Skip to main content
SpringerLink
Log in

From Population Monitoring to a Mathematical Model: A New Paradigm of Population Research

  • Published:
Biology Bulletin Reviews Aims and scope Submit manuscript

Abstract

The new paradigm of population research means formulating traditional and innovative themes of population theory in terms of a matrix model of the dynamics of the studied population with a discrete (age, stage, or other kind of) structure, analyzing the relevant properties of this model, interpreting them in biological terms, and obtaining objective quantitative characteristics. Our knowledge of the species biology and the method of population monitoring determine the life cycle graph of organisms, which, in turn, generates the pattern of the population projection matrix (PPM), the core of the matrix model, according to the standard rule of matrix theory and predetermines its further properties. Calibration of the PPM according to empirical data gives quantitative certainty to its elements, the population vital rates., whereby the needed properties and quantitative indicators of the population are obtained by the appropriate methods of matrix algebra. The survey gives an overview of the wide range of problems studied within the framework of the new paradigm and the broad abilities the matrix population models possess to solve those problems. The task and methodological difficulties of assessing the population viability based on data from long-term monitoring are considered in the greatest detail. Noted are some current directions in the development and application of the mathematical apparatus of matrix population models.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.
Fig. 6.
Fig. 7.

Similar content being viewed by others

Notes

  1. In the mathematical theory of matrices, a matrix A that has the property A2 = A, i.e., re-projection does not change the result of the first projection, is a projection matrix.

  2. As a consequence of (22), the system of algebraic equations for unknown elements of the matrix G turns out to be overdetermined (AKADEMIK, 2021; GUFO.ME, 2021) and does not have any exact solution.

  3. As a result of the reproductive uncertainty in the data (Logofet et al., 2016a), λ1(L(t)) and λ1(G) are determined only within a certain range of values.

  4. Access to the databases is open, and all models are digitized in the R environment (https://www.r-project.org/foundation/).

  5. Purely random coincidence.

  6. https://www.worldwildlife.org/ecoregions/nt0105.

  7. Horn and Johnson, 1990; Logofet and Ulanova, 2018, §7.

REFERENCES

  1. Abbott, R.J., Barton, N.H., and Good, J.M., Genomics of hybridization and its evolutionary consequences, Mol. Ecol., 2016, vol. 25, no. 11, pp. 2325–2332.

    Article  PubMed  Google Scholar 

  2. Akademik, 2021. https://dic.academic.ru/dic.nsf/ruwiki/214326.

  3. Andreeva, S.I. and Andreev, N.I., Evolyutsionnye preobrazovaniya dvustvorchatykh mollyuskov Aral’skogo morya v usloviyakh ekologicheskogo krizisa (Evolutionary Transformations of Bivalve Mollusks from the Aral Sea in Conditions of Ecological Crisis), Omsk: Omsk. Gos. Pedagog, Univ., 2003.

  4. Antoniou, A., Frantzis, A., Alexiadou, P., Paskou, N., and Poulakakis, N., Evidence of introgressive hybridization between Stenella coeruleoalba and Delphinus delphis in the Greek Seas, Mol. Phylogenet. Evol., 2019, vol. 129, pp. 325–337.

    Article  Google Scholar 

  5. Barraquand, F. and Gimenez, O., Integrating multiple data sources to fit matrix population models for interacting species, Ecol. Model., 2019, vol. 411, art. ID 108713.

  6. Baskin, L.M., Severnyi olen’. Upravlenie povedeniem i populyatsiyami. Olenevodstvo. Okhota (Reindeer: Behavior and Population Management. Reindeer Herding. Hunting), Moscow: KMK, 2009.

  7. Bearzi, S., Bonizzoni, S., Santostasi, N.L., Furey, B., Eddy, L., et al., Dolphins in a scaled-down Mediterranean: the Gulf of Corinth’s odontocetes, Mamm. Rev., 2016, vol. 33, nos. 3–4, pp. 297–331.

    Google Scholar 

  8. Belyakova, G.A., Garibova, L.V., D’yakov, Yu.T., Kamnev, A.N., Sidorova, I.I., et al., Botanika. Kurs al’gologii i mikologii (Botany. Course on Algology and Mycology), D’yakov, Yu.T., Ed., Moscow: Mosk. Gos. Univ., 2007.

  9. Bender, M.H., Baskin, J.M., and Baskin, C.C., Age of maturity and life span in herbaceous, polycarpic perennials, Bot. Rev., 2000, vol. 66, no. 3, pp. 311–349.

    Article  Google Scholar 

  10. Berman, A. and Plemmons, R.J., Nonnegative Matrices in the Mathematical Sciences, Philadelphia: Soc. Ind. Appl. Math., 1994.

    Book  Google Scholar 

  11. Bernardelli, H., Population waves, J. Burma Res. Soc., 1941, vol. 31, pp. 1–18.

    Google Scholar 

  12. Buckley, Y.M., Ramula, S., Blomberg, S.P., Burns, J.H., Crone, E.E., et al., Causes and consequences of variation in plant population growth rate: a synthesis of matrix population models in a phylogenetic context, Ecol. Lett., 2010, vol. 13, pp. 1182–1197.

