Bulletin of Mathematical Biology

, Volume 69, Issue 4, pp 1341–1354 | Cite as

A New Method for Calculating Net Reproductive Rate from Graph Reduction with Applications to the Control of Invasive Species

  • T. de-Camino-Beck
  • M. A. Lewis
Original Article


Matrix models are widely used for demographic analysis of age and stage structured biological populations. Dynamic properties of the model can be summarized by the net reproductive rate R 0. In this paper, we introduce a new method to calculate and analyze the net reproductive rate directly from the life cycle graph of the matrix. We show, with examples, how our method of analysis of R 0 can be used in the design of strategies for controlling invasive species.


Matrix models Net reproductive rate Invasion Biological control 


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  1. 1.
    Caswell, H., 1982a. Optimal life histories and the age-specific costs of reproduction. J. Theor. Biol. 98(3), 519–529.CrossRefGoogle Scholar
  2. 2.
    Caswell, H., 1982b. Optimal life histories and the maximization of reproductive value—a general theorem for complex life-cycles. Ecology 63(5), 1218–1222.CrossRefGoogle Scholar
  3. 3.
    Caswell, H., 1984. Optimal life histories and age-specific costs of reproduction—2 extensions. J. Theor. Biol. 107(1), 169–172.CrossRefGoogle Scholar
  4. 4.
    Caswell, H., 2001. Matrix population models: construction, analysis, and interpretation, 2nd edn. Sinauer Associates.Google Scholar
  5. 5.
    Chen, W., 1976. Applied graph theory; graphs and electrical networks, 2nd edn. North-Holland Pub. Co.Google Scholar
  6. 6.
    Cushing, J., Zhou, Y., 1994. The net reproductive value and stability in matrix population models. Nat. Res. Model. 8(4), 297–333.Google Scholar
  7. 7.
    Dinnetz, P., Nilsson, T., 2002. Population viability analysis of Saxifraga cotyledon, a perennial plant with semelparous rosettes. Plant Ecol. 159(1), 61–71.CrossRefGoogle Scholar
  8. 8.
    Hinz, H., 1996. Scentless chamomile, a target weed for biological control in Canada: Factors influencing seedling establishment. In: Proceedings of the IX International Symposium on Biological Control of Weeds, pp. 187–192.Google Scholar
  9. 9.
    Hinz, H., McClay, A., 2000. Ten years of scentless chamomile: Prospects for the biological control of a weed of cultivated land. In: Proceedings of the X International Symposium on Biological Control of Weeds, pp. 537–550.Google Scholar
  10. 10.
    Horn, R., Johnson, C., 1985. Matrix Analysis. Cambridge University Press.Google Scholar
  11. 11.
    Hubbell, S., Werner, P., 1979. Measuring the intrinsic rate of increase of populations with heterogeneous life histories. Am. Nat. 113(2), 277–293.CrossRefMathSciNetGoogle Scholar
  12. 12.
    Krivan, V., Havelka, J., Feb 2000. Leslie model for predatory gall-midge population. Ecol. Modell. 126, 73–77.CrossRefGoogle Scholar
  13. 13.
    Lewis, E., 1977. Network models in population biology. Springer-Verlag.Google Scholar
  14. 14.
    Li, C., Schneider, H., 2002. Applications of Perron-Frobenius theory to population dynamics. J. Math. Biol. 44(5), 450–462.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Mason, S., Zimmermann, H., 1960. Electronic circuits, signals, and systems. Wiley.Google Scholar
  16. 16.
    Parker, I., 2000. Invasion dynamics of Cytisus scoparius: a matrix model approach. Ecol. Appl. 10(3), 726–743.CrossRefGoogle Scholar
  17. 17.
    Shea, K., Kelly, D., 1998. Estimating biocontrol agent impact with matrix models: Carduus nutans in New Zealand. Ecol. Appl. 8(3), 824–832.CrossRefGoogle Scholar
  18. 18.
    van den Driessche, P., Watmough, J., Nov 2002. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29–48.zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Werner, P., Caswell, H., 1977. Population-growth rates and age versus stage-distribution models for teasel (Dipsacus sylvestris Huds). Ecology 58(5), 1103–1111.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  1. 1.Centre for Mathematical Biology and Department of Biological SciencesUniversity of AlbertaEdmontonCanada
  2. 2.Centre for Mathematical Biology and Department of Mathematical and Statistical SciencesUniversity of AlbertaEdmontonCanada

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