## 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.

Oscillation aside this is a wonderful collection of images! 🙂

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