QUESTION 1. Consider a population that grows according to the logistic updating

Question

QUESTION 1. Consider a population that grows according to the logistic updating function and is harvested at a linear rate h > = 0. The number of individuals of the species satisfies the DTDS Xt+1= xt(4-xt)-hxt. (a) Determine all equilibria of the DTDS. (b) Determine conditions on h such that all equilibria are biologically relevant. (e) Determine conditions on h such that the positive equilibrium is stable. (d) Determine conditions on h such that the positive equilibrium is unstable. (e) Determine the value of h that maximizes the yield and state the resulting maximum yield.

Explanation / Answer

AT THE OUT SET WE SEE THAT X(T) SHALL BE AN INTEGER > = 0 . X(T+1)=[X(T)][4-H]-[X(T)]^2……………………..1 THIS WILL BE POSITIVE ONLY IF 4-H > X(T) WHAT IS THE INITIAL VALUE OF X(T) AT T= 0 ? SINCE H > = 0 , FOR 4-H > = X(T) WE SHALL HAVE TO RESTRICT X(0) DEPENDING ON H OBVIOUSLY 0 < = X(0)

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