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Nonlinear Dynamics

, Volume 95, Issue 2, pp 875–892 | Cite as

Spatiotemporal dynamics of an NPZ model with prey-taxis and intratrophic predation

  • Evgeniya GirichevaEmail author
Original Paper
  • 142 Downloads

Abstract

Spatiotemporal dynamics of plankton community and the nutrients in a vertical column of water are considered. In this paper, the temporal model is first researched to determine the effect of the density-dependent death rate of predator on the system dynamics. Subsequently, the influence of intraspecific predation is investigated in a spatially distributed model. The effects of this factor and the predator’s search activity on the possibility of spatial pattern formation are also analyzed. Necessary conditions for the Turing and wave instabilities are obtained through local stability analysis around the spatially homogeneous equilibrium. Numerical analysis shows that for an unequal diffusion coefficient, increasing the search activity of zooplankton results in its homogeneous vertical distribution. In addition, increasing the diffusion causes spatial heterogeneity. The stabilizing role of intraspecific predation in the spatial model is obscure. Escalating this factor can also lead to a wave instability. A similar effect on the system stability is provided by taxis in the case of equal diffusion coefficients.

Keywords

NPZ model Intratrophic predation Prey-taxis Turing and wave instabilities 

Notes

Acknowledgements

This work is supported by the Russian Foundation for Fundamental Research (Grant No. 18-01-00213).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest concerning the publication of this manuscript.

