From the studies mentioned previously, changes in model dynamics can occur when the parameter values are perturbed during conventional parameter sensitivity analyses. However, much larger changes in system dynamics can result from changes in the structural formulation of process functions, despite these formulae producing similarly shaped curves (Wood and Thomas, 1999; Fussmann and Blasius, 2005; Levin and Lubchenco, 2008; Flora et al., 2011; Adamson and Morozov, 2013; Aldebert et al., 2016). This is because a change in formulation may affect the function shape, and this also affect the stable state of the system (Aldebert et al., 2016). This is called structural sensitivity (Wood and Thomas, 1999; Flora et al., 2011; Adamson and Morozov, 2013). A study by Fussmann and Blasius (2005), demonstrated that in a simple Rosenzwig-McArthur predator and prey model, using similarly shaped prey uptake functions could produce different predator and prey dynamics. In the study, the parameters are chosen so that the functions are phenomenologically similar, as shown in Figure 1.5.
1.4. Why structural sensitivity needs to be addressed 21
Figure 1.5: Three similarly shaped prey uptake functions from (Fussmann and Blasius, 2005). The rectangular hyperbolic (1.6) is shown in blue, hyperbolic tangent (1.7) is shown in red, and the ivlev (1.8) function is shown in black. Figure originally from Fussmann and Blasius (2005).
fH(x) = arhx 1 + brhx (1.6) fT(x) = ahttanh(bhtx) (1.7) fI(x) = ai(1 − exp(−bix)) (1.8)
Using equation 1.6, resulted in oscillations with high amplitude of (∼ 0.8), and predator and prey con- centrations dropping close to zero, conversely equations 1.8 produced much lower amplitude (∼ 0.2) of oscillation. Equation 1.7 produced the most striking difference as it produces a steady state, summarised in Figure 1.6. These differences in model dynamics show structural sensitivity due to model functional forms (hereafter structural sensitivity).
Structural sensitivity may be less significant in models built on well-tested mechanisms such as those in the physical sciences. However, in a biogeochemical model, any mathematical functions that describe a process are likely to be an oversimplified representation of that process. This is because biogeochemical processes are mostly complex, and involve many interactions between diverse individuals, across differ- ent regions and temporal resolutions in an environment that is changing rapidly (Adamson, 2015). This is even more problematic if the process itself is not well understood so that theoretical justification for the specific representation is weak (Adamson and Morozov, 2013). Often it is difficult to implement the functional relations that are observed in the laboratory into a large scale ecosystem with heterogeneous populations (Englund and Leonardsson, 2008). From simple predator-prey models, applying similarly shaped equations can also give completely different stability and oscillatory model dynamics (Fussmann
Figure 1.6: Different dynamics produced from similarly shaped prey uptake function 1.5. Rectangular hyperbolic (1.6) is shown in a, ivlev (1.8) is shown in b, and hyperbolic tangent (1.7) is shown in c. Figure is originally from Fussmann and Blasius (2005).
and Blasius, 2005; Roy and Chattopadhyay, 2007). Moreover, a specific functional form may not cap- ture all details of the biological processes, for example, the rectangular hyperbolic function for grazing, commonly known as the ’Holling Type II’, fails to correctly describe what happens to grazer’s move- ments when satiation has been reached (Flynn and Mitra, 2016). The formulation that is adequate from a theoretical point of view, does not necessarily describe the data quantitatively (Aldebert et al., 2018). In a recent study, although the complex model promotes the survival of a species, such as phytoplankton, the variability of dynamics in the foodweb is more strongly affected by the choice of functional forms (Aldebert et al., 2016).
Some studies have investigated the effects of different process formulations on biogeochemical models, e.g. Edwards and Yool (2000); Yool et al. (2011) have demonstrated that in a simple NPZ model and an intermediately complex model, linear density dependent mortality produces the most significant differ- ences when applied to diatoms, compared with sigmoidal, quadratic, or hyperbolic forms, as stated in section 1.2. The choice of zooplankton grazing equations can also affect phytoplankton concentration dramatically in a model with five plankton types, PlankTOM5.2 (Anderson et al., 2010),which has been discussed previously, and also in self-assembling ecosystem models (Prowe et al., 2012). The Holling type II grazing function produces 30% less total surface phytoplankton concentration compared to the Holling type III functions in the North Atlantic and North Pacific (Anderson et al., 2010). Nevertheless, not all processes give significantly different model output. Anderson et al. (2015) also shows that when two similarly shaped photosynthesis-irradiance curves, namely, Smith and the exponential function, were used in an NPZD model, the concentration of chlorophyll during the spring bloom was only slightly higher (0.2 mg m−3) for the exponential function (Anderson et al., 2015). For these studies, only one
1.4. Why structural sensitivity needs to be addressed 23
process function is altered in the model, despite multiple processes that might be structurally sensitive. Therefore, a thorough assessment on how structural sensitivity affects the results of an intermediately- complex or an operational model, with more biological processes that are varied, is needed (Aldebert and Stouffer, 2018).
There are models that avoid the issue of structural sensitivity by constructing a generalised bifurcation diagram in a generalised parameter space, to explore the possible model dynamics (Gross and Feudel, 2006). However, this kind of model does not address alternative stable states that can be affected by structural sensitivity (Aldebert et al., 2016). Further, the discrepancies reported from simple interaction models suggest that the dynamics of complex biogeochemical models need to be tested by altering their default functional forms (Anderson and Mitra, 2010; Anderson et al., 2010). To properly address the structural sensitivity in an operational biogeochemical model, it is possible to generate a model ensemble.
1.4.1 Generating ensembles to address uncertainty
In order to address the alternative simulations representing structural uncertainty, it is possible to generate an ensemble. Here, an ensemble means multiple model variations are run to acquire a range of future predictions, or simulations, with its uncertainties represented by the spread of outcomes. The ensemble approach has also been used to inform policy makers and planners to estimate uncertainty associated with physical model projection, so that appropriate strategies for adaptation could be identified (Murphy et al., 2007). This also means that the ensemble should represent the uncertainties that may also occurred in the true value from the observation.
In climate modelling, perturbed physics ensembles have been developed to investigate multiple param- eter uncertainty (Murphy et al., 2007; Tinker et al., 2016). The ensemble can be generated by varying uncertain parameters according to prior studies of relevant physical processes, or using an ensemble Kalman filter (Murphy et al., 2007). Another approach to ensemble modelling is by exploring multiple parameterization (functional) uncertainties (Subramanian and Palmer, 2017). Additionally, it is possible to generate an ensemble which consists of multiple climate models, or multimodel ensembles, although since some models share similar process representations and parameterisations, it is often that nominally different models might still have similar biases, because the models are not independent (Abramowitz et al., 2019). Although uncertainty is always present in every model, it is possible to utilise this so to im- prove the skill of climate models. The next section will introduce data assimilation, where uncertainties are used to make model more consistent with observation.