Population Model Homework: Determine Equilibrium Solutions

[tylersmith7690]

Homework Statement

The simplest useful model for fisher comes from the logistic model for population growth, together with a harvest h which is proportional to the current population P, that is,

h=EP,

where the constant E is called the effort. E measures the fraction of the population harvested, so that 0 <= E <= 1. This gives the model

dP/dt = kP(1-(P/a) - EP,

where P(t) is the number of these fish at time t year and k (the natural growth rate) and a( the carrying capacity) are constants for a particular fish polulation. In what follows take k= 1 and a= 4, for simplicity.

a) determine the equilibrium solutions for a given effort E.

my attempt.

dP/dt = P - P^2/4 - EP

dP/dt = ( P + E) (P -1/4)

equilibrium solutions are when dP/dt=0

so P(0)= -E, 1/4.

 

[ehild]

Homework Statement

The simplest useful model for fisher comes from the logistic model for population growth, together with a harvest h which is proportional to the current population P, that is,

h=EP,

where the constant E is called the effort. E measures the fraction of the population harvested, so that 0 <= E <= 1. This gives the model

dP/dt = kP(1-(P/a) - EP,

You miss a parentheses: dP/dt = kP(1-(P/a)) - EP

where P(t) is the number of these fish at time t year and k (the natural growth rate) and a( the carrying capacity) are constants for a particular fish polulation. In what follows take k= 1 and a= 4, for simplicity.

a) determine the equilibrium solutions for a given effort E.

my attempt.

dP/dt = P - P^2/4 - EP

dP/dt = ( P + E) (P -1/4)

The last equation is wrong.

equilibrium solutions are when dP/dt=0

so P(0)= -E, 1/4.

Because of the faulty factorisation, the result is not correct (and physically impossible).

ehild