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Biology Bulletin Reviews

, Volume 2, Issue 1, pp 89–104 | Cite as

Two paradigms in mathematical population biology: An attempt at synthesis

  • D. O. Logofet
  • N. G. Ulanova
  • I. N. Belova
Article

Abstract

However popular the slogan of nonlinear ecological interactions has been in theory, practical ecology professes the projection matrix paradigm, which is essentially linear, i.e., the linear matrix model paradigm for discrete structured population dynamics. The dominant eigenvalue λ1 of projection matrix L is considered the growth potential of a population. It provides for a quantitative measure of species adaptation to the given environment, with the measure being adequate and precise, given data of the “identified individuals” type. The case of “identified individuals with uncertain parents” gives rise to uncertainty in the status-specific reproduction rates, which is eliminated in a unique way (for a broad class of structures and life cycle graphs) by maximizing λ1(L) under the constraints ensuing from the data on and knowledge about species biology. The paradigm of linearity gives way to the nonlinear models when species interactions such as competition for shared resources are to be modeled and where the outcome of interaction depends on the population structure of the competitors. This circumstance dictates the need for synthesis of the two paradigms, which is achieved in nonlinear matrix operators as models of interaction between the species whose populations are discrete-structured.

Keywords

Matrix Model Projection Matrix Biology Bulletin Review Sandbar Shark Matrix Population Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Akçkaya, R.H., Burgman, M.A., and Ginzburg, L.R., Applied Population Ecology: Principles and Computer Exercises using RAMAS EcoLab 2.0, 2nd ed., Sunderland, MA: Sinauer, 1999.Google Scholar
  2. Begon, M., Harper, J.L., and Townsend, C.R., Ecology. Individuals, Populations and Communities. 3rd ed., Oxford: Blackwell Sci. Publ, 1996.Google Scholar
  3. Bernardelli, H., Population Waves, J. Burma Res. Soc., 1941, vol. 31, pp. 1–18.Google Scholar
  4. Brewster-Geisz, K.K. and Miller, T.J., Management of the Sandbar Shark, Carcharhinus plumbeus: Implications of a Stage-Based Model, Fish. Bull., 2000, vol. 98, pp. 236–249.Google Scholar
  5. Caswell, H., Matrix Population Models: Construction, Analysis, and Interpretation, Sunderland, MA: Sinauer Associates, 1989.Google Scholar
  6. Caswell, H., Matrix Population Models: Construction, Analysis, and Interpretation, 2nd ed., Sunderland, MA: Sinauer Associates, 2001.Google Scholar
  7. Chistyakova, A.A., Betala pendula, in Diagnozy i klyuchi vozrastnykh sostoyanii lesnykh rastenii. Derev’ya i kustarniki: metodicheskie razrabotki dlya studentov biologicheskikh spetsial’nostei (Diagnoses and Keys to the Age of Forest Plants. Trees and Shrubs: Methodological Development for Students of Biological Disciplines), Moscow: Prometei, 1989, pp. 69–76.Google Scholar
  8. Csetenyi, A.I. and Logofet, D.O., Leslie Model Revisited: Some Generalizations for Block Structures, Ecol. Model., 1989, vol. 48, pp. 277–290.CrossRefGoogle Scholar
  9. Cull, P. and Vogt, A., The Periodic Limits for the Leslie Model, Math. Biosci., 1974, vol. 21, pp. 39–54.CrossRefGoogle Scholar
