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, flower and produce more seeds. Of those that do not germinate, a fraction g germinate the second spring, flower 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 difference equation satisï¬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 ï¬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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