Re-evaluation of quadratic and exponential models of litter accumulation incorporating climatic and species-specific dependence

arising from M. Adams & M. Neumann. Nature Communications https://doi.org/10.1038/s41467-023-37166-9 (2023)

Models for the accumulation of litter fuels play a key role in the management of environmental systems, such as those related to forest flammability, soil development and carbon sequestration. Adams and Neumann¹ examined continental datasets of litter and litterfall and found that modified quadratic models provided the best explanation of the variation in the data. Here we question that finding and provide a re-analysis of the data considered by ref. ¹. Given the significance of the application, our results highlight the importance of ensuring that the choice of model and the methods employed in their implementation yield minimal estimation error.

Specifically¹, considered models of the form:

$$X\left(t\right)=f\left(t\right)+{cAI}+d{Q}_{{lf}}$$

(1)

where X(t) denotes the density of litter accumulated (g m⁻²) after time t (years), f(t) denotes the temporal variation in litter accumulation, AI denotes the aridity index and ${Q}_{{lf}}$ denotes a variable used to describe the quality of litter for decomposition. The coefficients c and d are model parameters.

Adams and Neumann¹ modelled the temporal variation of $X(t)$, described by the function $f(t)$ in Eq. (1), by fitting linear, quadratic, exponential and power-law functions. They found that when AI and ${Q}_{{lf}}$ were combined with time since fire t, models of the form (1) using a quadratic function for $f(t)$, were superior at describing litter accumulation, compared to those using other functions (based on AIC and goodness-of-fit; R²).

Litter accumulation: model selection and evaluation

Adams and Neumann¹ used the following quadratic and exponential models for $f(t)$ (ref. ¹, Supplementary Table 4):

$$f\left(t\right)=I+{at}+b{t}^{2}$$

(2)

$$f\left(t\right)=A\left(1-{e}^{-{kt}}\right)+{f}_{0}{e}^{-{kt}}$$

(3)

where I, a, b, A, k and f₀ are model parameters. The parameter ${f}_{0}\ge 0$ represents the initial amount of litter (¹ prescribed ${f}_{0}=0$). The model (3) is often referred to as the Olson model in the literature^2,3.

Adams and Neumann¹ determined model parameters for each of the datasets described in their Table 1 and listed the best-fit (modified quadratic) model parameters in their Table 2. Unfortunately, they did not include a figure that illustrates how the resulting models compared to litter accumulation data, such as those depicted in their Fig. 2. Here we provide such an illustration.

Table 1 Parameters defining the Quadratic and Olson models for All Eucalypt Forests and Representative Forests for all data and data satisfying t < 40

The aridity index (${AI}$) and the quality of litter for decomposition (${Q}_{{lf}}$) are both dependent on climatic conditions and the species under consideration, which in turn are time and space dependent. However, for assessment of the modified quadratic litter accumulation models of ref. ¹, it is sufficient to consider values of ${AI}$ and ${Q}_{{lf}}$ over an indicative range of values. This is provided by ref. ¹ (Fig. 4), which indicates that, for the species considered, values of ${AI}$ and ${Q}_{{lf}}$ satisfy $0.4\le AI \le 1.8$ and $0.2\le {Q}_{{lf}}\le 1.0$. This defines a domain we denote as ${{{\mathcal{C}}}}$. Considering the function $g(AI,{Q}_{lf})=cAI+d{Q}_{lf}$ over the domain ${{{\mathcal{C}}}}$, we find that:

$${g}_{{\min}} \le g(AI,{{Q}_{lf}}) \le {g}_{{\max}}$$

where ${g}_{\min }$ and ${g}_{\max }$ are attained somewhere on the boundary of ${{{\mathcal{C}}}}$. According to Eq. (1), this means that regardless of which specific model is used to determine the function $f\left(t\right)$, the modelled litter accumulation over time $X\left(t\right)$ must satisfy:

$$f\left(t\right)+{g}_{\min }\le X\left(t\right)\le f\left(t\right)+{g}_{\max }$$

(4)

This is illustrated in Fig. 1, using the quadratic model for the function $f\left(t\right)$ as determined using the best-fit parameter values from ref. ¹ (Table 2). The figure shows that when $(AI,{Q}_{lf})\in {{{\mathcal{C}}}}$, the modelled values of $X\left(t\right)$ are constrained to the grey shaded regions in the figures.

Fig. 1: Quadratic and Olson model comparison.

