Global Stability of a Lotka-Volterra Competition-Diffusion-Advection System with Different Positive Diffusion Distributions

: In this paper, the problem of a Lotka–Volterra competition–diffusion–advection system between two competing biological organisms in a spatially heterogeneous environments is investi-gated. When two biological organisms are competing for different fundamental resources, and their advection and diffusion strategies follow different positive diffusion distributions, the functions of speciﬁc competition ability are variable. By virtue of the Lyapunov functional method, we discuss the global stability of a non-homogeneous steady-state. Furthermore, the global stability result is also obtained when one of the two organisms has no diffusion ability and is not affected by advection.


Introduction
For researchers from the fields of biology and mathematics, advancing the exploration of dynamic systems is a long-term challenge (see [1][2][3]). The competitive system of two diffusive organisms is often used to simulate population dynamics in biomathematics; for an example, see [1,2,4]. The key to spatial heterogeneity has been discussed in a lot of work, such as [2,5] and its references. In 2020, by proposing a new Lyapunov functional, Ni et al. [6] first studied and proved the global stability of a diffusive, competitive twoorganism system, and then extended it to multiple organisms.
Since various methods in the reaction-diffusion-convection system cannot continue to work well, the global dynamics is far from being fully understood. In competitive diffusion advection systems, some progress has been made in [7][8][9][10][11]. Li et al. introduced the weighted Lyapunov functional related to the advection term to study global stability results in 2020 (see [12]), and studied the stability and bifurcation analysis of the model with the time delay term in 2021 (see [11]). Similarly, in 2021, Ma et al. described the overlapping characteristics of bifurcation solutions and studied the influence of advection on the stability of bifurcation solutions. Their results showed that the advection term may change its stability (see [13]). In 2021, Zhou et al. studied the global dynamics of a parabolic system using the competition coefficient (see [14]).
Motivated by the efforts of the aforementioned papers, we will investigate the global stability of a non-homogeneous steady-state solution of a Lotka-Volterra model between two organisms in heterogeneous environments, where two competing organisms have different intrinsic growth rates, advection and diffusion strategies, and follow different positive diffusion distributions.
Hence, we discuss the following advection system: Here, U(x, t) and V(x, t) are the population densities of biological organisms, location x ∈ Ω, time t > 0, which are supposed to be nonnegative. µ 1 (x), µ 2 (x) > 0 correspond to the dispersal rates of two competing biological organisms, respectively. R 1 (x), R 2 (x) > 0 correspond to the advection rates of two competing biological organisms, and B 1 (x), B 2 (x) ∈ C 2 (Ω) are the nonconstant functions and represent the advective directions. Two bounded functions λ 1 (x) and λ 2 (x) are the intrinsic growth rates of competing organisms , ρ 1 (x), ρ 2 (x) ∈ C 2 (Ω) are two positive diffusion distributions, respectively.
ij (x) > 0, i = 1, 2, j = 1, 2 show the strength of competition ability. The spatial habitat Ω ⊂ R N is a bounded smooth domain, 1 ≤ N ∈ Z; n denotes the outward unit normal vector on the boundary ∂Ω. No one can enter or leave the habitat boundary.
The following are our basic assumptions:

Hypothesis 2.
where c 1 and c 2 are constants.
To simplify the calculation, by letting , the system (1) converts into the following coupled system when c 1 = c 2 = 0, ρ 1 (x) = ρ 2 (x) = 1, the model (2) has been studied in Ni et al. [6]. (2) has been studied in Li et al. [12]. The rest of this article is arranged as follows. In Section 2, we carry out some preparatory work and give four lemmas, where some related properties of the system (1) are deduced from the properties of a single organism model (4). Using the Lyapunov functional method, we will provide and prove our main results in Section 3. In Section 4, one example is given to explain our conclusions.
on Ω, then the elliptic problem: has a unique positive solution, denoted by u θ .

Main Results
In this section, firstly, by utilizing the Lyapunov function method, the global stability of the model (5) is obtained, and we can see that the non-constant steady-state for (5) is equivalent to the solution u θ of (7). Theorem 1. Assume that u 0 (x) 0. If µ, ρ, c, λ, satisfy (6), then Equation (5) has a unique solution u(x, t) > 0 with lim t→∞ u(x, t) = u θ in C 2 (Ω).
Proof. According to the upper-lower solutions method [1,18], we obtain (5) with a unique solution u(x, t) > 0. Let M be a upper solution of (5), By applying Lemma 1, we can obtain that there exists a constant Λ > 0 such that Then, Φ(t) ≥ 0, t ≥ 0. By (2) and (4), we have We get By virtue of (8) Applying (8) Combining with (12) In addition, taking advantage of Lyapunov function method, the global stability results of (2) are obtained.
(ii) Let's define a function Φ : [0, ∞) → R, (4) and (21), we have The following discussion will refer to the part (i), then we will not repeat it.

Discussion
In this paper, by using the Lyapunov functional method, we mainly analyzed the global stability of non-homogeneous steady-state for the Lotka-Volterra competitiondiffusion-advection system between two competing biological organisms in heterogeneous environments, where two biological organisms are competing for different fundamental resources, their advection and diffusion strategies follow different positive diffusion distributions, and the functions of specific competition ability are variable. Moreover, we also obtained the global stability result when one of the two organisms has no diffusion ability and is not affected by advection.
At the end of this section, we propose an interesting research problem. To the best of our knowledge, for the Lotka-Volterra competition-diffusion-advection system between two competing biological organisms in heterogeneous environments, we did not obtain any results under the condition of cross-diffusion, such as the existence and stability of nontrivial positive steady state. We leave this challenge to future investigations.