We first consider a diffusive logistic model of a single species in a heterogeneous environment, with two parameters, r(x) for intrinsic growth rate and K(x) for carrying capacity. When r(x) and K(x) are proportional, i.e., \(r=cK\), it is proved by Lou (J Differ Equ 223(2):400–426, 2006) that a population diffusing at any rate will reach a higher total equilibrium biomass than the population in an environment in which the same total resources are distributed homogeneously. This paper studies another case when r(x) is a constant, i.e., independent of K(x). In such case, a striking result is that for any dispersal rate, the logistic equation with spatially heterogeneous resources will always support a total population strictly smaller than the total carrying capacity at equilibrium, which is just opposite to the case \(r = cK\). These two cases of single species models also lead to two different forms of Lotka–Volterra competition-diffusion systems. We then examine the consequences of the aforementioned difference on the two forms of competition systems. We find that the outcome of the competition in terms of the dispersal rates and spatial distributions of resources for the two forms of competition systems are again quite different. Our results indicate that in heterogeneous environments, the correlation between r(x) and K(x) has more profound impacts in population ecology than we had previously expected, at least from a mathematical point of view.

我们首先考虑异质环境中单个物种的扩散逻辑模型，具有两个参数，r（x）表示内在增长率，K（x）表示承载能力。当r（x）和K（x）成比例，即\（r = cK \）时，Lou（J Differ Equ 223（2）：400–426，2006）证明了人口以任何速率扩散在相同的总资源均匀分布的环境中，将达到比种群更高的总平衡生物量。当r（x）是一个常数，即独立于K（x）。在这种情况下，一个惊人的结果是，对于任何分散率，具有空间上异质性资源的逻辑方程将始终支持总人口严格小于平衡时的总承载能力，这与情况\（r = cK \ ）。单一物种模型的这两种情况也导致了Lotka–Volterra竞争扩散系统的两种不同形式。然后，我们研究上述差异对两种形式的竞争体系的影响。我们发现，就两种形式的竞争系统而言，竞争的结果在资源的分散率和空间分布方面都大不相同。我们的结果表明，在异构环境中，r（x）和K（x）至少在数学上对人口生态学的影响比我们以前预期的要深。