Prediction ranges (grey shading) given by Eq. (4) along with the All Eucalypt Forests and Representative Forests datasets of ref. ¹: a All Eucalypt Forests (all data); b All Eucalypt Forests (data with $t\, < \,40$); c Representative Forests (all data); d Representative Forests (data with $t\, < \,40$). The red and blue curves are the quadratic model prediction ${X}_{{Quad}}\left(t\right)$ (Eq. 5) and the Olson model prediction ${X}_{{Olson}}\left(t\right)$ (Eq. 6), respectively, determined using least-squares regression. The black curves represent the quadratic component (Eq. 2) of the model values calculated using the corresponding parameter values in ref. ¹, Table 2.

Apart from the case for the Representative Forests full dataset, the models of¹ suggest that litter density initially increases with time but then decreases. Specifically, Fig. 1b, d indicates that when $t\, < \,40$, litter density increases for about the first 20 years and decreases afterwards. Such behaviour is in direct contradiction to the remarks made by ref. ¹ implying a continual increase in litter accumulation. Furthermore, the broad-scale temporal behaviour illustrated in Fig. 1a, c for $t \, < \, 90$ is distinctly at odds with the temporal behaviour over the first 40 years post-fire shown in Fig. 1b, d. Indeed, extending the models in Fig. 1b, d to $t \, > \, 40$ would produce physically impossible negative litter mass predictions.

The results depicted in Fig. 1 also cast doubt on the finding of ref. ¹ that modified quadratic functions were superior to exponential functions in describing litter accumulation. To quantify the goodness-of-fit of the models, we calculated values of the modified quadratic model by randomly selecting $\left({AI},{Q}_{{lf}}\right)$ from a bivariate uniform distribution:

$$X\left(t\right)=I+{at}+b{t}^{2}+c\widetilde{{AI}}+d\widetilde{{Q}_{{lf}}}$$

with $\left({\widetilde{AI}},\widetilde{{Q}_{{lf}}}\right) \sim U({{{\mathcal{C}}}})$ and the community-specific parameter values taken from ref. ¹ (Table 2). These model values were combined with the data to calculate R² statistics. This process was repeated a million times and the largest R² value so obtained was recorded. For All Eucalypt Forests the maximum R² value obtained was −0.200, while for Representative Forests the maximum R² was 0.221. Note that a negative R² value indicates that the model performs worse than using the mean of the data as a descriptor. These results also cast considerable doubt on the R² values of 0.438 and 0.454 reported by ref. ¹ (Table 2) for All Eucalypt Forests and Representative Forests, respectively. Similarly perplexing results were found when the data were restricted to t < 40. In these cases, the maximum R² value obtained for All Eucalypt Forests (t < 40) was −0.558, while for Representative Forests (t < 40) the maximum R² was 0.046. Again, these figures are at odds with the figures of 0.593 and 0.650 reported by ref. ¹ and casts significant doubt on their conclusion that the modified quadratic model could account for over 60% of the variance in litter mass over a 40-year time span.

Re-analysis of the litter accumulation data

The All Eucalypt Forests and Representative Forests data of ref. ¹ were refitted using the following models for accumulated litter density:

$${X}_{{Quad}}\left(t\right)={p}_{1}{t}^{2}+{p}_{2}t+{p}_{3}$$

(5)

$${X}_{{Olson}}\left(t\right)={X}_{{ss}}\left(1-{e}^{-{kt}}\right)+{X}_{0}{e}^{-{kt}}$$

(6)

Figure 1 shows the best-fit models (5) and (6) using all the available data and the data restricted to $t\, < \,40$. The fitted parameter values are given in Table 1 along with the corresponding R² values. The R² values in Table 1 indicate that ${X}_{{Quad}}\left(t\right)$ and ${X}_{{Olson}}\left(t\right)$ have very similar explanatory power and that both are significantly better models than the modified quadratic models of ref. ¹. This contradicts the conclusions of ref. ¹.

Another issue with use of quadratic functions to model litter accumulation is illustrated in Fig. 2, which shows ${X}_{{Quad}}\left(t\right)$ and ${X}_{{Olson}}\left(t\right)$ refitted to the full All Eucalypt Forests dataset, but with the single data point at t = 250 years omitted. The quadratic model shows significant sensitivity to exclusion of the single data point, while the Olson model exhibits negligible change. In fact, the quadratic model derived from the censored data predicts an impossible litter density, of around −8000 g m⁻², when t = 250. This indicates that, in general, quadratic functions are not a good choice of model for describing temporal variation in litter accumulation, with the Olson model providing a more robust modelling alternative.

Fig. 2: Olson model ${X}_{{Olson}}\left(t\right)$ (blue) and Quadratic model ${X}_{{Quad}}\left(t\right)$ (red).

Both recalculated using all of the All Eucalypt Forests dataset, but with the single data point at $t=250$ years omitted. The black line is the model of ref. ¹ (fitted to the uncensored data).