Abstract
Dispersal-induced pattern formation is important from both fundamental and application points of view. Spatial pattern in an ecological system can strongly depend on exogenous factors like arrangement of the natural and artificial physical features of the habitat (topography) and distribution of resources. It is also influenced by endogenous factors (intrinsic biological forces) such as the ecological interactions of individuals. Here, we consider a discrete space–time host–parasitoid metapopulation model in the presence of both self-diffusion (due to endogenous factors) and cross-diffusion (due to exogenous factors). Dynamics of a metapopulation system consists of a dispersal stage and a reaction stage. In the dispersal stage, populations from an individual site can disperse to the nearest neighboring sites via dispersal and may cause variation in the host and parasitoid biomass of the node. In the reaction stage, hosts and parasitoids interact in each site and their local interaction is governed by the modified Nicholson–Bailey-type interaction. The conditions for the existence of pattern-forming instabilities (like Turing, Hopf and Hopf–Turing) in the reaction–diffusion discrete metapopulation model have been determined analytically, and the patterns have been visualized numerically. A wide range of complex spatiotemporal patterns (like periodic, quasi-periodic, chaotic) is observed with respect to the variation of diffusion coefficients and other local interacting parameters of the model.
Similar content being viewed by others
References
Lafferty, K.D.: Biodiversity loss decreases parasite diversity: theory and patterns. Philos. Trans. R. Soc. B 367(1604), 2814–2827 (2012)
Daszak, P., Cunningham, A.A., Hyatt, A.D.: Emerging infectious diseases of wildlife-threats to biodiversity and human health. Science 287(5452), 443–449 (2000)
Meier, C.M., Bonte, D., Kaitala, A., Ovaskainen, O.: Invasion rate of deer ked depends on spatiotemporal variation in host density. Bull. Entomol. Res. 104(3), 314–322 (2014)
Turing, A.M.: On the chemical basis of morphogenesis. Philos. Trans. R. Soc. B 237, 37–72 (1952)
Nicolis, G., Prigogine, I.: Self-organization in Nonequilibrium Systems. Wiley, New York (1977)
Winfree, A.T.: Spiral waves of chemical activity. Science 175(4022), 634–636 (1972)
Castets, V., Dulos, E., Boissonade, J., Kepper, P.D.: Experimental evidence of a sustained standing Turing-type non-equilibrium chemical pattern. Phys. Rev. Lett. 64(24), 2953 (1990)
Waddington, C.H., Perry, M.M.: The ultrastructure of the developing urodele notochord. Proc. R. Soc. B 156(965), 459–482 (1962)
Gierer, A., Meinhardt, H.: A theory of biological pattern formation. Kybernetik 12(1), 30–39 (1972)
Rogers, K.W., Schier, A.F.: Morphogen gradients: from generation to interpretation. Annu. Rev. Cell Dev. Biol. 27, 377–407 (2011)
Medvinsky, A.B., Petrovskii, S.V., Tikhonova, I.A., Malchow, H., Li, B.L.: Spatiotemporal complexity of plankton and fish dynamics. SIAM Rev. 44(3), 311–370 (2002)
Ghorai, S., Poria, S.: Turing patterns induced by cross-diffusion in a predator–prey system in presence of habitat complexity. Chaos Soliton Fractals 91, 421–429 (2016)
Ghorai, S., Poria, S.: Pattern formation in a system involving prey–predation, competition and commensalism. Nonlinear Dyn. 89(2), 1309–1326 (2017)
Sun, G.: Mathematical modeling of population dynamics with Allee effect. Nonlinear Dyn. 85, 1–12 (2016)
Ma, J., Xu, Y., Ren, G., Wang, C.: Prediction for breakup of spiral wave in a regular neuronal network. Nonlinear Dyn. 84(2), 497–509 (2016)
Xu, Y., Jin, W., Ma, J.: Emergence and robustness of target waves in a neuronal network. Int. J. Mod. Phys. B 29(23), 1550164 (2015)
Song, X., Wang, C., Ma, J., Ren, G.: Collapse of ordered spatial pattern in neuronal network. Phys. A 451, 95–112 (2016)
Ma, J., Xu, Y., Wang, C., Jin, W.: Pattern selection and self-organization induced by random boundary initial values in a neuronal network. Phys. A 461, 586–594 (2016)
