5 Discussion: Towards a dynamic landscape theory

5.1 Universality behind case studies

On the basis of the landscape and model characteristics summarized in the previous sections, we suggest that there exists a profound unity between landscape-modelling case studies. Patchy structures, network roles and multiscale behaviours justify searching for a unified view of dynamic landscapes. The landscape concept is generic and the heterogeneity concept is universal. The central paradigm of landscape ecology is that the spatial structure of a landscape has an effect on the underlying ecological processes (Forman and Godron, 1986; Turner and Gardner, 1991). Some properties of the landscape such as heterogeneity, connectivity or fragmentation strongly influence the exchange and flow of organisms, matter and energy between the different components. From the start, deciphering the structural and the functional aspects of connectivity has been a challenge to landscape ecologists (Tischendorf and Fahrig, 2000) and discussions are still on-going. Moreover, this branch of ecology insists on the importance of scales and integration of processes (Dungan et al., 2002).

Multiscale and network concepts, for example, are widespread and have recently appeared in the literature for a wide range of specific landscape studies (Caillault et al., 2013; Gaucherel, 2008*; Thenail and Baudry, 2004). They offer opportunities for renewing the usual landscape ecology concepts (e.g., heterogeneity) that are no longer sufficient to understand complex and dynamic systems. Lambin mentioned that “the most fundamental obstacle to progress in the understanding and prediction of human impacts on terrestrial ecosystems lies in the lack of comprehensive theory and land-use changes” (Lambin, 1997). This remark, made for human-driven landscapes, may also be addressed to more natural landscapes: Do we today have at our disposal a comprehensive theory of land-cover dynamics? (Bolliger et al., 2005*). We have therefore to search for a common theoretical framework proposing coherent concepts and adapted formalisms to improve our understanding and modelling of landscape dynamics. From a more practical point of view, the UML (Unified Modelling language) emerged to provide a simpler and more universal language to model, but it remained poorly used in landscape modelling (Degenne et al., 2009*). The theoretical statement should not conceal the important role of data acquisition, without which no LM could exist, and the role of model application and validation in specific sites and dates. We are also conscious of the limits of describing ecological processes from landscape patterns in this way (Schröder and Seppelt, 2006). A theoretical framework for all types of terrestrial ecosystems, if any, should bear in mind such a debate and offer generic tools to bridge the gap between them.

5.2 Complexity theory as a conceptual basis

Awareness that a certain complexity was present in most landscapes has led to a holistic approach to landscape (Gallagher and Appenzeller, 1999; Looijen, 1998). The theory of complexity has shown that the state and behaviour of certain systems does not result from the simple linear combination of the state and behaviour of its separate parts. Thus new properties are seen to emerge, which originate in the non-linearity and non-equilibrium of these systems (in thermodynamic terms). The study of these complex systems suggests holistic strategies that apply to landscape modelling as the landscape is a complex object (Bolliger et al., 2005; Saltré et al., 2013), which often has complex dynamics (Bürgi et al., 2004). It shows emerging behaviours/patterns that are the result of complex interactions, such as diffusion or dispersal, crop allocation system or land-cover transition (time patterns), heterogeneous and multiscale growth of areas (spatial patterns), between its elementary/single constituents. If such processes seem very different, they may all appear for various situations and highly different landscape types. It is thus relevant to study them in unusual situations in order to trigger a possible unity hidden behind landscape dynamics of distinct types.

Landscapes are dynamical, a property that often remained understudied compared to the landscape pattern in itself. Some diachronic works intended to compare successive dates of these patterns (Kolb et al., 2013; Mackey, 2000; Paudel and Yuan, 2012*), but it was a preliminary task compared to complex, superimposed, multiscale and long term dynamics that are at play in every landscape. We now need to analyse these dynamics for themselves and ask related questions: Is a landscape stationary (i.e., with constant statistical moments) or does it present some discontinuities in time (Gaucherel et al., 2012*)? Is a landscape dynamic chaotic? Is it ergodic (i.e., with the same behaviour averaged over time as averaged over space)? What are its asymptotical states, if there are some (Gaucherel, 2011*)? Does the landscape gather some conflicts (i.e., opposed changes of the same unit), for example between dynamics operating at different scales? We today need a conceptual shift to address these new issues.

