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1. Bogdanov‐Takens Bifurcation of Codimension 4 in a Leslie‐Gower Predator‐Prey Model With Allee Effect and Intraspecific Competition

   https://doi.org/10.1002/mma.11200  

   Liang, Chenyu; Xu, Yancong; Rong, Libin (July 2025, Mathematical Methods in the Applied Sciences)

   ABSTRACT Predator‐prey models, such as the Leslie‐Gower model, are essential for understanding population dynamics and stability within ecosystems. These models help explain the balance between species under natural conditions, but the inclusion of factors like the Allee effect and intraspecific competition adds complexity and realism to these interactions, enhancing our ability to predict system behavior under stress. To detect early indicators of population collapse, this study investigates the intricate dynamics of a modified Leslie‐Gower predator‐prey model with both Allee effect and intraspecific competition. We analyze the existence and stability of equilibria, as well as bifurcation phenomena, including saddle‐node bifurcations of codimension 2, Hopf bifurcations of codimension 2, and Bogdanov‐Takens bifurcations of codimension at least 4. Detailed transitions between bifurcation curves–specifically saddle‐node, Hopf, homoclinic, and limit cycle bifurcations–are also examined. We observe a novel transition phenomenon, where a system jumps from saddle‐node bifurcation to homoclinic and limit cycle bifurcations. This suggests that burst oscillations may serve as an early warning of system collapse rather than simply a tipping point. Our findings indicate that moderate levels of intraspecific competition or Allee effect support coexistence of both populations, while excessive levels may destabilize the entire biological system, leading to collapse. These insights offer valuable implications for ecological management and the early detection of risks in population dynamics. 

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2. Dynamics of a state-dependent delay-differential equation

   https://doi.org/10.14232/ejqtde.2025.1.37  

   Gedeon, Tomas; Humphries, Antony; Mackey, Michael; Walther, Hans-Otto; Wang, Zhao (January 2025, Electronic Journal of Qualitative Theory of Differential Equations)

   We present a detailed study of a scalar differential equation with threshold state-dependent delayed feedback. This equation arises as a simplification of a gene regulatory model. There are two monotone nonlinearities in the model: one describes the dependence of delay on state, and the other is the feedback nonlinearity. Both increasing and decreasing nonlinearities are considered. Our analysis is exhaustive both analytically and numerically as we examine the bifurcations of the system for various combinations of increasing and decreasing nonlinearities. We identify rich bifurcation patterns including Bautin, Bogdanov–Takens, cusp, fold, homoclinic, and Hopf bifurcations whose existence depend on the derivative signs of nonlinearities. Our analysis confirms many of these patterns in the limit where the nonlinearities are switch-like and change their value abruptly at a threshold. Perhaps one of the most surprising findings is the existence of a Hopf bifurcation to a periodic solution when the nonlinearity is monotone increasing and the time delay is a decreasing function of the state variable. 

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3. Bifurcation Modeling and Analysis for the Dynamics of Product Adoption and Abandonment

   https://doi.org/10.1142/S0218127426501932  

   Hernández-López, Eymard; Kong, Lingju; Wang, Jin (July 2026, International Journal of Bifurcation and Chaos)

   This paper provides a concise synthesis of a three-compartment deterministic adoption–abandonment model characterized by nonlinear adoption incidence and dual-channel (infectious and noninfectious) abandonment. By reducing the system to its invariant total-population manifold, we derive explicit algebraic criteria for codimension-1 bifurcations, saddle-node and Hopf, and establish the conditions for Bogdanov–Takens (BT) degeneracy. These analytic anchors, combined with robust numerical continuation, allow us to locate BT and Bautin organizing centers and to trace global bifurcation curves, including homoclinic orbits and limit points of cycles. The resulting bifurcation atlas delineates the parameter regions associated with monostability, bistability, and sustained oscillations in the model. Our analysis reveals that these global structures reconfigure the basins of attraction, often creating parameter regimes where a stable coexistence equilibrium is confined within fragile boundaries in phase space. We demonstrate that modest parameter perturbations can trigger abrupt and sometimes irreversible qualitative transitions in the system’s dynamics. The mathematical framework offers a rigorous foundation for understanding how nonlinear adoption–abandonment interactions generate complex regime structures, which may serve as heuristic analogs for abrupt transitions in technology diffusion contexts. 

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4. Widespread neuronal chaos induced by slow oscillating currents

   https://doi.org/10.1063/5.0248001  

   Scully, James; Hinsley, Carter; Bloom, David; Meijer, Hil_G E; Shilnikov, Andrey L (March 2025, Chaos: An Interdisciplinary Journal of Nonlinear Science)

   This paper investigates the origin and onset of chaos in a mathematical model of an individual neuron, arising from the intricate interaction between 3D fast and 2D slow dynamics governing its intrinsic currents. Central to the chaotic dynamics are multiple homoclinic connections and bifurcations of saddle equilibria and periodic orbits. This neural model reveals a rich array of codimension-2 bifurcations, including Shilnikov–Hopf, Belyakov, Bautin, and Bogdanov–Takens points, which play a pivotal role in organizing the complex bifurcation structure of the parameter space. We explore various routes to chaos occurring at the intersections of quiescent, tonic spiking, and bursting activity regimes within this space and provide a thorough bifurcation analysis. Despite the high dimensionality of the model, its fast–slow dynamics allow a reduction to a one-dimensional return map, accurately capturing and explaining the complex dynamics of the neural model. Our approach integrates parameter continuation analysis, newly developed symbolic techniques, and Lyapunov exponents, collectively unveiling the intricate dynamical and bifurcation structures present in the system. 

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5. Bifurcations and pattern formation in a host–parasitoid model with nonlocal effect

   https://doi.org/10.1017/prm.2024.24  

   Xiang, Chuang; Huang, Jicai; Lu, Min; Ruan, Shigui; Wang, Hao (March 2024, Proceedings of the Royal Society of Edinburgh: Section A Mathematics)

   In this paper, we analyse Turing instability and bifurcations in a host–parasitoid model with nonlocal effect. For a ordinary differential equation model, we provide some preliminary analysis on Hopf bifurcation. For a reaction–diffusion model with local intraspecific prey competition, we first explore the Turing instability of spatially homogeneous steady states. Next, we show that the model can undergo Hopf bifurcation and Turing–Hopf bifurcation, and find that a pair of spatially nonhomogeneous periodic solutions is stable for a(8,0)-mode Turing–Hopf bifurcationand unstable for a(3,0)-mode Turing–Hopf bifurcation. For a reaction–diffusion model with nonlocal intraspecific prey competition, we study the existence of the Hopf bifurcation, double-Hopf bifurcation, Turing bifurcation, and Turing–Hopf bifurcation successively, and find that a spatially nonhomogeneous quasi-periodic solution is unstable for a(0,1)-mode double-Hopf bifurcation. Our results indicate that the model exhibits complex pattern formations, including transient states, monostability, bistability, and tristability. Finally, numerical simulations are provided to illustrate complex dynamics and verify our theoretical results. 

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