Farid Bozorgnia: Numerical investigations for some systems of spatially segregation types

Abstract:

In this talk, we consider different models of Reaction-Diffusion systems with high competition rate. These models are as following:

• Strongly competing systems of Lotka-Volterra type: in this model particles or species annihilate on contact, and there is a common surface of separation [3];
• Segregation at distance: recently in [1] Caarelli et al. proposed a model that species keep a positive distance;
• A class of singularly perturbed elliptic systems: as the competition rate tends to infinity then the product of all components tends to zero, [2].

We review dierent aspects and properties of these models. Then, we show existence and uniqueness of the solution for each model. Moreover, we use properties of limiting problem to construct efficient numerical simulations for these systems. For the last model, we present an explicit solution in the limiting case.

References

[1] L. Caarelli, S. Patrizi and V. Quitalo, On a long range segregation model. Preprint.

[2] L. Caarelli and J. Roquejore, Uniform Hölder estimate in a class of elliptic systems and applications to singular limits in models for diffusion flames, Arch. Ration. Mech. Anal. 183, no. 3, (2007) 457-487.

[3] M. Conti, S. Terracini, and G. Verzini, Asymptotic estimate for spatial segregation of competitive systems. Advances in Mathematics. 195, (2005) 524-560.