Abstract
A new approach to loop analysis is presented in which decompositions of the total elasticity of a population projection matrix over a set of life history pathways are obtained as solutions of a constrained system of linear equations. In loop analysis, life history pathways are represented by loops in the life cycle graph, and the elasticity of the loop is interpreted as a measure of the contribution of the life history pathway to the population growth rate. Associated with the life cycle graph is a vector space—the cycle space of the graph—which is spanned by the loops. The elasticities of the transitions in the life cycle graph can be represented by a vector in the cycle space, and a loop decomposition of the life cycle graph is then defined to be any nonnegative linear combination of the loops which sum to the vector of elasticities. In contrast to previously published algorithms for carrying out loop analysis, we show that a given life cycle graph admits of either a unique loop decomposition or an infinite set of loop decompositions which can be characterized as a bounded convex set of nonnegative vectors. Using this approach, loop decompositions which minimize or maximize a linear objective function can be obtained as solutions of a linear programming problem, allowing us to place lower and upper bounds on the contributions of life history pathways to the population growth rate. Another consequence of our approach to loop analysis is that it allows us to identify the exact tradeoffs in contributions to the population growth rate that must exist between life history pathways.
Similar content being viewed by others
References
Avis D. and Fukuda K. (1992). A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra. Discrete Comput. Geom. 8: 295–313
Bollobás B. (1998). Modern Graph Theory. Springer, New York
Caswell H. (2001). Matrix population models: construction, analysis and interpretation, 2nd edn. Sinauer Associates, Sunderland
de Kroon H., Plaisier A., van Groenendael J. and Caswell H. (1986). Elasticity: the relative contribution of demographic parameters to population growth rate. Ecology 67: 1427–1431
de Kroon H., van Groenendael J. and Ehrlen J. (2000). Elasticities: a review of methods and model limitations. Ecology 81: 607–618
Güneralp B. (2007). An improved formal approach to demographic loop analysis. Ecology 88: 2124–2131
Horn R. and Johnson C. (1985). Matrix Analysis. Cambridge University Press, Cambridge
Murty K. (1983). Linear Programming. Wiley, New York
Sun L. and Wang M. (2007). An algorithm for a decomposition of weighted digraphs: with applications to life cycle analysis in ecology. J Math. Biol. 54: 199–226
Thomassen, C.: Parity, cycle space, and K 4-subdivisions in graphs. In: Lamb, J.D., Preece, D.A. (eds.) London Mathematical Society Lecture Note Series 267, Surveys in Combinatorics, 1999. Cambridge University Press, Cambridge (1999)
van Groenendael J., de Kroon H., Kalisz S. and Tuljapurkar S. (1994). Loop analysis: evaluating life history pathways in population projection matrices. Ecology 75: 2410–2415
Wardle G.M. (1998). A graph theory approach to demographic loop analysis. Ecology 79: 2539–2549
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was partially supported by NSF Grant No. 0436348 through Truman State University’s Mathematical Biology Initiative.
Rights and permissions
About this article
Cite this article
Adams, M.J. Graph decompositions for demographic loop analysis. J. Math. Biol. 57, 209–221 (2008). https://doi.org/10.1007/s00285-007-0152-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00285-007-0152-0
Keywords
- Population projection matrix
- Life cycle graph
- Elasticity analysis
- Loop analysis
- Graph theory
- Cycle space
- Linear programming
- Dipsacus sylvestris