Theory in Biosciences

, Volume 133, Issue 3–4, pp 135–143 | Cite as

Richards-like two species population dynamics model

  • Fabiano Ribeiro
  • Brenno Caetano Troca Cabella
  • Alexandre Souto Martinez
Original Paper

Abstract

The two-species population dynamics model is the simplest paradigm of inter- and intra-species interaction. Here, we present a generalized Lotka–Volterra model with intraspecific competition, which retrieves as particular cases, some well-known models. The generalization parameter is related to the species habitat dimensionality and their interaction range. Contrary to standard models, the species coupling parameters are general, not restricted to non-negative values. Therefore, they may represent different ecological regimes, which are derived from the asymptotic solution stability analysis and are represented in a phase diagram. In this diagram, we have identified a forbidden region in the mutualism regime, and a survival/extinction transition with dependence on initial conditions for the competition regime. Also, we shed light on two types of predation and competition: weak, if there are species coexistence, or strong, if at least one species is extinguished.

Keywords

Complex systems Population dynamics (ecology) Pattern formation ecological  

References

  1. Araujo RP, McElwain DLS (2004) A history of the study of solid tumour growth: the contribution of mathematical modelling. Bull Math Biol 66:1039–1091PubMedCrossRefGoogle Scholar
  2. Arruda TJ, González RS, Terçariol CAS, Martinez AS (2008) Arithmetical and geometrical means of generalized logarithmic and exponential functions: generalized sum and product operators. Phys Lett A 372:2578–2582CrossRefGoogle Scholar
  3. Barberis L, Condat C, Roman P (2011) Vector growth universalities. Chaos Solitons Fractals 44:1100–1105CrossRefGoogle Scholar
  4. Bettencourt LMA, Lobo J, Helbing D, Kuhnert C, West GB (2007) Growth, innovation, scaling, and the pace of life in cities. Proc Nat Acad Sc 104:7301–7306CrossRefGoogle Scholar
  5. Bomze I (1995) Lotka–Volterra equation and replicator dynamics: new issues in classification. Biol Cybern 72:447–453CrossRefGoogle Scholar
  6. Cabella BCT, Martinez AS, Ribeiro F (2011) Data collapse, scaling functions, and analytical solutions of generalized growth models. Phys Rev E 83:061902CrossRefGoogle Scholar
  7. Cabella BCT, Ribeiro F, Martinez AS (2012) Effective carrying capacity and analytical solution of a particular case of the Richards-like two-species population dynamics model. Phys A 391:1281–1286CrossRefGoogle Scholar
  8. Cavalieri LF, Kocak H (1995) Intermittent transition between order and chaos in an insect pest population. J Theor Biol 175:231–234CrossRefGoogle Scholar
  9. Cross SS (1997) Fractals in Pathology. J Pathol 182:1–8PubMedCrossRefGoogle Scholar
  10. dOnofrio A (2009) Fractal growth of tumors and other cellular populations: linking the mechanistic to the phenomenological modeling and vice versa. Chaos Solitons Fractals 41:875–880Google Scholar
  11. dos Santos LS, Cabella BCT, Martinez AS (2014) Generalized Allee effect model. Theory Biosci doi:10.1007/s12064-014-0199-6 PubMedGoogle Scholar
  12. Edelstein-Keshet L (ed) (2005) Mathematical models in Biology. SIAM, PhiladelphiaGoogle Scholar
  13. Espíndola AL, Bauch C, Cabella BCT, Martinez AS (2011) An agent-based computational model of the spread of tuberculosis. J Stat Mech 2011:P5003CrossRefGoogle Scholar
