Controlling chaos (OGY) implemented on a reconstructed ecological two-dimensional map

We numerically demonstrate a way to stabilize an unstable equilibrium in the ecological dynamics reconstructed from real-world time series data, namely, alternate bearing of citrus trees. The reconstruction of deterministic dynamics from short and noisy ecological time series has been a crucial issue since May’s historical work [May RM. Biological populations with nonoverlapping generations: stable points, stable cycles and chaos. Science 1974;186:645–7; Hassell MP, Lawton JH, May RM. Patterns of dynamical behavior in single species populations. J Anim Ecol 1976;45:471–86]. Response surface methodology, followed by the differential equation approach is recognized as a promising method of reconstruction [Turchin P. Rarity of density dependence or population with lags? Nature 1990;344:660–3; Turchin P, Taylor AD. Complex dynamics in ecological time series. Ecology 1992;73:289–305; Ellner S, Turchin P. Chaos in a noisy world: new method and evidence from time series analysis. Am Nat 1995;145(3):343–75; Turchin P, Ellner S. Living on the edge of chaos: population dynamics of fennoscandian voles. Ecology 2000;8(11):3116]. Here, the reconstructed ecological dynamics was described by a two-dimensional map derived from the response surface created by the data. The response surface created was experimentally validated in four one-year forward predictions in 2001, 2002, 2003 and 2004. Controlling chaos is very important when applying chaos theory to solving real-world problems. The OGY method is the first and most popular methodology for controlling chaos and can be used as an algorithm to stabilize an unstable fixed point by putting the state on a stable manifold [Ott E, Grebogi C, York JA. Controlling chaos. Phys Rev Lett 1990;64:1996–9]. We applied the OGY method to our reconstructed two-dimensional map and as a result were able to control alternate bearing in numerical simulations.