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dy(t) / dt = -k' y^{2}(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 = A_{1}y_{1} + A_{2}y_{2} 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)/y^{2}|_{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 y^{0}, to y^{1}, and y^{2}.
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 x_{m}, i.e., y = x/x_{m}, r refers to the rate of reproduction, and K denotes the the equilibrium size of population allowable by the environment, i.e., K = x_{eq}/x_{m}. 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)]} = r_{}dt ---------- (13) | |
Figure 06 Logistic Growth |
which yields: y/(1-y/K) = [y_{o}/(1-y_{o}/K)] e^{rt} or y = Ky_{o}e^{rt}/[K + y_{o}(e^{rt} - 1)] = y_{o} / {(y_{o}/K) + [1-(y_{o}/K)]e^{-rt}} ---------- (14) |
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). |
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 y_{n} term, and unity step size (see difference of the 2 versions in Figure 08, also compare to Figure 09,b). y_{n+1} = Ry_{n} (1 - y_{n})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, |
n = 1. The behavior of Eq.(15) is more complicated depending on the value of R as shown in Figure 09 (for y_{o} = 0.1, and 80 generations) and summarized below : |
Figure 10 Recursive Logistic, R=4 [view large image] |
See "Bifurcation" for further consequence, and y_{n} behavior dependent on R in "Logistic Map". |