Homework #2, IBS 8201

Assigned: March 3, 2014

Due: March 12, 2014

The stable isotope ratios of hydrogen, carbon, nitrogen, oxygen, and sulfur within biotacan 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 xFprey2 + 13Cprey3xFprey3

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; 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.