On average population levels for models with directed diffusion in heterogeneous environments

In 2006 (J. Differential Equ.), Lou proved that, once the intrinsic growth rate $r$ in the logistic model is proportional to the spatially heterogeneous carrying capacity $K$ ($r=K^1$), the total population under the regular diffusion exceeds the total of the carrying capacity. He also conjectured that the dependency of the total population on the diffusion coefficient is unimodal, increasing to its maximum and then decreasing to the asymptote which is the total of the carrying capacity. DeAngelis et al (J. Math. Biol. 2016) argued that the prevalence of the population over the carrying capacity is only observed when the growth rate and the carrying capacity are positively correlated, at least for slow dispersal. Guo et al (J. Math. Biol. 2020) justified that, once $r$ is constant ($r=K^0$), the total population is less than the cumulative carrying capacity. Our paper fills up the gap for when $r=K^{\lambda}$ for any real $\lambda$, disproving an assumption that there is a critical $\lambda^{\ast} \in (0,1)$ at which the tendency of the prevalence of the carrying capacity over the total population size changes, demonstrating instead that the relationship is more complicated. In addition, we explore the dependency of the total population size on the diffusion coefficient when the third parameter of the dispersal strategy $P$ is involved: the diffusion term is $d \Delta(u/P)$, not just $d \Delta u$, for any $\lambda$. We outline some differences from the random diffusion case, in particular, concerning the profile of the total population as a function of the diffusion coefficient.