Skip to main content
SpringerLink
Log in

Numerical Study of Bifurcations Occurring at Fast Timescale in a Predator–Prey Model with Inertial Prey-Taxis

  • Chapter
  • First Online:
Advanced Mathematical Methods in Biosciences and Applications

Abstract

Bifurcations occurring in a system of partial differential equations (PDEs) describing spatiotemporal dynamics of predator and prey populations with prey-taxis have been studied numerically. The model of the local kinetics of the system assumes logistic reproduction of the prey and a simplest Lotka–Volterra functional response of the predator. Since the model ignores relatively slow and rare demographic processes of birth and death in the population of predator, the predator abundance is kept constant under the considered zero-flux boundary conditions. The abundance of predator populations together with the predator taxis coefficient were used as bifurcation parameters in the numerical study that have been made with help of two qualitatively different techniques of discretization: the Bubnov–Galerkin method and grid method of lines. It has been shown that the considered simple model of prey-taxis in predator–prey system demonstrates complex bifurcation transitions leading to periodic, quasi-periodic and chaotic spatiotemporal dynamics.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. G.R. Ivanitskii, A.B. Medvinskii, M.A. Tsyganov, From disorder to order as applied to the movement of micro-organisms. Phys. Usp. 34(4), 289–316 (1991). https://doi.org/10.1070/PU1994v037n10ABEH000049

    Article  Google Scholar 

  2. G.R. Ivanitskii, A.B. Medvinskii, M.A. Tsyganov, From the dynamics of population autowaves generated by living cells to neuroinformatics. Phys. Usp. 37(10), 961–989 (1994). https://doi.org/10.1070/PU1994v037n10ABEH000049

    Article  Google Scholar 

  3. J.D. Murray, Mathematical Biology: I. An Introduction, vol I (Springer, New York, 2002), p. 576. https://doi.org/10.1007/b98868

    Book  Google Scholar 

  4. J.D. Murray, Mathematical Biology: II. Spatial Models and Biomedical Applications, vol II (Springer, New York, 2003), p. 811. https://doi.org/10.1023/A:1025805822749

    Book  MATH  Google Scholar 

  5. A. Okubo, S. Levin, Diffusion and Ecological Problems: Modern Perspectives, vol 467 (Springer, New York, 2001). https://doi.org/10.1007/978-1-4757-4978-6

    Book  MATH  Google Scholar 

  6. A.F.G. Dixon, Insect Predator-Prey Dynamics: Ladybird Beetles and Biological Control (Cambridge University Press, Cambridge, 2000), p. 257. ISBN 0-521-62203-4

    Google Scholar 

  7. O.V. Kovalev, V.V. Vechernin, Description of a new wave process in population with reference to introduction and spread of the leaf beetle Zygogramma suturalis F. (Coleoptera, Chrysomelidae). Entomol. Rev. 65(3), 93–112 (1986)

    Google Scholar 

  8. P.J. Moran, C.J. DeLoach, T.L. Dudley, J. Sanabria, Open field host selection and behavior by tamarisk beetles (Diorhabda spp. ) (Coleoptera: Chrysomelidae) in biological control of exotic saltcedars (Tamarix spp.) and risks to non-target athel (T. aphylla) and native Frankenia spp. Biol. Control 50, 243–261 (2009)

    Article  Google Scholar 

  9. L. Winder, C.J. Alexander, J.M. Holland, W.O. Symondson, J.N. Perry, C. Woolley, Predatory activity and spatial pattern: the response of generalist carabids to their aphid prey. J. Anim. Ecol. 74(3), 443–454 (2005). https://doi.org/10.1111/j.1365-2656.2005.00939.x

    Article  Google Scholar 

  10. L. Winder, C.J. Alexander, J.M. Holland, C. Woolley, J.N. Perry, Modelling the dynamic spatio-temporal response of predators to transient prey patches in the field. Ecol. Lett. 4(6), 568–576 (2001). https://doi.org/10.1046/j.1461-0248.2001.00269.x

    Article  Google Scholar 

  11. A.B. Medvinsky, B.V. Adamovich, A. Chakraborty, E.V. Lukyanova, T.M. Mikheyeva, N.I. Nurieva, N.P. Radchikova, A.V. Rusakov, T.V. Zhukova, Chaos far away from the edge of chaos: a recurrence quantification analysis of plankton time series. Ecol. Complex. 23, 61–67 (2015). https://doi.org/10.1016/j.ecocom.2015.07.001

    Article  Google Scholar 

  12. N.B. Petrovskaya, ‘Catch me if you can’: Evaluating the population size in the presence of a spatial pattern. Ecol. Complex. 34, 100–110 (2018). https://doi.org/10.1016/j.ecocom.2017.03.003

