System dynamics is an umbrella term for those approaches aiming to understand the behaviour of network-like systems over time. What makes system dynamics different from other methods about complex systems is the use of feedback loops, stocks and ?ows which allow to model how network-like systems can display strong nonlinear behaviours. Fuzzy Cognitive Maps (FCM) are semi-quantitative networks that can be regarded as a system dynamics method. Here I suggest 4 kinds of modifications to FCM: (1) if-then-else stop option; (2) piecewise option; (3) non-monotonicity option; (4) non-linearity option. These improvements to FCM might allow for fitter simulations of ecological and biological systems over time. I also present an  applicative example to illustrate the proposed modifications

Alessandro Ferrarini

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Network Biology, 2011, 1(1):72-78   I A E E S                                                                                     www.iaees.orgShort Communication  Some steps forward in semi-quantitative networks modelling  Alessandro Ferrarini Department of Evolutionary and Functional Biology, University of Parma, Via G. Saragat 4, I-43100 Parma, Italy E-mail: sgtpm@libero.it, alessandro.ferrarini@unipr.it  Received 1 April 2011; Accepted 1 May 2011; Published online 6 May 2011 IAEES  Abstract System dynamics is an umbrella term for those approaches aiming to understand the behaviour of network-like systems over time. What makes system dynamics different from other methods about complex systems is the use of feedback loops, stocks and ﬂows which allow to model how network-like systems can display strong nonlinear behaviours. Fuzzy Cognitive Maps (FCM) are semi-quantitative networks that can be regarded as a system dynamics method. Here I suggest 4 kinds of modifications to FCM: (1) if-then-else stop option; (2) piecewise option; (3) non-monotonicity option; (4) non-linearity option. These improvements to FCM might allow for fitter simulations of ecological and biological systems over time. I also present an  applicative example to illustrate the proposed modifications.  Keywords  semi-quantitative networks; fuzzy cognitive maps; if-then-else stops; non-linearity; non-monotonicity; piecewise model.   1 Introduction Semi-quantitative networks are halfway between qualitative networks (e.g., loop analysis; Puccia and Levins, 1985) and quantitative ones (e.g., ecological network analysis; Ulanowicz, 1986).   Fuzzy cognitive maps (FCM; Kosko, 1986) are semi-quantitative networks which describe the behaviour of a system in terms of concepts (nodes); each concept represents a variable (e.g., species' abundances) or a characteristic of the system. Values of nodes change over time, and take values in the interval [0, 100]: a value of 0 means that the factor is absent, a value of 100 means that the factor is present to the maximum possible extent, while a value of 50 represents the actual level of system variables.  The causal links between nodes are represented by directed weighted edges that illustrate how much one concept inﬂuences the interlinked concepts, and the causal weights of the interconnections belong to the [-1, +1] interval. The strength of the weight wij indicates the degree of inﬂuence between concept Ci and concept Cj. The value of each concept at every simulation step is calculated as follows:        () ( 1 ) ( 1 ) * jj i i jiAt At At w        (1) where Aj(t) is the value of concept Cj at time t, Aj(t-1) is the value of concept Cj at time t-1, wij is the weight of the interconnection from concept Ci to concept Cj.  FCM are particularly useful for applications where relationships between concepts cannot be expressed in Network Biology, 2011, 1(1):72-78   I A E E S                                                                                     www.iaees.orgexact mathematical equations. Much human expert knowledge is expressed in natural language, which is characterised by this type of uncertainty and imprecision. Biological and environmental quantities and their causal interactions are often described in relative and vague terms. A large proportion of the ecological information is represented in this way, and cannot be used as an input to data-driven mathematical or statistical models. The main advantage of FCM relies in its ability to represent such fuzziness. Recent applications can be found in Kok (2009), Prigent et al. (2008), Ramsey and Veltman (2005).  2 An Example Let's consider an environmental system with 4 interacting variables (e.g., 4 species in a plant community). The matrix below depicts the existing interactions among variables. Fig. 1 shows the simulation results for the depicted system  Table 1 On the left, initial values. On the right interaction matrix among variables.    Fig. 1 Results of the application of FCM to the environmental system of Table 1.     73Network Biology, 2011, 1(1):72-78   I A E E S                                                                                     www.iaees.orgThere a 2 winning species (A and D) and 2 losers (B and C). While A and C respectively always increases and decreases, species B and D present a more complex behavior, sometimes increasing and sometimes decreasing. The system reaches a steady final state with B and C imploding to 0 (i.e. they disappear), while A and D reach their maximum possible extent (i.e., 100).  3 Some Steps Forward The previous example shows how linear feedbacks can produce dynamics that are strongly non-linear. The ability to capture the effect of feedbacks and produce non-linear system changes is exactly where the strength of FCM lies. Nonetheless, I suggest here 4 kinds of modifications to FCM: a) if-then-else stop option; b) piecewise option; c) non-monotonicity option; d) non-linearity option. 