How generalist predation shapes a predator-prey model: A bifurcation analysis.

We investigate the dynamical properties of cusp bifurcations in max-plus dynamical systems derived from continuous differential equations through the tropical discretization and the ultradiscrete limit. A general relationship between cusp bifurcations in continuous and corresponding discrete systems is formulated as a proposition. For applications of this proposition, we analyze the Ludwig and Lewis models, elucidating the dynamical structure of their ultradiscrete cusp bifurcations obtained from the original continuous models. In the resulting ultradiscrete max-plus systems, the cusp bifurcation is characterized by piecewise linear representations, and its behavior is examined through the graph analysis.

Vibration is a critical issue in aerospace structures, where lightweight design, high flexibility, and complex operational environments often lead to pronounced nonlinear dynamic responses. Excessive vibrations induced by harmonic excitations, aerodynamic loads, or onboard equipment can significantly degrade structural integrity, control accuracy, and service life. Consequently, advanced passive vibration suppression techniques with strong robustness and broadband effectiveness are of great importance in aerospace engineering applications. The bifurcation boundary and vibration suppression performance of Piezoelectric–Monostable Nonlinear Energy Sink (PMNES) are crucial for evaluating its effectiveness on the main structure. To simplify the analysis of flexible aerospace structures, a reduced-order model is derived by modal truncation in the low-frequency range, which is then treated as a two-degree-of-freedom main structure. To focus on the underlying nonlinear dynamic mechanisms, an equivalent two-degree-of-freedom lumped-parameter system is adopted as a generic representation of the dominant low-frequency dynamics of flexible aerospace structures. In this work, the electromechanical coupling control equations of the system of a two-degree-of-freedom main structure coupled with PNES are derived through the application of Newton’s second law and Kirchhoff’s voltage law. The methods of complexification-averaging (CX-A) and Runge–Kutta (RK) are employed to assess the vibration suppression performance and stability characteristics of the system under harmonic excitation. The approximate solution is validated through numerical solutions. The approximate solutions of the system are employed to derive the Saddle Node (SN) bifurcation and codimension-two cusp bifurcation points, while the enhanced algorithm is employed to ascertain the most unfavorable amplitude at each external excitation circular frequency and to determine whether the mark represents a Hopf Bifurcation (HB) point. The generalized transmissibility is utilized to assess the efficacy of vibration suppression. The various vibration suppression efficiency regions are created by superimposing the vibration suppression efficiency maps and bifurcation maps. The influence of PNES parameters on the vibration suppression region is investigated. The results indicate that this method can effectively evaluate the bifurcation boundary and vibration suppression performance of PMNES.

Population dynamics, mainly the predator–prey interactions, are fundamental to ecological systems. In this work, we focus on a predator–prey model with Allee effects in both populations. By selecting key parameters, we explore diverse bifurcations, including transcritical bifurcation, saddle-node bifurcation, Hopf bifurcation, Bautin bifurcation, cusp bifurcation, and Bogdanov–Takens bifurcation. To account for environmental variability, stochastic perturbations are introduced into prey growth rates and harvesting intensities, represented by stochastic differential equations with multiplicative white noise. At higher noise intensity, we observe a rare and intriguing phenomenon: simulations under identical parameters and noise intensity yield three distinct trajectory behaviors. These include all trajectories escaping to extinction, some trajectories dividing between extinction and persistence, and all trajectories converging to persistence. This variability, absent in the deterministic framework, underscores the critical role of noise in driving unpredictable ecological outcomes. We use Stochastic Sensitivity Function (SSF) technique and confidence ellipse method to find exact noise intensity of a transition. These findings provide new insights into the interplay of deterministic and stochastic forces in ecological systems, emphasizing the importance of noise in shaping population dynamics and informing conservation strategies under environmental uncertainty.

Abstract We present a framework for analyzing collections of interacting hysterons through the lens of catastrophe theory. By modeling hysteron dynamics as a gradient system, we show how to construct hysteron transition graphs by characterizing the fold bifurcations of the dynamical system. Transition graphs represent the sequence of hysterons switching states, providing critical insights into the collective behavior of driven disordered media. Extending this analysis to higher codimension bifurcations, such as cusp bifurcations and crossings of fold curves, allows us to map out how the topology of transition graphs changes with variations in system parameters. This approach can suggest strategies for designing metamaterials capable of encoding targeted memory and computational functionalities, but it also highlights the rapid increase of design complexity with system size, further underscoring the computational challenges of controlling large hysteretic systems.

