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Exploring dynamical complexity in diffusion driven predator–prey systems: Effect of toxin producing phytoplankton and spatial heterogeneities

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  • Upadhyay, Ranjit Kumar
  • Kumari, Nitu
  • Rai, Vikas

Abstract

In this paper, dynamical complexities in two reaction–diffusion (RD) model systems are explored. A spatial heterogeneity in the form of linear spatial gradient in the reproductive growth rate of the phytoplankton is incorporated in both the model systems. Extra mortality of the zooplankton due to toxin production by the phytoplankton is included in the second reaction diffusion model system. Effect of toxin production and spatial heterogeneity in the model systems are studied. Toxin production does not seem to have an appreciable effect on the asymptotic dynamics of the model systems. On the other hand, spatial heterogeneity does influence the dynamics. In particular, it increases the frequency of occurrence of chaos as evident from two dimensional parameter scans. Both these model systems display short term recurrent chaos [Rai V. Chaos in natural populations: edge or wedge? Ecol Complex 2004;1: 127–38] as they reside on ‘edges of chaos’ (EOC) [Rai V, Upadhyay RK. Evolving to the edge of chaos: chance or necessity? Chaos, Solitons and Fractals 2006;30:1074–87]. This suggests that the ecological systems have a tendency to evolve to EOC. The study corroborates the inferences drawn from an earlier study by Rai and Upadhyay [Rai V, Upadhyay RK. Evolving to the edge of chaos: chance or necessity? Chaos, Solitons and Fractals 2006;30:1074–87]. The system’s dynamics is largely unpredictable and admits bursts of short-term predictability.

Suggested Citation

  • Upadhyay, Ranjit Kumar & Kumari, Nitu & Rai, Vikas, 2009. "Exploring dynamical complexity in diffusion driven predator–prey systems: Effect of toxin producing phytoplankton and spatial heterogeneities," Chaos, Solitons & Fractals, Elsevier, vol. 42(1), pages 584-594.
    • Handle: RePEc:eee:chsofr:v:42:y:2009:i:1:p:584-594
    DOI: 10.1016/j.chaos.2009.01.027
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    References listed on IDEAS

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    1. Xu, Rui & Chaplain, M.A.J. & Davidson, F.A., 2006. "Travelling wave and convergence in stage-structured reaction–diffusion competitive models with nonlocal delays," Chaos, Solitons & Fractals, Elsevier, vol. 30(4), pages 974-992.
    2. Li, Wan-Tong & Wu, Shi-Liang, 2008. "Traveling waves in a diffusive predator–prey model with holling type-III functional response," Chaos, Solitons & Fractals, Elsevier, vol. 37(2), pages 476-486.
    3. Rai, Vikas & Upadhyay, Ranjit Kumar, 2006. "Evolving to the edge of chaos: Chance or necessity?," Chaos, Solitons & Fractals, Elsevier, vol. 30(5), pages 1074-1087.
    4. Jiang, Guirong & Lu, Qishao & Qian, Linning, 2007. "Complex dynamics of a Holling type II prey–predator system with state feedback control," Chaos, Solitons & Fractals, Elsevier, vol. 31(2), pages 448-461.
    5. Upadhyay, Ranjit Kumar & Kumari, Nitu & Rai, Vikas, 2009. "Wave of chaos in a diffusive system: Generating realistic patterns of patchiness in plankton–fish dynamics," Chaos, Solitons & Fractals, Elsevier, vol. 40(1), pages 262-276.
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    1. Jana, Debaldev & Pathak, Rachana & Agarwal, Manju, 2016. "On the stability and Hopf bifurcation of a prey-generalist predator system with independent age-selective harvesting," Chaos, Solitons & Fractals, Elsevier, vol. 83(C), pages 252-273.
    2. Banerjee, Malay & Zhang, Lai, 2014. "Influence of discrete delay on pattern formation in a ratio-dependent prey–predator model," Chaos, Solitons & Fractals, Elsevier, vol. 67(C), pages 73-81.

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