  10. Cushing, J.M. and Yicang, Z., The Net Reproductive Value and Stability in Matrix Population Models, Natur. Res. Model., 1994, vol. 8, pp. 297–333.Google Scholar
  11. Cushing, J.M., Henson, S.M., and Blackburn, C.C., Multiple Mixed-Type Attractors in Competition Models, J. Biol. Dynam., 2007, vol. 1, no. 4, pp. 347–362.CrossRefGoogle Scholar
  12. Cushing, J.M., Le Varge, S., Chitnis, N., and Henson, S.M., Some Discrete Competition Models and the Competitive Exclusion Principle, J. Diff. Equat. Applic., 2004, vol. 10, nos. 13–15, pp. 1139–1151.CrossRefGoogle Scholar
  13. Falin-ska, K., Plant Demography in Vegetation Succession, Dortrecht: Kluwer Acad. Publ., 1991.CrossRefGoogle Scholar
  14. Feigenbaum, M.J., Quantitative Universality for a Class of Nonlinear Transformations, J. Stat. Phys., 1978, vol. 19, no. 1, p. 25.Google Scholar
  15. Feigenbaum, M.J., The Universal Metric Properties of Nonlinear Transformations, J. Stat. Phys., 1979, vol. 21, no. 6, p. 669.CrossRefGoogle Scholar
  16. Gantmakher, F.R., Teoriya matrits (The Theory of Matrices), Moscow: Nauka, 1967.Google Scholar
  17. Geramita, J.M. and Pullman, N.J., An Introduction to the Application of Nonnegative Matrices to Biological Systems, Queen’s Papers in Pure and Applied Mathematics no. 68, Kingston, Ontario, Canada: Queen’s Univ., 1984.Google Scholar
  18. Goel, N.S., Maitra, S.C., and Montroll, E.W., On the Volterra and Other Nonlinear Models of Interacting Populations, Rev. Modern Phys., 1971, vol. 43, pp. 231–276.CrossRefGoogle Scholar
  19. 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.PubMedCrossRefGoogle Scholar
  20. Hansen, P.E., Leslie Matrix Models: a Mathematical Survey, in Papers on Mathematical Ecology, Csetenyi, A.I., Ed., Budapest: Karl Marx Univ. Economics, 1986, pp. 54–106.Google Scholar
  21. Harary, F., Norman, R.Z., and Cartwright, D., Structural Models: an Introduction to the Theory of Directed Graphs, New York: John Wiley, 1965.Google Scholar
  22. Horn, R.A. and Johnson, C.R., Matrix Analysis, Cambridge: Cambridge Univ. Press, 1990.Google Scholar
  23. Impagliazzo, J., Deterministic Aspects of Mathematical Demography: An Investigation of Stable Population Theory Including an Analysis of the Population Statistics of Denmark. Biomathematics, Berlin: Springer, 1985, vol. 13.Google Scholar
  24. Jørgensen, S.E. and Bendoricchio, G., Fundamentals of Ecological Modelling, 3rd ed., Amsterdam: Elsevier, 2001.Google Scholar
  25. Jørgensen, S.E., Fundamentals of Ecological Modelling. Developments in Environmental Modelling, Amsterdam: Elsevier, 1986, vol. 9.Google Scholar
  26. Jury, E.I., Inners and Stability of Dynamic Systems, New York: Wiley, 1974. Translated under the title Innory i ustoichivost’ dinamicheskikh system, Moscow: Nauka, 1979.Google Scholar
  27. Klochkova, I.N., Generalization of the Theorem on the Reproductive Potential for Logofet Matrices, Vestn. Mosk. Univ., Ser. 1: Matem. Mekhan. 2004, no. 3, pp. 45–48.Google Scholar
  28. Kon, R., Permanence of Discrete-Time Kolmogorov Systems for Two Species and Saturated Fixed Points, J. Math. Biol., 2004, vol. 48, pp. 57–81.PubMedCrossRefGoogle Scholar
  29. Korn, G. and Korn, T., Spravochnik po matematike dlya nauchnykh rabotnikov i inzhenerov (Mathematical Handbook for Scientists and Engineers), Moscow: Nauka, 1973.Google Scholar