References

  1. 1.
    Volterra, V.: Variations and fluctuations of the number of individuals in animal species living together. In: Chapman, R.N. (ed.) Animal Ecology. McGraw-Hill, New York (1926)Google Scholar
  2. 2.
    Rosenzweig, M.L., MacArthur, R.H.: Graphical representation and stability conditions of predator–prey interactions. Am. Nat. 97(895), 209–223 (1963)Google Scholar
  3. 3.
    Arditi, R., Ginzburg, L.R.: Coupling in predator–prey dynamics: ratio dependence. J. Theor. Biol. 139, 311–326 (1989)Google Scholar
  4. 4.
    Berezovskaya, F.S., Karev, G., Arditi, R.: Parametric analysis of the ratiodependent predator–prey model. J. Math. Biol. 43, 221–246 (2001)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Hassell, M.P.: Arthropod Predator–Prey Systems. Princeton University Press, Princeton (1978)zbMATHGoogle Scholar
  6. 6.
    Bazykin, A.D.: Nonlinear Dynamics of Interacting Populations. World Scientific, Singapore (1998)Google Scholar
  7. 7.
    Svirezhev, Yu.M, Logofet, D.O.: Stability of Biological Communities. MIR Publication, Moscow (1983)Google Scholar
  8. 8.
    Wangersky, P.J.: Lotka–Volterra population models. Ann. Rev. Ecol. Syst. 9, 189–218 (1978)Google Scholar
  9. 9.
    Chow, P.L., Tam, W.C.: Periodic and traveling wave solutions to Volterra–Lotka equations with diffusion. Math. Biol. 38(6), 643–658 (1976)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Dunbar, S.R.: Traveling wave solutions of diffusive Lotka–Volterra equations. J. Math. Biol. 17, 11–32 (1983)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Owen, M.R., Lewis, M.A.: How predation can slow, stop or reverse a prey invasion. Math. Biol. 63, 655–684 (2001)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Polis, G.A.: The evolution and dynamics of intratrophic predation. Ann. Rev. Ecol. Syst. 12, 225–251 (1981)Google Scholar
  13. 13.
    Ruan, S., Ardito, A., Ricciardi, P., DeAngelis, D.L.: Coexistence in competition models with density-dependent mortality. C. R. Biol. 330, 845–854 (2007)Google Scholar
  14. 14.
    Kuang, Y., Fagan, W.F., Loladze, I.: Biodiversity, habitat area, resource growth rate and interference competition. Bull. Math. Biol. 65, 497–518 (2003)zbMATHGoogle Scholar
  15. 15.
    Lobry, C., Harmand, J.: A new hypothesis to explain the coexistence of n species in the presence of a single resource. C. R. Biol. 329, 40–46 (2006)Google Scholar
  16. 16.
    Kohlmeier, C., Ebenhoh, W.: The stabilizing role of cannibalism in a predator–prey system. Bull. Math. Biol. 57, 401–411 (1995)zbMATHGoogle Scholar
  17. 17.
    Pitchford, J., Brindley, J.: Intratrophic predation in simple predator–prey models. Bull. Math. Biol. 60, 937–953 (1998)zbMATHGoogle Scholar
  18. 18.
    Jang, S., Baglama, J., Seshaiyer, P.: Droop models of nutrient–plankton interaction with intratrophic predation. Appl. Math. Comput. 169(2), 1106–1128 (2005)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Zaret, T.M., Suffern, J.S.: Vertical migration in zooplankton as a predator avoidance mechanism. Limnol. Oceanogr. 21, 804–813 (1976)Google Scholar
  20. 20.
    Bollens, S.M., Frost, B.W.: Predator induced diel vertical migration in a marine planktonic copepod. J. Plankton Res. 11, 1047–1065 (1989)Google Scholar
  21. 21.
    McLaren, J.A.: Effect of temperature on growth of zooplankton and the adaptive value of vertical migration. J. Fish. Res. Board Can. 20, 685–727 (1963)Google Scholar
  22. 22.
    Han, B.P., Straskraba, M.: Modeling patterns of zooplankton diel vertical migration. J. Plankton Res. 20, 1463–1487 (1998)Google Scholar
  23. 23.
    Keller, E.F., Segel, L.A.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399–415 (1970)zbMATHGoogle Scholar
  24. 24.
    Horstmann, D.: From 1970 until present:the Keller–Segel model in chemotaxis and its consequences I. Jahresber. DMV 105(3), 103–165 (2003)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Ivanitsky, G., Medvinsky, A., Tsyganov, M.: From disorder to order as applied to the movement of microorganisms. Adv. Phys. Sci. Am. Inst. Phys. 34(4), 289–316 (1991)Google Scholar
  26. 26.
    Kareiva, P., Odell, G.: Swarms of predators exhibit prey taxis if individual predators use area restricted search. Am. Nat. 130, 233–270 (1987)Google Scholar