    Article  PubMed  Google Scholar 

  13. Caswell, H., Matrix Population Models: Construction, Analysis, and Interpretation, Sunderland, MA: Sinauer, 1989.

    Google Scholar 

  14. Caswell, H., Matrix Population Models: Construction, Analysis, and Interpretation, Sunderland, MA: Sinauer, 2001, 2nd ed.

    Google Scholar 

  15. Caswell, H., Life table response experiment analysis of the stochastic growth rate, J. Ecol., 2010, vol. 98, no. 2, pp. 324–333.

    Article  Google Scholar 

  16. Caswell, H., Matrix models and sensitivity analysis of populations classified by age and stage: a vec-permutation matrix approach, Theor. Ecol., 2012, vol. 5, no. 3, pp. 403–417.

    Article  Google Scholar 

  17. Caswell, H. and Salguero-Gómez, R., Age, stage and senescence in plants, J. Ecol., 2013, vol. 101, pp. 585–595.

    Article  PubMed  PubMed Central  Google Scholar 

  18. Caswell, H., Naiman, R.J., and Morin, R., Evaluating the consequences of reproduction in complex salmonid life cycles, Aquaculture, 1984, vol. 43, pp. 123–134.

    Article  Google Scholar 

  19. Che-Castaldo, J., Jones, O., Kendall, B.E., Burns, J.H., Childs, D.Z., et al., Comments to “Persistent problems in the construction of matrix population models,” Ecol. Model., 2020, vol. 416, art. ID 108913. https://doi.org/10.1016/j.ecolmodel.2019.108913

    Article  Google Scholar 

  20. Chesnova, L.V. and Striganova, B.R., Pochvennaya zoologiya—nauka XX veka (Soil Zoology—A Science of the 20th Century), Dobrovol’skii, G.V., Ed., Moscow: Yanus-K, 1999.

  21. Cheteni, A.I., Matrichnaya model’ populyatsii severnykh olenei s uchetom ekofiziologicheskikh pokazatelei (Matrix Model of the Reindeer Population Taking into Account Ecophysiological Indicators), Moscow: Akad. Nauk SSSR, 1988.

  22. Cochran, M.E. and Ellner, S., Simple methods for calculating age-based life history parameters for stage-structured populations, Ecol. Monogr., 1992, vol. 62, no. 3, pp. 345–364.

    Article  Google Scholar 

  23. Cohen, J.E., Ergodicity of age structure in populations with Markovian vital rates, I: Countable states, J. Am. Stat. Assoc., 1976, vol. 71, pp. 335–339.

    Article  Google Scholar 

  24. Cohen, J.E., Comparative statics and stochastic dynamics of age-structured populations, Theor. Popul. Biol., 1979, vol. 16, no. 2, pp. 159–171.

    Article  CAS  PubMed  Google Scholar 

  25. COMADRE, 2021. https://www.compadre-db.org/Data/Comadre

  26. COMPADRE, 2021. https://compadre-db.org/Data/Compadre

  27. Coste, C.F.D. and Pavard, S., Analysis of a multitrait population projection matrix reveals the evolutionary and demographic effects of a life history trade-off, Ecol. Model., 2020, vol. 418, art. ID 108915.

  28. Coste, C.F.D., Austerlitz, F., and Pavard, S., Trait level analysis of multitrait population projection matrices, Theor. Popul. Biol., 2017, vol. 116, pp. 47–58.

    Article  PubMed  Google Scholar 

  29. Csetenyi, A.I. and Logofet, D.O., Leslie model revisited: some generalizations for block structures, Ecol. Model., 1989, vol. 48, pp. 277–290.

    Article  Google Scholar 

  30. Cushing, J.M. and Yicang, Z., The net reproductive value and stability in matrix population models, Nat. Resour. Model., 1994, vol. 8, no. 4, pp. 297–333. https://doi.org/10.1111/j.1939-7445.1994.tb00188.x

    Article  Google Scholar 

  31. Dunham, K., Dinkelacker, S., and Miller, J., A stage-based population model for American alligators in northern latitudes, J. Wildl. Manage., 2014, vol. 78, pp. 440–447.

    Article  Google Scholar 

  32. Dzerzhinskii, F.Ya., Vasil’ev, B.D., and Malakhov, V.V., Zoologiya pozvonochnykh (Zoology of Vertebrates), Moscow: Akademiya, 2013.

  33. Falińska, K., Plant Demography in Vegetation Succession, Dordrecht: Kluwer, 1991.

    Book  Google Scholar 

  34. Fryxell, D.C., Weiler, D.E., Kinnison, M.T., and Palkovacs, E.P., Eco-evolutionary dynamics of sexual dimorphism, Trends Ecol. Evol., 2019, vol. 34, no. 7, pp. 591–594.

    Article  PubMed  Google Scholar 

  35. Furstenberg, H. and Kesten, H., Products of random matrices, Ann. Math. Stat., 1960, vol. 31, pp. 457–469.

    Article  Google Scholar 

  36. Gantmacher, F., Teoriya matrits (The Theory of Matrices), Moscow: Nauka, 1967.

  37. Goodman, L.A., An elementary approach to the population projection-matrix, to the population reproductive value, and to related topics in the mathematical theory of population growth, Demography, 1968, vol. 5, pp. 382–409.

