URGENT Second order difference equations and general solution

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March 17th, 2023

Im studying and need help with a Mathematics question to help me learn.

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This is a question on my homework and I keep getting the wrong answers. Please help me with detailed answers to all four parts.

Consider an annual plant which germinates in the spring, blooms in the summer and produces S seeds in the autumn.

A fraction f of these seeds germinate the next spring, ?ower and produce more seeds. Of those that do not germinate, a fraction g germinate the second spring, ?ower and produce more seeds. There are no seeds that survive more than two winters.

(a) Let Pn be the plant population in year n. Write a second order linear di?erence equation satis?ed by Pn. Explain the role of various terms occurring in the equation.

(b) Find the general solution of this equation.

(c) Suppose that in the ?rst summer you have 80 plants and no extra seeds. Determine the solution Pn after n years as a function of S, f and g.

(d) Let f = 0.014, g = 0.006. Discuss the limiting behavior of the general solution as a function of S. How large should S be to ensure that the plant population eventually increases in size?

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