Original Research Article Spatio-temporal dynamics and quantification of daisyworld in two-dimensional coupled map lattices Dharani Punithan, Dong-Kyun Kim, RI (Bob) McKay * Structural Complexity Laboratory, Department of Computer Science and Engineering, Seoul National University, Seoul, South Korea 1. Introduction Watson and Lovelock (1983) proposed ‘daisyworld’ as a demonstration of homeostasis by and for the biosphere, and a mathematical underpinning of the Gaia hypothesis (Lovelock, 1972). The original model was based on a zero-dimensional world. It was extended to a one-dimensional (1D) world by Adams et al. (2003), to a two-dimensional (2D) world by von Bloh et al. (1997) and to a curved 2D world by Ackland et al. (2003). For further reading, see Wood et al. (2008). A wide range of fascinating examples have been observed in nature, portraying the interaction between life and its environ- ment. Examples include the formation of lakes, dams and canals by those so-called ‘ecosystem engineers’, beavers; chemical changes in soil by earthworms; cloud formation by marine phytoplankton; carbon sinks in terrestrial plants via photosynthesis and in the oceans via physicochemical and biological processes; stromatolite formation by microbial mats, etc. Daisyworld emphasises this fundamental interaction between the biota and the abiotic environment. However the daisyworld model, an example of ecological homeostasis, has received only limited attention in mainstream ecology (see Section 2.1.1), and most daisyworld models have been published in non-ecological journals (Wilk- inson, 2003). Nevertheless, the daisyworld concept integrates many aspects of ecological studies such as physiological ecology, population ecology, community ecology, evolutionary ecology, spatial ecology and landscape ecology. Hence, there is much scope for ecological studies in daisyworld, and for researchers proposing alternative daisyworld models. Daisyworld models in the literature lack detailed population dynamics, owing to their primary focus on environmental regulation. But population dynamics, with density dependent factors such as resource limitation and influence, crowding, dispersion, competition, disease, predation, etc., may lead to homeostasis at an ecosystem level. The regulation due to density dependence is a well known emergent ecological phenomenon. For example, Maddock (1991) studied population models in 1D daisyworlds, unfortunately receiving limited attention. Spatially extended systems are often modelled by partial differential equations (PDE), ordinary differential equations (ODE), individual based models (IBM), cellular automata (CA) and coupled map lattices (CML). Most 2D spatial extensions of daisyworld in the literature are based on CA (von Bloh et al., 1997; Lenton and Van Oijen, 2002; Ackland et al., 2003; Ackland and Wood, 2010). In these CA based models, the new state of each cell is determined by applying one of the predetermined set of rules to the previous state of that cell and a randomly chosen cell in the neighbourhood. The discrete set of state values represents the presence or absence of a species at a cell. In almost any real ecosystem, the population dynamics include both temporal reproduction processes and spatial diffusion processes. Hence we propose the use of CML as convenient for Ecological Complexity xxx (2012) xxx–xxx A R T I C L E I N F O Article history: Received 16 May 2012 Received in revised form 26 August 2012 Accepted 16 September 2012 Available online xxx Keywords: Daisyworld Coupled map lattice (CML) Logistic(Verhulst) growth model Laplacian diffusion Permutation entropy Moran’s I A B S T R A C T We spatially extend the daisyworld model on a two-dimensional toroidal coupled map lattice (CML – a generalisation of cellular automata). We investigated whether this tightly coupled system of local nonlinear dynamics with bi-directional life-environment feedback can generate a specific kind of behaviour, characterised by global stability coexisting with local instability. We introduce appropriate metrics to measure the spatio-temporal dynamics of the daisyworld system. Specifically, we evaluate spatial autocorrelation using Moran’s I, and local and global temporal fluctuation through the permutation entropy and the temporal standard deviation. We categorise a range of different behaviours that can arise in such scenarios, and relate them through a parameter analysis. ß 2012 Elsevier B.V. All rights reserved. * Corresponding author. E-mail address: rimsnucse@gmail.com (R.(. McKay). G Model ECOCOM-364; No. of Pages 15 Please cite this article in press as: Punithan, D., et al., Spatio-temporal dynamics and quantification of daisyworld in two-dimensional coupled map lattices. Ecol. Complex. (2012), http://dx.doi.org/10.1016/j.ecocom.2012.09.004 Contents lists available at SciVerse ScienceDirect Ecological Complexity jo ur n al ho mep ag e: www .elsevier .c om /lo cate/ec o co m 1476-945X/$ – see front matter ß 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ecocom.2012.09.004