Previously in [14], we considered a diffusive logistic equation with two parameters, $ r(x) $ – intrinsic growth rate and $ K(x) $ – carrying capacity. We investigated and compared two special cases of the way in which $ r(x) $ and $ K(x) $ are related for both the logistic equations and the corresponding Lotka-Volterra competition-diffusion systems. In this paper, we continue to study the Lotka-Volterra competition-diffusion system with general intrinsic growth rates and carrying capacities for two competing species in heterogeneous environments. We establish the main result that determines the global dynamics of the system under a general criterion. Furthermore, when the ratios of the intrinsic growth rate to the carrying capacity for each species are proportional — such ratios can also be interpreted as the competition coefficients — this criterion reduces to what we obtained in [18]. We also study the detailed dynamics in terms of dispersal rates for such general case. On the other hand, when the two ratios are not proportional, our results in [14] show that the criterion in [18] cannot be fully recovered as counterexamples exist. This indicates the importance and subtleties of the roles of heterogeneous competition coefficients in the dynamics of the Lotka-Volterra competition-diffusion systems. Our results apply to competition-diffusion-advection systems as well. (See Corollary 5.1 in the last section.)

中文翻译：

---

异构环境中一般Lotka-Volterra竞争扩散系统的整体动力学

以前在[14]，我们考虑了具有两个参数的扩散对数方程，$ r（x）$ –内在增长率和$ K（x）$ –承载能力。我们研究并比较了两种特殊情况，其中逻辑方程和相应的Lotka-Volterra竞争扩散系统的$ r（x）$和$ K（x）$相关。在本文中，我们将继续研究Lotka-Volterra竞争扩散系统，该系统具有普遍的内在增长率和在异质环境中两个竞争物种的承载能力。我们建立了确定一般条件下系统全局动力学的主要结果。此外，18岁]。我们还研究了这种一般情况下的分散速度方面的详细动态。另一方面，当两个比率不成比例时，我们的结果为[14]表示[18岁由于存在反例，因此无法完全恢复。这表明了异质竞争系数在Lotka-Volterra竞争扩散系统动力学中的作用的重要性和微妙之处。我们的结果也适用于竞争扩散对流系统。（请参阅最后一部分的推论5.1。）