In the realm of ecology, the interplay between cooperative behaviours and the Allee effect represents a crucial nexus in understanding natural dynamics. These elements elucidate various mechanisms governing prey-predator intraspecific interactions. In this study, we introduce a two-dimensional prey-predator model that incorporates a strong Allee effect impacting on prey growth function and facilitating hunting cooperation among predators. Our proposed model innovatively integrates a density-dependent death rate term referred to as predator intraspecific competition, denoted as D > 0, and employs a Holling type III functional response to enhance our analysis. The primary focus of this research is to explore how hunting cooperation influences the dynamics of the model, specially whenever the intraspecific predation affecting the predator populations. Furthermore, we meticulously document the system's dynamical behaviours through an extensive bifurcation analysis, using both one-parameter and twoparameter approaches. Through the rigorous local and global bifurcation analysis, we uncover a spectrum of significant bifurcations, including saddle-node bifurcation, Hopf bifurcation, Bogdanov-Takens (BT) bifurcation, and cusp bifurcation points (CP), alongside generalized Hopf (GH) bifurcations as local bifurcations, and a homoclinic bifurcation as a global bifurcation. This comprehensive examination not only contributes to the theoretical understanding of ecological interactions but also underscores the profound implications of cooperation and density-dependent effects in shaping ecosystem dynamics.

Consider the nongeneric family of 3D piecewise linear differential systems, with a discontinuity plane that has two parallel invisible tangency lines, such that the region between them is the sliding region. It is known that the change of stability of the sliding region gives rise to a Crossing Limit Cycle (CLC). The stability of the CLC that emerges from this bifurcation mechanism is characterized by two control parameters, which are considered as bifurcation parameters. One of them is associated with the existence of saddle-node bifurcation points for CLCs, while the other one is associated with the existence of cusp bifurcation points for CLCs. By putting together these three bifurcation mechanisms in a three-parametric unfolding, we obtain a codimension-3 bifurcation point, which we will call the pseudo-cusp bifurcation. The 3D bifurcation diagram is characterized by regions with zero, one, two and three CLCs.

A liquid drop connected to thin pliable object produces a deflection. If the volume of the liquid is sufficiently large, the deflection is modest and its shape is unique. Decreasing the volume from this state, however, leads to one of two possible minimal-volume outcomes, depending on the balance of rigidity and surface tension: The structure either returns to its original unstrained configuration or fully collapses to create a stable, highly stressed shape. Using a simple three-dimensional setup arising from self-assembly applications where the deflection is restricted to a single hinge, we characterize all the possible equilibrium shapes and show that the two possibilities arise as a consequence of a codimension-2 cusp bifurcation.

Mathematical modeling is essential for understanding infectious disease dynamics and guiding public health strategies. We study the global dynamics of a susceptible-infectious-recovered-susceptible (SIRS) model with a generalized nonlinear incidence function, revealing a rich array of bifurcation phenomena, including saddle-node, cusp, forward and backward bifurcations, Bogdanov-Takens bifurcations, saddle-node bifurcation of limit cycles, subcritical and supercritical Hopf bifurcations, generalized Hopf bifurcations, homoclinic and degenerate homoclinic bifurcations, as well as isola bifurcation. Using normal form theory, we show that the Hopf bifurcation reaches codimension three, resulting in up to three small-amplitude limit cycles. The involvement of the recovered population enables coexistence of these limit cycles, leading to bistable and tristable dynamics. We employ a one-step transformation method to analyze codimension two and three Bogdanov-Takens bifurcations, confirming a maximum codimension of three. In particular, we identify isolas of limit cycles in an SIRS model involving double exposure, introducing a mechanism for generating limit cycles centered on the isola. The findings may have important public health implications, highlighting how nonlinearities in transmission and immunity can produce recurrent outbreaks or persistent infection despite interventions. The existence of multiple limit cycles suggests that small changes in transmission rates or immune response could cause abrupt shifts in outbreak patterns, emphasizing the need for adaptive and flexible intervention strategies.

