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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 negative feedback from y increasingly slows down the process as the contribution changed from y^{0}, to y^{1}, and y^{2}.

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)]} = 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)] ---------- (14) |

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).

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: y _{n+1} = Ry_{n} (1 - y_{n}) ---------- (15)where R = 1 + r, and K = 1, which rescales the population size y to between 0 and 1. Thus, | |

## Figure 07 Logistic Equation |
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 y_{o} = 0.1, and 80 generations): |

- 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).
- 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.

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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