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Mathematical Minimum (for Aspiring Physicists)


Nonlinear Differential Equations

In a nonlinear differential equation such as:

dy(t) / dt = -k' y2(t) ---------- (10)

with solution in the form:

1/y - 1/yo = k't     or     y = 1 / [k't+(1/yo)] ---------- (11)

(see formula for differentiation of function with power n in footnote), it is obvious that the combined solution y = A1y1 + A2y2 is not a solution of the same differential equation any more. The blue line in Figure 02 is the decay curve described by Eq.(10) with k' = -(dy/dt)/y2|t=0 ~ 0.07/day-gm. The red line represents the case with a constant decay rate, i.e., dy/dt = -ko. Comparison shows that the negative feedback from y increasingly slows down the process as the contribution changed from y0, to y1, and y2.
Logistic Growth The logistic equation for the growth of population is a little bit more complicated but is still solvable in closed form:

dy/dt = ry(1 - y/K) ---------- (12)

where y is the population size, r refers to the rate of reproduction, and K denotes the carrying capacity (the maximum size of population allowable by the environment). Following similar procedure for solving Eq.(2) in footnote §, we can derive the integration in the form:

dy/y(1-y/K) = d{ln[y/(1-y/K)]} = rdt ---------- (13)

Figure 06 Logistic Growth
[view large image]

which yields: y/(1-y/K) = [yo/(1-yo/K)] ert   or    y = Kyoert/[K + yo(ert - 1)] ---------- (14)

In the limit t , the population size converges to the equilibrium x = K; while for the case K (unlimited population size), it reverts back to exponential growth as opposed to the exponential decay in Eq.(2) (see Figure 06). This is an example to demonstrate clearly the effect of negative feedback in controlling runaway process.

If we redefine the meaning of the parameter such that r = (a - b) and re-label the variable y with y + x = 1, Eq.(12) can be used to describe the process of natural selection, where a and b are the reproductive rates for species A and B, y and x are the corresponding population numbers for K = 1. Without further ado, it is clear that if a > b, then A will become more abundant than B. Eventually A will take over the entire population; B will become extinct (and vice versa).

Logistic Equation Eq.(12) can also be recast to the form of difference equation with discrete steps, e.g., using the generation number n as running variable with n = 1 instead of the time t with t0:

yn+1 = Ryn (1 - yn) ---------- (15)

where R = 1 + r, and K = 1, which rescales the population size y to between 0 and 1. Thus,

Figure 07 Logistic Equation
[view large image]

R < 1 means r < 0, which implies decay instead of growth. The behavior of Eq.(15) is more complicated depending on the value of R as shown in Figure 07 and summarized below (for yo = 0.1, and 80 generations):

As shown in Figures 05 and 06, the behavior of the logistic differential and difference equations is drastically at odds with each other. The crucial difference is the absence of the k parameter, which can keep minimize the negative feed back. Nonlinear differential equations appear in "General Relativity", "Fluid Dynamics", ..., and the Lorenz systems (in chaos theory), which is described by a set of coupled differential equations:

dx/dt = (y - x),    dy/dt = rx - y -xz,    dz/dt = xy - bz ---------- (16)

where t is time (the independent variable), x, y, z are the dependent variables, and , r, b are the parameters. A characteristic of nonlinear differential equation(s) is the mixing of many functions (dependent variables) or the function acting on itself. While it seems to be more realistic, the solution for such holistic approach is more difficult to find. Sometimes the nonlinear equations can be linearized because some variables (or their derivatives) are small in comparison with the others. See Chaos Theory for more about nonlinear equations.

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