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.
The rates at which individuals memorize and forget new environmental information are closely linked to their spatial navigation and resulting patterns of space use. In neuroscience, studies have revealed a relationship between hippocampal neurogenesis and roaming patterns via roaming entropy, a measure of active territory coverage [10,16]. A recent study found that patients with Alzheimer's disease show significantly different roaming entropy compared with healthy older adults [11]. In particular, [3] links spatial navigation and the formation of cognitive maps through roaming entropy, highlighting the importance of accounting for both individual memory variability and the structure of the environment.
Although neuroscientific insights into learning, forgetting, and roaming have rarely been extended to population-level ecology, recent mathematical models formalize spatial memory using cognitive map frameworks [24,25,26,27] and examine its consequences for home range formation and aggregation [20,28,31]. In addition, [30] outlines open modeling and analytical challenges for cognitive movement models. In many of these studies, cognitive movement is represented as diffusion and advection with a nonlocal advection term, where a nonlocal memory field biases movement. This nonlocality is thought to be biologically natural because organisms sense their surroundings beyond a single point through vision or smell, so models introduce various types of spatial kernels with a finite perceptual radius, such as the top-hat kernel and bump functions. As a result, much of the literature investigates how the kernel, especially its radius and shape, affects aggregation, home range size, and spatial patterning [9,14,20,23]. Those models have been studied under periodic boundary condition on the domain, which makes it difficult to fully capture how habitat boundaries influence the information animals perceive.
We hypothesize that memory-based navigation with nonlocal perception can generate heterogeneous patterns of space use and residence times even in nearly homogeneous landscapes. For example, even when food availability is uniform, limited visual range and habitat structure, such as obstacles or cover, create heterogeneity in the cognitive map, consistent with evidence that environmental geometry shapes cognitive maps [3]. On such a heterogeneous cognitive map, low-retention environmental cues are forgotten before they can guide movement, shifting use toward strongly remembered locations; therefore the rates of learning and forgetting are critical. These ideas lead to a central question about how learning and forgetting rates shape long-term population distributions in ecological systems. It is expected that insufficient memory yields diffusion-like wandering and poor patch use, whereas excessive memory yields maladaptive persistence to stale information. We also expect that there exists a moderate balance that maximizes residency near favorable regions. We refer to this effect as lingering.
Weaker memory of habitat
We formalize these ideas by coupling population density u(x, t) on a bounded domain Ω ⊂ R n with a cognitive map m(x, t) that encodes spatial memory of habitat density s(x). We work with a dimensionless formulation. The quantity u(x, t) denotes a scaled population density and s(x) represents the corresponding scaled habitat quality or carrying capacity, so that (s(x) -u)u has the form of a logistic growth term with spatially varying capacity. The nonlocal quantity s(x) represents the perceived habitat quality obtained by spatially averaging s around x. Thus s, s and m are all dimensionless and measured on the same scale. The movement and memory dynamics are
where γ(m) > 0 renders memory-dependent mobility, α > 0 represents the rate at which individuals memorize (or learning strength), and µ > 0 is the memory decay rate, that is, how quickly they forget their memory. The update term (αs -m)u implements “learn-where-you-are’’: memory is reinforced (or down-weighted) proportionally to local presence u, depending on the perceived favorability αs.
The memory equation for m provides a mechanistic way to construct a cognitive map. The similar dynamics for cognitive map is demonstrated in literature [20,30]. We define the nonlocal quantity
where Z R (x) = B R (x) J R (x -y)χΩ(y)dy and J R encodes a perception kernel whose support is a ball B R (0) centered at 0 with radius R. χΩ is a characteristic function, that is, it is equal to 1 in Ω and otherwise it is 0, and we illustrate its use in Figure 2. We will consider a normalized and a not-normalized version of the integral operator in (3). We assume that γ is a positive, decreasing function of m. In other words, individuals move more slowly in regions where the memory variable m is high, tending to remain longer in locations they remember as favorable. Formally expanding the diffusion term generates a natural advection term induced by gradients of m, and since γ
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