An island supports a population of finches. The birth rate for these finches is

Question

An island supports a population of finches. The birth rate for these finches is .69 births per finch per year; the death rate is .49 deaths per finch per year. Each year 12 finches fly to this island from nearby islands and 8 flinches fly off to neighboring islands.

(a) Write down the differential equation governing this population.

(b) If there were 500 finches initially, what is the population after 20 years?

4. A national park contains a population of 5,000 bears.

(a) Left alone, the population would grow exponentially, doubling in twenty years. Find the growth rate k.

(b) To control the popuation, a constant number of bears are removed each year, relocated to other parks and zoos. How many should be removed each year to keep the population constant?

5. An open tank is completely filled with 10,000 L of water with a concentration of 3g/L of salt. Water containing a concentration of 2g/Lof salt is pumped into the tank at a rate of 20L/hour; (well-mixed) water flows out over the sides at the same rate.

(a) What is the differential equation governing the process?

(b) What is the amount of salt present in the tank at time t?

6. In 1995, a wildlife preserve had a population of 50 tortoises. This population satisfies the differ ential equation:

dy/dt= 0.1y (1- y/500)

where y is the population after t years.

(a) What is the carrying capacity of the island?

(b) Determine y(t).

(c) How long does it take for the tortoise poplulation to reach 200?

7. An infectious disease has hit a town of 10,000. The disease is spreading logistically. There were four people initially infected; after 4 weeks, there were a hundred cases. How many weeks will it take for half the town to become infected?

8. 500 smallmouth bass are introduced into a lake capable of supporting a population of 25,000. Four years later, the population has reached 5,000. When does the population reach 10,000? 20,000?

9. Suppose a population of y0 experiences a faster-than-exponential growth satisfying

dy/dt=ky^1+a

where k, a are constants and a>0

(a) Determine the population at time t.

(b) At what time does the population explosion occur?

Explanation / Answer

(a)Rate of change of population =mumber of birth+Number of Immigration-Number of Death -Number of Emigration

dP/dt=.69P+12-.49P-8

dP/dt = 0.2P + 4

general solution is

Pe^-0.2t=-4e^-0.2t/0.2 + C,C is constant

Pe^-0.2t=-20e^-0.2t +C

P=-20+Ce^0.2t,

(b)At time t=0,P=500

so,500=-20+C

implies,C=520

Hence,P=-20+520e^0.2t

After 20 years,

P=-20+520e^4

(4)P(t)=Ae^kt

where A is the initial population

fiven,A=5000

and for t=20,P(t)=10000

so,2=e^20k

implies,20k=log2

implies,k=log2/20

(b) growth in first year=Ae^k -A

To keep the population constant ,

population to be removed in first year=Ae^k - A

Same population has to be removed each year.

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