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The application of mathematical equations in modeling is an excellent example of the applicaiton of Ockham's razor. Thus I present several appropriate functions or equations that can be applied in mathematical modeling. The order is from simplest to most complicated in format and in the curve represented.
1. Linear
y = mx + b
¥where m is the slope and b is the y intercept
2. Allometric
y = bx^a
¥where b is a constant and a is a parameter
¥at best, this is a simplified approximation, but is the most commonly used equation in modeling
3. VonBertalanffy's expansion of the allometric equation
du/dt = nu^a - ku
¥where u = biomass, n and k are rate constants, ^ means raised to the power of, a is a parameter, nu^a is the anabolic process, and ku is the catabolic process
¥VonB's equation in words says that the growth rate of a living system is equal to synthesis minus degradation
4. Exponential growth or decay
Q = Qo e^+/-rt
¥where Q = quantity of material, o represents original value, e = base of the naperian log system, ^ means raised to the power of, r = rate constant, and t = time
¥this is the equation (when the sign is negative) that describes the decay of radioactive particles
¥this is the equation (when the sign is positive) that describes the accumulation of money in a bank account when the interest is continuously compounded
5. Power series
y = ao + a1X + a2X^2 + a3X^3 + ... anX^n
¥where ao, a1, a2 ... an are sequential constants specific to the term and ^2...^n are integer exponents
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