Any help on this would be appreciated!! Even if its just one of the parts of the question or any help at all! (ps. please give as much explanation as possible!)

**If P(t) denotes a fish population at time t, then the model to be considered is** **where r, K, A, P are positive constants. This model is more realistic in two aspects: it has a fixed point at P=0 for all values of the parameters, and the rate at which fish are caight decreases with P. This is plausible- when fewer fish are available, it is harder to find them so the daily catch drops.** **(A) **Compare (by plotting) the harvesting rate in this model with the constant harvesting rate H and linear rate HP(t). Give a biological interpretation of the parameter A; what could it measure?

(**B**) In order to analyze this differential equation, first we reduce the number of parameters, by using the following dimensionless variables:

By applying the chain rule

, show that the system can be rewritten in dimensionless form as

. In this way, the new equivalent system has only two parameters. From now on, we are going to work with the dimensionless system.

**(C) **Assume h = 0.2 . Analyze the number of fixed points and their stability properties as parameter a varies. Explain the long term behavior of the population in each case. Plot in the

*ax* plane the fixed points vs.

*a *(bifurcation diagram). Can you pinpoint some values of

*a *where important changes in the behavior of the system happen? Such values of

*a *are called bifurcation points. What type of bifurcations do you notice (pitchfork, saddle-node, or transcritical)? Give an interpretation of this situation.

**(D)** Assume now that a = 0.2 . Analyze the number of fixed points and their stability properties as parameter

*h *is varied. Explain the long term behavior of the population in each case. Plot in the

*hx* plane the fixed points vs.

*h *(bifurcation diagram). Can you pinpoint some values of

*h *where important changes in the behavior of the system happen? What type of bifurcations do you notice? Give an interpretation of the situation.