Fragility of reaction-diffusion models with respect to competing advective processes

Abstract

We study the coupling of a Fisher-Kolmogorov-Petrovsky-Piskunov (FKPP) equation to a separate, advection-only transport process. We find that an infinitesimal coupling can cause a finite change in the speed and shape of the reaction front, indicating the fragility of the FKPP model with respect to such a perturbation. The front dynamics can be mapped to an effective FKPP equation only at sufficiently fast diffusion or large coupling strength. We also discover conditions when the front width diverges, and when its speed is insensitive to the coupling. At zero diffusion in our mean-field description, the downwind front speed goes to a finite value as the coupling goes to zero.

The role of advective transport mechanism in continental-scale transport of fungal plant epidemics

“…Logistic growth function is of that type, because a theory based on the pulled front assumption match with numerical results of the model with a non-linear term (growth limiting term), see Fig. 3 and [33].…”

“…We now summarize the key results of this basic model; details can be found in [33]. It will prove to be very instructive to first describe the behavior in the regime when diffusion is set to zero.…”

“…The actual, full solution of ρ and σ, which includes both adsorption and desorption, is given in [33].…”

“…On the other hand, observed invasions of fungal epidemics typically move tens of kilometers per day [1], [26] (see Chapter 14), [45]. Therefore, the simplistic theory in [33] is overestimating the front speed by an order of magnitude.…”

“…The new dispersion relation now obeys e −iωτ G(−ik + b + iω) + iabω + ω(a + G + iω)(ω − k − ib) = 0 (33) Note the presence of the term e −iωτ , which makes these equations transcendental. Following the same steps as before, we obtain -after setting k * r = 0 and ω * r = 0 -the following two equations 0 = k i −sbG − e sk i τ G(1 + s(bτ − 1)) + sk i (2(s − 1)a − 2G + 2s(b…”