Cusp Bifurcation Research Articles

Bifurcation analysis and exploration of noise-induced transitions of a food chain model with Allee effect

Seeking shelter from threats is a widespread instinct across species, specially employed by prey to avoid direct confrontations with predators. The present investigation centers on a three-species food chain model wherein the basal prey, characterized by logistic growth, seeks refuge to evade the intermediate predator, while the intermediate predator, in turn, seeks refuge to avoid encounters with the top predator. Additionally, our model assumes that the presence of the top predator induces a mate-finding Allee effect among the intermediate predator. We investigate how varying levels of refuge, along with other critical parameters such as the reproduction rate of the basal prey and the natural mortality rate of the top predator, influence system’s dynamics within the biparametric planes. Our model displays multistability and undergoes transcritical, saddle–node, Bogdanov–Takens and cusp bifurcations across different parameters. Moreover, the external environmental noise can induce interesting dynamics in the predator–prey system, resulting in noise-induced frequent transitions between distinct interior attractors or from interior to axial attractors. This phenomenon is particularly notable in scenarios where the deterministic model exhibits tristability. In summary, our findings offer potential new avenues for developing control strategies within the realm of community ecology in constant as well as fluctuating environments.

Analysis of an intra- and interspecific interference model with allelopathic competition

In this paper, we propose an original model of two-microbial species competing for a single nutrient in the chemostat including general intra- and interspecific density-dependent growth rates with allelopathic interactions. Each species produces a toxin that affects the growth of other species as well as its own growth. The removal rates are distinct and include the specific death rate and the autotoxicity of each species. We establish an in-depth mathematical analysis by determining the multiplicity of all steady states of the three-dimensional system and their necessary and sufficient conditions of existence and local stability according to the operating parameters, which are the dilution rate and the inflowing concentration of the substrate. To describe the asymptotic behavior of the process according to these control parameters, we first determine theoretically the operating diagram. Using MATCONT software, these theoretical results are validated numerically but it reveals the cusp bifurcation that occurs by varying two parameters. The one-parameter bifurcation diagram in the dilution rate shows that there can be either transcritical or saddle-node bifurcations. Finally, we apply our results to a particular model in the literature without intra- and interspecific interference but with only allelopathic effects of the second species on the first species. We show that one of the coexistence steady states is locally exponentially stable when it exists, whereas in the literature they have not been able to demonstrate that the stability condition of this steady state is always fulfilled.

Extracting and analyzing the governing model for plastic deformation of metallic glasses

This paper first detects hidden system from plastic deformation of metallic glasses by sparse identification. The extracted model simulates four types of stress-time curves and displays the prediction of serrated events. This interpretation effectively explains various experimental phenomena of repeated yielding. Further, in terms of parametric sensitivity analysis to the model, two parameters are taken as bifurcation parameter, and the analysis of codimension-one and codimension-two bifurcation are carried out to excavate the causes of dynamic transformation, including saddle–node bifurcation, Hopf bifurcation, Bogdanov–Takens bifurcation and cusp bifurcation. Different bifurcation points correspond different types of stress-time curves. The homologous phase diagrams including periodic orbit, unstable orbit and chaotic behavior are presented to show the dynamics diversity of the model. In addition to dynamic analysis, statistical analysis for plasticity values is also applied to excavate the crossover between periodic and chaotic plastic dynamic transitions. Our results provide a novel perspective on the deformation of metallic glasses from the viewpoint of dynamic model and are also important for evaluating the plastic deformation properties of metallic glasses in practical applications.

Unraveling Chaos, Transition and Dynamical Complexities in a Generalized Predator–Prey Model with Cooperation-Induced Fear and Gestation Delay

This research investigates the interaction between a generalist predator and a prey, where the predator exhibits cooperative behavior during hunting, inducing fear into the prey population. Additionally, both the prey and predator populations are subject to harvesting. The study establishes the positivity and boundedness of the model’s solutions, ensuring the existence of the population. Analyzing the system, we explore its feasible steady states and their stability, along with various types of bifurcations, including Hopf with direction of stability, Saddle-node, Transcritical, Homoclinic, Bogdanov–Takens, and Cusp bifurcation. We also demonstrate the stability and bifurcation behavior of a delayed system. These findings are verified through one-parameter and two-parameter bifurcation structures, complemented by respective phase portraits. Notably, the system displays transition between different equilibria and bistability. Furthermore, numerical investigations reveal the impact of gestation delay, indicating chaotic behavior in the system due to this time delay.

