Bounds on the total population for species governed by reaction-diffusion equations in arbitrary two-dimensional regions

✍ Scribed by Gerald Rosen; Richard G. Fizell

• Publisher: Springer
• Year: 1975
• Tongue: English
• Weight: 367 KB
• Volume: 37
• Category: Article
• ISSN: 1522-9602
• DOI: 10.1016/s0092-8240(75)80008-5

✦ Synopsis

Analysis based on the integration of differential inequalities is employed to derive upper and lower bounds on the total population N(t) = ~R 0(xl, x2, t)dxl dx2 of a biological species with an area-density distribution function 0 = 0(Xl, x2, t) (>~0) governed by a reaction-diffusion equation of the form 00lOt = DV20 + fO -gO n+l where D (>0), n ( > 0), f and g are constant parameters, {? = 0 at all points on the boundary ~R of an (arbitrary) two-dimensional region/~, and the initial distribution 0(xl, x2, 0) is such that iV(0) is finite. For g >i 0 with R the entire two-dimensional Euclidean space, a lower bound on N(t) is obtained, showing in particular that 2/(oo) is bounded below b y a finite positive quantity for f >~ 0 and n > 1. An upper bound on N(t) is obtained for arbitrary (bounded or unbounded) R with n = 1, f and g negative, and ~ O(xl, x2, 0) 2 dxl dx2 sufficiently small in magnitude, implying that the population goes to extinction with increasing values of the time, l~(oo) = 0. For g ~> 0 and /~ of finite area, the analysis yields upper bounds on N(t), predicting eventual extinction of the population if either ] ~< 0 or if the area of R is less than a certain grouping of the parameters in cases for which / is positive. These results are directly applicable to biological species with distributions satisfying the Fisher equation in two spatial dimensions and to species governed by certain specialized population models.