Cross, M.C., Hohenberg, P.C.: Pattern formation outside of equilibrium. Rev. Mod. Phys. 65(3), 851–1112 (1993)
Li, K., Vandermeer, J.H., Perfecto, I.: Disentangling endogenous versus exogenous pattern formation in spatial ecology: a case study of the ant Azteca sericeasur in southern Mexico. R. Soc. Open Sci. 3(5), 160073 (2016)
Baurmann, M., Gross, T., Feudel, U.: Instabilities in spatially extended predator–prey systems: spatio-temporal patterns in the neighborhood of Turing–Hopf bifurcations. J. Theor. Biol. 245(2), 220–229 (2007)
Chakraborty, B., Bairagi, N.: Complexity in a prey–predator model with prey refuge and diffusion. Ecol. Complex. 37, 11–23 (2019)
Wang, W., Liu, Q.X., Jin, Z.: Spatiotemporal complexity of a ratio-dependent predator–prey system. Phys. Rev. E 75(5), 051913 (2007)
Aly, S., Kim, I., Sheen, D.: Turing instability for a ratio-dependent predator–prey model with diffusion. Appl. Math. Comput. 217(17), 7265–7281 (2011)
Sun, G.Q., Zhang, G., Jin, Z., Li, L.: Predator cannibalism can give rise to regular spatial pattern in a predator–prey system. Nonlinear Dyn. 58(1–2), 75–84 (2009)
Ghorai, S., Poria, S.: Emergent impacts of quadratic mortality on pattern formation in a predator–prey system. Nonlinear Dyn. 87(4), 2715–2734 (2017)
Wang, W., Liu, S., Liu, Z.: Spatiotemporal dynamics near the Turing–Hopf bifurcation in a toxic-phytoplankton–zooplankton model with cross-diffusion. Nonlinear Dyn. 98(1), 27–37 (2019)
Sun, G.Q., Wang, C.H., Wu, Z.Y.: Pattern dynamics of a Gierer–Meinhardt model with spatial effects. Nonlinear Dyn. 88(2), 1385–1396 (2017)
Sun, G.Q.: Mathematical modeling of population dynamics with Allee effect. Nonlinear Dyn. 85(1), 1–12 (2016)
Wang, T.: Pattern dynamics of an epidemic model with nonlinear incidence rate. Nonlinear Dyn. 77(1–2), 31–40 (2014)
Dagbovie, A.S., Sherratt, J.A.: Absolute stability and dynamical stabilisation in predator–prey systems. J. Math. Biol. 68(6), 1403–1421 (2014)
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)
Bennett, J.J., Sherratt, J.A.: Long-distance seed dispersal affects the resilience of banded vegetation patterns in semi-deserts. J. Theor. Biol. 481, 151–161 (2019)
Tang, S., Xiao, Y., Cheke, R.A.: Multiple attractors of host–parasitoid models with integrated pest management strategies: eradication, persistence and outbreak. Theor. Popul. Biol. 73(2), 181–197 (2008)
Singh, B.K., Rao, J.S., Ramaswamy, R., Sinha, S.: The role of heterogeneity on the spatiotemporal dynamics of host–parasite metapopulation. Ecol. Model. 180(2–3), 435–443 (2004)
Hassell, M.P., Comins, H.N., May, R.M.: Spatial structure and chaos in insect population dynamics. Nature 353(6341), 255–258 (1991)
Hassell, M.P., Comins, H.N., May, R.M.: Species coexistence and self-organising spatial dynamics. Nature 370(6487), 290–292 (1994)
Nicholson, A.J., Bailey, V.A.: The balance of animal populations. Part I. Proc. Zool. Soc. Lond. 105(3), 551–598 (1935)
Hassell, M.P., May, R.M.: Stability in insect host–parasite models. J. Anim. Ecol. 42(3), 693–726 (1973)
Anderson, R.M., May, R.M.: Coevolution of hosts and parasites. Parasitology 85(2), 411–426 (1982)
Savill, N.J., Rohani, P., Hogeweg, P.: Self-reinforcing spatial patterns enslave evolution in a host–parasitoid system. J. Theor. Biol. 188(1), 11–20 (1997)
Kang, Y., Sasmal, S.K., Bhowmick, A.R., Chattopadhyay, J.: A host–parasitoid system with predation-driven component Allee effects in host population. J. Biol. Dyn. 9, 213–232 (2015)
Comins, H.N., Hassell, M.P., May, R.M.: The spatial dynamics of host–parasitoid systems. J. Anim. Ecol. 61, 735–748 (1992)
Li, M., Han, B., Xu, L., Zhang, G.: Spiral patterns near Turing instability in a discrete reaction diffusion system. Chaos Soliton Fractals 49, 1–6 (2013)
Sun, G.Q., Jusup, M., Jin, Z., Wang, Y., Wang, Z.: Pattern transitions in spatial epidemics: mechanisms and emergent properties. Phys. Life Rev. 19, 43–73 (2016)