Self-similar approaches in particular are in the vein of complexity theory, as they manage multiscale, non-linear and often out-of-equilibrium systems (Pascual and Guichard, 2005; Solé and Bascompte, 2006*). Neutral and multiscale models are recent attempts to detect and quantify self-organization, while patchy mosaics and networks are well adapted to such attempts, too (He et al., 2013; Paudel and Yuan, 2012). This explains why several scientists have begun to adapt these approaches to landscape studies (see Appendix A.2 for details), as mentioned in the previous section. Patchy and network systems are complex as they handle non-linear behaviour in space and time, for fluxes or movements that they shelter (Gaucherel, 2008*; Proulx et al., 2005). Hence, the four model characteristics detailed in this paper are well adapted to further develop a complex and unified view of terrestrial landscapes. They are probably not yet exhaustive, as specific temporal dynamics, driving forces, and spatial interactions of neighbouring or level of integration may be explored in more detail and compared between landscapes.

To go a step further, it has been mentioned that self-organization is a kind of optimization of the structure (D’Souza et al., 2007). We propose here the hypothesis that every landscape is a structure optimized to better use (dissipate) its incomes (of energy and matter). Neutral LM may help to test this working hypothesis, by assembling various local and path-dependent processes (Brown et al., 2005), i.e., depending on the previous time step (or steps) only, acting on patches, linear networks and point-patterns, and being finally optimized in order to fulfil the various constraints imposed to the landscape. Some recent illustrations of such landscape optimisations may be found in the literature (Gaucherel, 2011*) and we recently explored this issue with the optimal control theory (Whittle, 1996*) to summarize a landscape by defining its Hamiltonien function. This attempt proposes an appealing framework to improve our description of very different (yet, with common properties) landscapes. Such work is in the continuity of landscape ecology, as landscape heterogeneity may then be interpreted as a consequence of such optimization of land uses and/or land covers. In a sense, the landscape exhibits some properties close to that of living organisms, with changing, scaling, emerging and stabilizing patterns (heterogeneity properties) depending on its driving forces.

5.3 Mathematical formalism and modelling as a toolkit

To imagine new concepts is a prerequisite for developing a new dynamic landscape theory, but it will be necessary to simultaneously develop a mathematical formalism, in order to generically handle these concepts over a wide range of landscape types. There are very few attempts, to our knowledge, to formalize landscapes in terms of equations. The literature provides powerful Markov chains to manage landscape composition (Tepley and Thomann, 2012), under stationary and sometimes first order neighbourhood hypotheses (Le Ber et al., 2006), while reaction-diffusion equations are sometimes able to manage landscape configuration, for natural and non-patchy lattices (Couteron and Lejeune, 2001*). Yet, the coupling of landscape composition/configuration is rarely managed and agricultural or forested mosaics have not been set in equations. It is today a challenge for example to mathematically describe spatial autocorrelation of patchy and highly discontinuous patterns (Gaucherel, 2011*; Levin et al., 1993*), in order to reveal hidden processes, to integrate them, to determine asymptotical behaviour or to predict landscape behaviour in alternative conditions. Here are the urgent issues of this formalization objective. Furthermore, the dynamics of such structures are still far from being mathematically described. For example, it is a challenging question to understand whether some landscapes are at equilibrium or not. The self-organization or optimization concepts have the advantage of offering coherent formalisms to handle various landscape types such as relatively continuous patterns. Statistical mechanics and optimal control theory (Whittle, 1996), may fulfil this requirement to open the way to holistic studies of landscapes.

A recent approach based on formal grammars, and in particular those related to graph rewriting-rules opens another track in this direction (Gaucherel et al., 2012*, 2010*), especially for highly non-stationary (discontinuous in space and time) mosaics. Models based on formal languages, such as L-systems used in linguistics and biology (Lindenmayer, 1968a,b), develop objects using grammar-linked components to simulate automata: landscape patches are considered as words whose dynamics are described by a succession of rules that a grammar helps to formalize. It has been demonstrated that most patchy landscape compositional as well as configurational changes may be formalized by a combination of eight rules only: a landscape patch rotation, merge, split, dilation, erosion, appearance, disappearance and no change (Gaucherel et al., 2012*, 2010). This so-called landscape language has a flavour of universality and appropriately illustrates the possible unification of a wide range of landscape dynamics, be they patch, linear network or natural gradient changes.

Such kind of formalisms has the potential to widely spread into environmental disciplines. In a sense, they offer bottom-up approaches that require symmetrical top-down approaches, such as global optimization procedures, to complete our understanding of landscape dynamics (Castella et al., 2007; Houet et al., 2010). This is why it is still important to develop mechanistic models and to model contingent and specific processes (Gustafson, 2013), with local and Markovian rules. As a corollary, another challenge for landscape dynamic studies consists in developing model designs able to conciliate the rich ecological and socioeconomic knowledge we have with the requirement for formal models amenable to generalization across landscape types. Adequate methodologies such as platforms and domain-specific languages are at present under development to help this purpose (Figure 2*b) (de Coligny, 2006; Degenne et al., 2009; Fall and Fall, 2001; Gaucherel et al., 2012).

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