Decoding cancer dynamics: The role of miRNA in E2F/Myc networks and bifurcation points.

In this paper, we study the high codimension bifurcation problems of an SIR epidemic model with a nonlinear incidence rate [Formula: see text]. Through bifurcation analysis of the system, it was found that the model can undergo high codimension bifurcation phenomena, including the codimension 3 cusp-type Bogdanov–Takens bifurcation, codimension 3 focus-type Bogdanov–Takens bifurcation, codimension 2 Bogdanov–Takens bifurcation, codimension 2 generalized Hopf bifurcation and codimension 2 cusp bifurcation of equilibrium points. In particular, considering the basic reproduction number [Formula: see text] as the bifurcation parameter, we found that the model exhibits complex dynamic behaviors during parameter variations, such as forward bifurcation with hysteresis, endemic bubble phenomena, mutual nesting of hysteresis and bubbles, multistability, and multi-periodicity. Our results show that high order nonlinear incidence rates may enable epidemic models to generate novel complex dynamic characteristics such as multiple steady states and large amplitude periodic oscillations. This study provides a new research perspective for analyzing the complex dynamic behaviors of epidemic systems with nonlinear incidence rates and their research achievements are of great significance for the prevention and control of infectious diseases.

Abstract To further promote the research in the field of discrete memristors, a discrete memristor is introduced into a two-dimensional chaotic map, which is a discrete model describing a second-order digital filter. A novel two-dimensional chaotic map is coined. The investigations on the local dynamics reveal the distribution patterns of dynamic, which are characterised on two-dimensional parameter planes. Two typical nonlinear phenomena, namely, resonance entrainments and cusp bifurcations, are identified explicitly by employing isocline method. Furthermore, the evolutions of basins of attraction are presented based on the tangency of stable and unstable manifolds. The critical curves are employed to derive global bifurcations, and further to demonstrate that the chaotic attractors are confined within specific boundaries. To show more evidences on the realizability of the map, several typical chaotic attractors are implemented on the DSP hardware platform.

The prey–predator interaction significantly influences ecosystems. Prey species perform anti-predator activities while foraging and balance their food and safety requirements to improve their survival chances. Several prey species use strategies to mitigate predation risk, including relocating to lower-risk and less lucrative habitats, increasing vigilance and modifying reproductive strategy. Certain prey species use group defenses to diminish the possibility of predator’s predation. This study examines a model involving one prey and two predators, where the prey species exhibits group defense against predator species. While studying we have got several codimension-one bifurcations (Transcritical, Saddle node, Hopf bifurcation) and codimension-two bifurcations (Bogdanov–Takens, generalized Hopf, Cusp bifurcation) exhibited by the proposed system. We have conducted both one-parametric and two-parametric bifurcation analyses of our proposed model. We have investigated the complex dynamical characteristics of the system over several parametric planes using bifurcation and Lyapunov exponent diagrams. We have observed several types of organized periodic structures, including both period-doubling and period-halving routes to chaos. Additionally, we have discussed the variation in the population density of each species. This study examines the influence of toxicity parameters on the dynamics of the system and their roles in the survival of the species.

Augmenting flexibility: mutual inhibition between inhibitory neurons expands functional diversity.

This paper introduces a novel computer-assisted method for detecting and constructively proving the existence of cusp bifurcations in differential equations. The approach begins with a two-parameter continuation along which a tool based on the theory of Poincaré index is employed to identify the presence of a cusp bifurcation. Using the approximate cusp location, Newton's method is then applied to a given augmented system (the cusp map), yielding a more precise numerical approximation of the cusp. Through a successful application of a Newton-Kantorovich type theorem, we establish the existence of a non-degenerate zero of the cusp map in the vicinity of the numerical approximation. Employing a Gershgorin circles argument, we then prove that exactly one eigenvalue of the Jacobian matrix at the cusp candidate has zero real part, thus rigorously confirming the presence of a cusp bifurcation. Finally, by incorporating explicit control over the cusp's location, a rigorous enclosure for the normal form coefficient is obtained, providing the explicit dynamics on the center manifold at the cusp. We show the effectiveness of this method by applying it to four distinct models.