Identifying numerical bifurcation structures of codimensions 1 and 2 in interacting species system

The present study has focused on the examination of a Holling type III nonlinear mathematical model that incorporates the influence of fear and Michaelis–Menten‐type predator harvesting. The incorporation of fear with respect to the prey species has been observed to result in a reduction in the survival probability of the prey population and a concurrent reduction in the reproduction rate of the prey species. The existence and stability of ecologically significant equilibria have been ascertained through mathematical analysis. Emphasis within the proposed model primarily centers on numerical bifurcations of codimensions 1 and 2. Numerical validation has been performed on all simulated outcomes within the feasible range of parametric values. Dynamical characteristics of the model have subsequently undergone investigation through a series of numerical simulations, successfully revealing various forms of local and global bifurcations. In addition to the identification of saddle‐node, Hopf, Bogdanov–Takens, transcritical, cusp, homoclinic, and limit point cycle (LPC) bifurcations, the model has also demonstrated bistability and global asymptotic stability. These bifurcation phenomena serve as illustrative examples of the intricate dynamical behavior inherent to the model. Numerical validation through graphical representations has been utilized to elucidate the effects of factors such as fear, nonlinear predator harvesting, and predation rate on the dynamics of the interacting species under different parametric conditions.

Exploring cooperative hunting dynamics and PRCC analysis: insights from a spatio-temporal mathematical model

The proposed mathematical model explores the intricate dynamics of a predator-prey system involving prey infection and cooperative hunting of predators. The model incorporates habitat complexity, emphasizing its influence on ecological interactions. The well-posedness of the system has rigorously been examined in a temporal setting and also conducted stability analysis. The bifurcation analysis reveals the existence of several local bifurcations on the system, namely transcritical bifurcation, saddle-node bifurcation, and Hopf bifurcation. Furthermore, these investigations delineate the two-dimensional bifurcations including Bogdanov–Takens and cusp bifurcations for different parametric combinations. With suitable choices of parameter values, the proposed model exhibits diverse dynamic phenomena, including bistable and tri-stable behavior. Latin hypercube sampling is utilized to conduct uncertainty analysis on input parameters, aiming to observe their effects on population dynamics. Subsequently, Kendall’s tau and Spearman’s rank correlation coefficients are also computed to investigate the impact of these uncertainties on the population. In the later part, a spatio-temporal system is proposed with two-dimensional diffusion terms to obtain the conditions for Turing instability. Numerical simulations have been conducted to observe the emergence of spatial patterns and the impact of predator cooperation in these patterns. The study provides valuable insights into the dynamics of complex ecological systems, emphasizing the interplay of spatial and temporal factors in shaping population dynamics and predator-prey interactions.

Trade-off and chaotic dynamics in a two-prey one-predator model with refuge, environmental noise and seasonal effects

In nature, poisonous prey species have developed toxins as an attack against predators. However, predator’s decisions to consume aposematic toxic prey are not solely determined by their toxicity. Predators often weigh the costs of ingesting toxins against the benefits of obtaining nutrients, with body size acting as a significant cue. In terms of avoiding predation, prey commonly respond by seeking refuge, a behavior that fluctuates considerably due to seasonality and environmental variability. In this study, we integrate these factors into the dynamics of two prey and one predator model. We explore the cost of fear to illustrate the trade-off dynamics of prey, and we adjust the predator’s death rate by integrating a function that serves predator loss resulting from encounters with toxic prey. Our model demonstrates bistability as a consequence of hysteresis effect and undergoes various bifurcations, including saddle–node, Hopf, Bautin, and cusp bifurcations. We observe that increasing prey refuge and reducing the level of fear contribute to system stabilization, yet chaotic behavior persists with respect to both parameters. While seasonality can introduce stability and periodicity, it may also give rise to complex and unpredictable dynamics. We comprehend that the system might undergo transitions through multiple periodic cycles as a result of changes in the degree of seasonality, yet it eventually settles into a periodic solution. Furthermore, our analysis using the stochastic model reveals that low noise levels maintain species densities close to deterministic equilibrium, whereas high noise levels result in broader dispersion or even extinction of species. The information presented in this article may help readers understand the biological knowledge gained by investigating interactions between predators and prey.