Li, L.: Patch invasion in a spatial epidemic model. Appl. Math. Comput. 258, 342–349 (2015)
Xu, L., Zhang, G., Cui, H.: Dependence of initial value on pattern formation for a logistic coupled map lattice. PLoS ONE 11(7), e0158591 (2016)
Sole, R.V., Valls, J., Bascomte, J.: Spiral waves, chaos and multiple attractors in lattice models of interacting populations. Phys. Lett. A 166(2), 123–128 (1992)
Parekh, N., Sinha, S.: Controlling spatiotemporal dynamics in excitable systems. Phys. Rev. E 65, 036227–1 (2002)
May, R.M.: Simple mathematical models with very complicated dynamics. Nature 261, 459–467 (1976)
Khan, A.Q., Qureshi, M.N.: Dynamics of a modified Nicholson–Bailey host–parasitoid model. Adv. Differ. Equ. 2015(23), 1–15 (2015)
Smith, M.J., Sherratt, J.A., Lambin, X.: The effects of density-dependent dispersal on the spatiotemporal dynamics of cyclic populations. J. Theor. Biol. 254(2), 264–274 (2008)
Dubey, B., Das, B., Hussain, J.: A predator–prey interaction model with self and cross-diffusion. Ecol. Model. 141(1–3), 67–76 (2001)
Vanag, V.K., Epstein, I.R.: Cross-diffusion and pattern formation in reaction–diffusion systems. Phys. Chem. Chem. Phys. 11(6), 897–912 (2009)
Elaydi, S.N.: Discrete Chaos with Applications in Science and Engineering. Chapman and Hall, New York (2007)
Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer, New York (1983)
Bai, L., Zhang, G.: Nontrivial solutions for a nonlinear discrete elliptic equation with periodic boundary conditions. Appl. Math. Comput. 210, 321–333 (2009)
Xu, L., Zhang, G., Han, B., Zhang, L., Li, M.F., Han, Y.T.: Turing instability for a two-dimensional logistic coupled map lattice. Phys. Lett. A 374, 3447–3450 (2010)
Bascompte, J., Solé, R.V.: Spatially induced bifurcations in single-species population dynamics. J. Anim. Ecol. 63, 256–264 (1994)
Xu, L., Liu, J., Zhang, G.: Pattern formation and parameter inversion for a discrete Lotka–Volterra cooperative system. Chaos Solitons Fractals 110, 226–231 (2018)
Ghorai, S., Poria, S.: Pattern formation and control of spatiotemporal chaos in a reaction diffusion prey–predator system supplying additional food. Chaos Soliton Fractals 85, 57–67 (2016)
Ghorai, S., Poria, S.: Impacts of additional food on diffusion induced instabilities in a predator–prey system with mutually interfering predator. Chaos Soliton Fractals 103, 68–78 (2017)
Wang, W., Zhang, L., Wang, H., Li, Z.: Pattern formation of a predator-prey system with Ivlev-type functional response. Ecol. Model. 221(2), 131–140 (2010)
Kapral, R.: Pattern formation in two-dimensional arrays of coupled, discrete-time oscillators. Phys. Rev. A 31(6), 3868 (1985)
Boerlijst, M.C., Lamers, M.E., Hogeweg, P.: Evolutionary consequences of spiral waves in a host–parasitoid system. Proc. R. Soc. Lond. B 253(1336), 15–18 (1993)
Guo, Z.G., Song, L.P., Sun, G.Q., Li, C., Jin, Z.: Pattern dynamics of an SIS epidemic model with nonlocal delay. Int. J. Bifurc. Chaos Appl. Sci. Eng. 29(02), 1950027 (2019)
Hagos, Z., Stankovski, T., Newman, J., Pereira, T., McClintock, P.V., Stefanovska, A.: Synchronization transitions caused by time-varying coupling functions. Philos. Trans. R. Soc. A 377(2160), 20190275 (2019)
Acknowledgements
Research of Santu Ghorai is supported by Dr. D. S. Kothari Postdoctoral Fellowship under University Grants Commission scheme (Ref. No. F.4-2/2006 (BSR)/MA/18-19/0004, dated 26th December, 2018). Research of Nandadulal Bairagi is supported by RUSA 2.0, Jadavpur University, Ref No. R-11/725/19, dated 26.06.2019. We are thankful to the anonymous reviewers for their valuable comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Ghorai, S., Chakraborty, P., Poria, S. et al. Dispersal-induced pattern-forming instabilities in host–parasitoid metapopulations. Nonlinear Dyn 100, 749–762 (2020). https://doi.org/10.1007/s11071-020-05505-w
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-020-05505-w