Skip to main content

Projection matrices revisited: a potential-growth indicator and the merit of indication


The mathematics of matrix models for age- and/or stage-structured population dynamics substantiates the use of the dominant eigenvalue λ 1 of the projection matrix L as a measure of the growth potential, or of adaptation, for a given species population in modern plant or animal demography. The calibration of L = T +F on the “identified-individuals-of-unknown-parents” kind of empirical data determines precisely the transition matrix T, but admits arbitrariness in the estimation of the fertility matrix F. We propose an adaptation principle that reduces calibration to the maximization of λ 1(L) under the fixed T and constraints on F ensuing from the data and expert knowledge. A theorem has been proved on the existence and uniqueness of the maximizing solution for projection matrices of a general pattern. A conjugated maximization problem for a “potential-growth indicator” under the same constraints has appeared to be a linear-programming problem with a ready solution, the solution testing whether the data and knowledge are compatible with the population growth observed.

This is a preview of subscription content, access via your institution.


  1. 1.

    R. H. Akcakaya, M. A. Burgman, and L. R. Ginzburg, Applied Population Ecology: Principles and Computer Exercises Using RAMAS EcoLab 2.0, Sinauer, Sunderland (1999).

  2. 2.

    H. Bernardelli, “Population waves,” J. Burma Res. Soc., 31, 1–18 (1941).

    Google Scholar 

  3. 3.

    K. K. Brewster-Geisz and T. J. Miller, “Management of the sandbar shark, Carcharhinus plumbeus: implications of a stage-based model,” Fish. Bull., 98, 236–249 (2000).

    Google Scholar 

  4. 4.

    T. de Camino-Beck and M. A. Lewis, “On net reproductive rate and the timing of reproductive output,” Am. Naturalist, 172, No. 1, 128–139 (2008).

    Article  Google Scholar 

  5. 5.

    H. Caswell, Matrix Population Models: Construction, Analysis, and Interpretation, Sinauer, Sunderland (1989).

    Google Scholar 

  6. 6.

    H. Caswell, Matrix Population Models: Construction, Analysis, and Interpretation, Sinauer, Sunderland (2001).

    Google Scholar 

  7. 7.

    A. I. Csetenyi and D. O. Logofet “Leslie model revisited: some generalizations for block structures,” Ecol. Model., 48, 277–290 (1989).

    Article  Google Scholar 

  8. 8.

    P. Cull and A. Vogt, “The periodic limits for the Leslie model,” Math. Biosci., 21, 39–54 (1974).

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    J. M. Cushing and Z. Yicang, “The net reproductive value and stability in matrix population models,” Nat. Resour. Model., 8, 297–333 (1994).

    Google Scholar 

  10. 10.

    G. M. Fikhtengol’ts, A Course in Differential and Integral Calculus [in Russian], Vol. 1, Nauka, Moscow (1970).

  11. 11.

    F. R. Gantmakher, Theory of Matrices [in Russian], Nauka, Moscow (1967).

    Google Scholar 

  12. 12.

    J. M. Geramita and N. J. Pullman, An Introduction to the Application of Nonnegative Matrices to Biological Systems, Queen’s Papers Pure Appl. Math., No. 68. Queen’s Univ., Kingston, Ontario, Canada (1984).

  13. 13.

    L. A. Goodman, “The analysis of population growth when the birth and death rates depend upon several factors,” Biometrics, 25, 659–681 (1969).

    MathSciNet  Article  Google Scholar 

  14. 14.

    P. E. Hansen, “Leslie matrix models: A mathematical survey,” in: A. I. Csetenyi, ed., Papers on Mathematical Ecology, I, Karl Marx Univ. of Economics, Budapest (1986), pp. 54–106.

    Google Scholar 

  15. 15.

    F. Harary, R. Z. Norman, and D. Cartwright, Structural Models: An Introduction to the Theory of Directed Graphs, Wiley, New York (1965).

    MATH  Google Scholar 

  16. 16.

    R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge Univ. Press, London (1990).

    MATH  Google Scholar 

  17. 17.

    J. Impagliazzo, Deterministic Aspects of Mathematical Demography: An Investigation of Stable Population Theory Including an Analysis of the Population Statistics of Denmark, Biomathematics, Vol. 13, Springer, Berlin (1985).

  18. 18.

    I. N. Klochkova, “Expansion of the reproductive potential theorem for Logofet matrices,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 3, 45–48 (2004).

  19. 19.

    G. Korn and T. Korn, Mathematical Handbook for Scientists and Engineers [in Russian], Nauka, Moscow (1973).

    Google Scholar 

  20. 20.

    L. P. Lefkovitch, “The study of population growth in organisms grouped by stages,” Biometrics, 21, 1–18 (1965).

    Article  Google Scholar 

  21. 21.

    P. H. Leslie, “On the use of matrices in certain population mathematics,” Biometrika, 33, 183–212 (1945).

    MathSciNet  MATH  Article  Google Scholar 

  22. 22.

    E. G. Lewis, “On the generation and growth of a population,” Sankhya: Indian J. Stat., 6, 93–96 (1942).

    Google Scholar 

  23. 23.

