12.1 General Systems Theory

141

where r greater than 0r > 0 and upper K greater than 0K > 0 are constants. Retaining terms up to g 1g1 gives simple expo-

nential growth,

g 1 left parenthesis t right parenthesis equals g 1 left parenthesis 0 right parenthesis e Superscript r tg1(t) = g1(0)e^rt

(12.4)

where g 1 left parenthesis 0 right parenthesisg1(0) is the quantity of g 1g1 at t equals 0t = 0. Retaining terms up to g 1 squaredg²

1 ^gives

g 1 left parenthesis t right parenthesis equals StartFraction upper K Over 1 plus e Superscript minus r left parenthesis t minus m right parenthesis Baseline EndFraction commag1(t) =

K

1 + e^{−r(t−m) ,}

(12.5)

the so-called logistic equation, which is sigmoidal with a unique point of inflexion at

t equals m comma g 1 equals upper K divided by 2t = m, g1 = K/2 at which the tangent to the curve is rr, and asymptotes g 1 equals 0g1 = 0 and

g 1 equals upper Kg1 = K.rr is called the growth rate andupper KK is called the carrying capacity in ecology. ⁴

Consider now two objects,

StartLayout 1st Row d g 1 slash d t equals a 11 g 1 plus a 12 g 2 plus a 111 g 1 squared plus midline horizontal ellipsis 2nd Row d g 2 slash d t equals a 21 g 1 plus a 22 g 2 plus a 211 g 1 squared plus midline horizontal ellipsis EndLayout right brace^{dg1/dt = a11g1 + a12g2 + a111g2}

1 ^{+ · · ·}

dg2/dt = a21g1 + a22g2 + a211g²

1 ^{+ · · ·}

)

(12.6)

in which the functionsscript upper GG are now given explicitly in terms of their coefficientsaa (a 11a11,

for example, gives the time in which an isolated upper G 1G1 returns to equilibrium after a

perturbation). The solution is

StartLayout 1st Row g 1 left parenthesis t right parenthesis equals g 1 Superscript asterisk Baseline minus h 11 e Superscript lamda 1 t Baseline minus h 12 e Superscript lamda 2 t Baseline minus h 111 e Superscript 2 lamda 1 t Baseline minus midline horizontal ellipsis 2nd Row g 2 left parenthesis t right parenthesis equals g 2 Superscript asterisk Baseline minus h 21 e Superscript lamda 1 t Baseline minus h 22 e Superscript lamda 2 t Baseline minus h 211 e Superscript 2 lamda 1 t Baseline minus midline horizontal ellipsis EndLayout right brace^{g1(t) = g∗}

1 ^{−h11eλ1t −h12eλ2t −h111e2λ1t −· · ·}

g2(t) = g^∗

2 ^{−h21eλ1t −h22eλ2t −h211e2λ1t −· · ·}

)

(12.7)

where the starred quantities are the stationary values, obtained by setting d g 1 slash d t equals d g 2 slash d t equals 0dg1/dt =

dg2/dt = 0, and thelamdaλs are the roots of the characteristic equation, which is (ignoring

all but the first two terms of the right-hand side of Eq. 12.6)

StartEnclose left right StartLayout 1st Row 1st Column a 11 minus lamda 2nd Column a 12 2nd Row 1st Column a 21 2nd Column a 11 minus lamda EndLayout EndEnclose equals 0 period ^{a11 −λ}

a12

a21

a11 −λ ^{= 0 .}

(12.8)

Depending on the values of theaa coefficients, the phase diagram (i.e., a plot ofg 1g1 v.

g 2g2) will tend to a point (alllamdaλ are negative), or a limit cycle (thelamdaλ are imaginary, hence

there are periodic terms), or there is no stationary state (lamdaλ are positive). Regarding

the last case, it should be noted that however large the system, a single positive lamdaλ

will make one of the terms in (12.7) grow exponentially and hence rapidly dominate

all the other terms.

Although this approach can readily be generalized to any number of variables,

the equations can no longer be solved analytically and indeed the difficulties become

4 Unrelated to the previousupper KK (Sects. 6.1.3 and 11.5). We retain the same symbol here because of

“upper KK-selection” (Sect. 14.9.4), an expression well anchored in the literature of ecology. Yet another

unrelated use ofupper KK is in Kauffman’supper N upper KN K model (Sect. 12.2.3).