sides. Let's go step by step:
Step 1: Separate the variables.
dP / (3P - (P^2)/6000) = dt
Step 2: Integrate both sides.
∫ [1 / (3P - (P^2)/6000)] dP = ∫ dt
Step 3: Evaluate the integrals.
This step involves solving the integral on the left side and the integral on the right side, which will provide us with the solution.
Step 4: Solve for P(t).
Once we have integrated both sides, we can solve for P(t) to find the population at time t.
Step 5: Determine if the solution is increasing or decreasing.
To determine whether the solution is increasing or decreasing, we can analyze the sign of the derivative of P(t).
By following these steps, we can find the solution to the logistic differential equation and determine whether it is increasing or decreasing.
To understand what the problem is asking for, let's break it down step by step.
1. The problem gives a logistic differential equation: dP/dt = 3P - (P^2)/6000. This equation describes the rate of change of the fish population (P) with respect to time (t).
2. The equation has two terms. The first term, 3P, represents the natural growth rate of the fish population. The second term, (P^2)/6000, represents the effect of limited resources on the population growth. As the population grows, this term becomes more negative, which slows down the growth.
3. The problem also provides an initial condition: P(0) = 4000. This means that at the beginning (t = 0), the fish population is 4000.
Now, let's find the solution and determine whether the solution is increasing or decreasing.
To solve a differential equation like this, we can use separation of variables. Here's how you can do it:
1. Separate the variables by moving all terms involving P to one side of the equation:
dP / (3P - (P^2)/6000) = dt
2. Integrate both sides with respect to their respective variables:
∫ dP / (3P - (P^2)/6000) = ∫ dt
3. Solve the integrals:
(1/3) * ln|3P - (P^2)/6000| = t + C
where C is the constant of integration.
4. Exponentiate both sides to eliminate the logarithm:
e^((1/3) * ln|3P - (P^2)/6000|) = e^(t + C)
3P - (P^2)/6000 = ke^t
Note that e^C = k, where k is another constant.
5. Rearrange the equation to isolate P:
P^2 - 3P + (6000 * ke^t) = 0
This is a quadratic equation in P. You can use the quadratic formula or factorization to solve for P.
Once you find the solution for P, you can analyze its behavior to determine whether it is increasing or decreasing. You can do this by examining the sign of dP/dt for different values of P.
If dP/dt > 0, then the solution is increasing.
If dP/dt < 0, then the solution is decreasing.
Plug in some values of P into dP/dt = 3P - (P^2)/6000 to see whether the result is positive or negative. This will give you an idea of how the solution behaves.
Note: The specific solution to this logistic differential equation may require additional calculations because of the initial condition P(0) = 4000.