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intro - Biomathematics What is it and how does it work...

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Unformatted text preview: Biomathematics: What is it and how does it work? Richard Bertram Department of Mathematics and Programs in Neuroscience and Molecular Biophysics Florida State University Tallahassee, Florida 32306 1 2 How can math be used in biology? Population Biology An early model for population growth was developed by Thomas Malthus , who lived from 1766 to 1834. This is based on the observation that a species can increase in numbers according to a geometric series: N 1 = N R (1) N 2 = N 1 R (2) = N R 2 (3) N 3 = N R 3 (4) . . . (5) N j = N R j (6) This is a linear discrete dynamical system ( N j = RN j- 1 ) that is simple to solve. It also yields a testable prediction , that N j → , if R < 1 ∞ , if R > 1 (7) 3 Pierre Verhulst (1838) developed a model that better re ects the biology. His logistic model assumes that the growth rate R declines as the population increases: R = R 1- N κ (8) Then, N 1 = N R = N R 1- N κ (9) or N 1 = RN- R κ N 2 (10) and in general N j = RN j- 1- R κ N 2 j- 1 . (11) For this nonlinear difference equation the asymptotic behavior is bounded when R > 1: N j → κ ( R- 1) R (12) Moral of the story : (A) Linear models can provide insight and can typically be solved analytically. However, the mostand can typically be solved analytically....
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intro - Biomathematics What is it and how does it work...

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