Thomas Craig's A Treatise on Linear Differential Equations, Volume I: PDF

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5. 2 that it appears in a natural way as a discretization of well-established population models. The logistic equation. The Beverton–Holt models are best applied to semelparous insect populations but have been also used in fisheries models. 18) so that the per capita increase in the population is given by F (N ) = R0 S(N ). 14) due to the fact that in semelparous populations we have no adult survivals from one season to another. , by introducing a factor d < 1 in front of the first Nk ). As before, the function F can have different forms but should satisfy the following requirements: • Due to overcrowding, F (N ) must decrease as N increases until N equals the carrying capacity K; then F (K) = 0 and, as above, N = K stops changing.

The rates of change typically are expressed as derivatives and thus continuous time modelling leads to differential equations which involve the derivatives of the function describing the state of the system. In what follows we shall derive several differential equation models trying to provide continuous counterparts of some discrete systems described above. 5) and compare the results. 1 Continuously compounded interest and loan repayment Many banks advertise continuous compounding of interest. 1 tends to zero so that the interest is added to the account on a continual basis.

1 Compound interest Compound interest is relevant to loans or deposits made over long periods. The interest is added to the initial sum at regular intervals, called the conversion periods, and the new amount, rather than the initial one, is used for calculating the interest for the next conversion period. The fraction of a year occupied by the conversion period is denoted by α. Thus the conversion period of 1 month is given by α = 1/12. Instead of saying that the conversion period is 1 month we also say that the interest is compounded monthly.

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A Treatise on Linear Differential Equations, Volume I: Equations with Uniform Coefficients by Thomas Craig

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