  14. Espíndola AL, Girardi D, Penna TJP, Bauch C, Martinez AS, Cabella BCT (2012) Exploration of the parameter space in an agent-based model of tuberculosis spread: emergence of drug resistance in developing vs developed countries. Int J Mod Phy C 23:12500461–12500469CrossRefGoogle Scholar
  15. Espíndola AL, Girardi D, Penna TJP, Bauch C, Cabella BCT, Martinez AS (2014) An antibiotic protocol to minimize emergence of drug-resistant tuberculosis. Phys A 400:80–92CrossRefGoogle Scholar
  16. Fowler CW (1981) Density dependence as related to life history strategy. Ecology 62:602–610CrossRefGoogle Scholar
  17. Gavrilets S, Hastings A (1995) Intermittency and transient chaos from simple frequency- dependen selection. Proc R Soc B Biol Sci 261:233–238CrossRefGoogle Scholar
  18. Gompertz B (1825) On the nature of the function expressive of the law of human mortality, and on the new mode of determining the value of life contingencies. Phil Trans Royal Soc Lond A 115:153Google Scholar
  19. Gould H, Tobochnik J, Christian W (2006) An introduction to computer simulation methods. Addison-WesleyGoogle Scholar
  20. Guiot C, Degiorgis PG, Delsanto PP, Gabriele P, Deisboeck TS (2003) Does tumor growth follow a “universal law”? J Theoretical Biol 225:147–151PubMedCrossRefGoogle Scholar
  21. Harrison M (2001) Dynamical mechanism for coexistence of dispersing species. J Theoretical Biol 213:53–72PubMedCrossRefGoogle Scholar
  22. Hastings A (2004) Transients: the key to long-term ecological understanding? Trends Ecol Evol 19(1):39–45PubMedCrossRefGoogle Scholar
  23. Imre AR, Bogaert J (2004) The fractal dimension as a measure of the quality of habitats. Acta Biotheor 52(1):41–56PubMedCrossRefGoogle Scholar
  24. Kaitala V (1999) Dynamic complexities in host-parasitoid interaction. J Theor Biol 197:331–341PubMedCrossRefGoogle Scholar
  25. Kozusko F, Bourdeau M (2007) A unified model of sigmoid tumour growth based on cell proliferation and quiescence. Cell Prolif 40:824–834PubMedCrossRefGoogle Scholar
  26. Lai Y (1995a) Persistence of supertransients of spatiotemporal chaotic dynamical-systems in noisy environment. Phy Lett A 200:418–422CrossRefGoogle Scholar
  27. Lai Y (1995b) Unpredictability of the asymptotic attractors in phasecoupled oscillators. Phys Rev E 51:2902–2908CrossRefGoogle Scholar
  28. Lai Y, Winslow R (1995) Geometric-properties of the chaotic saddle responsible for supertransients in spatiotemporal chaotic systems. Phys Rev Lett 74:5208–5211PubMedCrossRefGoogle Scholar
  29. Martinez AS, González RS, Espíndola AL (2009) Generalized exponential function and discrete growth models. Phys A 388:2922–2930CrossRefGoogle Scholar
  30. Martinez AS, González RS, Terçariol CAS (2008) Continuous growth models in terms of generalized logarithm and exponential functions. Phys A 387:5679–5687CrossRefGoogle Scholar
  31. Mombach JCM, Lemke N, Bodmann BEJ, Idiart MAP (2002a) A mean-field theory of cellular growth. Europhy Lett 59:923–928CrossRefGoogle Scholar
  32. Mombach JCM, Lemke N, Bodmann BEJ, Idiart MAP (2002b) A mean-field theory of cellular growth. Europhy Lett 60:489–489CrossRefGoogle Scholar
  33. Murray JD (2002) Mathematical biology I: an introduction. Springer, New YorkGoogle Scholar
  34. Motoike IN, Adamatzky A (2005) Three-valued logic gates in reaction-diffusion excitable media. Chaos Solitons Fractals 24:107–114CrossRefGoogle Scholar
  35. Novozhilov AS, Berezovskaya FS, Koonin EV, Karev GP (2006) Mathematical modeling of tumor therapy with oncolytic viruses: regimes with complete tumor elimination within the framework of deterministic models. Biol Dir 1:6CrossRefGoogle Scholar
