Derivation of cross-diffusion models in population dynamics: dichotomy, time-scales and fast-reaction
speaker: Cinzia Soresina (Università di Trento )abstract: In population dynamics, cross-diffusion describes the influence of one species on the diffusion of another one. In the context of competing species, the cross-diffusion SKT model was proposed to account for stable inhomogeneous steady states exhibiting spatial segregation. Even though the reaction part does not present the activator-inhibitor structure, the cross-diffusion terms are the key ingredient for the appearance of spatial patterns 1. Furthermore, from the modelling perspective, cross-diffusion terms naturally appear in the fast-reaction limit of a "microscopic'' model (in terms of time scales) presenting only standard diffusion and fast-reaction terms, thus incorporating processes occurring on different time scales 4. This approach can be exploited in several contexts, e.g., predator-prey interactions, plant ecology and epidemiology. Here we consider the auto-toxicity effect in plant growth dynamics 2, i.e. negative plant-soil feedback due to the decomposed biomass of the plant on its own growth. The "macroscopic'' model presents a cross-diffusion term that allows the formation of spatial patterns without introducing water as a variable 3. A deeper understanding of the conditions required for non-homogeneous steady states to exist is provided by combining a detailed linear analysis with advanced numerical bifurcation methods via the continuation software pde2path and numerical simulations.
1 Breden, M., Kuehn, C., Soresina, C. (2021). On the influence of cross-diffusion in pattern formation. Journal of Computational Dynamics 8(2):213--240. 2 Cartenì, F., Marasco, A., Bonanomi, G., Mazzoleni, S., Rietkerk, M., & Giannino, F. (2012). Negative plant soil feedback explaining ring formation in clonal plants. Journal of Theoretical Biology 313:153--161. 3 Giannino, F., Iuorio, A., Soresina, C. (in preparation). The effect of auto-toxicity in plant-growth dynamics: a cross-diffusion model. 4 Kuehn, C., Soresina, C. (2020). Numerical continuation for a fast-reaction system and its cross-diffusion limit. SN Partial Differential Equations and Applications 1:7. ----
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