Oscillations

One of the last exercises in an introductory course to programming I teach is to implement a straight-forward approach for modeling population growth over discrete time steps with a logistic growth function: The population x of a species at time t+1 is determined as x(t+1) = r * x(t) * (1-x(t)) where x(t) is the population at time t, and r is a fixed reproduction parameter. The choice of r influences the long term behaviour of the resulting time series – thus, the growth of the species population; for example, for r < 1, the series tends towards zero – the species goes extinct. However, for r > 3 the series oscillates – it exhibits a periodic behaviour (for some values of r the series even becomes seemingly random without a fixed period, see e.g. here). The length of the period depends upon r, but it never reaches an equilibrium; like a pendulum, swinging around its only stable position in the middle. Like life pulsating between non-steady positions, but never reaching a balanced state.

Oscillations are present constantly. The term (1-x(t)) models the environmental restrictions that prohibit unlimited growth. Restrictions which prevent us to come to a rest. The fantasy of a steady state is a futile one. There are times where a stable position seems in reach; until external restraints pull us back into another direction. At the moment, it’s the direction of work; and hence, photography and blog posts are somewhat neglected. Winter already fades again, making way for summer. Left are only some solitary pictures of oscillating camera movements and colorless nature.