A new approach to the solution of the linear mixing model for a single isotope: application to the case of an opportunistic predator

Mixing models are used to determine diets where the number of prey items are greater than one, however, the limitation of the linear mixing method is the lack of a unique solution when the number of potential sources is greater than the number (n) of isotopic signatures +1. Using the IsoSource program all possible combinations of each source contribution (0-100%) in preselected small increments can be examined and a range of values produced for each sample analysed. We propose the use of a Moore Penrose (M-P) pseudoinverse, which involves the inverse of a 2x2 matrix. This is easily generalized to the case of a single isotope with (p) prey sources and produces a specific solution. The Antarctic leopard seal (Hydrurga leptonyx) was used as a model species to test this method. This seal is an opportunistic predator, which preys on a wide range of species including seals, penguins, fish and krill. The M-P method was used to determine the contribution to diet from each of the four prey types based on blood and fur samples collected over three consecutive austral summers. The advantage of the M-P method was the production of a vector of fractions f for each predator isotopic value, allowing us to identify the relative variation in dietary proportions. Comparison of the calculated fractions from this method with 'means' from IsoSource allowed confidence in the new approach for the case of a single isotope, N.