  27. 27.
    Turchin, P.: Quantitative Analysis of Movement: Measuring and Modeling Population Redistribution in Animals and Plants. Sinauer Associates, Sunderland (1998)Google Scholar
  28. 28.
    Berezovskaya, F.S., Karev, G.P.: Bifurcations of travelling waves in population taxis models. Phys. Usp. 42, 917–929 (1999)Google Scholar
  29. 29.
    Turing, A.M.: The chemical basis of the morphogenesis. Philos. Trans. R. Soc. Lond. B 237, 37–72 (1952)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Segel, L.F., Jackson, J.L.: Dissipative structure. An explanation and an ecological example. J. Theor. Biol. 37, 345–359 (1972)Google Scholar
  31. 31.
    Levin, S.A., Segel, L.A.: Hypothesis for origin of plankton patchiness. Nature 259, 659 (1976)Google Scholar
  32. 32.
    Malchow, H.: Spatio-temporal pattern formation in nonlinear nonequilibrium plankton dynamics. Proc. R. Soc. Lond. B. 251, 103–109 (1993)Google Scholar
  33. 33.
    Malchow, H.: Flow-and locomotion-induced pattern formation in nonlinear population dynamics. Ecol. Model. 82, 257–264 (1995)Google Scholar
  34. 34.
    Malchow, H.: Motional instability in prey–predator system. J. Theor. Biol. 204, 639–647 (2000)Google Scholar
  35. 35.
    Sun, G.Q., Jin, Z., Zhao, Y.G., Liu, Q.X., Li, L.: Spatial pattern in a predator–prey system with both self- and cross-diffusion. Int. J. Mod. Phys. 20(1), 71–84 (2009)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Zhang, G.,Wang, X.: Effect of diffusion and cross-diffusion in a predator-prey model with a transmissible disease in the predator species. Abstr. Appl. Anal. 2014, 167856 (2014)Google Scholar
  37. 37.
    Tello, J.I., Wrzosek, D.: Predator–prey model with diffusion and indirect prey-taxis. Math. Models Methods Appl. Sci. 26(11), 2129–2162 (2016)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Tyutyunov, Y.V., Titova, L.I., Senina, I.N.: Prey-taxis destabilizes homogeneous stationary state in spatial Gause–Kolmogorov-type model for predator–prey system. Ecol. Complex. 31, 170–180 (2017)Google Scholar
  39. 39.
    Wang, X., Wang, W., Zhang, G.: Global bifurcation of solutions for a predator–prey model with prey-taxis. Math. Methods Appl. Sci. 38, 431–443 (2015)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Wang, Q., Song, Y., Shao, L.J.: Nonconstant positive steady states and pattern formation of 1D prey-taxis systems. Nonlinear Sci. 27(1), 71–97 (2017)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Petrovskii, S.V., Malchow, H.: A minimal model of pattern formation in a prey–predator system. Math. Comput. Model. 29, 49–63 (1999)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Petrovskii, S.V., Malchow, H.: Wave of chaos: new mechanism of pattern formation in spatio-temporal population dynamics. Theor. Popul. Biol. 59, 157–174 (2001)zbMATHGoogle Scholar
  43. 43.
    Steele, J.H., Henderson, E.W.: A simple model for plankton patchiness. J. Plankton Res. 14, 1397–1403 (1992)Google Scholar
  44. 44.
    Steele, J.H., Henderson, E.W.: The role of predation in plankton models. J. Plankton Res. 14, 157–172 (1992)Google Scholar
  45. 45.
    Shi, J., Shivaji, R.: Persistence in reaction diffusion models with weak Allee effect. J. Math. Biol. 52, 807–829 (2006)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Lee, J.M., Hillen, T., Lewis, M.A.: Pattern formation in prey-taxis systems. J. Biol. Dyn. 3(6), 551–573 (2009)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Sun, G.Q.: Mathematical modeling of population dynamics with Allee effect. Nonlinear Dyn. 85(1), 1–12 (2016)MathSciNetGoogle Scholar
  48. 48.
    Edwards, A.M., Brindley, J.: Oscillatory behaviour in a three component plankton population model. Dyn. Stabil. Syst. 11, 347–370 (1996)zbMATHGoogle Scholar
  49. 49.
    Dueria, S., Dahllöf, I., Hjorth, M., Marinova, D., Zaldívara, J.M.: Modeling the combined effect of nutrients and pyrene on the plankton population: validation using mesocosm experiment data and scenario analysis. Ecol. Model. 220, 2060–2067 (2009)Google Scholar
  50. 50.