    Article  Google Scholar 

  38. Goodman, L.A., The analysis of population growth when the birth and death rates depend upon several factors, Biometrics, 1969, vol. 25, pp. 659–681.

    Article  CAS  PubMed  Google Scholar 

  39. Goodman, L.A., On the sensitivity of the intrinsic growth rate to changes in the age-specific birth and death rates, Theor. Popul. Biol., 1971, vol. 2, no. 3, pp. 339–354.

    Article  Google Scholar 

  40. Govaert, L., Fronhofer, E.A., Lion, S., et al., Eco-evolutionary feedbacks—Theoretical models and perspectives, Funct. Ecol., 2019, vol. 33, no. 1, pp. 13–30.

    Article  Google Scholar 

  41. GUFO.ME, 2021. https://gufo.me/dict/mathematics_encyclopedia/Переопределенная_Система

  42. Hansen, P.E., Leslie matrix models: a mathematical survey, in Papers on Mathematical Ecology, Csetenyi, A.I., Ed., Budapest: Karl Marx Univ. Econ., 1986, vol. 1, pp. 54–106.

    Google Scholar 

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

    Google Scholar 

  44. Hart, S.P., Turcotte, M.M., and Levine, J.M., Effects of rapid evolution on species coexistence, Proc. Natl. Acad. Sci. U.S.A., 2019, vol. 116, no. 6, pp. 2112–2117.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  45. Hendry, A., Eco-Evolutionary Dynamics, Princeton: Princeton Univ. Press, 2017.

    Book  Google Scholar 

  46. Horn, R.A. and Johnson, C.R., Matrix Analysis, Cambridge: Cambridge Univ. Press, 1990.

    Google Scholar 

  47. How short the short-lived perennial lives, or averaging problems in non-autonomous matrix population models, Russian Foundation for Basic Research project report no. 16-04-00832-a, 2018. https://istina.msu.ru/projects/38954655/.

  48. Hunter, C.M. and Caswell, H., The use of the vec-permutation matrix in spatial matrix population models, Ecol. Model., 2005, vol. 188, pp. 15–21.

    Article  Google Scholar 

  49. Janczyk-Weglarska, J., An ex situ ecological experiment on the morphological and developmental variation of Calamagrostis epigejos (Poaceae), Fragm. Florist. Geobot., 1997, vol. 42, no. 2, pp. 239–247.

    Google Scholar 

  50. Kazantseva, E.S., Population dynamics and seed productivity of short-lived alpine plants in the North-West Caucasus, Cand. Sci. (Biol.) Dissertation, Moscow: Moscow State Univ., 2016.

  51. Kazantseva, E.S., Onipchenko, V.G., and Kipkeev, A.M., Age of the first flowering of herbaceous alpine perennials in the Northwestern Caucasus, Byull. Mosk. O-va. Ispyt. Prir., Otd. Biol., 2016, vol. 121, no. 2, pp. 73–80.

    Google Scholar 

  52. Kemeny, J.G. and Snell, J.L., Finite Markov Chains, Berlin: Springer-Verlag, 1976.

    Google Scholar 

  53. Kendall, B.E., Fujiwara, M., Diaz-Lopez, J., Schneider, S., Voigt, J., and Wiesner, S., Persistent problems in the construction of matrix population models, Ecol. Model., 2019, vol. 406, pp. 33–43.

    Article  Google Scholar 

  54. Keyfitz, N., Introduction to the Mathematics of Population, Reading, MA: Addison-Wesley. 1968.

  55. Klevezal’, G.A., Printsipy i metody opredeleniya vozrasta mlekopitayushchikh (Principles and Methods for Determination of Age of Mammals), Moscow: KMK, 2007.

  56. Klimas, C.A., Cropper, W.P., Jr., Kainer, K.A., and Wadt, L.H.O., Viability of combined timber and non-timber harvests for one species: a Carapa guianensis case study, Ecol. Model., 2012, vol. 246, pp. 147–156.

    Article  Google Scholar 

  57. Körner, C., Alpine Plant Life: Functional Plant Ecology of High Mountain Ecosystems, Berlin: Springer-Verlag, 2003, 2nd ed.

    Book  Google Scholar 

  58. Krasnaya kniga Krasnodarskogo kraya (Rasteniya i griby) (The Red Data Book of the Krasnodar Krai: Plants and Fungi), Litvinskaya, S.A., Ed., Krasnodar: Dizainerskoe Byuro No. 1, 2007, 2nd ed.

  59. Krasnaya kniga Respubliki Adygeya: Redkie i nakhodyashchiesya pod ugrozoi ischeznoveniya ob”ekty zhivotnogo i rastitel’nogo mira (The Red Data Book of the Republic of Adygea: Rare and Endangered Species of Fauna and Flora), in 2 parts, Maikop: Kachestvo, 2012.

  60. Kroon de, H., van Groenendael, J., and Ehrlen, J., Elasticities: a review of methods and model limitations, Ecology, 2000, vol. 81, pp. 607–618.