    Article  Google Scholar 

  13. Y. Dolak, T. Hillen, Cattaneo models for chemosensitive movement numerical solution and pattern formation. J. Math. Biol. 46, 153–170 (2003). https://doi.org/10.1007/s00285-003-0221-y

    Article  MathSciNet  Google Scholar 

  14. H.C. Berg, Motile behavior of bacteria. Phys. Today 53(1), 24–29 (2000). https://doi.org/10.1063/1.882934

    Article  Google Scholar 

  15. H.C. Berg, E. coli in Motion (Springer, New York, 2004), p. 133. https://doi.org/10.1007/b97370

    Book  Google Scholar 

  16. E.O. Budrene, H.C. Berg, Complex patterns formed by motile cells of Escherichia coli. Nature 349(6310), 630–633 (1991). https://doi.org/10.1038/349630a0

    Article  Google Scholar 

  17. R. Tyson, S.R. Lubkin, J.D. Murray, Model and analysis of chemotactic bacterial patterns in a liquid medium. J. Math. Biol. 38, 359–375 (1999). https://doi.org/10.1007/s002850050153

    Article  MathSciNet  MATH  Google Scholar 

  18. N. Sapoukhina, Y. Tyutyunov, R. Arditi, The role of prey-taxis in biological control: a spatial theoretical model. Am. Nat. 162(1), 61–76 (2003). https://doi.org/10.1086/375297

    Article  Google Scholar 

  19. Y.V. Tyutyunov, O.V. Kovalev, L.I. Titova, Spatial demogenetic model for studying phenomena observed upon introduction of the ragweed leaf beetle in the South of Russia. Math. Model. Nat. Phenom. 8(6), 80–95 (2013). https://doi.org/10.1051/mmnp/20138606

    Article  MathSciNet  MATH  Google Scholar 

  20. Y. Tyutyunov, I. Senina, R. Arditi, Clustering due to acceleration in the response to population gradient: a simple self-organization model. Am. Nat. 164(6), 722–735 (2004). https://doi.org/10.1086/425232

    Article  Google Scholar 

  21. Y.V. Tyutyunov, N.Y. Sapoukhina, I. Senina, R. Arditi, Explicit model for searching behavior of predator. Zh. Obshch. Biol. 63(2), 137–148 (2002). (in Russian)

    Google Scholar 

  22. Y.V. Tyutyunov, L.I. Titova, I.N. Senina, Prey–taxis destabilizes homogeneous stationary state in spatial Gause–Kolmogorov-type model for predator–prey system. Ecol. Complex. 31, 170–180 (2017). https://doi.org/10.1016/j.ecocom.2017.07.001

    Article  Google Scholar 

  23. Y.V. Tyutyunov, L.I. Titova, Simple models for studying complex spatiotemporal patterns of animal behavior. Deep-Sea Res. II Top. Stud. Oceanogr. 140, 193–202 (2017). https://doi.org/10.1016/j.dsr2.2016.08.010

    Article  Google Scholar 

  24. Y.V. Tyutyunov, A.D. Zagrebneva, F.A. Surkov, A.I. Azovsky, Microscale patchiness of the distribution of copepods (Harpacticoida) as a result of trophotaxis. Biophysics 54(3), 355–360 (2009). https://doi.org/10.1134/S000635090903018X

    Article  Google Scholar 

  25. Y.V. Tyutyunov, A.D. Zagrebneva, F.A. Surkov, A.I. Azovsky, Derivation of density flux equation for intermittently migrating population. Oceanology 50(1), 67–76 (2010). https://doi.org/10.1134/S000143701001008X

    Article  Google Scholar 

  26. R. Arditi, Y. Tuytyunov, A. Morgulis, V. Govorukhin, I. Senina, Directed movement of predators and the emergence of density-dependence in predator-prey models. Theor. Popul. Biol. 59(3), 207–221 (2001). https://doi.org/10.1006/tpbi.2001.1513

    Article  MATH  Google Scholar 

  27. V.N. Govorukhin, A.B. Morgulis, I.N. Senina, Y.V. Tyutyunov, Modelling of active migrations for spatially distributed populations. Surv. Appl. Ind. Math. 6(2), 271–295 (1999). (in Russian)

    MATH  Google Scholar 

  28. V.N. Govorukhin, A.B. Morgulis, Y.V. Tyutyunov, Slow taxis in a predator–prey model. Dokl. Math. 61(3), 420–422 (2000). ISSN 1064-5624

    MATH  Google Scholar 

  29. A.J. Lotka, Elements of Physical Biology (Williams & Wilkins, Baltimore, 1925), p. 460

    MATH  Google Scholar 

  30. V. Volterra, Fluctuations dans la lutte pour la vie: leurs lois fondamentales et de réciproctié Bulletin de la SMF, vol 67 (1939), pp. 135–151

