Question

Q2- Fish Population In this question, we will use differential equations to study the fish population in a certain lake. An acceptable model for fish population change should take into account the birth rate, death rate, as well as harvesting rate Let P(t) denote the living fish population (measured in tonnes) at time t (measured in year) Then the net rate of change of the fish population in tonnes of fish per year is P'(t): P'(t) birth rate - death rate - harvesting rate where we measure all the rates in tonnes per year Close observation of many species over many years suggests that birth rate is proportional to the size of the population, whereas the death rate is affected by natural mortality as well as overcrowding factorWe have: birth rate at time t = bP(t) death rate at time t = mP(t)+cP2(t) where b, m, andc are nonnegative proportionality constants. Now let's pull all of these bits and pieces together and create a model. Suppose b 1.55, m 0.35 (a) First, suppose c 0.1 and harvesting rate is 3.5 (tonnes per year) (i) Write a differential equation for P(t). (ii) Is the differential equation in (i) separable? Autonomous? Pure-time? Explain your answer. (b) Now, ignore overcrowding factor (suppose c = 0) and assume harvesting has been set to 0.3 (tonnes per year) (i) Suppose P(0) = 0.25. What is P(t)? Why? (ii) suppose P(0) 5 and write an initial value problem for P(t) and solve it using the technique of separating variables.

Answer 1

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