    Article  Google Scholar 

  61. Law, R., A model for the dynamics of a plant population containing individuals classified by age and size, Ecology, 1983, vol. 64, pp. 224–230.

    Article  Google Scholar 

  62. Lefkovitch, L., The study of population growth in organisms grouped by stages, Biometrics, 1965, vol. 21, no. 1, pp. 1–18.

    Article  Google Scholar 

  63. Leslie, P.H., On the use of matrices in certain population mathematics, Biometrika, 1945, vol. 33, no. 3, pp. 183–212.

    Article  CAS  PubMed  Google Scholar 

  64. Leslie, P.H., Some further notes on the use of matrices in population mathematics, Biometrika, 1948, vol. 35, nos. 3–4, pp. 213–245.

    Article  Google Scholar 

  65. Lewis, E.G., On the generation and growth of a population, Sankhya, 1942, vol. 6, pp. 93–96.

    Google Scholar 

  66. Li, C.-K. and Schneider, H., Application of Perron–Frobenius theory to population dynamics, J. Math. Biol., 2002, vol. 44, pp. 450–462.

    Article  PubMed  Google Scholar 

  67. Logofet, D.O., The indecomposability and imprimitivity of nonnegative matrices of block structure, Dokl. Akad. Nauk SSSR, 1989, vol. 308, no. 1, pp. 46–49.

    Google Scholar 

  68. Logofet, D.O., Towards a theory of matrix models for the dynamics of populations with the age and the additional structures, Zh. Obshch. Biol., 1991, vol. 52, no. 6, pp. 793–804.

    Google Scholar 

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

    Google Scholar 

  70. Logofet, D.O., Three sources and three constituents of the formalism for a population with discrete age and stage structures, Mat. Model., 2002, vol. 14, no. 12, pp. 11–22.

    Google Scholar 

  71. Logofet, D.O., Convexity in projection matrices: projection to a calibration problem, Ecol. Model., 2008, vol. 216, no. 2, pp. 217–228.

    Article  Google Scholar 

  72. Logofet, D.O., Svirezhev’s substitution principle and matrix models for dynamics of populations with complex structures, Zh. Obshch. Biol., 2010, vol. 71, no. 1, pp. 30–40.

    CAS  PubMed  Google Scholar 

  73. Logofet, D.O., Projection matrices revisited: a potential-growth indicator and the merit of indication, J. Math. Sci., 2013a, vol. 193, no. 5, pp. 671–686.

    Article  Google Scholar 

  74. Logofet, D.O., Projection matrices in variable environments: λ1 in theory and practice, Ecol. Model., 2013b, vol. 251, pp. 307–311.

    Article  Google Scholar 

  75. Logofet, D.O., Calamagrostis model revisited: matrix calibration as a constraint maximization problem, Ecol. Model., 2013c, vol. 254, pp. 71–79.

    Article  Google Scholar 

  76. Logofet, D.O., Complexity in matrix population models: polyvariant ontogeny and reproductive uncertainty, Ecol. Complexity, 2013d, vol. 15, pp. 43–51.

    Article  Google Scholar 

  77. Logofet, D.O., Aggregation may or may not eliminate reproductive uncertainty, Ecol. Model., 2017, vol. 363, pp. 187–191.

    Article  Google Scholar 

  78. Logofet, D.O., Averaging the population projection matrices: heuristics against uncertainty and nonexistence, Ecol. Complexity, 2018, vol. 33, no. 1, pp. 66–74.

    Article  Google Scholar 

  79. Logofet, D.O., Does averaging overestimate or underestimate population growth? It depends, Ecol. Model., 2019, vol. 411, art. ID 108744.

  80. Logofet, D.O. and Belova, I.N., Nonnegative matrices as a tool to model population dynamics: classical models and contemporary expansions, J. Math. Sci., 2008, vol. 155, no. 6, pp. 894–907.

    Article  Google Scholar 

  81. Logofet, D.O. and Klochkova, I.N., Mathematics of Lefkovitch model: reproductive potential and asymptotic cycles, Mat. Model., 2002, vol. 14, no. 10, pp. 116–126.

    Google Scholar 

  82. Logofet, D.O. and Salguero-Gómez, R., Novel challenges and opportunities in the theory and practice of matrix population modelling: an editorial for the special feature “Theory and Practice in Matrix Population Modeling,” Ecol. Model., 2021, vol. 443, art. ID 109457. https://doi.org/10.1016/j.ecolmodel.2021.109457

    Article  Google Scholar 

  83. Logofet, D.O. and Ulanova, N.G., Matrichnye modeli v populyatsionnoi biologii. Uchebnoye posobie (Matrix Models in Population Biology: Manual), Moscow: MAKS Press, 2018, 2nd ed. https://elibrary.ru/item.asp?id=32701104.

  84. Logofet, D.O., Ulanova, N.G., and Belova, I.N., Two paradigms in mathematical population biology: An attempt at synthesis, Biol. Bull. Rev., 2012, vol. 2, no. 1, pp. 89–104.

