Traveling waves in delayed reaction-diffusion equations in biology

Infection spreading in cell culture as a reaction-diffusion wave

“…This condition is related to the location of the essential spectrum, and we do not know whether the waves exist if it not satisfied. It is possible that oscillating waves emerge instead of monotone waves as it is the case for some other models [18].…”

“…Existence of waves described by monotone reaction-diffusion systems is studied in the case without time delay (see [20] and the references therein). In general, the introduction of time delay essentially complicates the analysis of wave existence, as reviewed in [18]. In some cases, when the maximum principle remains applicable, the wave existence can be proved by conventional methods, though it may be more technically difficult and some additional conditions can be necessitated to be imposed.…”

Periodic recurrent waves of Covid-19 epidemics and vaccination campaign

“…Periodic recurrent waves occur in out of equilibrium dynamics for propagation into unstable states in viral epidemics [1] , [2] , [3] , [4] , in population dynamics, ecology, cyclic predator-prey systems, spreading fires, ferroelectrics and membranes [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] , [19] , [20] . The viral pandemic Influenza A appearing in 1918 (called Spanish flu) has shown three main waves in its first two years [1] followed by seasonal waves of influenza variants for many years [2] .…”

Spatial dynamics of a nonlocal bistable reaction diffusion equation

“…Alfaro et al [3] researched the case of the nonlinearities with the form of u(u−θ)(1−φ * u). For more results about bistable reaction-diffusion equation can be referred to [8,25,26,30]. It should be pointed out that the difficulties caused by different nonlinear term (the integral located at different place) are different.…”