Abstract
Differential inequality methods are developed for establishing upper and lower bounds on the total particle numberN(t)=∫θ(x,t) d3 x associated with solutions to nonlinear reaction-diffusion equations of the form ∂θ/∂t=D∇2θ+fθ-gθn+1, whereD(>0),n(>0),f andg are constant parameters. If finite in a neighborhood oft=0,N(t) is bounded below for allt≥0 by a certain derived function oft for equations withg≥0. An upper bound onN(t) is obtained for equations withn=1,f<0 andg<0. These results provide general preservation and extinction criteria for the total particle number.
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Rosen, G. Bounds on the total particle number for species governed by reaction-diffusion equations in the infinite spatial domain. Bltn Mathcal Biology 36, 589–593 (1974). https://doi.org/10.1007/BF02463270
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DOI: https://doi.org/10.1007/BF02463270