    Article  Google Scholar 

  85. Logofet, D.O., Ulanova N.G., and Belova, I.N., Adaptation on the ground and beneath: Does the local population maximize its λ1? Ecol. Complexity, 2014, vol. 20, pp. 176–184.

    Article  Google Scholar 

  86. Logofet, D.O., Ulanova, N.G., and Belova, I.N., Polyvariant ontogeny in woodreeds: novel models and new discoveries, Biol. Bull. Rev., 2016, vol. 6, no. 5, pp. 365–385.

    Article  Google Scholar 

  87. Logofet, D.O., Ulanova, N.G., and Belova, I.N., From uncertainty to an exact number: developing a method to estimate the fitness of a clonal species with polyvariant ontogeny, Biol. Bull. Rev., 2017a, vol. 7, no. 5, pp. 387–402.

    Article  Google Scholar 

  88. Logofet, D.O., Belova, I.N., Kazantseva, E.S., and Onipchenko, V.G., Local population of Eritrichium caucasicum as an object of mathematical modeling. I. Life cycle graph and a nonautonomous matrix model, Biol. Bull. Rev., 2017b, vol. 7, no. 5, pp. 415–427.

    Article  Google Scholar 

  89. Logofet, D.O., Belova, I.N., Kazantseva, E.S., and Onipchenko, V.G., Local population of Eritrichium caucasicum as an object of mathematical modeling. II. How short does the short-lived perennial live? Biol. Bull. Rev., 2018a, vol. 8, no. 3, pp. 193–202.

    Article  Google Scholar 

  90. Logofet, D.O., Kazantseva, E.S., Belova, I.N., and Onipchenko, V.G., How long does a short-lived perennial live? A modeling approach, Biol. Bull. Rev., 2018b, vol. 8, no. 5, pp. 406–420.

    Article  Google Scholar 

  91. Logofet, D.O., Kazantseva, E.S., Belova, I.N., and Onipchenko, V.G., Local population of Eritrichium caucasicum as an object of mathematical modeling. III. Population growth in the random environment, Biol. Bull. Rev., 2019, vol. 9, no. 5, pp. 453–464.

    Article  Google Scholar 

  92. Logofet, D.O., Kazantseva, E.S., Belova, I.N., and Onipchenko, V.G., Disappointing survival forecast for a local population of Androsace albana in a random environment, Biol. Bull. Rev., 2020a, vol. 10, no. 3, pp. 202–214.

    Article  Google Scholar 

  93. Logofet, D.O., Kazantseva, E.S., and Onipchenko, V.G., Seed bank as a persistent problem in matrix population models: from uncertainty to certain bounds, Ecol. Model., 2020b, vol. 438, art. ID 109284.

  94. Logofet, D.O., Kazantseva, E.S., Belova, I.N., and Onipchenko, V.G., Backward prediction confirms the conclusion on local plant population viability, Biol. Bull. Rev., 2021, vol. 11, no. 5, pp. 462–475.

    Article  Google Scholar 

  95. Lopes, C., Péry, A.R.R., Chaumot, A., and Charles, S., Ecotoxicology and population dynamics: using DEBtox models in a Leslie modeling approach, Ecol. Model., 2005, vol. 251, pp. 307–311.

    Google Scholar 

  96. Lotka, A.J., Elements of Physical Biology, Baltimore: Williams and Wilkins, 1925. https://archive.org/details/elementsofphysic017171mbp.

    Google Scholar 

  97. Marescot, L., Gimenez, O., Duchamp, C., Marboutinc, E., and Chapron, G., Reducing matrix population models with application to social animal species, Ecol. Model., 2012, vol. 232, pp. 91–96.

    Article  Google Scholar 

  98. Marcus, M. and Mink, H., A Survey of Matrix Theory and Matrix Inequalities, Boston, MA: Allyn and Bacon, 1964.

    Google Scholar 

  99. Maslov, A.A. and Logofet, D.O., Joint population dynamics of Vaccinium myrtillus and V. vitis-idaea in the protected postfire Cladina–Vaccinium pine forest. Markov model with averaged transition probabilities, Biol. Bull. Rev., 2021, vol. 11, no. 5, pp. 438–450.

    Article  Google Scholar 

  100. Masterov, V.B. and Romanov, M.S., Tikhookeanskii orlan Haliaeetus pelagicus: ekologiya, evolyutsiya, okhrana (Steller’s Sea Eagle Haliaeetus pelagicus: Ecology, Evolution, and Conservation), Moscow: KMK, 2014.

  101. MathWorks, 2021. https://www.mathworks.com/help/matlab/ref/eig.html.

  102. McDonald, J.J., Paparella, P., Michae, J., and Tsatsomeros, M.J., Matrix roots of eventually positive matrices, Linear Algebra Appl., 2014, vol. 456, pp. 122–137.

    Article  Google Scholar 

  103. Molnár, E. and Bokros, S., Studies on the demography and life history of Taraxacum serotinum (Waldst. et Kit.) Poir, Folia Geobot., 1996, vol. 31, pp. 453–464.

    Article  Google Scholar 

  104. Monitoring of alpine juvenile populations: long-term forecast of survival by stochastic growth rate, Russian Foundation for Basic Research project no. 19-04-01227-a, 2019. https://istina.msu.ru/projects/261137811/.

