Global generalized solutions to a forager–exploiter model with superlinear degradation and their eventual regularity properties

Abstract

<jats:p> In this paper, we consider a cascaded taxis model for two proliferating and degrading species which thrive on the same nutrient but orient their movement according to different schemes. In particular, we assume the first group, the foragers, to orient their movement directly along an increasing gradient of the food density, while the second group, the exploiters, instead track higher densities of the forager group. Specifically, we will investigate an initial boundary-value problem for a prototypical forager–exploiter model of the form [Formula: see text] in a smoothly bounded domain [Formula: see text], where [Formula: see text], [Formula: see text] is nonnegative and the functions [Formula: see text] are assumed to satisfy [Formula: see text], [Formula: see text] as well as [Formula: see text] respectively, with constants [Formula: see text], [Formula: see text] and [Formula: see text] and [Formula: see text]. </jats:p><jats:p> Assuming that [Formula: see text], [Formula: see text] and that [Formula: see text] satisfies certain structural conditions, we establish the global solvability of this system with respect to a suitable generalized solution concept and then, for the more restrictive case of [Formula: see text] and [Formula: see text], investigate an eventual regularity effect driven by the decay of the nutrient density [Formula: see text]. </jats:p>

Global boundedness of solutions to the two-dimensional forager-exploiter model with logistic source

“…where Ω ⊂ R 2 is a bounded open domain with smooth boundary ∂Ω, and the parameters µ, µ 1 , µ 2 , λ, m and l are some positive constants. Moreover, we suppose that 2 ≤ m < 3 and l ≥ 3,…”

“…For instance, the shearwater flocks and kittiwake foragers, the death of individuals at high population densities should not be neglected. Thus, as n = 2, Black in [2] discussed the existence of the global generalized solutions to the forager-exploiter model with logistic sources f (u) and g(v), and supposed that…”

“…In [2], the author showed that the global solvability of this system with respect to a suitable generalized solution concept were established with α > 1 + √ 2, β > min{α, β} > α+1 α and r(x, t) satisfies certain structural conditions. Specially, as α, β > 1 + √ 2, an eventual regularity effect driven by the decay of the nutrient density w was completed.…”

Global generalized solutions to the forager-exploiter model with logistic growth

“…However, unlike the simple structure in the minimal Keller-Segel system or the prey-taxis model, the different taxis strategies in the system (1.3) produce more mathematical challenges and make the analysis work more complicate and difficult. Motivated by the arguments in previous studies [4,32,37,40,41], we proved that system (1.3) admits at least a generalized solution for any choice of m > 1 + √ 2 and l > m+1 m−1 , where the small initial value conditions (1.6) and (1.7) in [37] is not necessary anymore. Now, we state our main result.…”

“…When n ≥ 1 and η 1 = η 2 = 0, by taking into account the volume-filling effect and 0 ≤ u 0 , v 0 ≤ 1, the global existence of the classical solutions was built in [18]. When n = 2 and η 1 = η 2 > 0, Black [4] showed that (1.3) has at least one global generalized solution if m > √ 2 + 1 and min{m, l} > (m + 1)/(m − 1), and proved that the generalized solution actually becomes a classical one after some waiting time under conditions: m, l > √ 2 + 1, r ≥ 0 satisfies (1.4) and r ∈ L 1 ((0, ∞); L ∞ (Ω)). In the same case, Wang and Wang [32]…”