Use the mathematical models discussed in this section to solve the given problem. In each case, assume that the conditions under which the models are constructed are satisfied.

1. The birth and death rates of two non-competing populations are 1 = 1/(t + 1), 1 = 1/10 and 2 = 2/(t + 2), 2 = 1/10, respectively. If their starting sizes are P₁₀ = 2 and P₂₀ = 1 units, find out how long it takes the second population to (i) become larger than the first one, and (ii) be twice as large as the first population.

2. A bank pays an annual rate of interest of 5%, and an annual fixed loyalty bonus of 1/100 of a money unit. Modify the compound interest model accordingly, then compute the value of one money unit after 5 years. Also, find out what interest rate would need to be paid by the bank to generate the same amount at that time if there was no loyalty bonus.

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