  105. Morris, W.F., Tuljapurkar, S., Haridas, C.V., Menges, E.S., Horvitz, C.C., and Pfister, C.A., Sensitivity of the population growth rate to demographic variability within and between phases of the disturbance cycle, Ecol. Lett., 2006, vol. 9, pp. 1331–1341.

    Article  CAS  PubMed  Google Scholar 

  106. Morris, W.F., Tuljapurkar, S., Haridas, C.V., Menges, E.S., Horvitz, C.C., and Pfister, C.A., A stage-based matrix population model of invasive lionfish with implications for control, Biol. Invasions, 2011, vol. 13, pp. 7–12.

    Article  Google Scholar 

  107. Nakhutsrishvili, G.S. and Gamtsemlidze, Z.G., Zhizn’ rastenii v extremal’nykh usloviyakh vysokogorii: na primere Tsentral’nogo Kavkasa (Plant Life in Extreme High-Altitude Conditions by Example of the Central Caucasus), Leningrad: Nauka, 1984.

  108. Nguyen, V., Buckley, Y.M., Salguero-Gomez, R., and Wardle, G.M., Consequences of neglecting cryptic life stages from demographic models, Ecol. Model., 2019, vol. 408, art. ID 108723. https://doi.org/10.1016/j.ecolmodel.2019.108723

    Article  Google Scholar 

  109. Notov, A.A. and Zhukova, L.A., The concept of ontogenesis polyvariance and modern evolutionary morphology, Biol. Bull. (Moscow), 2019, vol. 46, no. 1, pp. 47–55.

    Article  Google Scholar 

  110. Onipchenko, V.G. and Komarov, A.S., Population dynamics and life history features of three alpine plant species in the Northwest Caucasus, Zh. Obshch. Biol., 1997, vol. 58, no. 6, pp. 64–75.

    Google Scholar 

  111. On the ground and beneath: the limits of adaptation in the local population of a clonal plant with multivariant ontogeny, Russian Foundation for Basic Research project no. 13-04-01836-a, 2015. https://istina.msu.ru/projects/8473479/.

  112. Oseledets, V.I., A multiplicative ergodic theorem: Lyapunov characteristic parameters for dynamical systems, Tr. Mosk. Matem. O-va, 1968, vol. 19, pp. 179–210.

    Google Scholar 

  113. Ozgul, A., Childs, D.Z., Oli, M.K., Armitage, K.B., Blumstein, D.T., et al., Coupled dynamics of body mass and population growth in response to environmental change, Nature, 2010, vol. 466, pp. 482–485.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  114. Paniw, M., Ozgul, A., and Salguero-Gómez, R., Interactive life-history traits predict sensitivity of plants and animals to temporal autocorrelation, Ecol. Lett., 2018, vol. 21, no. 2, pp. 275–286.

    Article  PubMed  Google Scholar 

  115. Pathikonda, S., Ackleh, A.S., Hasenstein, K.H., and Mopper, S., Invasion, disturbance, and competition: modeling the fate of coastal plant populations, Conserv. Biol., 2008, vol. 23, no. 1, pp. 164–173.

    Article  PubMed  Google Scholar 

  116. Pavlov, V.N. and Onipchenko, V.G., High altitude vegetation, Itogi Nauki Tekh., Ser.: Bot., 1987, vol. 7, pp. 3–38.

    Google Scholar 

  117. Pinto, C.F., Torrico-Bazoberry, D., Flores-Pradod, L., Bustamante, R.O., and Niemeyer, H.M., Demographic and performance effects of alternative host use in a Neotropical treehopper (Hemiptera: Membracidae), Ecol. Model., 2020, vol. 416, art. ID 108905.

  118. Politi, T. and Popolizio, M., On stochasticity preserving methods for the computation of the matrix pth root, Math. Comput. Simul., 2015, vol. 110, pp. 53–68.

    Article  Google Scholar 

  119. Polivariantnost’ razvitiya organizmov, populyatsii i soobshchestv (Polyvariant Development of Organisms, Populations, and Communities), Yoshkar-Ola: Mariisk. Gos. Univ., 2006.

  120. Pollard, J.H., On the use of the direct matrix product in analysing certain stochastic population models, Biometrika, 1966, vol. 53, pp. 397–415.

    Article  Google Scholar 

  121. Populyatsionno-ontogeneticheskoe napravlenie v Rossii i blizhnem zarubezh’e (Population-Ontogenetic Studies in Russia and Neighbor Countries), Tver: Tversk. Gos. Univ., 2018.

  122. Poshkurlat, A.P., Structure and development of chia turf, Uch. Zap. Mosk. Gos. Pedagog. Inst. im. V.I. Lenina, 1941, vol. 30, no. 1, pp. 101–151.

    Google Scholar 

  123. Protasov, V.Yu. and Logofet, D.O., Rank-one corrections of nonnegative matrices, with an application to matrix population models, SIAM J. Matrix Anal. Appl., 2014, vol. 35, no. 2, pp. 749–764.

