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Complex pattern formations induced by the presence of cross-diffusion in a generalized predator–prey model incorporating the Holling type functional response and generalization of habitat complexity effect

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  • Souna, Fethi
  • Belabbas, Mustapha
  • Menacer, Youssaf

Abstract

The interaction between predator and their prey is one of the most complex processes in ecosystems due to its structure and nature. Many extrinsic and intrinsic factors can affect the population dynamics. In this manuscript, we have focused our study on the process of Turing-pattern formation in a generalized predator–prey system with cross-diffusion and habitat complexity effects. By applying the linear stability theory, sufficient conditions for the occurrence of Hopf bifurcation and Turing-driven instability are successfully established. Moreover, the amplitude equations are derived near the critical value of Turing instability according to the standard multiple-scale analysis. From the mathematical view, our investigation reveals that the whole patterns generated by the system are governed by two control parameters, which are the cross-diffusion parameter δ21 and the predator mortality rate m. Complex but interesting dynamical pattern formations are specified theoretically and numerically such as spot pattern, stripe pattern, and spot-strip pattern in terms of varying the two control parameters. To support the theoretical results, a series of numerical tests are carried out, especially, to illustrate the diverse patterns near the codimension- Turing–Hopf bifurcation point (δ21T,m(0)H) and the significant impact of δ21 and m on the distribution of the two species.

Suggested Citation

  • Souna, Fethi & Belabbas, Mustapha & Menacer, Youssaf, 2023. "Complex pattern formations induced by the presence of cross-diffusion in a generalized predator–prey model incorporating the Holling type functional response and generalization of habitat complexity e," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 204(C), pages 597-618.
    • Handle: RePEc:eee:matcom:v:204:y:2023:i:c:p:597-618
    DOI: 10.1016/j.matcom.2022.09.004
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    References listed on IDEAS

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    1. Souna, Fethi & Lakmeche, Abdelkader & Djilali, Salih, 2020. "Spatiotemporal patterns in a diffusive predator-prey model with protection zone and predator harvesting," Chaos, Solitons & Fractals, Elsevier, vol. 140(C).
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    6. Bentout, Soufiane & Djilali, Salih & Kumar, Sunil, 2021. "Mathematical analysis of the influence of prey escaping from prey herd on three species fractional predator-prey interaction model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 572(C).
    7. Yan, Shuixian & Jia, Dongxue & Zhang, Tonghua & Yuan, Sanling, 2020. "Pattern dynamics in a diffusive predator-prey model with hunting cooperations," Chaos, Solitons & Fractals, Elsevier, vol. 130(C).
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