Fig 1.
Representing technology as a converter of material resources into product.
The set of resources R1, …, Rm, which can be raw materials, semiproducts, parts, or other commodities, are jointly transformed into a product P under the “catalytic” action of the set of funds ϕ1, …, ϕm′. The arrow labels stand for respective flows.
Fig 2.
Diffusion of an isolated technology feeding on an exhaustible limiting resource.
The nondimensional variables are used (ref. (20)). Solid S-curves depict time profiles of population of firms, dashed lines represent kinetics of resource. The position of an inflexion point (indicated by a circle) on concrete S-curve is determined by the value of the parameter ϰ of the model. In particular, ϰ = 1 yields the logistic growth law.
Fig 3.
Diffusion curves, generated by model (21), in the plane .
The value of y, which delivers zero to , corresponds to the inflection point (indicated by dashed line) in the domain of admissible values. The curves are right-skewed for ϰ > 1, and left-skewed for ϰ < 1. The case ϰ = 1 features a logistic growth curve.
Fig 4.
Apple’s iPod penetration chart (2002–2014) in the context of the model (21).
A raw time series, , representing quarterly sales of iPod (in million units) over a period covering 49 consecutive quarters can be found on website [73]. It is the following set of figures: {0.125, 0.057, 0.054, 0.14, 0.219, 0.078, 0.304, 0.336, 0.733, 0.807, 0.86, 2.016, 4.58, 5.311, 6.155, 6.451, 14, 8.526 8.111, 8.729, 21, 10.549, 9.815, 10.2, 22.121, 10.644, 11.011, 11.022, 22.722, 11.01, 10.2, 10.2, 21, 10.89, 9.41, 9.051, 19.45, 9.02, 7.535, 6.62, 15.397, 7.673, 4.02, 5.344, 12.679, 5.633, 4.569, 3.498, 6.049}. This series first was subject to smoothing by taking the moving average over a four-quarter sliding window, and then fitted by the Eq (21).
Fig 5.
Phase portrait of the system (19).
The unpopulated steady state F1, given by (23a), is a saddle point, while the positive equilibrium, F2, given by (23b), is a stable node. The steady-states lie on the intersections of the nullclines. The vertical nullcline is the hyperbola N = R(r − dR)/((dK + γq)R − Kr) with the asymptote R = Kr/(dK + γq) (shown by the dotted line). The horizontal nullcline
consists of two lines, N = 0 and N = R(γ − D)/(DK).
Fig 6.
Bifurcation diagram for system (19).
The solid lines depict stable behavior and the dotted lines depict unstable behavior. For D < γ, there is an unstable fixed point F1 (ref. Eq (23a)) and a stable fixed point F2 (ref. Eq (23b)). As D increases toward γ, fixed point F2 approaches fixed point F1, and coalesces with it when D = γ. Finally, when D > γ, fixed point F2 has become unstable, and fixed point F1 is now stable. There is an exchange of stabilities between the two fixed points.
Fig 7.
A hypothetical water-limited development of grassed acreage simulated with model (19).
The total sown area tends via an S-curve to a steady-state size 148.4 ha. The initial conditions are R(0) = 0 and N(0) = 1 ha; water supply velocity is chosen to be r = 500 ton/yr. The other parameters of the model are given in the text.
Fig 8.
Growth dynamics of five technologies competing for one resource as predicted by Eq (28).
Parameters of simulation (arb. units): K = (1.02 0.9 1.09 1.17 0.91)⊤, γ = (0.95 1.1 0.81 0.88 0.98)⊤, q = (1.04 0.83 0.97 0.84 1.2)⊤, D = (0.62 0.5 0.61 0.54 0.53)⊤, r = 2, d = 0.2, R(0) = 0, and N(0) = (0.1 0.1 0.1 0.1 0.1)⊤. Because happens to be the smallest, technology 2 competitively displaces all other rivals by reducing the resource to a level at which they cannot maintain their business.