# Dfq Prob, using Logisitc Equation

• Naeem
In summary, we need to develop the logistic equation and solve for the population size P as a function of days t, given the initial conditions P(0) = 100, P(1) = 200, and Pmax = 10,000. Using partial fractions and integration, we can find the equation P(t) = Ce^kt / Pmax - P, where C is determined using the initial condition and k is determined using the other given values. Then, to find the number of flies present after 3 days, we can plug in the value of t = 3 into the equation.
Naeem
Q. Develop the model of the logisitic equation and use it to solve the following. At first there are 100 fruit flies. After one day there are 200 fruit flies.The maximum population is 10,000 fruit flies.

a) Determine the population size P as a function of days t.

We know,

P(0) = 100

P(1) = 200

P max = 10,000

Logisitc equation:

dP/dt = kP ( 1-P/Pmax)

Integrating both sides:

Integral ( dP/P(1-P/m) = Integral kdt

Using Partial fractions, we get:

PMax - P^2 = Ce^kt

Therefore,

C = PM - P^2 / e^kt

P(t) = Ce^kt / Pmax - P

Is this correct,

b. How many flies are present after 3 days?

For, this we can find C, using the initial condition: P(0) = 100

To find k, we can use P(1) = 200, and Pmax = 10,000

After finding all this,

We can do, P(3) = Ce^kt / Pmax - P

Any suggestions / ideas, Please help

Please note, that M , in the previous parts, is actually referring to Pmax

.

Yes, your approach is correct. To find the value of C, we can use the initial condition P(0) = 100:

C = PM - P^2 / e^kt
C = 10,000 - 100^2 / e^k(0)
C = 10,000 - 10,000 / 1
C = 0

Now, we can substitute the values of C, k, and Pmax into the equation we derived earlier to find the population size after 3 days:

P(3) = Ce^kt / Pmax - P
P(3) = 0 * e^k(3) / 10,000 - 0
P(3) = 0 / 10,000
P(3) = 0

Therefore, after 3 days, there are no fruit flies present. This is because the population reaches its maximum size of 10,000 and then starts to decline due to limited resources and competition. This can also be seen in the graph of the logistic equation, where the population initially grows rapidly, reaches a peak, and then starts to decline.

## 1. What is the DFq Prob using Logistic Equation?

The DFq Prob using Logistic Equation is a mathematical model that is used to describe the growth of a population over time. It takes into account factors such as birth rate, death rate, and carrying capacity to predict the population size at any given time.

## 2. How is the Logistic Equation used in DFq Prob?

The Logistic Equation is used in DFq Prob to model the population growth rate. It takes into consideration the initial population size, the intrinsic growth rate, and the carrying capacity of the environment to predict the population size at any given time.

## 3. What is the difference between DFq Prob and Logistic Equation?

DFq Prob is a broader concept that refers to the study of population dynamics and how it changes over time. The Logistic Equation, on the other hand, is a specific mathematical model that is used in DFq Prob to predict population growth.

## 4. What are the assumptions made in using the Logistic Equation for DFq Prob?

The Logistic Equation makes several assumptions, such as a constant carrying capacity, a constant intrinsic growth rate, and a lack of external factors that could affect the population growth. It also assumes that the population is closed, meaning there is no migration in or out of the population.

## 5. Can the Logistic Equation be applied to any population?

The Logistic Equation can be applied to any population as long as the assumptions hold true. However, it is important to note that different populations may have different carrying capacities and intrinsic growth rates, which would affect the accuracy of the predictions.

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