  36. Nowak MA, Anderson RM, McLean AR, Wolfs TF, Goudsmit J, May RM (1991) Antigenic diversity thresholds and the development of aids. Science 254:963–969PubMedCrossRefGoogle Scholar
  37. Pereira MA, Martinez AS (2010) Pavlovian prisoner’s dilemma analytical results, the quasi-regular phase and spatio-temporal patterns. J Theretical Biol 265:346–358CrossRefGoogle Scholar
  38. Pereira MA, Martinez AS, Espíndola AL (2008) Prisoner’s dilemma in one-dimensional cellular automata: visualization of evolutionary patterns. Int J Mod Phy C 19:187–201CrossRefGoogle Scholar
  39. Ribeiro F (2014) A non-phenomenological model to explain population growth behaviors. http://arxiv.org/abs/1402.3676. Accessed 8 Aug 2014
  40. Richards FJ (1959) A flexible growth functions for empirical use. J Exp Bot 10:290–300CrossRefGoogle Scholar
  41. Saether BE, Engen Matthysen SE (2002) Demographic characteristics and population dynamical patterns of solitary birds. Science 295:2070–2073Google Scholar
  42. Savageau MA (1979) Growth of complex systems can be related to the properties of their underlying determinants. Proc Natl Acad Sci USA 76(11):5413–5417PubMedCentralPubMedCrossRefGoogle Scholar
  43. Silby RM, Barker D, Denham MC, Hone J, Pagel M (2005) On the regulation of populations of mammals, birds, fish, and insects. Science 309:607–610CrossRefGoogle Scholar
  44. Sibly RM, Hone J (2002) Population growth rate and its determinants: an overview. Philos Trans R Soc Lond Ser B 357:1153–1170CrossRefGoogle Scholar
  45. Strzalka D (2009) Connections between von Foerster coalition growth model and Tsallis \(q\)-exponential. Acta Physica Polonica B 40:41–47Google Scholar
  46. Strzalka D, Grabowski F (2008) Towards possible \(q\)-generalizations of the Malthus and Verhulst growth models. Phys A 387:2511–2518CrossRefGoogle Scholar
  47. Tokeshi M, Arakaki S (2012) Habitat complexity in aquatic systems: fractals and beyond. Hydrobiologia 685:2747CrossRefGoogle Scholar
  48. Tsallis C (1988) Possible generalization of Boltzmann-Gibbs statistics. J Stat Phy 52:479–487CrossRefGoogle Scholar
  49. Tsallis C (1994) What are the numbers experiments provide? Química Nova 17:468–471Google Scholar
  50. Tsoularis A, Wallace J (2002) Analysis of logistic growth models. Math Biosci 179:21–55PubMedCrossRefGoogle Scholar
  51. von Foerster H, Mora PM, Amiot LW (1960) Doomsday: friday, 13 November, A.D. 2026. Science 132(3436):1291–1295CrossRefGoogle Scholar
  52. West GB, Brown JH, Enquist BJ (2001) A general model for ontogenetic growth. Nature 413:628–631PubMedCrossRefGoogle Scholar
  53. Wodarz D (2001) Viruses as antitumor weapons: defining conditions for tumor remission. Cancer Res 61(8):3501–3507PubMedGoogle Scholar
  54. Wodarz D, Komarova N (2005) Computational biology of cancer: lecture notes and mathematical modeling. Scientific Publishing Company, SingaporeCrossRefGoogle Scholar
  55. Yeomans JM (1992) Statistical mechanics of phase transitions. Oxford Science Publications.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Fabiano Ribeiro
    • 1
  • Brenno Caetano Troca Cabella
    • 2
  • Alexandre Souto Martinez
    • 3
    • 4
  1. 1.Departamento de Ciências Exatas (DEX)Universidade Federal de Lavras (UFLA)LavrasBrazil
  2. 2.Sapra AssessoriaSão CarlosBrazil
  3. 3.Faculdade de Filosofia, Ciências e Letras de Ribeirão Preto (FFCLRP)Universidade de São Paulo (USP)Ribeirão PretoBrazil
  4. 4.Instituto Nacional de Ciência e Tecnologia em Sistemas Complexos (INCT-SC)Rio de Janeiro Brazil

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