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Seminarium i matematik

Diffusion-convection reaction equations with sign-changing diffusivity and population dynamics with biased movements

Välkommen till föreläsningen i seminarieserien i matematik.

In this talk we deal with a diffusion-convection reaction equation in one space variable whose diffusivity assumes both positive and negative values. We are then faced with forward-backward parabolic equations. We discuss both the cases when the reaction term is monostable and when it models a strong Allee effect, i.e., it is of a so called bistable type. Besides the study of population dynamics, these models meet in several other contexts such as, for instance, vehicular traffic and crowds dynamics.

We focus on traveling-wave solutions, that is constant profile solutions, that connect two equilibria and cross both regions where the diffusivity is positive and regions where it is negative.  We prove the existence of these solutions and their possible multiplicity; we also show their qualitative properties such as the sharpness of their profiles when the diffusivity degenerates. We provide estimates for their admissible speeds. 

We perform this investigation by nonlinear analysis techniques and benefits of a suitable order reduction combined with an upper and lower solutions method.

We apply the obtained results and the techniques developed to describe the movements of a biological population formed by isolated and grouped organisms. We introduce biases in their movements and then obtain a diffusion-convection reaction equation. We show that traveling-wave solutions do exist, for some significant choices of the parameters, and study the sign of their speeds, which provides information on the long-term behavior of this population, namely, its survival or extinction.

 

References

D. Berti – A. Corli – L. Malaguti, The role of convection in the existence of wavefronts for biased movements. Math. Meth. Appl. Sci. (2023), 1–25, DOI 10.1002/mma.9667.

D. Berti - A. Corli - L. Malaguti, Wavefronts in forward-backward parabolic equations and applications to biased movements. In: U. Kähler, M. Reissig, I. Sabadini, J.Vindas  (eds) Analysis, Applications, and Computations. Trends in Mathematics. Birkhäuser, Cham, in press.

D. Berti – A. Corli – L. Malaguti, Diffusion-convection reaction equations with sign-changing diffusivity and bistable reaction term. Nonlinear Anal. Real World Appl. 67 (2022), Paper No. 103579

D. Berti - A. Corli – L. Malaguti, Wavefronts for degenerate diffusion-convection reaction equations with sign-changing diffusivity.  Discrete and Continuous Dynamical Systems A, 41(2021), 6023-6046.

 

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