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

1. Problem Statement

The paper addresses the gap between neuroscientific insights into memory (learning and forgetting rates) and population-level ecology. Specifically, it investigates how nonlocal perception and cognitive maps influence spatial movement and population distribution.

• Core Question: How do the rates of learning ($\alpha$) and forgetting ($\mu$) shape long-term population distributions?
• Hypothesis: Memory-based navigation with nonlocal perception can generate heterogeneous space-use patterns (residence times) even in nearly homogeneous landscapes. The authors hypothesize a "lingering" effect where intermediate forgetting rates maximize residency near favorable locations, while insufficient memory leads to diffusion-like wandering and excessive memory leads to maladaptive persistence on stale information.
• Context: Previous models often used periodic boundary conditions, failing to capture how habitat boundaries influence perceived information. This study aims to clarify the role of domain geometry and memory dynamics on population patterns.

2. Mathematical Model

The authors propose a coupled reaction-diffusion system describing population density $u(x,t)$ and a cognitive map $m(x,t)$ on a bounded domain $\Omega \subset \mathbb{R}^n$.

The System:
$$\begin{cases} \partial_t u = \nabla \cdot (\gamma(m) \nabla u) + \nabla \cdot (\gamma'(m) u \nabla m) + f(s, u) & \text{in } \Omega \times (0, T] \\ \partial_t m = (\alpha \bar{s}(x) - m)u - \mu m & \text{in } \Omega \times (0, T] \\ \nabla \cdot (\gamma(m) u) \cdot n = 0 & \text{on } \partial \Omega \times (0, T] \end{cases}$$

Key Components:

• Cognitive Map ($m$): Represents spatial memory. It is updated locally based on presence ($u$) and perceived habitat quality ($\bar{s}$), with learning rate $\alpha$ and forgetting rate $\mu$.
• Perceived Quality ($\bar{s}$): A nonlocal quantity obtained by convolving the actual habitat quality $s(x)$ with a perception kernel $K(x,y)$ over a finite perceptual radius $R$. The paper considers both normalized (compensating for boundary truncation) and non-normalized kernels.
• Movement: Individuals move via a Fokker-Planck type diffusion $\nabla \cdot (\gamma(m) \nabla u + \gamma'(m) u \nabla m)$. The motility function $\gamma(m)$ is positive and decreasing, meaning individuals move slower in regions with high memory (favorable locations), creating an advection term that attracts them to high-memory peaks.
• Growth: The term $f(s, u)$ represents logistic growth, typically $u(s(x) - u)$.

3. Methodology

The authors employ a combination of rigorous mathematical analysis and numerical simulations:

1. Movement-Only Analysis (Section 2):

   • The growth term is removed ($f=0$) to isolate the effects of memory-driven movement.
   • Heterogeneity Analysis: They demonstrate that non-normalized kernels create artificial heterogeneity near boundaries (due to truncated perception), whereas normalized kernels eliminate this, allowing the study of intrinsic lingering on heterogeneous landscapes.
   • Steady-State Analysis: They analyze the steady-state equations to characterize the "lingering phenomenon" (non-monotonic dependence of peak density on forgetting rate).

2. Full Population Dynamics (Section 3):

   • Well-posedness: Proving local existence and uniqueness of solutions using fixed-point arguments (Schauder's theorem) and parabolic regularity theory.
   • Persistence: Proving the extinction equilibrium $(0,0)$ is linearly unstable and establishing the existence of a positive steady state using Schaefer's Fixed Point Theorem.
   • Uniqueness: Deriving conditions on the motility function $\gamma$ to ensure the uniqueness of the positive steady state.
   • Small $\alpha$ Analysis: Using perturbation theory (Inverse Function Theorem) to analyze the system when learning rates are small.

3. Numerical Simulations:

   • Simulations are conducted on 1D domains with unimodal resource distributions to visualize the trade-off between peak density (lingering strength) and total population mass under varying $\alpha$ and $\mu$.

4. Key Contributions and Results

A. The Lingering Phenomenon

• Definition: A phenomenon where the peak population density at a favorable location is maximized at an intermediate forgetting rate ($\mu$), given a high learning rate ($\alpha$).
• Mechanism:
  • Low $\mu$ (High Memory): Individuals remember too much, potentially getting "stuck" or failing to adapt to the current landscape structure, leading to flatter distributions.
  • High $\mu$ (Low Memory): Individuals forget too quickly, behaving like random walkers (diffusion), leading to uniform distribution.
  • Intermediate $\mu$: Balances retention of favorable locations with the ability to update the map, causing a sharp concentration (lingering) around the resource peak.
• Theorem 2.1: Rigorously proves that for a unimodal cognitive map, if $\alpha$ is sufficiently large, the peak density $u(0)$ first increases and then decreases as $\mu$ increases.

B. Boundary Effects and Kernels

• Non-normalized Kernels: In homogeneous environments, these create artificial heterogeneity near boundaries because the perceptual radius is truncated, leading to lower perceived quality at edges. This causes population accumulation in the center even without resource gradients.
• Normalized Kernels: Compensate for boundary truncation, ensuring that in a homogeneous environment, the perceived quality remains uniform. The study adopts normalized kernels to focus on intrinsic memory effects.

C. Population Persistence and Steady States

• Instability of Extinction: The extinction equilibrium is proven to be linearly unstable, ensuring population persistence.
• Existence: A positive steady-state solution exists under specific conditions on the motility function $\gamma$.
• Uniqueness: A unique positive steady state is guaranteed if the motility function satisfies a specific bound involving the ratio of its derivative to itself (Eq. 22).

D. Trade-offs in Population Size (The "Cost" of Lingering)

• Classical vs. Cognitive: In classical random diffusion models, the total population mass always exceeds the total resource capacity ($\int u > \int s$).
• Cognitive Deviation: In the cognitive model, lingering can reduce total population size.
  • Numerical results show that in the regime of strongest lingering (intermediate $\mu$, high $\alpha$), the total mass $\int u$ can drop below the total resource $\int s$.
  • Reason: Strong lingering causes overcrowding in specific patches, intensifying intraspecific competition (logistic term) and reducing the overall carrying capacity utilization.
• Optimal Strategy: The regime that maximizes peak density (lingering) is not the regime that maximizes total population size. The latter is achieved with faster forgetting rates that allow better distribution across the habitat.

5. Significance and Implications

• Ecological Insight: The study challenges the assumption that memory always benefits a population. It suggests a trade-off: while memory allows individuals to find and stay in good spots (lingering), it can lead to overcrowding and reduced total population size if the forgetting rate is not tuned correctly.
• Mathematical Advancement:
  • Extends Fokker-Planck diffusion theory to include nonlocal memory fields.
  • Provides rigorous proofs for well-posedness and steady-state existence in coupled nonlocal PDE systems with Neumann boundary conditions.
  • Demonstrates that nonlocal perception can generate spatial patterns even in homogeneous environments (via boundary effects).
• Future Directions: The paper identifies the need for global existence proofs for the full coupled system and suggests future work on evolutionary game theory (competition between species with different cognitive abilities).

In summary, the paper establishes that cognitive maps driven by nonlocal perception create a "lingering" effect, where intermediate forgetting rates maximize local density. However, this strategy comes with a demographic cost: the regime of strongest lingering can lead to a total population size smaller than the available resources, contrasting sharply with classical diffusion models.