Tipping points, multistability, and stochasticity in a two-dimensional traffic network dynamics

Mitigating traffic jams is a critical step for the betterment of the urban transportation system, which comprises a large number of interconnected routes to form an intricate network. To understand distinct features of vehicular traffic flow on a network, a macroscopic two-dimensional traffic network model is proposed incorporating intra-nodal and inter-nodal vehicular interaction. Utilizing the popular techniques of nonlinear dynamics, we investigate the impact of different parameters like occupancy, entry rates, and exit rates of vehicles. The existence of saddle-node, Hopf, homoclinic, Bogdanov–Takens, and cusp bifurcations have been shown using single or biparametric bifurcation diagrams. The occurrences of different multistability (bistability/tristability) phenomena, stochastic switching, and critical transitions are explored in detail. Further, we calculate the possibility of achieving each alternative state using the basin stability metric to characterize multistability. In addition, critical transitions from free flow to congestion are identified at different magnitudes of stochastic fluctuations. The applicability of critical slowing down based generic indicators, e.g., variance, lag-1 autocorrelation, skewness, kurtosis, and conditional heteroskedasticity are investigated to forewarn the critical transition from free flow to traffic congestion. It is demonstrated through the use of simulated data that not all of the measures exhibit sensitivity to rapid phase transitions in traffic flow. Our study reveals that traffic congestion emerges because of either bifurcation or stochasticity. The result provided in this study may serve as a paradigm to understand the qualitative behavior of traffic jams and to explore the tipping mechanisms occurring in transport phenomena.

Prey group defense and hunting cooperation among generalist-predators induce complex dynamics: a mathematical study.

Group defense in prey and hunting cooperation in predators are two important ecological phenomena and can occur concurrently. In this article, we consider cooperative hunting in generalist predators and group defense in prey under a mathematical framework to comprehend the enormous diversity the model could capture. To do so, we consider a modified Holling-Tanner model where we implement Holling type IV functional response to characterize grazing pattern of predators where prey species exhibit group defense. Additionally, we allow a modification in the attack rate of predators to quantify the hunting cooperation among them. The model admits three boundary equilibria and up to three coexistence equilibrium points. The geometry of the nontrivial prey and predator nullclines and thus the number of coexistence equilibria primarily depends on a specific threshold of the availability of alternative food for predators. We use linear stability analysis to determine the types of hyperbolic equilibrium points and characterize the non-hyperbolic equilibrium points through normal form and center manifold theory. Change in the model parameters leading to the occurrences of a series of local bifurcations from non-hyperbolic equilibrium points, namely, transcritical, saddle-node, Hopf, cusp and Bogdanov-Takens bifurcation; there are also occurrences of global bifurcations such as homoclinic bifurcation and saddle-node bifurcation of limit cycles. We observe two interesting closed 'bubble' form induced by global bifurcations due to change in the strength of hunting cooperation and the availability of alternative food for predators. A three dimensional bifurcation diagram, concerning the original system parameters, captures how the alternation in model formulation induces gradual changes in the bifurcation scenarios. Our model highlights the stabilizing effects of group or gregarious behaviour in both prey and predator, hence supporting the predator-herbivore regulation hypothesis. Additionally, our model highlights the occurrence of "saltatory equilibria" in ecological systems and capture the dynamics observed for lion-herbivore interactions.

Bifurcations and mixed mode oscillations in a bi-stable plasma model with slow parametric excitation.