    Article  Google Scholar 

  124. Rabonov, T.A., Life cycle of perennial herbaceous plants in meadow phytocoenoses, Tr. Bot. Inst., Akad. Nauk SSSR, Ser. 3. Geobot., 1950, no. 6.

  125. Razzhevaikin, V.N. and Tyrtyshnikov, E.E., On Stability Indicators of Nonnegative Matrices, Dokl. Math., 2020, vol. 101, no. 1, pp. 43–45.

    Article  Google Scholar 

  126. Rees, M. and Ellner, S.P., Integral projection models for populations in temporally varying environments, Ecol. Monogr., 2009, vol. 79, pp. 575–594.

    Article  Google Scholar 

  127. Romanov, M.S. and Masterov, V.B., Matrix model of Steller’s sea eagle Haliaeetus pelagicus population in Sakhalin, Mat. Biol. Bioinf., 2008, vol. 3, no. 2, pp. 36–49.

    Article  Google Scholar 

  128. Romanov, M.S. and Masterov, V.B., Low breeding performance of the Steller’s sea eagle (Haliaeetus pelagicus) causes the populations to decline, Ecol. Model., 2020, vol. 420, art. ID 108877.

  129. Roth, G. and Caswell, H., Hyperstate matrix models: extending demographic state spaces to higher dimensions, Methods Ecol. Evol., 2016, vol. 7, pp. 1438–1450.

    Article  Google Scholar 

  130. Salguero-Gómez, R. and Casper, B.B., Keeping plant shrinkage in the demographic loop, J. Ecol., 2010, vol. 98, no. 2, pp. 312–323.

    Article  Google Scholar 

  131. Salguero-Gómez, R., Jones, O.R., Archer, C.R., Buckley, Y.M., et al., The COMPADRE Plant Matrix Database: an open online repository for plant demography, J. Ecol., 2015, vol. 103, pp. 202–218.

    Article  Google Scholar 

  132. Salguero-Gómez, R., Jones, O.R., Archer, C.R., Buckley, Y.M., et al., COMADRE: a global data base of animal demography, J. Anim. Ecol., 2016, vol. 85, pp. 371–384.

    Article  PubMed  PubMed Central  Google Scholar 

  133. Santostasi, N.L., Ciucci, P., Bearzi, G., Bonizzoni, S., and Gimenez, O., Assessing the dynamics of hybridization through a matrix modelling approach, Ecol. Model., 2020, vol. 431, art. ID 109120. https://doi.org/10.1016/j.ecolmodel.2020.109120

    Article  Google Scholar 

  134. Sanz, L., Conditions for growth and extinction in matrix models with environmental stochasticity, Ecol. Model., 2019, vol. 411, art. ID 108797.

  135. Sanz, L. and Bravo de la Parra, R., Stochastic matrix metapopulation models with fast migration: re-scaling survival to the fast scale, Ecol. Model., 2019, vol. 418, art. ID 108829. https://doi.org/10.1016/j.ecolmodel.2019.108829

    Article  Google Scholar 

  136. Severtsov, A.S., Biogeocenotic control of natural selection and evolution rate. To the 150th anniversary of the first edition of the book “The Origin of Species…” by Ch. Darwin, Zool. Zh., 2009, vol. 88, no. 9, pp. 1027–1038.

  137. Shefferson, R.P. and Salguero-Gómez, R., Eco-evolutionary dynamics in plants: interactive processes at overlapping time-scales and their implications, J. Ecol., 2015, vol. 103, pp. 789–797.

    Article  Google Scholar 

  138. Sosnová, M., Herben, T., Martínkov, J., and Klimešová, J., To resprout or not to resprout? Modeling population dynamics of a root-sprouting monocarpic plant under various disturbance regimes, Plant Ecol., 2014, vol. 215, pp. 1245–1254.

    Article  Google Scholar 

  139. Svirezhev, Yu.M. and Logofet, D.O., Ustoichivost’ biologicheskikh soobshchestv (Resistance of Biological Communities), Moscow: Nauka, 1978.

  140. Takada, T. and Shefferson, R., The long and winding road of evolutionary demography: preface, Popul. Ecol., 2018, vol. 60, pp. 3–7.

    Article  Google Scholar 

  141. Tsenopopulyatsii rastenii (osnovnye ponyatiya i struktura) (Plant Cenopopulations: General Terms and Structure), Smirnov, O.V., Zaugol’nova, L.B., et al., Eds., Moscow: Nauka, 1976.

  142. Tsenopopulyatsii rastenii (ocherki populyatsionnoi biologii) (Plant Cenopopulations: Essays on Population Biology), Zaugol’nova, L.B., Zhukova, L.A., Komarov, A.S., and Smirnov, O.V., Eds., Moscow: Nauka, 1988.

    Google Scholar 

  143. Tuljapurkar, S.D., Demography in stochastic environments. II. Growth and convergence rates, J. Math. Biol., 1986, vol. 24, pp. 569–581.

    Article  CAS  PubMed  Google Scholar 

  144. Tuljapurkar, S.D., Population Dynamics in Variable Environments, New York: Springer-Verlag, 1990.

    Book  Google Scholar 

  145. Tveraa, T., Fauchald, P., Henaug, C., et al., An examination of a compensatory relationship between food limitation and predation in semi-domestic reindeer, Oecologia, 2003, vol. 137, pp. 370–376.

