Sustainability 2011 , 3 2350
The initial boundary conditions (assuming steady state) arePnp(t 0 ) =P 0 (12)C(t 0 ) =λ(1−h)ψβ
q
P 0.The most positive eigenvalue of the state transition matrix,A, sets the upper bound on the rate of
energy supply growth attainable for a given reinvestment factor,β. The eigenvalues,α, ofAare solutions
of the quadratic equation:
det|A−αI|= 0 (13)det∣∣
∣∣
∣∣
∣∣
∣−
1
T
−α
1
λ
(1−h)ψβ
q
−
1
λ
−α∣∣
∣∣
∣∣
∣∣
∣= 0or
α^2 +α(
1
λ
+
1
T
) +
1
λT
−
1
λ(1−h)ψβ
q
= 0 (14)This equation can be solved for the eigenvalues,α, using the quadratic formula:α=−(^1
λ
+^1
T
)±√
(^1
λ
+^1
T
)^2 +^4
λT[
(1−h)ψTβ
q
− 1]2
(15)There are two eigenvalues. By inspection, at least one is positive if
(1−h)ψTβ
q >^1. Calling the
most positive eigenvalueα∗, the persisting solution of the state equations is simple exponential growth
at an annual rate ofα∗, starting from the steady state initial condition:
Pnp(t) =P 0 eα∗t (16)C(t) =λ
(1−h)ψβ
q P^0 eα∗tThe doubling time figure of meritτ 2 – applicable to growing infrastructures under an energy plowback
constrained is defined as
τ 2 =
ln 2
α∗ (17)