Homework #2, IBS 8201
Assigned: March 3, 2014
Due: March 12, 2014
The stable isotope ratios of hydrogen, carbon, nitrogen, oxygen, and sulfur within biota
can be used to uncover biogeochemical processes, trace energy and nutrient flows
through food webs, and study movements and migrations of organisms (Peterson and Fry
1987). An important advance in the field of food web ecology was made when linear
mixing models were shown to be a valid for estimating the contribution of different prey
to a consumer using stable isotopes (Phillips and Gregg 2001). The linear mixing model
can be readily solved when there are no more prey than one plus the number of stable
isotope ratios (i.e., where two stable isotope ratios are used, the maximum number of
prey is three). The model takes the following form for two stable isotope ratios (carbon:
13C, nitrogen: 15N), where  is the trophic enrichment factor, l is the number of trophic
levels, and F is the contribution of each prey:
13Cconsumer – Δ13C * l = 13Cprey1 x Fprey1 + 13Cprey2 x Fprey2 + 13Cprey3 x Fprey3
15Nconsumer – Δ15N * l = 15Nprey1 x Fprey1 + 15Nprey2 x Fprey2 + 15Nprey3 x Fprey3
1 = Fprey1 + Fprey2 + Fprey3
However, it is not uncommon to have many potential prey (i.e., more than one plus the
number of stable isotope ratios). In this instance, the linear mixing model is
indeterminate; that is, there is not a unique solution to the problem.
Phillips and Gregg (2003) proposed an iterative solution to the indeterminate model; prey
contribution estimates are incrementally varied and multivariate solutions that fit the
model constraint (all sources sum to one) within a desired tolerance are determined
(IsoSource software). The solution includes a distribution of feasible solutions for each
prey.
Exercise: Construct the distribution of feasible solutions for each prey under the
following three scenarios using the IsoSource software (v1.3.1;
http://www.epa.gov/wed/pages/models/stableIsotopes/isosource/isosource.htm). Use the
default increment and tolerance values – they are appropriate for these scenarios.
For all scenarios, there are four prey items (“sources”) with the following values:
Prey 1: -30‰ 13C, 0‰ 15N
Prey 2: -30‰ 13C, 10‰ 15N
Prey 3: -10‰ 13C, 0‰ 15N
Prey 4: -10‰ 13C, 10‰ 15N
The consumer values change for each scenario. The following values have been corrected
for trophic fractionation:
Scenario 1. Consumer: -20‰ 13C, 5‰ 15N
Scenario 2. Consumer: -10‰ 13C, 5‰ 15N
Scenario 3. Consumer: -25‰ 13C, 8‰ 15N
Answer the following questions:
1. Phillips and Gregg (2003) argue that the model output should be reported with
information regarding the variation of each prey contribution – what measure of
central tendency (mean or median) and variation (range, standard deviation,
quartiles) would be most useful?
2. Which scenarios yield a “diffuse” solution (terminology from Phillips and Gregg
2003)? Which yield a “constrained” solution?
3. Can we interpret the distribution of feasible results for any single prey source to
represent a probability distribution? See Semmens et al. (2013) and associated
papers.
As with Homework #1, you may work in groups.
References
Peterson, Bruce J., and Brian Fry. 1987. Stable isotopes in ecosystem studies. Annual
Review of Ecology and Systematics 18: 293-320.
Phillips, Donald L., and Jillian W. Gregg. 2001. Uncertainty in source partitioning using
stable isotopes. Oecologia 127: 171-179.
Phillips, Donald L., and Jillian W. Gregg. 2003. Source partitioning using stable isotopes:
coping with too many sources. Oecologia 136: 261-269.
Semmens, Brice X., Eric J. Ward, Andrew C. Parnell, Donald L. Phillips, Stuart Bearhop,
Richard Inger, Andrew Jackson, and Jonathan W. Moore. 2013. Statistical basis
and outputs of stable isotope mixing models: Comment on Fry (2013). Marine
Ecology Progress Series 490: 285-289.