# Population growth model

In some populations, the amount of births is directly proportional to the population at any given point in time and the amount of deaths is directly proportional to the square of the population at any given point in time.

1. Write an equation that models the change in a population that fits the above description. Make sure to carefully label what each variable represents.

2. Your equation from (1) should be separable. Separate the equation by hand but use your TI calculator to perform the appropriate integration (don't forget the constant of integration). Then use the solve function to solve this equation for the population.

3. Use the deSolve function to solve your original equation from (1). Is the answer different from the one you find in (2)? If so, how?

4. Assume the population at time zero is N. Use your calculator to plug in and give the population model in terms of N (that is, make sure your equation doesn't have any undefined constants).

Use the models/equations you developed in questions #1 and #4 to answer question #5.

5. Consider a population of goblins. Goblins are rather virile and so breed at the prodigious rate of 1 goblin per 10 goblins per month. Fortunately, goblins are also quite adventurous (not to mention stupid). So, they migrate out of a population or die at a rate of 1 goblin per 1200 goblins2 per month.

a) Graph the direction field that models the population of the goblins. What is the equilibrium point?

b) After two years, how many goblins will there be in a population that begins with 50?

#### Solution Preview

Please see the attached file.

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#### Solution Summary

This solution shows how to solve the differential equation arises from population growth model, and discusses the effects of the different parameters on the population.