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

TL;DR

The paper addresses how total population in a logistic diffusion model with directed diffusion depends on diffusion strength and dispersal strategy in spatially heterogeneous environments, focusing on $r(x)=\alpha\left(\frac{K(x)}{P(x)}\right)^\lambda$. It develops a three-parameter framework with $P$, $K$, and $r$, derives sharp limits as $d\to0^+$ and $d\to\infty$, and identifies conditions (notably $P\propto K/r$ with non-constant $r$) that guarantee population abundance beyond carrying capacity for all $d>0$. It also analyzes how small-diffusion tendencies relate to gradients via $\int_\Omega \nabla\left(\frac{K}{P}\right)\cdot\nabla r\,dx$, and investigates the power-law case $r=\alpha\left(\frac{K}{P}\right)^\lambda$, showing a monotone increase of the far-field population with $\lambda$ and a transition in the relation between endpoints at $\lambda=1$. Through analytical results and numerical examples, the work demonstrates that unimodality of $M(d)$ is not universal when the dispersal strategy is nontrivial, and it discusses implications for management and harvesting in heterogeneous habitats.

Abstract

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 $r=K^λ$ for any real $λ$, disproving an assumption that there is a critical $λ^{\ast} \in (0,1)$ 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 $d Δ(u/P)$, not just $d Δu$, for any $λ$. 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.

Figures (5)

• Figure 1: Population decline when $\int_\Omega\nabla(K/P)\cdot\nabla r\,dx=0$, for $K(x)=(\cos(2\pi x)+2)^2$, $P(x)=\sqrt{K(x)}$, and $r(x)=\cos(\pi x)+2$.
• Figure 2: Population growth when $\int_\Omega\nabla(K/P)\cdot\nabla r\,dx=0$, for $K(x)=(2x^3-3x^2+3)e^{2x^3-3x^2}$, $P(x)=e^{2x^3-3x^2}$ and $r(x)=\cos(2\pi x)+3$.
• Figure 3: Different behaviours for the total population when $\int_\Omega \nabla\left(\frac{K}{P}\right)\cdot \nabla r\,dx>0$, for $K(x)=2+\cos(\pi x)$, $P(x)=1+\frac{1}{5}\cos(\pi x)$ and (a) $r(x)=\frac{5}{4}+\frac{1}{4}\cos(\pi x)$, (b) $r(x)=\exp(4\cos(\pi x))$.
• Figure 4: Non-unimodal total population curve under spatially heterogeneous dispersal strategy, for $K(x)=0.1+\cos(\pi x)+5\cos^2(\pi x)-2\cos^3(\pi x)$, $P(x)=1.5-3\cos(\pi x)+\cos^2(\pi x)+3\cos^6(\pi x)$ and $r(x)=\frac{K(x)}{P(x)}$.
• Figure 5: Total population curves for $K(x)=2+\cos(\pi x)$, $P(x)=2-\cos(2\pi x)$, and $r(x)=\left(\frac{K(x)}{P(x)}\right)^\lambda$, where $\lambda=-1$, $0$, $0.5$, $1$, $1.4$ and $2.3$

Theorems & Definitions (34)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Corollary 2.3
• proof
• Theorem 3.1
• proof
• Remark 3.2
• Theorem 3.3