    Article  PubMed  Google Scholar 

  146. Ulanova, N.G., Population structure and dynamics of woodreeds in various ecological conditions, in Trudy Okskogo gosudarstvennogo prirodnogo biosfernogo zapovednika (Transactions of the Oka State Nature Biosphere Reserve), Ryazan: Golos Gubernii, 2015, no. 34, pp. 201–207.

  147. Ulanova, N.G., Demidova, A.N., Klochkova, I.N., and Logofet, D.O., Structure and dynamics of Calamagrostis canescens coenopopulation: a model approach, Zh. Obshch. Biol., 2002, vol. 63, no. 6, pp. 509–521.

    CAS  PubMed  Google Scholar 

  148. Ulanova, N.G., Belova, I.N., and Logofet, D.O., Competition among populations with discrete structure: the population dynamics of woodreed and birch growing together, Zh. Obshch. Biol., 2008, vol. 69, no. 6, pp. 478–494.

    Google Scholar 

  149. Uranov, A.A., Age spectrum of phytocoenopopulations as a function of time and energetic wave processes, Biol. Nauki, 1975, no. 2, pp. 7–34.

  150. Volterra, V., Leçons Sur la Théorie Mathématique de la Lutte Pour la Vie, Paris: Gauthiers-Villars, 1931.

    Google Scholar 

  151. Vries de, C., Desharnais, R.A., and Caswell, H., A matrix model for density-dependent selection in stage-classified populations, with application to pesticide resistance in Tribolium, Ecol. Model., 2020, vol. 416, art. ID 108875. https://doi.org/10.1016/j.ecolmodel.2019.108875

    Article  Google Scholar 

  152. Wikberg, S. and Svensson, B.M., Ramet demography in a ring-forming clonal sedge, J. Ecol., 2003, vol. 91, pp. 847–854.

    Article  Google Scholar 

  153. Williams, H.J., Jacquemyn, H., Ochocki, B.M., Brys, R., and Miller, T.E.X., Life history evolution under climate change and its influence on the population dynamics of a long-lived plant, J. Ecol., 2015, vol. 103, pp. 798–808.

    Article  Google Scholar 

  154. World of mathematics, 2021. https://matworld.ru/posledovatelnosti/chislovye-posledovatelnosti.php.

  155. Zhmylev, P.Yu., Alekseev, Yu.E., Karpukhina, E.A., and Balandin, S.A., Biomorfologiya rastenii: illyustrirovannyi slovar’. Uchebnoe posobie (Plant Biomorphology: An Illustrated Dictionary. Manual), Moscow: Grif i K, 2005, 2nd ed.

  156. Zhukova, L.A., Ontogenies and reproduction cycles of plants, Zh. Obshch. Biol., 1983, vol. 44, no. 3, pp. 361–374.

    Google Scholar 

  157. Zhukova, L.A., Polyvariance of the meadow plants, in Zhiznennye formy v ekologii i sistematike rastenii (Life Forms in Ecology and Plant Systematics), Moscow: Mosk. Gos. Pedagog. Inst., 1986, pp. 104–114.

  158. Zhukova, L.A., Populyatsionnaya zhizn’ lugovykh rastenii (Population Life of Meadow Plants), Yoshkar-Ola: Lanar, 1995.

  159. Zhukova, L.A. and Komarov, A.S., Polyvariance of ontogenesis and dynamics of plant populations, Zh. Obshch. Biol., 1990, vol. 51, no. 4, pp. 450–461.

    Google Scholar 

  160. Zhukova, L.A. and Komarov, A.S., Quantitative analysis of dynamic polyvariance in cenopopulations of the broadleaf plantain at different planting densities, Biol. Nauki, 1991, no. 8, pp. 51–67.

Download references

ACKNOWLEDGEMENTS

The authors are grateful to M.S. Romanov, who worked out the text of the manuscript in detail and made valuable comments.

Funding

The work was supported by the Russian Foundation for Basic Research, project no. 20-14-50311.

Author information

Authors and Affiliations

  1. A.M. Obukhov Institute of Atmospheric Physics, Russian Academy of Sciences, 119017, Moscow, Russia

    D. O. Logofet

  2. M.V. Lomonosov Moscow State University, Biological Faculty, Department of Plant Ecology and Geography, 119234, Moscow, Russia

    N. G. Ulanova

Authors
  1. D. O. Logofet
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. N. G. Ulanova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to D. O. Logofet or N. G. Ulanova.

Ethics declarations

Conflict of interest. The authors declare no conflict of interest.

Statement on the welfare of humans or animals. This article does not contain any studies involving humans or animals performed by any of the authors.

Additional information

Translated by M. Shulskaya

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Logofet, D.O., Ulanova, N.G. From Population Monitoring to a Mathematical Model: A New Paradigm of Population Research. Biol Bull Rev 12, 279–303 (2022). https://doi.org/10.1134/S2079086422030057

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S2079086422030057

Search

Navigation