The predator-prey interaction in mathematical ecology is a basic phenomenon in nature that has an important impact on community organization and on preserving the ecological diversity. In this research work, we have developed an unique aquatic ecological model to investigate the interaction between Microcystis aeruginosa and filter-feeding fish in presence of toxicity. This model specifically focuses on describing the phenomenon of Microcystis aeruginosa aggregation and the effect of toxin producing Microcystis aeruginosa blooms on filter-feeding fishes. Holling type II and Holling type III functional responses are used in our proposed model. Here, we have analyzed the model parameters to examine the stability of all equilibrium points in our system. Our system shows local bifurcations, including transcritical, saddle-node, Hopf, generalized Hopf, Cusp bifurcation and Bogdanov-Takens. Further, we have seen global bifurcation, particularly homoclinic bifurcation. Additionally, we have provided evidence of the hysteresis phenomena and basins of attraction to support the existence of bi-stability. Multiple numerical examples support each of these theoretical findings.

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.

Abstract This study explores the dynamics of a simple mechanical oscillator involving a magnet on a spring constrained to an axis; this magnet is additionally subject to the attractive force from a second magnet, which is placed on a parallel offset axis. The moments of both magnets remain aligned. The dynamics of the first magnet is first analysed in isolation for an unforced situation in which the second magnet is static and its position is taken as a parameter. We find codimension-1 saddle-node bifurcations, as well as a codimension-2 cusp bifurcation. The system has a region of bistability which increases in size with increasing force ratio. Next, the parametrically forced situation is considered, in which the second magnet moves sinusoidally. A comprehensive analysis of the forced oscillator behaviour is presented from the dynamical-systems standpoint. The solutions are shown to include periodic, quasiperiodic and chaotic trajectories. Resonances are shown to exist and the effect of weak damping is explored. Layered stroboscopic maps are used to produce cross-sections of the chaotic attractor as the parametric forcing frequency is varied. The strange attractor is found to disappear for a narrow window of forcing frequencies near the natural frequency of the spring.

This study presents a comprehensive model of predator–prey interactions within a toxic environment, with a particular focus on the effect of toxicant compounds on the development of populations. By incorporating environmental disturbances, the dynamics of the model are investigated to enhance the system’s authenticity. Analytical explanations have been provided for the deterministic system solutions, including positivity, uniform boundedness and persistence. The deterministic portion of the investigation entails a comprehensive examination of occurrence and stability criteria pertaining to every possible equlibria. The bifurcation studies conducted on the system exhibit the appearance of local bifurcations, including transcritical, saddle-node and Hopf bifurcations. Moreover, these evaluations establish the parametric region in which Bautin, Bogdanov–Takens and cusp bifurcation occur. Under a relevant selection of parametric values, the suggested system has the capacity to manifest a wide range of dynamic phenomena, such as bi-stable behavior, emergence of limit cycles, and presence of homoclinic loops. Furthermore, in a stochastic environment, the use of Lyapunov functions explains the existence of a global positive solution. It has additionally been argued that the proposed system exhibits ultimate stochastic boundedness. Subsequently, specific and adequate criteria demonstrate the eradication of both species as well as the long-term survival of prey communities. We have also investigated the impact of the exogenous input rate of toxic substances and the coefficient of toxic substances in both species on the behavior of the whole system, both in deterministic and stochastic scenarios. Theoretical findings have been confirmed by various numerical investigations.

A family of dissipative two-dimensional nonlinear mappings is considered. The mapping is described by the angle and action variables and parameterized by ε controlling nonlinearity, δ controlling the amount of dissipation, and an exponent γ is a dynamic free parameter that enables a connection with various distinct dynamic systems. The Lyapunov exponents are obtained for different values of the control parameters to characterize the chaotic attractors. We investigated the time evolution for the stationary state at period-doubling bifurcations. The convergence to the stationary state is made using a robust homogeneous and generalized function at the bifurcation parameter, which leads us to obtain a set of universal critical exponents. The parameter space of the mapping is investigated, and tangent, period-doubling, pitchfork, and cusp bifurcations are found, and a street of saddle-area and spring-area structures is observed.