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

Abstract

Analysis based on the integration of differential inequalities is employed to derive upper and lower bounds on the total populationN(t) = ∫ _R θ(x ₁,x ₂,t) dx ₁ dx ₂ of a biological species with an area-density distribution function θ=θ(x ₁,x ₂,t) (≥0) governed by a reaction-diffusion equation of the form ∂θ/∂t =D∇²θ +fθ −gθⁿ⁺¹ whereD (>0),n (>0),f andg are constant parameters, θ=0 at all points on the boundary ∂R of an (arbitrary) two-dimensional regionR, and the initial distribution (θ(_{x 1},x ₂, 0) is such thatN(0) is finite. Forg≥0 withR the entire two-dimensional Euclidean space, a lower bound onN(t) is obtained, showing in particular thatN(∞) is bounded below by a finite positive quantity forf≥0 andn>1. An upper bound onN(t) is obtained for arbitrary bounded or unbounded)R withn=1,f andg negative, and ∫ _R θ(x ₁,x ₂, 0)² dx ₁ dx ₂ sufficiently small in magnitude, implying that the population goes to extinction with increasing values of the time,N(∞)=0. Forg≥0 andR of finite area, the analysis yields upper bounds onN(t), predicting eventual extinction of the population if eitherf≤0 or if the area ofR is less than a certain grouping of the parameters in cases for whichf 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.