The lingering phenomenon and pattern formation in a nonlocal population model with cognitive map

TL;DR

The paper addresses how learning and forgetting dynamics, mediated by nonlocal perception, shape population patterns via memory-driven diffusion. It develops a memory-based Fokker-Planck type model with a cognitive map updated by local presence and analyzed under normalized and truncated kernels; analytical and numerical results reveal a lingering phenomenon where intermediate forgetting optimizes residency near favorable memory peaks, even without growth, and persists under logistic growth. The study proves local well-posedness and persistence for the growth model, demonstrates the instability of extinction and existence of a positive steady state (with uniqueness under a motility-condition), and uncovers a trade-off between lingering strength and total population, including regimes where the total mass falls below the resource, contrasting classical diffusion models. These findings underscore the impact of cognitive-map structure and memory rates on long-term space use and population viability, with potential implications for understanding movement strategies in heterogeneous environments.

Abstract

The rates at which individuals memorize and forget environmental information strongly influence their movement paths and long-term space use. To understand how these cognitive time scales shape population-level patterns, we propose and analyze a nonlocal population model with a cognitive map. The population density moves by a Fokker--Planck type diffusion driven by a cognitive map that stores a habitat quality information nonlocally. The map is updated through local presence with learning and forgetting rates, and we consider both truncated and normalized perception kernels. We first study the movement-only system without growth. We show that finite perceptual range generates spatial heterogeneity in the cognitive map even in nearly homogeneous habitats, and we prove a lingering phenomenon on unimodal landscapes: for the fixed high learning rate, the peak density near the best location is maximized at an intermediate forgetting rate. We then couple cognitive diffusion to logistic growth. We establish local well-posedness and persistence by proving instability of the extinction equilibrium and the existence of a positive steady state, with uniqueness under an explicit condition on the motility function. Numerical simulations show that lingering persists under logistic growth and reveal a trade-off between the lingering and total population size, since near the strongest-lingering regime the total mass can fall below the total resource, in contrast to classical random diffusive--logistic models.

Figures (10)

• Figure 1: Schematic illustration of lingering in a one-dimensional habitat. The green curve represents perceived habitat quality (a cognitive map), with darker shading indicating locations that are more strongly remembered. Squirrels depict where individuals tend to reside. In the left panel, animals occupy two high-memory-retention regions and possibly move between the peaks. In the right panel, individuals linger in a single strongly remembered core region, leaving other peak areas underused.
• Figure 2: The blue curve represents the boundary of the habitat $\Omega$ occupied by the population. Dashed regions represent the nonlocal neighborhoods perceived under a finite perceptual radius at different locations.
• Figure 3: The left first figure is different kernel shapes of $J_R$ for radius $R=0.5,1,1.5$ and 2. The second plot illustrates $\bar{s}$ corresponding to kernels in the left figure. The black dotted line is $s(x)\equiv 1$. $\bar{s}(x)\approx s$ in the interior but decreases near the boundary, more strongly for larger $R$. The figure on the right depicts the numerical solution $u(x,t_{\text{final}})$ at a large time $t=500$. As $R$ increases, $u(x,t_{\text{final}})$ becomes more concentrated in the middle due to the artificial heterogeneity.
• Figure 4: The left panel shows normalized kernels with $R=0.5,1,1.5,$ and $2$. In the second figure, the environment is given as $s(x) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2} + 0.1$. $\bar{s}(x)$ closely tracks $s(x)$, with mild variation across $R$. In the last figure, population distributions are similar across $R$. The main shape of the profiles comes from $s(x)$, not the boundary truncation.
• Figure 5: Snapshots of $u(x,t_{\mathrm{final}}=500)$ for different values of the forgetting rate $\mu$. The left panel corresponds to $\alpha=1$ and the right panel to $\alpha=10$. The initial condition is $u_0=s$ and we use the bump kernel \ref{['bump']} with radius $R=1.5$.

Theorems & Definitions (18)

• Theorem 2.1: Lingering phenomenon on unimodal $\bar{s}(x)$
• proof
• Remark 1
• Remark 2
• Theorem 3.1
• proof
• Lemma 3.2: Unstable extinction equilibrium
• proof
• Theorem 3.3: Schaefer's Fixed Point Theorem EvansPDE
• Lemma 3.4