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The differential equations for constant harvest and constant

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The differential equations for constant harvest and constant effort models for sustainable harvesting can both be written in a nicer factorised form as follows:

dP/dt = r/K (P - PU )(PS - P)

where P(t) is the population size, r is the intrinsic reproduction rate, K is the carrying capacity, PU is the unstable equilibrium and PS is the stable equilibrium. In this assignment you will derive a formula for the general solution and then use it to calculate the time required for a population to recover from a disaster.

(a) Use separation of variables and partial fractions to determine the general solution to the above differential equation. Leave your solution in implicit form, you do not need to make P the subject.

(b) Now assume that PU < P < PS and show that the general solution can be written in the form

a ln(P-PU /PS-P) = r/Kt + C where C is an arbitrary constant. Give an expression for the coffecient a.

(c) The constant harvest model has r = 0:3 per year, K = 3000 kilotons, PS = 1600 kilotons and PU = 1400 kilotons. A disaster has reduced the population close to the unstable equilibrium. Assuming P = 1410 when t = 0 calculate the time taken for the population to reach P = 1500.

(d) The constant effort model also has r = 0:3 per year, K = 3000 kilotons and PS = 1600 kilotons but PU = 0 kilotons. Assuming the same initial condition of P = 1410 when t = 0 calculate the time taken for the population to reach P = 1500 for this model.

The differential equations for constant harvest and constant effort models for sustainable harvesting can both be written in a nicer factorised form as follows:

dP/dt = r/K(P-PU )(PS - P)

where P(t) is the population size, r is the intrinsic reproduction rate, K is the carrying capacity, PU is the unstable equilibrium and PS is the stable equilibrium. In this assignment you will derive a formula for the general solution and then use it to calculate the time required for a population to recover from a disaster.

(a) Use separation of variables and partial fractions to determine the general solution to the above differential equation. Leave your solution in implicit form, you do not need to make P the subject.

(b) Now assume that PU < P < PS and show that the general solution can be written in the form

a ln(P-PU/PS-P) = r/Kt + C

where C is an arbitrary constant. Give an expression for the coefficient a.

(c) The constant harvest model has r = 0:3 per year, K = 3000 kilotons, PS = 1600 kilotons and PU = 1400 kilotons. A disaster has reduced the population close to the unstable equilibrium. Assuming P = 1410 when t = 0 calculate the time taken for the population to reach P = 1500.

(d) The constant effort model also has r = 0:3 per year, K = 3000 kilotons and PS = 1600 kilotons but PU = 0 kilotons. Assuming the same initial condition of P = 1410 when t = 0 calculate the time taken for the population to reach P = 1500 for this model.

Note: A fishing authority would want a plan that allows fast recovery of the fish population while allowing a viable fishing industry. If you have done the calculations correctly you will discover that the recovery time for one of the models is a few years and for the other is over a century.