Flashcards in 11.2.2 Logistic Growth Deck (13):

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## Logistic Growth

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• Distinguish between exponential growth models for short-term population growth and logistic growth models for long-term population growth that take into account the carrying capacity of the environment.

• Analyze solutions to logistic growth differential equations using direction fields.

• Solve logistic growth equations numerically using Euler's Method.

• Use the separable nature of logistic growth differential equations to solve them analytically.

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## note

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- A simple population model could assume exponential growth. This is a reasonable assumption if the population is small, because there are no constraints restricting

its growth.

- But in the long run, exponential population growth cannot be sustained due to naturally occurring constraints on resources.

- The logistic growth model takes this into account. When the population is small, the population growth rate is nearly exponential.

- When the population nears its carrying capacity, or the

maximum population that the environment can sustain in the long run, the growth rate approaches zero.

- The solutions all approach the carrying capacity of 1200.

- When the population is below 1200, the population increases to the carrying capacity. When the population is above 1200, the population decreases to the carrying capacity.

- When the population is 600, the population is increasing the most rapidly. That is, the slopes are the steepest.

- Recall that Euler's Method is a numerical method for solving differential equations by approximating the solution by a piecewise linear function.

- Given an initial condition, y 0 = y(x 0 ), the next point is given by x 1 = x 0 + h and y 1 = y 0 + hF(x 0 , y 0 ), where h is the step size. This process then repeats with (x 1 , y 1 ), the new starting point.

- With a step size of 25, the population after 50 time units is computed to be approximately 872. The population after 100 time units is computed to be approximately 1184.

- Notice that after 75 time units, the population exceeded its carrying capacity and, as would be expected by the logistic growth model, the population decreased at the next iteration.

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## note 2

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- The logistic differential equation can also be solved

analytically by relying on the fact that it is separable. A

differential equation is separable if it can be written in the form N(y)dy = M(x)dx. The solution can be obtained by integrating both sides of the equation.

- The analytical solution to the logistic differential equation reveals that the population approaches its carrying capacity in the long run.

- The exact solution can be used to find the actual populations (based on the model) after 50 time units and after 100 time units.

- Substitute 50 for t in the solution to obtain a population of 961 after 50 time units.

- Substitute 100 for t in the solution to obtain a population of 1,185 after 100 time units.

- Notice that the solutions obtained with Euler's method are reasonable approximations. In fact, at t = 100, the

approximate solution and the exact solutions only differ by one.

- To find when the population reaches 1,100, substitute 1,100 for P in the exact solution. Then solve for t.

- To solve for t after isolating the exponential expression that contains t, it is necessary to take the natural log of both sides of the equation.

- Solving for t reveals that 67 time units must pass before the population reaches 1,100.

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## Which of the following could be the logistic growth differential equation associated with the given direction field?

### dP/dt=0.8P(1−P/400)

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## Which of the following could be the logistic growth initial value problem associated with the given curve for the given direction field?

### dP/dt=0.8P(1−P/400), P(0)=600

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## Which of the following is the exact population P(10) rounded to the nearest unit, obtained from the given initial value problem?dP/dt=0.5P(1−P/500), P(0)=100

### P(10)≈487

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### dP/dt=0.5P(1−P/500)

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## Which of the following is an estimate of the population P(10) using Euler's method with a step size of 5, where P is the solution of the given initial value problem?dP/dt=0.5P(1−P/500), P(0)=100

### P(10)≈600

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## Use the exact solution to the given initial value problem to determine at which of the following times the population will reach 400.dP/dt=0.5P(1−P500), P(0)=100

### t=5.55

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## Which of the following is the exact population P(4) rounded to the nearest unit, obtained from the given initial value problem?dP/dt=0.8P(1−P400), P(0)=600

### P(4)≈406

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### dP/dt=0.5P(1−P/500), P(0)=100

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## Which of the following is an estimate of the population P(4) using Euler's method with a step size of 2, where P is the solution of the given initial value problem?dP/dt=0.8P(1−P400), P(0)=600

### P(4)≈254

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