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Regularity, Asymptotic Solutions and Travelling Waves Analysis in a Porous Medium System to Model the Interaction between Invasive and Invaded Species

Author

Listed:
  • José Luis Díaz Palencia

    (Escuela Politécnica Superior, Universidad Francisco de Vitoria, 28223 Madrid, Spain)

  • Julián Roa González

    (Department of Education, Universidad a Distancia de Madrid, 28400 Madrid, Spain)

  • Saeed Ur Rahman

    (Department of Mathematics, COMSATS University Islamabad, Abbottabad 22060, Pakistan)

  • Antonio Naranjo Redondo

    (Technology Programs, Schiller International University, 28002 Madrid, Spain)

Abstract

This work provides an analytical approach to characterize and determine solutions to a porous medium system of equations with views in applications to invasive-invaded biological dynamics. Firstly, the existence and uniqueness of solutions are proved. Afterwards, profiles of solutions are obtained making use of the self-similar structure that permits showing the existence of a diffusive front. The solutions are then studied within the Travelling Waves (TW) domain showing the existence of potential and exponential profiles in the stable connection that converges to the stationary solutions in which the invasive species predominates. The TW profiles are shown to exist based on the geometry perturbation theory together with an analytical-topological argument in the phase plane. The finding of an exponential decaying rate (related with the advection and diffusion parameters) in the invaded species TW is not trivial in the nonlinear diffusion case and reflects the existence of a TW trajectory governed by the invaded species runaway (in the direction of the advection) and the diffusion (acting in a finite speed front or support).

Suggested Citation

  • José Luis Díaz Palencia & Julián Roa González & Saeed Ur Rahman & Antonio Naranjo Redondo, 2022. "Regularity, Asymptotic Solutions and Travelling Waves Analysis in a Porous Medium System to Model the Interaction between Invasive and Invaded Species," Mathematics, MDPI, vol. 10(7), pages 1-19, April.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:7:p:1186-:d:787214
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    References listed on IDEAS

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    1. Anwar Shahid & Hulin Huang & Muhammad Mubashir Bhatti & Lijun Zhang & Rahmat Ellahi, 2020. "Numerical Investigation on the Swimming of Gyrotactic Microorganisms in Nanofluids through Porous Medium over a Stretched Surface," Mathematics, MDPI, vol. 8(3), pages 1-18, March.
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    Cited by:

    1. Alexander S. Bratus & Nicholas Leslie & Michail Chamo & Dmitry Grebennikov & Rostislav Savinkov & Gennady Bocharov & Daniil Yurchenko, 2022. "Mathematical Model of Pancreatic Cancer Cell Dynamics Considering the Set of Sequential Mutations and Interaction with the Immune System," Mathematics, MDPI, vol. 10(19), pages 1-12, September.

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