The answer to 1.2 a) is y\'=ky(a-y)-b Using this please describe the real world

Question

The answer to 1.2 a) is y'=ky(a-y)-b

Using this please describe the real world meaning of the terms in this equation and solve a) to g)

Thanks!

One of the differential equations in problem 1.1 de- scribes the growth of a population with limited resources and harvesting, such as the population of fish in a lake that is being fished by humans 1.2. (a) Which one? Explain briefly the real-world meaning of the terms in the equation (b) Consider the case where k-0.01, a 1000, b- 900 Find the equilibrium values of y. What do they rep- resent in real-world terms? (c) Sketch a direction field for this differential equa- tion (d) There is a qualitative difference between the two equilibria-what is it? Hint: What happens if you increase harvesting temporarily for one week? (e) * One of the equilibrium values is lower than b Explain why this might at first seem paradoxical in view of the real-world meaning of these val ues. Nevertheless the model makes sense even with these values for the constants. How can this be? (f) Other constants remaining as above, what is the highest harvesting rate that can be maintained without eventually depleting the fish population (assuming that the initial population is large enough)? (g) Draw the direction fields for this rate of fishing, and for a higher rate of fishing. Explain what these pic- tures show

Explanation / Answer

a.The given differential equation is y'=ky(a-y)-b

The real world meaning of this equation is that this equation shows that the rate at which the fish population in the lake is being harvested by human beings i,e, the relation between the rate at which fish is being produced in the lake and its rate of cosumption by human beings is taking place.

b. Equilibrium values means y'=0 that is the rate at which fishes are being produced in the lake is equal to the rate at which it is being consumed by human beings .

So equation becomes

ky(a-y)-b=0

given values of k,a and b we can solve quadratic equation to find value of y which is equal to

y=900 or y= 100 which are equilibrium values

d. The quallitative difference between two equlibrium values is that if for one week harvesting is increased(y=900) then its consumption by human will become more until it reaches to lower equlibrium value (y=100)and vice versa

e.

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