Suppose that a population develops according to the logistic equation
dP/dt = 0.06P−0.0001P^2
where t is measured in weeks.
1) The carrying capacity is .
2) The growth rate k is .
Use your calculator to sketch a direction field for this equation. Sketch the
solutions for the initial populations 200,600, and 800.
3) Which of these initial population(s) decrease?
4) Which solution(s) have an inflection point?
5) What are the non-zero equilibrium solution(s)?
since the curve is a parabola, there is no inflection point. Since #1 #2 do not provide any useful data, the other items are hard to figure.
To answer these questions, we need to perform some calculations and analyze the logistic equation.
1) The carrying capacity is the value at which the population stabilizes in the long run. In the given logistic equation, the carrying capacity is not explicitly given. However, it can be determined by finding the equilibrium solutions of the equation. An equilibrium solution occurs when dP/dt = 0.
So, let's set dP/dt = 0 and solve for P to find the equilibrium solutions:
0.06P - 0.0001P^2 = 0
Factorizing the equation, we have:
P(0.06 - 0.0001P) = 0
This equation is satisfied when P = 0 or 0.06 - 0.0001P = 0.
Solving the second equation, we get:
0.06 - 0.0001P = 0
0.0001P = 0.06
P = 0.06/0.0001
P = 600
Therefore, the carrying capacity is 600.
2) The growth rate k in the logistic equation can be determined by finding the value of dP/dt when P = carrying capacity. Substituting P = 600 into the logistic equation:
dP/dt = 0.06P - 0.0001P^2
dP/dt = 0.06(600) - 0.0001(600^2)
dP/dt = 36 - 0.0001(360000)
dP/dt = 36 - 36
dP/dt = 0
Therefore, the growth rate k is 0.
3) To determine which initial populations decrease, we can plot the direction field for the given logistic equation on a graph. This will allow us to see how the population changes with time.
To sketch a direction field, we need to select some points in the P-t plane and evaluate dP/dt at each of those points. Then, we can draw short line segments in the direction of the calculated values of dP/dt at those points.
However, as an AI text-based bot, I cannot create visual content directly. But you can use online graphing tools or software packages like MATLAB or Python to create the direction field. To create the direction field, select values of P and t, evaluate dP/dt, and draw arrows on the graph based on those values.
4) To determine which solutions have an inflection point, we need to find the second derivative of the logistic equation and analyze its behavior. The second derivative represents the rate at which the rate of change of population is changing.
Take the first derivative of the logistic equation with respect to t:
dP/dt = 0.06P - 0.0001P^2
Differentiating again with respect to t:
d^2P/dt^2 = 0.06 - 2(0.0001)(P)(dP/dt)
To find the inflection points, we need to set d^2P/dt^2 = 0 and solve for P.
0.06 - 2(0.0001)(P)(dP/dt) = 0
Simplifying:
0.06 = 2(0.0001)(P)(dP/dt)
Since we already know that dP/dt = 0.06P - 0.0001P^2, substitute this value into the equation above:
0.06 = 2(0.0001)(P)(0.06P - 0.0001P^2)
Simplifying further will involve solving a quadratic equation, and the resulting values of P can be identified as the solutions having inflection points.
5) The non-zero equilibrium solution(s) are the values of P where the population stabilizes and does not change. From the equation dP/dt = 0.06P - 0.0001P^2, we have already found that the equilibrium solution(s) is/are P = 0 and P = 600. As for non-zero equilibrium solutions, there is only one, which is P = 600.
Remember to check the calculations and equations for any errors as I am an AI bot and not immune to making mistakes.