  30. 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.CrossRefGoogle Scholar
  31. Lefkovitch, L.P., The Study of Population Growth in Organisms Grouped by Stages, Biometrics, 1965, vol. 21, pp. 1–18.CrossRefGoogle Scholar
  32. Leslie, P.H., On the Use of Matrices in Certain Population Mathematics, Biometrika, 1945, vol. 33, pp. 183–212.PubMedCrossRefGoogle Scholar
  33. Leslie, P.H., Some Further Notes on the Use of Matrices in Population Mathematics, Biometrika, 1948, vol. 35, pp. 213–245.Google Scholar
  34. Lewis, E.G., On the Generation and Growth of a Population, Sankhya. Indian J. Statistics, 1942, vol. 6, pp. 93–96.Google Scholar
  35. Li, C.-K. and Schneider, H., Application of Perron-Frobenius Theory to Population Dynamics, J. Math. Biol., 2002, vol. 44, no. 5, pp. 450–462.PubMedCrossRefGoogle Scholar
  36. Li, T.-Y. and Yorke, J.A., Period Three Implies Chaos, Amer. Math. Monthly, 1975, vol. 82, pp. 982–985.CrossRefGoogle Scholar
  37. Lieffers, V.J., Macdonald, S.E., and Hogg, E.H., Ecology of and Control Strategies for Calamagrostis canadensis in Boreal Forest Sites, Can. J. For. Res., 1993, vol. 23, pp. 2070–2077.CrossRefGoogle Scholar
  38. Logofet, D.O. and Belova, I.N., Nonnegative Matrices as a Tool for Modeling Population Dynamics: Classical Models and Contemporary Expansions, Fundam. Prikl. Matem., 2007, vol. 13, no. 4, pp. 145–164.Google Scholar
  39. Logofet, D.O. and Klochkova, I.N., Mathematics of the Lefkovich Model: Reproductive Potential and Asymptotic Cycles, Mat. Model., 2002, vol. 14, no. 10, pp. 116–126.Google Scholar
  40. Logofet, D.O., Matrices and Graphs: Stability Problems in Mathematical Ecology, Boca Raton, FL: CRC Press, 1993.Google Scholar
  41. Logofet, D.O., Convexity in Projection Matrices: Projection to a Calibration Problem, Ecol. Model., 2008, vol. 216, no. 2, pp. 217–228.CrossRefGoogle Scholar
  42. Logofet, D.O., On the Indecomposability and Imprimitivity of Non-Negative Matrices of a Block Structure, Dokl. Akad. Nauk, 1989, vol. 308, no. 1, pp. 46–49.Google Scholar
  43. Logofet, D.O., Once Again about the Nonlinear Leslie Model: Asymptotic Behavior of a Trajectory, Dokl. Akad. Nauk, 1991a, vol. 318, no. 5, pp. 1077–1081.Google Scholar
  44. Logofet, D.O., Paths and Cycles in the Digraph as Tools for Characterization of Certain Classes of Matrices, Dokl. Akad. Nauk, 1999, vol. 367, no. 3, pp. 295–298.Google Scholar
  45. Logofet, D.O., Svirezhev’s Principle of Substitution and Matrix Models of the Dynamics of Populations with a Complex Structure, Zh. Obshch. Biol., 2010, vol. 71, no. 1, pp. 30–40.PubMedGoogle Scholar
  46. Logofet, D.O., The Theory of Matrix Models of Population Dynamics with Age and Additional Structures, Zh. Obshch. Biol., 1991b, vol. 52, no. 6, pp. 793–804.Google Scholar
  47. Logofet, D.O., Three Sources and Three Component Parts of the Formalism of a Population with Discrete Stage and Age Structures, Mat. Model., 2002, vol. 14, no. 12, pp. 11–22.Google Scholar
  48. Logofet, D.O., Ulanova, N.G., Klochkova, I.N., and Demidova, A.N., Structure and Dynamics of a Clonal Plant Population: Classical Model Results in a Non-Classic Formulation, Ecol. Modell., 2006, vol. 192, pp. 95–106.CrossRefGoogle Scholar
