Table 1.
List of acronyms and notations used.
Fig 1.
Flowchart representing the procedure for estimating the bias () generated by applying inferential methods (SMMt, SMMΔt).
δc(t) is simulated with the mechanistic DMM under different scenarios of forcing diet ({δs(a); δs(b)}, ω) and λ values. In the inferential framework, SMMt and SMMΔt provide respectively instantaneous and integrated estimations of .
is computed by comparing the output of SMMs (i.e.,
) with the initial forced diet (i.e., ps(a) input of the DMM).
Fig 2.
Simulated isotopic values of consumer (δc(t)) during 500 d, for contrasted isotopic turnover rates (λ solid lines) and experiencing a variable diet (δd(t) dashed line) resulting from one (a) or four diet-switches (b) between two food sources (δs(a) = 0 ‰; δs(b) = 10 ‰).
The λ values were constant and low λ = 2.10−3 d-1 (red), intermediate λ = 2.10−2 d-1 (blue), and high λ = 2.10−1 d-1 (green) corresponding to the range of values of Thomas & Crowther (2015) [31]. The frequencies of diet shift are ω = 0.002 d-1 (a) and ω = 0.008 d-1 (b). The ratios ω/λ were respectively 1, 0.1 and 0.01 (a) 4, 0.4 and 0.04 (b). Note that for each of the food source, the trophic discrimination factors (Δs(i)) were set to 1 ‰.
Fig 3.
Simulated isotopic values of consumer (δc(t)) over T = 500 d, for contrasted isotopic turnover rates (constant λ values in red, blue and green lines and ontogenetic λ in orange line) and experiencing a variable diet (δd(t) in dashed dark line) resulting from four diet-switches (ω = 0.008 d-1) between two food sources ({δs(a); δs(b)} in thin dashed grey lines) as simulated with Brownian trajectories.
The ratios ω/λ were respectively 4, 0.4 and 0.04 for constant λ values and ranging from 6 to 0.04 for ontogenetic λ. Note that for each of the food source, the trophic discrimination factors (Δs(i)) were set to 1 ‰.
Fig 4.
Estimated contributions of source a ( solid line) to a consumer’s diet compared to reference (ps(a) dashed line).
are inferred from isotopic composition of consumer (δc(t)) simulated using DMM and forcing food sources (δs(a)(t); δs(b)(t) and ps(a)) over time.
First row (a-c) represents estimated instantly from the SMMt (pink lines), second row (d-f) represents integrative estimation of
from SMMΔt (orange lines). The reference diet (ps(a)) (turquoise dashed lines) corresponds to the forcing diet as input of DMM, at each t for SMMt (a-c) or averaged over the time window (Δt)—equated to twice the isotopic half-life (i.e., Δt = 2 ln(2)/λ and equals 69 d for these simulations) for SMMΔt (d-f). For SMMΔt the
values start at the 70th day by integrating the sources over previous 69 d. The columns represent different scenarios of the experimental design: ω was (a, d) 0.002 d-1 (one diet switch), (b, c, e, f) 0.008 d-1 (four diet switches), and the isotopic values of the food sources were (a, b, d, e) constant or, (c, f) variable. In the three scenarios λ is constant and set at an intermediate value (λ = 2.10−2 d-1).
Fig 5.
Bias estimates for the two static approaches (SMMt and SMMΔt, pink and orange points respectively).
The estimated bias () for each ratio ω/λ is the result of the difference between forcing ps(a) and inferred
. ω/λ ratio values come from combination of ω (2.10−4, 1.10−2, 2.10−2, 3.10−2, 4.10−2, 5.10−2, 6.10−2, 7.10−2, 8.10−2 d-1) with constant and intermediate λ value (2.10−2 d-1) to obtain a sequence of ω/λ between 0 and 4.
Fig 6.
Real dataset and case study application.
Data from Marín-Leal et al., (2008) [29], considering only one stable isotope (carbon). δ13C values of the consumer (Pacific oyster) and the two main sources (phytoplankton, PhyOM and microphytobenthos, MPB) once corrected by their respective Δs(i) (here equal to 1 ‰) and their linear interpolation between sampling dates (a). Estimated λ values for each sampling dates (b). Estimates of microphytobenthos contributions () to the oyster’s diet, according to different methods (c). Instantaneous SMMt is represented by pink dots at sampling date, integrated SMMΔt used averaged sources over a time window of two half-lives, and therefore estimates
constant over this time window. Note that with decreasing λ, the time window increases, resulting in longer orange bars (constant
estimated over a longer period). Furthermore, since the last window is a bit larger than the sampling period by 106 d, the entire period of sampling was considered.
is estimated through DMM for each period between sampling dates (turquoise bars).
Fig 7.
Decision tree of the most suitable method for estimating source proportions to the diet of one consumer considering the dynamics of λ and δ values, to reduce the bias induced by the isotopic equilibrium assumption.
The diagram highlights the key aspects to be considered on the estimation of λ, the sampling of consumer and food source dynamics in order to determine which model can be applied (SMMs or DMMs).