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Non-linear Waves of Interacting Populations with Density-Dependent Diffusion

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Advanced Computing in Industrial Mathematics (BGSIAM 2018)

Part of the book series: Studies in Computational Intelligence ((SCI,volume 961))

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Abstract

In this study we consider a mathematical model based on diffusion-reaction type PDEs to describe the spatio–temporal behaviour of interacting populations. We extend the generalized reaction–diffusion model for spatio–temporal dynamics of interacting (competing) populations, proposed in [N. K. Vitanov, I. P. Jordanov, Z. I. Dimitrova, Applied Mathematics and Computation 215, 2950–2964 (2009)], by adding density–dependent diffusive terms to the equations therein. The new density–dependent terms describe the intra– and inter–specific spatial interactions among individuals leading to formation of aggregation structures or individual repulsion. We are interested in the particular case of one population migrating in one spatial direction. Then the generalized model is reduced to one PDE of diffusion–reaction kind. We extract a general analytical solution of the considered equation applying the modified method of simplest equation. The ordinary differential equation of Bernoulli is used as simplest equation. Numerical study of the obtained analytical solution is made. We show that the population density wave can vary in its profile depending on assumptions made about density–depending diffusion coefficients.

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Acknowledgements

This work was supported by the UNWE project for scientific researchers with grant agreement No. NID NI–21/2019.

Author information

Authors and Affiliations

  1. Institute of Mechanics, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Elena V. Nikolova, Denislav Z. Serbezov & Nikolay K. Vitanov

  2. University of National and World Economy, Sofia, Bulgaria

    Ivan P. Jordanov

  3. Max-Planck Institute for the Physics of Complex Systems, Dresden, Germany

    Nikolay K. Vitanov

Authors
  1. Elena V. Nikolova
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  2. Denislav Z. Serbezov
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  3. Ivan P. Jordanov
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  4. Nikolay K. Vitanov
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Corresponding author

Correspondence to Elena V. Nikolova .

Editor information

Editors and Affiliations

  1. Institute of Information and Communication Technologies and Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Ivan Georgiev

  2. Institute of Mathematics and Informatics, Bulgarian Academy of Science, Sofia, Bulgaria

    Hristo Kostadinov

  3. Institute of Information and Communication Technologies, Bulgarian Academy of Science, Sofia, Bulgaria

    Elena Lilkova

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Nikolova, E.V., Serbezov, D.Z., Jordanov, I.P., Vitanov, N.K. (2021). Non-linear Waves of Interacting Populations with Density-Dependent Diffusion. In: Georgiev, I., Kostadinov, H., Lilkova, E. (eds) Advanced Computing in Industrial Mathematics. BGSIAM 2018. Studies in Computational Intelligence, vol 961. Springer, Cham. https://doi.org/10.1007/978-3-030-71616-5_29

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