    C.-K. Li and H. Schneider, “Application of Perron–Frobenius theory to population dynamics,” J. Math. Biol., 44, 450–462 (2002).

    MathSciNet  MATH  Article  Google Scholar 

  24. 24.

    D. O. Logofet, “The theory of matrix models for population dynamics with age and additional structures,” Zh. Obshch. Biol., 52, No. 6, 793–804 (1991).

    Google Scholar 

  25. 25.

    D. O. Logofet, Matrices and Graphs: Stability Problems in Mathematical Ecology, CRC Press, Boca Raton (1993).

    Google Scholar 

  26. 26.

    D. O. Logofet, “Paths and cycles as tools for characterizing some classes of matrices,” Dokl. Math., 60, No. 1, 46–49 (1999).

    Google Scholar 

  27. 27.

    D. O. Logofet, “Three sources and three constituents of the formalism for a population with discrete age and stage structures,” Mat. Model., 14, 11–22 (2002).

    MathSciNet  MATH  Google Scholar 

  28. 28.

    D. O. Logofet, “Convexity in projection matrices: projection to a calibration problem,” Ecol. Model., 216, 217–228 (2008).

    Article  Google Scholar 

  29. 29.

    D. O. Logofet, “Svirezhev’s substitution principle and matrix models for the dynamics of populations with complex structures,” Zh. Obshch. Biol., 71, No. 1, 30–40 (2010),

    Google Scholar 

  30. 30.

    D. O. Logofet and I. N. Belova, “Nonnegative matrices as a tool to model population dynamics: classical models and contemporary expansions,” J. Math. Sci., 155, No. 6, 894–907 (2008).

    MathSciNet  MATH  Article  Google Scholar 

  31. 31.

    D. O. Logofet and I. N. Klochkova, “Mathematics of the Lefkovitch model: the reproductive potential and asymptotic cycles,” Mat. Model., 14, 116–126 (2002).

    MathSciNet  MATH  Google Scholar 

  32. 32.

    D. O. Logofet, N. G. Ulanova, I. N. Klochkova, and A. N. Demidova, “Structure and dynamics of a clonal plant population: Classical model results in a non-classic formulation,” Ecol. Model., 192, 95–106 (2006).

    Article  Google Scholar 

  33. 33.

    J. S. Maybee, D. D. Olesky, P. van den Driessche, and G. Wiener, “Matrices, digraphs, and determinants,” SIAM J. Matrix Anal. Appl., 10, 500–519 (1989).

    MathSciNet  MATH  Article  Google Scholar 

  34. 34.

    V. Romanovsky, Un théorème sur les zeros des matrices non négatives, Bull. Soc. Math. Fr., 61, 213–219 (1933).

    MathSciNet  Google Scholar 

  35. 35.

    R. Salguero-Gómez and B. B. Casper, “Keeping plant shrinkage in the demographic loop,” J. Ecol., 98, No. 2, 312–323 (2010).

    Article  Google Scholar 

  36. 36.

    R. Salguero-Gómez and H. de Kroon, “Matrix projection models meet variation in the real world,” J. Ecol., 98, 250–254 (2010).

    Article  Google Scholar 

  37. 37.

    E. Seneta, Non-Negative Matrices and Markov Chains, Springer, New York (1981).

    MATH  Book  Google Scholar 

  38. 38.

    Yu. M. Svirezhev and D. O. Logofet, Stability of Biological Communities, Mir, Moscow (1983).

    Google Scholar 

  39. 39.

    N. G. Ulanova, “Plant age stages during succession in woodland clearing in central Russia,” in: Vegetation Science in Retrospect and Perspective, Opulus, Uppsala (2000), pp. 80–83.

  40. 40.

    N. G. Ulanova, I. N. Belova, and D. O. Logofet, “On the competition among discrete-structured populations: a matrix model for population dynamics of woodreed and birch growing together,” Zh. Obshch. Biol., 69, No. 6, 478–494 (2008).

    Google Scholar 

  41. 41.

    N. G. Ulanova and A. N. Demidova, “The population biology of woodreed (Calamagrostis canescens (Web.) Roth) in spruce clearings of the southern taiga,” Byull. Mosk. Obshch. Ispyt. Prir., Otd. Biol., 106, No. 5, 51–58 (2001).

    Google Scholar 

  42. 42.

    N. G. Ulanova, A. N. Demidova, D. O. Logofet, and I. N. Klochkova, “Structure and dynamics of the woodreed Calamagrostis canescens coenopopulation: a model approach,” Zh. Obshch. Biol., 63, No. 6, 509–521 (2002).

    Google Scholar 

  43. 43.

    V. V. Voyevodin and Yu. A. Kuznetsov, Matrices and Calculations [in Russian], Nauka, Moscow (1984).

    Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Dmitrii O. Logofet.

Additional information

Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 17, No. 6, pp. 41–63, 2011/12.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Logofet, D.O. Projection matrices revisited: a potential-growth indicator and the merit of indication. J Math Sci 193, 671–686 (2013).

Download citation


  • Matrix Model
  • Vertex Versus
  • Projection Matrice
  • Nonnegative Matrice
  • Sandbar Shark