Mathematical Minimum (for Aspiring Physicists)

Nonlinear Differential Equations

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

with solution in the form:

1/y - 1/y_o = k't     or     y = 1 / [k't+(1/y_o)] ---------- (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 = -k_o. Comparison shows that the feedback from y increasingly slows down the process as the contribution changed according to the power of y from y⁰, to y¹, and y².

The logistic equation for the growth of population is a little bit more complicated (or interesting) but is still solvable in closed form : dy/dt = ry(1 - y/K) ---------- (12) where y represents the ratio of existing population x to the maximum population xm, i.e., y = x/xm, r refers to the rate of reproduction, and K denotes the the equilibrium size of population allowable by the environment, i.e., K = xeq/xm. Following similar procedure for solving Eq.(30) 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)] = yo / {(yo/K) + [1-(yo/K)]e-rt} ---------- (14)

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

	Figure 07,a shows the application of the logistic equation to natural selection of two kinds of rose with different alleles (A and B) by imposing the condition A + B = 1 (for K = 1). In can be shown that the rate of reproduction has to be the reciprocal of each other. In this example, B will take over the entire population while A will become extinct eventually.
Figure 07 Logistics Equation, Examples [view large image]	Another variation involving change of r or/and K at certain point in time would create even more novel applications of the Logistic equation. For examples, a new medical technology or/and expansion of territory would enable additional increment of population size (see Figure 07,b and compare with Figure 06).

When the feedback is positive, i.e., dy/dt = ry(1 + y/K) and y = y_o / {-(y_o/K) + [1+(y_o/K)]e^-rt}, the growth is exponential approaching infinity as the time t (1/r)ln(1+K/y_o), it reaches the equilibrium size y = K earlier at t = (1/r)ln(1+K/y_o) - (1/r)ln(2).

Eq.(12) can be expressed in the discrete form :

y_n+1 = y_n + ry_n (1 - y_n)t

		Its solution is identical to the continuous version. However, the recurrence relation as shown in Eq.(15) yields entirely different solution with the absence of the yn term, and unity step size (see difference of the 2 versions in Figure 08, also compare to Figure 09,b). yn+1 = Ryn (1 - yn)n ---------- (15) where R denotes reproduction/generation, the t is replaced by increment of generation
Figure 08 Logistics Equation, 2 Version [view large image]	Figure 09 Logistic Equation, Recurrence Relation [view large image]	n = 1. The behavior of Eq.(15) is more complicated depending on the value of R as shown in Figure 09 (for yo = 0.1, and 80 generations) and summarized below :

• For 0 < R < 1 - The population will go extinct.
• For 1 < R < 3 - The equilibrium population size is given by y* = (R-1)/R (by setting y_n+1 = y_n in Eq.(15)). All trajectories starting from any initial y_o will converge to this value. The point y* is called global attractor.

For 3 < R < 4 - The equilibrium y* becomes unstable. For values of R slightly above 3, we find a stable oscillation of period 2. As R increases, the period 2 oscillator is replaced by period 4, then by 8, and so on. For R = 3.57 there are infinitely many even periods. For R = 3.6786 the first odd periods appear. For 3.82 < R < 4 all periods occur. For R = 4 - The dynamics becomes unpredictable in the sense that the trajectories starting from two slightly different initial points will diverge completely. This is called deterministic chaos, and is one of the characteristic of nonlinear equation(s). The solution for this case is : yn = sin2(2n), where = sin-1(y01/2). Comparison of Figures 10, 09f shows the divergence caused by minor difference in processing. For R > 4 - Values of y > 1 can be generated, and therefore negative y in the following year, thus the equation becomes unphysical (unbiological) beyond this point.

Figure 10 Recursive Logistic, R=4 [view large image]

See "Bifurcation" for further consequence, and yn behavior dependent on R in "Logistic Map".


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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