The research, published in Frontiers in Ecology and Evolution by Dayasagar Parida, Reenu Rani, and Anuraj Singh, builds a predator–prey model specifically designed around the constraints of central place foraging. Unlike classical spatial models, where prey and predators migrate freely between patches, this framework imposes a strict asymmetry: the prey, modeled as a schooling fish such as capelin, stays confined to a single foraging ground, while the predators, modeled as seabirds, are geographically anchored to a breeding colony and must commute rapidly between the colony and the sea. The model divides the world into two zones: a foraging ground inhabited by both prey and a fraction of the predator population, and a breeding colony reserved exclusively for the remaining predators.

Technically, the system is governed by a three-dimensional set of ordinary differential equations featuring slow–fast dynamics. The key insight is a separation of timescales: the birds’ daily commuting flights occur on a much faster timescale than demographic processes such as reproduction, mortality, and predation. To make the mathematics tractable, the authors use the method of aggregation, replacing the fast commuting variables with their quasi-equilibrium distributions. This reduces the three-dimensional slow–fast system to a two-dimensional, single-timescale system that still permanently embeds the behavioral parameters of the daily commute, namely the outward foraging effort and the prey-dependent return rate, within its nonlinear structure.

The model incorporates two well-established ecological formulations. Predator–prey interactions follow a Beddington–DeAngelis functional response, which accounts for mutual interference among predators during hunting, a mechanism that classical Holling-type responses ignore. Predator growth follows a modified Leslie–Gower scheme, which captures how seabirds rely on alternative food resources, such as zooplankton or less optimal forage fish, when their preferred prey becomes scarce. The return migration rate is prey-density-dependent: the more abundant the fish, the faster a bird can fill its beak and fly back to provision its chick, mirroring the biological reality that provisioning time is shaped by local prey availability.

After establishing the fundamental well-posedness of the aggregated system, including positivity, boundedness, and permanence of solutions, the authors derive conditions for the existence of four biologically feasible equilibrium points: simultaneous extinction, predator-free coexistence, prey-free survival on alternative food, and full coexistence. Linear stability analysis of the Jacobian matrix reveals that the extinction state is always unstable, meaning both species cannot go extinct simultaneously from small perturbations, and that predator extinction is unstable in the presence of abundant prey. A coexistence equilibrium exists whenever the prey’s intrinsic growth rate exceeds the maximum per-capita predation pressure that mutual interference among predators allows.

The heart of the study is an in-depth bifurcation analysis, which identifies the qualitative regime shifts the system can undergo. Near the prey-free equilibrium, a transcritical bifurcation establishes a critical threshold for the prey’s intrinsic growth rate: if environmental degradation pushes the growth rate below this threshold, the prey cannot sustain predation pressure and goes extinct, forcing the seabirds to survive entirely on alternative food. At the coexistence equilibrium, saddle-node and Hopf bifurcations appear, and the system experiences stability switching through two successive Hopf bifurcations as the predator’s capture rate varies, oscillating periodically between the two thresholds before settling into a stable but severely depleted state.

Two-parameter continuation reveals even richer topological structures, including Bogdanov–Takens, generalized Hopf, and cusp bifurcations. The Bogdanov–Takens point, characterized by a double-zero eigenvalue, acts as an organizing center from which saddle-node and Hopf bifurcation curves emerge, marking an extreme tipping point where ecosystem resilience is completely compromised. The generalized Hopf, or Bautin, bifurcation separates supercritical regimes of predictable population cycling from subcritical domains of extreme fragility, where minor parameter drifts trigger high-amplitude oscillations that raise the extinction risk of at least one species. All analytical findings are validated through numerical simulations using MATLAB and MatCont.

Perhaps the most consequential finding is the cusp bifurcation, which exposes a dangerous bistability trap. Inside the cusp region, the ecosystem exhibits alternative stable states: a high-population-density coexistence and a severely depleted, low-population-density survival state, separated by a saddle equilibrium that acts as a tipping point. Which state the system settles into depends entirely on history and initial conditions. Crucially, once a severe environmental shock forces the populations across the saddle boundary, merely restoring favorable environmental parameters is not enough to recover the ecosystem; conservation efforts must actively push the population densities back across the separatrix threshold.

The authors emphasize that this framework differs fundamentally from standard non-spatial models, where population collapses are typically attributed to macro-environmental degradation such as declining carrying capacity or increased predator mortality. Here, catastrophic regime shifts can be induced solely by micro-scale behavioral constraints, specifically the commuting effort of the foragers. Stable environmental conditions, in other words, do not guarantee population survival if the physical and temporal costs of commuting cross critical bifurcation thresholds. The model also provides an early warning signal: the emergence of stable limit cycles before a saddle-node collapse indicates that the ecosystem is dangerously close to a catastrophic regime shift.

While the thresholds derived in the study are theoretical rather than empirically validated predictions, they offer conceptual guidance for conservation, particularly for central place foragers like marine seabirds. The transcritical threshold underscores the need for strict, data-driven fishing quotas in seabird foraging zones to maintain a sustainable baseline prey population. The bistability trap revealed by the cusp bifurcation conceptually mirrors observed ecological transitions, such as sudden seabird colony abandonments where populations fail to recover even after local fish stocks are artificially replenished. The authors suggest that management efforts could benefit from monitoring the predators’ outward foraging effort and ensuring sufficient local prey density to optimize return rates, keeping these commuting parameters within the safe operational bandwidths identified in the analysis as a theoretical early warning system against sudden, irreversible regime shifts.

Subject of Research: Mathematical ecology

Article Title: Complex dynamics in a slow–fast modified Leslie–Gower model: the effects of predator migration under central place foraging framework

Article References: Parida, D., Rani, R., & Singh, A. (2026). Complex dynamics in a slow–fast modified Leslie–Gower model: the effects of predator migration under central place foraging framework. Frontiers in Ecology and Evolution, 14, Article 1896471. https://doi.org/10.3389/fevo.2026.1896471

Image Credits: AI Generated

DOI: 10.3389/fevo.2026.1896471

Keywords: central place foraging, predator-prey dynamics, bifurcation analysis, seabird ecology, slow-fast systems, regime shifts, Leslie-Gower model, Beddington-DeAngelis, Complex, dynamics, slow, fast