    Google Scholar 

  31. Y.V. Tyutyunov, N.Y. Sapoukhina, A.B. Morgulis, V.N. Govorukhin, Mathematical model of active migrations as a foraging strategy in trophic communities. Zh. Obshch. Biol. 62(3), 253–262 (2001). (in Russian)

    Google Scholar 

  32. M.A. Tsyganov, V.N. Biktashev, J. Brindley, A.V. Holden, G.R. Ivanitsky, Waves in systems with cross-diffusion as a new class of nonlinear waves. Phys. Uspekhi. 50(3), 263–286 (2007). https://doi.org/10.1070/PU2007v050n03ABEH006114

    Article  Google Scholar 

  33. F.S. Berezovskaya, A.S. Isaev, G.P. Karev, R.G. Khlebopros, The role of taxis in dynamics of forest insects. Dokl. Biol. Sci. 365(1–6), 148–151 (1999). ISSN 0012-4966

    Google Scholar 

  34. F.S. Berezovskaya, G.P. Karev, Bifurcations of travelling waves in population taxis models. Phys. Usp. 42(9), 917–929 (1999). https://doi.org/10.1070/PU1999v042n09ABEH000564

    Article  Google Scholar 

  35. F.S. Berezovskaya, A.S. Novozhilov, G.P. Karev, Families of traveling impulses and fronts in some models with cross-diffusion. Nonlinear Anal. Real World Appl. 9(5), 1866–1881 (2008). https://doi.org/10.1016/j.nonrwa.2007.06.001

    Article  MathSciNet  MATH  Google Scholar 

  36. T. Hillen, K.J. Painter, A user's guide to PDE models for chemotaxis. J. Math. Biol. 58(1–2), 183–217 (2009). https://doi.org/10.1007/s00285-008-0201-3

    Article  MathSciNet  MATH  Google Scholar 

  37. P. Kareiva, G. Odell, Swarms of predators exhibit preytaxis if individual predators use are-restricted search. Am. Nat. 130(2), 233–270 (1987). https://doi.org/10.1086/284707

    Article  Google Scholar 

  38. V. Rai, Spatial Ecology: Patterns and Processes, vol 138 (Bentham Science Publishers, Sharjah, 2013). https://doi.org/10.2174/97816080549091130101

    Book  Google Scholar 

  39. V. Rai, R.K. Upadhyay, N.K. Thakur, Complex population dynamics in heterogeneous environments: effects of random and directed animal movements. Int. J. Nonlin. Sci. Num. Simulat. 13(3–4), 299–309 (2012). https://doi.org/10.1515/ijnsns-2011-0115

    Article  MathSciNet  MATH  Google Scholar 

  40. J.I. Tello, D. Wrzosek, Predator–prey model with diffusion and indirect prey-taxis. Math. Model. Meth. Appl. Sci. 26(11), 2129–2162 (2016)

    Article  MathSciNet  Google Scholar 

  41. N.K. Thakur, R. Gupta, R.K. Upadhyay, Complex dynamics of diffusive predator–prey system with Beddington–DeAngelis functional response: The role of prey-taxis. Asian-Eur. J. Math. 10(3), 1750047 (2017). https://doi.org/10.1142/S1793557117500474

    Article  MathSciNet  MATH  Google Scholar 

  42. Y. Tyutyunov, L. Titova, R. Arditi, A minimal model of pursuit-evasion in a predator–prey system. Math. Model. Nat. Phenom. 2(4), 122–134 (2007). https://doi.org/10.1051/mmnp:2008028

    Article  MathSciNet  MATH  Google Scholar 

  43. I. Hataue, Spurious numerical solutions in higher dimensional discrete systems. AIAA J. 33(1), 163–164 (1995). https://doi.org/10.2514/3.12350

    Article  Google Scholar 

  44. S.M. Garba, A.B. Gumel, J.M.-S. Lubuma, Dynamically-consistent non-standard finite difference method for an epidemic model. Math. Comput. Model. 53(1–2), 131–150 (2011). https://doi.org/10.1016/j.mcm.2010.07.026

    Article  MathSciNet  MATH  Google Scholar 

  45. L. Chen, A. Jüngel, Analysis of a parabolic cross-diffusion population model without self-diffusion. J. Differ. Equ. 224(1), 39–59 (2006). https://doi.org/10.1016/j.jde.2005.08.002

    Article  MathSciNet  MATH  Google Scholar 

  46. A. Wolf, J.B. Swift, H.L. Swinney, J.A. Vastano, Determining Lyapunov exponents from a time series. Phys. D Nonlinear Phenom. 16(3), 285–317 (1985). https://doi.org/10.1016/0167-2789(85)90011-9

