In this paper, we consider the following Lotka–Volterra competition system with dynamical resources and density-dependent diffusion

in a bounded smooth domain \(\Omega \subset {{\mathbb {R}}^2}\) with homogeneous Neumann boundary conditions, where the parameters \(\mu \), \(a_{i}\), \(b_{i}\), \( c_{i}\) (\(i=1,2\)) are positive constants, m(x) is the prey’s resource, and the dispersal rate function \(d_{i}(w)\) satisfies the the following hypothesis:

• \(d_{i}(w)\in C^2([0,\infty ))\), \(d_{i}'(w)\le 0\) on \([0,\infty )\) and \(d(w)>0\).

When m(x) is constant, we show that the system (*) with has a unique global classical solution when the initial datum is in functional space \(W^{1,p}(\Omega )\) with \(p>2\). By constructing appropriate Lyapunov functionals and using LaSalle’s invariant principle, we further prove that the solution of (*) converges to the co-existence steady state exponentially or competitive exclusion steady state algebraically as time tends to infinity in different parameter regimes. Our results reveal that once the resource w has temporal dynamics, two competitors may coexist in the case of weak competition regardless of their dispersal rates and initial values no matter whether there is explicit dependence in dispersal or not. When the prey’s resource is spatially heterogeneous (i.e. m(x) is non-constant), we use numerical simulations to demonstrate that the striking phenomenon “slower diffuser always prevails” (cf. Dockery et al. in J Math Biol 37(1):61–83, 1998; Lou in J Differ Equ 223(2):400–426, 2006) fails to appear if the non-random dispersal strategy is employed by competing species (i.e. either \(d_1(w)\) or \(d_2(w)\) is non-constant) while it still holds true if both d(w) and \(d_2(w)\) are constant.

在本文中，我们考虑以下具有动态资源和密度依赖扩散的 Lotka-Volterra 竞争系统

在具有齐次 Neumann 边界条件的有界平滑域\(\Omega \subset {{\mathbb {R}}^2}\) 中，其中参数\(\mu \) , \(a_{i}\) , \ (b_{i}\) , \( c_{i}\) ( \(i=1,2\) ) 是正常数，m ( x ) 是猎物的资源，扩散率函数\(d_{i }(w)\)满足以下假设：

• \(d_{i}(w)\in C^2([0,\infty ))\) , \(d_{i}'(w)\le 0\) on \([0,\infty )\ )和\(d(w)>0\)。

当m ( x ) 为常数时，我们证明当初始数据在函数空间\(W^{1,p}(\Omega )\) with \(p >2\)。通过构造合适的李雅普诺夫泛函并使用拉萨尔不变原理，我们进一步证明了 (*) 的解在代数上收敛于共存稳态或竞争排斥稳态，因为时间在不同参数范围内趋于无穷大。我们的结果表明，一旦资源w具有时间动态性，两个竞争者在弱竞争的情况下可能共存，而不管他们的分散率和初始值如何，无论是否存在分散的显式依赖。当猎物的资源在空间上是异质的（即m ( x ) 是非常数的）时，我们使用数值模拟来证明“缓慢的扩散器总是盛行”的惊人现象（参见 Dockery 等人在 J Math Biol 37(1) 中） :61–83, 1998; Lou in J Differ Equ 223(2):400–426, 2006) 如果竞争物种采用非随机分散策略（即\(d_1(w)\)或\(d_2(w)\)是非常数），但如果d ( w ) 和\(d_2(w)\)是常数。