  49. May, R.M., Stability and Complexity in Model Ecosystems, Princeton: N.J.: Princeton Univ. Press, 1973.Google Scholar
  50. May, R.M., Biological Populations Obeying Difference Equations: Stable Points, Stable Cycles and Chaos, J. Theor. Biol., 1975, vol. 51, no. 2, pp. 511–524.PubMedCrossRefGoogle Scholar
  51. Maybee, J.S., Olesky, D.D., Van Den Driessche, P., and Wiener, G., Matrices, Digraphs, and Determinants, SIAM J. Matrix Anal. Appl., 1989, vol. 10, pp. 500–519.CrossRefGoogle Scholar
  52. Mirkin, B.M. and Naumova, L.G., Nauka o rastitel’nosti: istoriya i sovremennoe sostoyanie osnovnykh kontseptsii (The Science about Vegetation: The History and Current State of the Main Concepts), Ufa: Gilem, 1998.Google Scholar
  53. Mokrova, O.V., The Nonlinear Model of Competition between Two Populations with a Structure: The Search for Balance, Diploma Work, Moscow: Moscow State University, 2010.Google Scholar
  54. Nedorezov, L.V., Sadykov, A.M., and Sadykova, D.L., The Dynamics of Abundance of the Green Oak Leaf Roller: The Use of Discrete-Continuous Models with Density-Dependent Nonmonotonic Birth Rate, Zh. Obshch. Biol., 2010, vol. 71, no. 1, pp. 41–51.PubMedGoogle Scholar
  55. Pimm, S.L., Food Webs, London: Chapman Hall, 1982.Google Scholar
  56. Pyšek, P., Pattern of Species Dominance and Factors Affecting Community Composition in Areas Deforested Due to Air Pollution, Vegetatio, 1994, vol. 112, pp. 45–56.CrossRefGoogle Scholar
  57. Rabotnov, T.A., About Coenotical Populations of Plant Species Belonging to the Plant Communities Which Succeed Each Other in Successions, Bot. Zh., 1995, vol. 80, no. 7, pp. 67–72.Google Scholar
  58. Rebele, F. and Lehmann, C., Biological Flora of Central Europe: Calamagrostis epigejos (L.) Roth., Flora, 2001, vol. 196, pp. 325–344.Google Scholar
  59. Salguero-Gómez, R. and Casper, B.B., Keeping Plant Shrinkage in the Demographic Loop, J. Ecol., 2010, vol. 98, no. 2, pp. 312–323.CrossRefGoogle Scholar
  60. Seneta, E., Non-Negative Matrices and Markov Chains, 2nd ed., New York: Springer-Verlag, 1981, Chap. 3.Google Scholar
  61. Shapiro, A.P. and Luppov, S.P., Rekurrentnye uravneniya v teorii populyatsionnoi biologii (Recurrence Equations in the Theory of Population Biology), Moscow: Nauka, 1983.Google Scholar
  62. Sharkovskii, A.N., Coexistence of Cycles with a Continuous Mapping of a Straight Line in Itself, Ukr. Matem. Zh., 1964, vol. 16, pp. 61–71.Google Scholar
  63. Smirnova, O.V., Bobrovskii, M.V., and Khanina, L.G., Assessment and Prediction of Successional Processes in Forest Cenoses on the Basis of Demographic Methods, Byul. MOIP. Otd. Biol., 2001, vol. 106, no. 5, pp. 25–33.Google Scholar
  64. Smirnova, O.V., Chistyakova, A.A., Zaugolnova, L.B., Evstigneev, O.I., Popadiouk, R.V., and Romanovsky, A.M., Ontogeny of Tree, Bot. Zh., 1999, vol. 84, no. 12, pp. 8–20.Google Scholar
  65. Svirezhev, Yu.M. and Logofet, D.O., Ustoichivost’ biologicheskikh soobshchestv (Stability of Biological Communities), Moscow: Nauka, 1978.Google Scholar
  66. Svirezhev, Yu.M., Vito Volterra and Modern Mathematical Ecology, in Vol’terra V. Matematicheskaya teoriya bor’by za sushchestvovanie (V. Volterra. Mathematical Theory of the Struggle for Existence), Moscow: Nauka, 1976, pp. 245–286.Google Scholar