    Charria, G., Dadou, I., Llido, J., Drevillon, M., Garcon, V.: Importance of dissolved organic nitrogen in the North Atlantic ocean in sustaining primary production: a 3D modeling approach. Biogeosciences 5, 1437–1455 (2008)Google Scholar
  51. 51.
    Fasham, M.J.R., Ducklow, H.W., McKelvie, S.M.: A nitrogen-based model of plankton dynamics in the oceanic mixed layer. J. Mar. Res. 48, 591–639 (1990)Google Scholar
  52. 52.
    Montoya, J.P., Voss, M., Capone, D.G.: Spatial variation in N2-fixation rate and diazotroph activity in the tropical Atlantic. Biogeosciences 4, 369–376 (2007)Google Scholar
  53. 53.
    Zhichao, Pu, Cortez, Michael H., Jiang, Lin: Predator-prey coevolution drives productivity-richness relationships in planktonic systems. Am. Nat. 189, 28–42 (2017)Google Scholar
  54. 54.
    Gabric, A., Murray, N., Stone, L., Kohl, M.: Modelling the production of dimethylsulfide during a phytoplankton bloom. J. Geophys. Res. 98, 22805–22816 (1993)Google Scholar
  55. 55.
    Edwards, C.A., Batchelder, H.P., Powell, T.M.: Modelling microzooplankton and macrozooplankton dynamics within a coastal upwelling system. J. Plankton Res. 22, 1619–1648 (2000)Google Scholar
  56. 56.
    Edwards, C.A., Brinley, J.: Zooplankton mortality and the dynamical behavior of plankton population models. Bull. Math. Biol. 61, 303–339 (1999)zbMATHGoogle Scholar
  57. 57.
    Malchow, H., Petrovskii, S.V., Venturino, E.: Spatiotemporal Patterns in Ecology and Epidemiology—Theory, Models, and Simulation, Mathematical and Computational Biology Series. Chapman & Hall, Boca Raton (2008)zbMATHGoogle Scholar
  58. 58.
    Goursat, E.: Cows d’Analyse Mathimatique, vol. 11, 5th edn. Gauthier Villars, Paris (1933)Google Scholar
  59. 59.
    Rovinsky, A.B., Menzinger, M.: Chemical instability induced by a differential flow. Phys. Rev. Lett. 69, 1193–1196 (1992)Google Scholar
  60. 60.
    Scheffer, M.: Fish and nutrients interplay determines algal biomass: a minimal model. OIKOS 62, 271–282 (1991)Google Scholar
  61. 61.
    Medvinskii, A.B., Petrovskii, S.V., Tikhonova, I.A., Tikhonov, D.A., Li, B.-L., Venturino, E., Malchow, H., Ivanitskii, G.R.: 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)Google Scholar
  62. 62.
    Ozmidov, R.V.: Admixture Diffusion in the Ocean. Gidrometeoizdat, Leningrad (1986)Google Scholar
  63. 63.
    de Roos, A.M., McCauley, E., Wilson, W.G.: Mobility versus density-limited predator–prey dynamics of different spatiall scales. Proc. R. Soc. Lond. B 246, 117–122 (1991)Google Scholar
  64. 64.
    Cross, M.C., Hohenberg, P.C.: Pattern formation outside of equilibrium. Rev. Mod. Phys. 65(3), 851 (1993)zbMATHGoogle Scholar
  65. 65.
    Banerjee, M., Abbas, S.: Existence and non-existence of spatial patterns in a ratio-dependent predator–prey model. Ecol. Complex. 21, 199–214 (2015)Google Scholar
  66. 66.
    Huisman, J., Weissing, F.J.: Biodiversity of plankton by species oscillations and chaos. Nature 402, 407–410 (1999)Google Scholar
  67. 67.
    Petrovskii, S., Li, B.L., Malchow, H.: Transition to spatiotemporal chaos can resolve the paradox of enrichment. Ecol. Complex. 1(1), 37–47 (2004)Google Scholar
  68. 68.
    Murray, J.D.: Mathematical Biology II: Spatial Models and Biomedical Applications. Springer, Berlin (2003)zbMATHGoogle Scholar
  69. 69.
    Sun, G.Q., Wang, C.H., Chang, L.L., Wu, Y.P., Li, L., Jin, Z.: Effects of feedback regulation on vegetation patterns in semi-arid environments. Appl. Math. Model. 61, 200–215 (2018)MathSciNetGoogle Scholar
  70. 70.
    Sun, G.Q., Wu, Z.Y., Wang, Z., Jin, Z.: Influence of isolation degree of spatial patterns on persistence of populations. Nonlinear Dyn. 83(1–2), 811–819 (2016)MathSciNetGoogle Scholar
  71. 71.
    Guin, L.N.: Existence of spatial patterns in a predator–prey model with self-and cross-diffusion. Appl. Math. Comput. 226, 320–335 (2014)MathSciNetzbMATHGoogle Scholar
  72. 72.
    Banerjee, M., Volpert, V.: Spatio-temporal pattern formation in Rosenzweig–MacArthur model: effect of nonlocal interactions. Ecol. Complex. 30, 2–10 (2017)Google Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Institute of Automation and Control Processes of the FEB RASVladivostokRussia
  2. 2.Far Eastern Federal UniversityVladivostokRussia

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