Mathematical models in biology: an introduction by Elizabeth S. Allman, John A. Rhodes

By Elizabeth S. Allman, John A. Rhodes

Concentrating on discrete versions throughout various organic subdisciplines, this introductory textbook contains linear and non-linear versions of populations, Markov versions of molecular evolution, phylogenetic tree development from DNA series info, genetics, and infectious illness types. Assuming no wisdom of calculus, the improvement of mathematical issues, akin to matrix algebra and easy chance, is influenced through the organic versions. computing device learn with MATLAB is included all through in routines and extra broad tasks to supply readers with genuine event with the mathematical types.

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For certain fields, such as physiology (modeling such things as blood flow through the heart) and most of physics, where things really do seem to change continuously, they are still the natural choice. Difference equations are more appropriate in situations in which there are natural discrete time steps. An example would be in modeling insect populations, which tend to have rather rigid life histories, with well-defined development stages and life spans. Now that computers are readily available, difference equations can be studied through numerical experiments.

3) or xt+1 = Pxt . We’ve really gotten a bit ahead of ourselves here in our zeal to express the model in the simple form xt+1 = Pxt , which looks so much like the linear models we considered in the last chapter. What we have neglected to do is to make sure we know what we mean by writing Pxt , a matrix times a vector. We will define Pxt to be whatever is necessary, so that Eq. 3) means the same thing as Eq. 2). 0125Bt . 9875Bt This leads us to define multiplication by: Definition. The product of a 2 × 2 matrix and a vector in R2 is defined by ab cd x y = ax + by .

This system of three difference equations is a model of the insect population. Note that because the equations involve no terms more complicated than those that appear in the equation of a line, it is justifiable to refer to this as a linear model. Also note that, if we wish to use this model to predict future populations, we need three initial values, E 0 , L 0 , and A0 , one for each stage class. Because the three equations are coupled (the population of one developmental stage appears in the formula giving the future population of a different stage), this system of difference equations is slightly more complicated than the linear models in the last chapter.

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