Consider the following population model: dP = .01P (P - 50)(P - 20) dt a) What a

Question

Consider the following population model: dP = .01P (P - 50)(P - 20) dt

a) What are the equilibrium solutions?

b) What is the rate of population change at the initial populations of P(0) = 1,P(0) =
30, P (0) = 60 ?

c) What happens to the population with an initial population P0 where 20 < P0 < 50 as
time increases?

d) Consider an initial population of 30. Will this population ever be exactly 20? Why or
why not?

e) Use your answers from a-d to make a rough sketch of the slope field for 0 < t < 10 and
0 < P < 60

f) Use the dfield software to confirm your answer in e). See http://math.rice.edu/ ~dfield/dfpp.html

Explanation / Answer

a) Equilibrium means that dP=0 so .01P(P-50)(P-20)=0. Solving this equation gives you three solutions for equilibrium: 0, 50 and 20. (If the population has any of these values, dP will be 0 so P will not change.) b) Here you just need to plug each value of P into the equation for dP. For example, for P=1, you get dP=.01(1-50)(1-20)=.01*49*19. c) The population has to evolve continuously: it cannot jump from 21 to 19; it must pass through 20. But, if the population ever reaches 20 or 50, it reaches equilibrium and all changes stop. Thus, all changes are confined to the interval (20,50) - when one of the bounds is reached, dynamics stop. In order to see if a population 20

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