On the diffusion of biological populations

Positive periodic solutions and optimal control for a distributed biological model of two interacting species

“…Otherwise, inequality (14) would imply that the solutions u are unbounded in L 2 (Q T ) as ranges in (0, 1/2) and ρ in (0, 1], against our assumption (10). Of course, an analogous inequality holds for v .…”

“…In particular, system (1) is a possible model for the evolution of two biological species living in a common territory Ω where u(x, t) and v(x, t) are the respective densities of population at time t located at x ∈ Ω, while a(x, t) and b(x, t) are the growth rates at x and time t of the two populations in absence of any intra-and inter-interferences. In this framework the nonlinear terms ∆u m and ∆v m , m > 1, are proposed for instance in [10], [11], [18] and [20] instead of ∆u and ∆v in order to represent the tendency of the populations to avoid crowding. The nonlocal terms Ω K i (ξ, t)u 2 (ξ, t − τ i )dξ and Ω K i (ξ, t)v 2 (ξ, t − τ i )dξ evaluate a weighted fraction of individuals that actually interact at time t > 0.…”

“…Now, all the pairs (u, v) such that (1, u, v) ∈ S are T -periodic solutions of (3) with (u, v) = (0, 0) and, hence, satisfy (10). Since the L 2 -norm is continuous with respect to the L ∞ -norm and S is a continuum, for every ν > 0 there is…”

Numerical solution of two dimensional time fractional-order biological population model

“…Gurtin and Maccamy [16] described the scattering of a race in a biological model in zone B assuming the functions of spot X = (x, y) in B and time t; where p(x, y, t) is the density of population, v(x, y, t) is the velocity of diffusion and f (x, y, t) is the cause of population.…”

“…In this relationn is the normal unit vector outward side of boundary of the region R. This law presents that the changing rate of population in the region R plus the rate at which animals leave the boundary of R should be equal to the rate at which animals are right away to region R. Gurtin and Maccamy [16] proved it by assuming the conditions:…”