  67. Takada, T. and Hara, T., The Relationship between the Transition Matrix Model and the Diffusion Model, J. Math. Biol., 1994, vol. 32, pp. 789–807.CrossRefGoogle Scholar
  68. Ulanova, N.G., Calamagrostis epigeios, in Biologicheskaya flora Moskovskoi oblasti (Biological Flora of Moscow Region), Moscow: Izd. Mosk. Univ. and “Argus”, 1995, vol. 10, pp. 4–19.Google Scholar
  69. Ulanova, N.G., Belova, I.N., and Logofet, D.O., Competition among Populations with Discrete Structure: The Dynamics of Populations of Reed and Birch Growing Together, Zh. Obshch. Biol., 2008, vol. 69, no. 6, pp. 478–494.Google Scholar
  70. Ulanova, N.G., Demidova, A.N., Logofet, D.O., and Klochkova, I.N., Structure and Dynamics of Calamagrostis canescens Coenopopulation: A Model Approach, Zh. Obshch. Biol., 2002, vol. 63, no. 6, pp. 509–521.PubMedGoogle Scholar
  71. Ulanova, N.G., Plant Age Stages during Succession in Woodland Clearing in Central Russia, in Vegetation Science in Retrospect and Perspective, White, P.S., Mucina, L., Leps, J., and Maarel Van Der, E., Eds., Uppsala: Opulus, 2000, pp. 80–83.Google Scholar
  72. Ulanova, N.G., Zhukovskaya, O.V., Kuksina, N.V., and Demidova, A.N., Structure and Dynamics of Populations of White Birch (Betula pendula Roth.) in Calamagrostis epigeios Phytocenoses of Clear Cuttings of Spruce Forests of Kostroma Region, Byul. MOIP. Otd. Biol., 2005, vol. 110, no. 5, pp. 27–35.Google Scholar
  73. Usher, M.B., Developments in the Leslie Matrix Models, in Mathematical Models in Ecology, Jeffre, J.N.R., Ed., Oxford: Blackwell, 1972, pp. 29–60.Google Scholar
  74. Vasilevich, V.I., Ocherki teoreticheskoi fitotsenologii (Essays in Theoretical Phytocenology), Leningrad: Nauka, 1983.Google Scholar
  75. Voevodin, V.V. and Kuznetsov, Yu.A., Matritsy i vychisleniya (Matrices and Computation), Moscow: Nauka, 1984.Google Scholar
  76. Volterra, V., Lecons sur la Theorie Mathematique de la Lutte pour la Vie, Paris: Gauthier-Villars, 1931.Google Scholar
  77. Werner, P.A. and Caswell, H., Population Growth Rates and Age Versus Stage-Distribution Models for Teasel (Dipsacus sylvestris Huds.), Ecology, 1977, vol. 58, pp. 1103–1111.CrossRefGoogle Scholar
  78. Wikan, A., From Chaos to Chaos. An Analysis of a Discrete Age-Structured Prey-Predator Model, J. Math. Biol., 2001, vol. 43, pp. 471–500.PubMedCrossRefGoogle Scholar
  79. Yakobson, M.V., The Properties of One-Parameter Family of Dynamical Systems xAx*exp(-x), Ukr. Matem. Zh., 1976, vol. 31, no. 2, pp. 239–240.Google Scholar
  80. Zhdanova, O.L. and Frisman, E.Ya., Nonlinear Dynamics of Population Abundance: The Effect of Complexitwy of the Age Structure on the Scenario of Transition to Chaos, Zh. Obshch. Biol., 2011, vol. 72, no. 3, pp. 214–228.PubMedGoogle Scholar
  81. Zuidema, P.A., Brienen, R.J.W., During, H.J., and Guneralp, B., Do Persistently Fast-Growing Juveniles Contribute Disproportionately to Population Growth? A New Analysis Tool for Matrix Models and Its Application to Rainforest Trees, Am. Natur., 2009, vol. 174, no. 5, pp. 709–719.PubMedCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Laboratory of Mathematical Ecology, Obukhov Institute of Atmospheric PhysicsRussian Academy of SciencesMoscowRussia
  2. 2.Department of Geobotany, Biological FacultyMoscow State UniversityMoscowRussia

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