    Article  MathSciNet  MATH  Google Scholar 

  47. V.N. Govorukhin, Package MATDS (2004), http://kvm.math.rsu.ru/matds/

  48. P. Turchin, Complex Population Dynamics: A Theoretical/Empirical Synthesis, vol 35 (Princeton University Press, Princeton, 2003). 450 p

    MATH  Google Scholar 

  49. A.B. Medvinsky, S.V. Petrovskii, I.A. Tikhonova, D.A. Tikhonov, B.-L. Li, E. Venturino, H. Malchow, G.R. Ivanitsky, Spatio-temporal pattern formation, fractals, and chaos in conceptual ecological models as applied to coupled plankton-fish dynamics. Phys. Usp. 45(1), 27–57 (2002). https://doi.org/10.1070/PU2002v045n01ABEH000980

    Article  Google Scholar 

  50. S.V. Petrovskii, H. Malchow, Wave of chaos: New mechanism of pattern formation in spatio-temporal population dynamics. Theor. Popul. Biol. 59(2), 157–174 (2001). https://doi.org/10.1006/tpbi.2000.1509

    Article  MATH  Google Scholar 

  51. A. Chakraborty, M. Singh, D. Lucy, P. Ridland, Predator-prey model with prey-taxis and diffusion. Math. Comput. Model. 46(3–4), 482–498 (2007). https://doi.org/10.1016/j.mcm.2006.10.010

    Article  MathSciNet  MATH  Google Scholar 

  52. A. Chakraborty, M. Singh, D. Lucy, P. Ridland, A numerical study of the formation of spatial patterns in twospotted spider mites. Math. Comput. Model. 49(9), 1905–1919 (2009). https://doi.org/10.1016/j.mcm.2008.08.013

    Article  MathSciNet  MATH  Google Scholar 

  53. A. Chakraborty, M. Singh, P. Ridland, Effect of prey–taxis on biological control of the two-spotted spider mite—a numerical approach. Math. Comput. Model. 50(3–4), 598–610 (2009). https://doi.org/10.1016/j.mcm.2009.01.005

    Article  MathSciNet  MATH  Google Scholar 

  54. A.B. Medvinsky, N.I. Nurieva, A.V. Rusakov, B.V. Adamovich, Deterministic chaоs and the problem of predictability in population dynamics. Biophysics 62(1), 92–108 (2017). https://doi.org/10.1134/S0006350917010122

    Article  Google Scholar 

Download references

Acknowledgments

The research was funded by the project 0259-2014-0004 (state reg.no. 01201363188) of SSC RAS “Development of GIS-based methods of modelling marine and terrestrial ecosystems” (Tyutyunov), by the basic part of the state assignment research, project 1.5169.2017/8.9 of the Southern Federal University “Fundamental and applied problems of mathematical modelling” (Titova), and RFBR grant 18-01-00453 “Multistable spatiotemporal scenarios for population models” (Tyutyunov, Zagrebneva, Govorukhin).

Author information

Authors and Affiliations

  1. Federal Research Center The Southern Scientific Centre of the Russian Academy of Sciences (SSC RAS), Rostov-on-Don, Russia

    Yuri V. Tyutyunov

  2. Southern Federal University, Rostov-on-Don, Russia

    Yuri V. Tyutyunov

  3. Department of Computer and Computer-Based System Software, Faculty of IT Systems and Technologies, Don State Technical University, Rostov-on-Don, Russia

    Anna D. Zagrebneva

  4. Vorovich Institute of Mathematics, Mechanics and Computer Sciences, Southern Federal University, Rostov-on-Don, Russia

    Vasiliy N. Govorukhin & Lyudmila I. Titova

Authors
  1. Yuri V. Tyutyunov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Anna D. Zagrebneva
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Vasiliy N. Govorukhin
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Lyudmila I. Titova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yuri V. Tyutyunov .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Howard University, Washington, DC, USA

    Faina Berezovskaya

  2. Department of Mathematics, Howard University, Washington, DC, USA

    Bourama Toni

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Tyutyunov, Y.V., Zagrebneva, A.D., Govorukhin, V.N., Titova, L.I. (2019). Numerical Study of Bifurcations Occurring at Fast Timescale in a Predator–Prey Model with Inertial Prey-Taxis. In: Berezovskaya, F., Toni, B. (eds) Advanced Mathematical Methods in Biosciences and Applications. STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health. Springer, Cham. https://doi.org/10.1007/978-3-030-15715-9_10

Download citation

Publish with us

Policies and ethics

Search

Navigation