An underdetermined linear inverse problem: estimating diet composition of Southeast Alaskan Mink.

Description

Input data for assessing the diet composition of mink in southeast Alaska, using C and N isotope ratios (d13C and d15N).

The data consist of

1. the input matrix Prey, which contains the C (1st row) and N (2nd row) isotopic values of the prey items (columns), corrected for fractionation.

2. the input vector Mink, with the C and N isotopic value of the predator, mink

There are seven prey items as food sources:

• fish

• mussels

• crabs

• shrimp

• rodents

• amphipods

• ducks

The d13C and d15N for each of these prey items, and for mink (the predator) was assessed. The isotopic values of the preys were corrected for fractionation.

The problem is to find the diet composition of mink, e.g. the fraction of each of these food items in the diet.

Mathematically this is by solving an lsei (least squares with equalities and inequalities) problem: Ex=f subject to Gx>h.

The equalities Ex=f:

d13CMink = p1*d13Cfish+p2*d13Cmussels + .... + p7*d13Cducks

d15NMink = p1*d15Nfish+p2*d15Nmussels + .... + p7*d15Nducks

1 = p1+p2+p3+p4+p5+p6+p7

and inequalities Gx>h:

pi >= 0

are solved for p1,p2,...p7.

The first two equations calculate the isotopic ratio of the consumer (Mink) as a weighted average of the ratio of the food sources

Equation 3 assures that the sum of all fraction equals 1.

As there are 7 unknowns and only 3 equations, the model is UNDERdetermined, i.e. there exist an infinite amount of solutions.

This model can be solved by various techniques:

1. least distance programming will select the "simplest" solution. See ldei.

2. the remaining uncertainty ranges of the fractions can be estimated using linear programming. See xranges

3. the statistical distribution of the fractions can be estimated using an MCMC algorithm which takes a sample of the solution space. See xsample

Usage

Minkdiet

Format

a list with matrix Prey and vector Mink.

• Prey contains the isotopic composition (13C and 15N) of the 7 possible food items of Mink

• Mink contains the isotopic composition (13C and 15N) of Mink

columnnames of Prey are the food items, rownames of Prey (=names of Mink) are the names of the isotopic elements.

Author(s)

Karline Soetaert <karline.soetaert@nioz.nl>

References

Ben-David M, Hanley TA, Klein DR, Schell DM (1997) Seasonal changes in diets of coastal and riverine mink: the role of spawning Pacific salmon. Canadian Journal of Zoology 75:803-811.

See Also

ldei to solve for the parsimonious solution

xranges to solve for the uncertainty ranges

xsample to sample the solution space

Examples

# 1. visualisation of the data
plot(t(Minkdiet$Prey), xlim = c(-25, -13), xlab = "d13C", ylab = "d15N",
     main = "Minkdiet", sub = "Ben-David et al. (1979)")

text(t(Minkdiet$Prey)-0.1, colnames(Minkdiet$Prey))   

points(t(Minkdiet$Mink), pch = 16, cex = 2)
text(t(Minkdiet$Mink)-0.15, "MINK", cex = 1.2)   
legend("bottomright", pt.cex = c(1, 2), pch = c(1, 16),
       c("food", "predator"))

# 2. Generate the food web model input matrices
# the equalities: 
E <- rbind(Minkdiet$Prey, rep(1, 7))
F <- c(Minkdiet$Mink, 1)

# the inequalities (all pi>0)
G <- diag(7)
H <- rep(0, 7)

# 3. Select the parsimonious (simplest) solution
parsimonious <- ldei(E, F, G = G, H = H)

# 4. show results
data.frame(food = colnames(Minkdiet$Prey),
           fraction = parsimonious$X)

dotchart(x = as.vector(parsimonious$X), labels = colnames(Minkdiet$A),
         main = "Estimated diet composition of Mink",
         sub = "using ldei and xranges", pch = 16)

# 5. Ranges of diet composition
iso   <- xranges(E, F, ispos = TRUE)
segments(iso[,1], 1:ncol(E), iso[,2], 1:ncol(E))
legend  ("topright", pch = c(16, NA), lty = c(NA, 1),
          legend = c("parsimonious", "range"))

pairs (xsample(E = E, F = F, G = diag(7), H = rep(0, 7), iter = 1000)$X,
       main = "Minkdiet 1000 solutions, using xsample")