Consider a population that grows according to the Beverton-Holt updating function and is harvested at a linear rateh0. The number of individuals per square kilometer of the species satisfies the DTDS

xt+1=11xt1+xthxt

fort=0,1,2,...

a)Give the updating function for this DTDS. Your answer will include the parameterh.

Answer:f(x)=

b)Find the fixed points of this DTDS. Separate each value with asemi-colon.

Note that one point should depend on the harvesting rateh.

Answer:

c)For which values ofhis there a positive fixed point?

Answer

d)The yield of the harvest at equilibrium is the product of the harvesting rate and the equilibrium population. Write the formula for the yield of the positive equilibrium point.

Answer:Y(h)=

e)Compute the derivative ofY(h).

Answer:Y(h)=

f)Which harvesting rateh=Hmaximizes the number of individuals harvested at the positive equilibrium point? Justify your answer.

Give an exact answer, or else an approximation valid to at least three decimal places.

Answer:H=

-because (select one) Y'(h)>0 for 0OR Y'(h)<0 for 0

-(select one) or/and.

-(select one) Y'(h)<0 for h>H OR Y'(h)>0 for h>H

g)Compute the derivative of the updating function at the positive equilibrium pointpthat you have found in (b). Your answer will be in terms of the parameterh.

Answer: general formula forf(p)=

h)Now computef(p)assuming thath=H, the value you obtained in (f). Your answer should be either exact, or accurate to two decimal places.

Answer:f(p)=

i)The positive fixed pointpwith the valueh=Hobtained in (f) is (choose one):

		unstable
		stable

because (choose one):

		|f(p)|>1
		f(p)=0
		f(p)<0
		|f(p)|=1
		|f(p)|<1
		f(p)>0