3.1 If-then-else stops Let's suppose that species B is important from a preservation viewpoint, and that management actions start when species B reaches about one-half (i.e. 25) its initial abundance in order to keep it almost constant. Hence, FCM should be able to incorporate an if-then-else stop condition (Fig. 2) so that system dynamics stop when species B reaches a value equal to about 25.    Fig. 2 Conceptual model of if-then-else stop option for FCM.   If applied to the starting example, the final (if-then-else stopped) state would be: A= 49; B= 20; C= 28, D= 47. We could consider that if-then-else stop may act on multiple conditions (Fig. 3), for instance that a stop condition happens when both species A and D are above 60 (i.e. 120% of the initial vales). In this case, the final (if-then-else stopped) state would be: A= 82; B= 15; C= 0, D= 63.  74Network Biology, 2011, 1(1):72-78   I A E E S                                                                                     www.iaees.org Fig. 3 Conceptual model of multiple if-then-else stops option for FCM.   3.2 Piecewise option Piecewise option (Fig. 4) may simulate real-world events. For instance, the fishing pressure on a particular species might change depending on the abundance of the species itself. Also the hunting pressure on invasive species might change based on the abundance of such species. This often happens in the real world, where predator-prey interactions change over time depending on the abundance of both the prey and the predator. Let's consider that the negative effect of A on C decreases from -0.6 to -0.3 when C goes below 40 (i.e., 80% of its initial abundance).    Fig. 4 Conceptual model of piecewise option for FCM.  75Network Biology, 2011, 1(1):72-78   I A E E S                                                                                     www.iaees.orgIf applied to the starting example, the system dynamics would be like in Fig. 5. Although the final state is the same as in Fig. 1, the intermediate states are very different.   Fig. 5 Results of the application of a piecewise option to the environmental system of Table 1.   3.3 Non-monotonicity option Non-monotonicity option for FCM (Fig. 6) is similar to piecewise one, but it's more drastic since it requires that the effect of a variable on another one changes during simulation time. For instance, let's suppose that the effect of species A on species C becomes positive (+0.2) as C goes below 40 (i.e., 80% of its initial abundance). If applied to the starting example, the system dynamics would be like in Fig. 7. Both the final and intermediate states are very different from Fig. 1.    Fig. 6 Conceptual model of non-monotonicity option for FCM.  76Network Biology, 2011, 1(1):72-78   I A E E S                                                                                     www.iaees.org    Fig. 7 Results of the application of a non-monotonicity option to the system of Table 1.    It's interesting to note that species B reaches high values (>90), but then it quickly decreases when species A disappears from the system. 3.4 Non-linearity option Last, I conceive that the updating rule for one or more species may become non-linear (Fig. 8). For instance, let's simulate that the update of the positive effect of species A on species B may become quadratic after C goes below 10 (i.e., 20% of its initial abundance). If applied to the starting example, the system dynamics would be like in Fig. 9.      Fig. 8 Conceptual model of non-linearity option for FCM.  77Network Biology, 2011, 1(1):72-78   I A E E S                                                                                     www.iaees.org    Fig. 9 Results of the application of a non-linearity option to the system of Table 1.    It's interesting to note that species B reaches its maximum possible value (100), but then it rapidly decreases when species A disappears from the system.  4 Discussion and Conclusions FCM are a valuable tool for the simulation of biological and ecological systems over time. They: a) provide a lucid representation of complex systems, b) focus on indirect effects, c) force users to be explicit on relationship strengths, d) are reasonably parsimonious with regard to model building complexity, e) produce dynamics that are strongly non-linear. Nonetheless, I have suggested here four improvements to FCM to fitter represent the complexity of real-world systems. I am aware that the proposed improvements make FCM less parsimonious, but I also claim that from a benefit-cost viewpoint the proposed improvements are rich in significance and implications.    References Kok K. 2009. The potential of Fuzzy Cognitive Maps for semi-quantitative scenario development, with an   example from Brazil. Global Environmental Change, 19: 122-133 Kosko B. 1986. Fuzzy cognitive maps. International Journal of Man-Machine Studies, 24: 65-75 Prigent M, Fontenelle G, Rochet M, Trenkel VM. 2008. Using cognitive maps to investigate ﬁshers’  ecosystem objectives and knowledge. Ocean & Coastal Management, 51: 450-462 Puccia CJ, Levins R. 1985. Qualitative Modelling of Complex Systems: an Introduction to Loop Analysis   and Time Averaging. Harvard University Press, Cambridge, UK Ramsey D, Veltman C. 2005. Predicting the effects of perturbations on ecological communities: what can   qualitative models offer? Journal of Animal Ecology, 74: 905-916 Ulanowicz RE. 1986. Growth and Development: Ecosystems Phenomenology. Springer-Verlag, New York,   USA 78

Some steps forward in semi-quantitative networks modelling

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