In this study, considering a bi-stable plasma model with slow parametric excitation, the bifurcation of periodic and chaotic responses as well as the resulting fast-slow motions is discussed analytically and numerically. For a nonautonomous fast sub-system, the generalized harmonic balancing method is utilized to obtain an averaged system. Bifurcation analysis about the averaged system shows that the critical manifolds form a S-shape structure. Meanwhile, supercritical and subcritical period doubling (PD) occurs on the upper branch simultaneously. As the frequency of the external excitation changes, bifurcation points on the limit cycle manifolds can present different relative locations. Moreover, an additional bi-stable structure induced by Cusp bifurcation emanates from the upper branch. On the other hand, the existence of a chaotic attractor and the corresponding boundary crisis phenomenon are verified using the Melnikov method and the basin of attraction. The structures of the numerical bifurcation diagram show good agreements with the analytical results. Considering two cases of low-frequency excitation, the corresponding fast-slow dynamics are discussed. It is found that, when the fast-slow flow passing the subcritical PD point, a low frequency with different magnitudes will lead to two patterns of bifurcation delay, i.e., the typical one and the excessive delay, which suppress the PD. As for the boundary crisis point, the slow passage effects show no distinct influence. Thus, three transition mechanisms based on two cases of the bifurcation structure are explained, including "fold of cycle-fold of cycle" type, "fold of cycle-delayed subcritical PD" type, and "fold of cycle-boundary crisis" type.

Bifurcation Analysis and Spatiotemporal Dynamics in a Diffusive Predator–Prey System Incorporating a Holling Type II Functional Response

This study aims to investigate a diffusive predator–prey system incorporating additional food for predators, prey refuge, fear effect, and its carry-over effects. For the temporal model, the well-posedness and persistence of the system have been discussed. We investigated the existence and the stability behavior of the various equilibria. Furthermore, we explored the bifurcations of codimension-1 including transcritical, saddle-node, and Hopf, concerning the crucial parameters. The system also presents codimension-2 bifurcations such as Bogdanov–Takens and cusp bifurcation along with the global homoclinic bifurcation. We observed the bubbling phenomena, which illustrate the fluctuations in the amplitudes of the periodic oscillations. For the spatiotemporal system, we established the non-negativity and boundedness of the solutions. We derived the conditions for the diffusion-driven instabilities in a confined region with Neumann boundary conditions. Extensive numerical simulations have been conducted to depict the various stationary patterns in Turing space. It is observed that incorporating cross-diffusion divides the bi-parametric plane into various sub-regions and dynamic patterns are analyzed in these different regions. The intricate spatiotemporal dynamics exhibited by prey–predator interactions are crucial for unraveling the intricacies within ecological systems.

Dynamics of a stage-structured predator-prey system with fear-induced group defense in autonomous and nonautonomous settings.

In this investigation, we construct a predator-prey model that distinguishes between immature and mature prey, highlighting group defense strategies within the mature prey. First, we embark on exploring the positivity and boundedness of the solution, unraveling sustainable equilibrium points, and deducing their stability conditions. Upon further investigation, we observe that the system exhibits diverse bifurcations, including Hopf, saddle-node, transcritical, generalized Hopf, cusp, and Bogdanov-Takens bifurcations. The results reveal that heightened fear decreases mature prey density, potentially causing prey extinction beyond a certain threshold. Increased maturation rates lead to the coexistence of immature and mature prey populations and higher predator density. Stronger group defense boosts mature prey density, while weaker defense results in weak persistence. Lower values of the maturation rate of prey and the decline rate of predators sustain only the predator population, reliant on resources other than focal prey. Furthermore, our model demonstrates intriguing and diverse dynamical phenomena, including various forms of bistability across distinct bi-parameter planes. We also explore the dynamics of a related nonautonomous system, where certain parameters are considered to vary with time. In the seasonally forced model, we set out to define criteria regarding the existence and stability of positive periodic solutions. Numerical investigations into the seasonally forced model uncover a spectrum of dynamics, ranging from simple periodic solutions to higher periodicities, bursting patterns, and chaotic behavior.

Bifurcation analysis of a two infection SIR-SIR epidemic model with temporary immunity and disease enhancement

In this paper we study a two infection SIR-SIR compartmental model, considering biological features described in dengue fever epidemiology. Due to a progressive loss of protective antibodies there is waning immunity in the first infection stage and disease enhancement or protection effects by the second infection stage. Bifurcation analysis reveals two codim-2 bifurcations as organizing centers. The unfolding of a cusp bifurcation describes the transition of the disease-free equilibrium into an endemic equilibrium by varying a parameter. These equilibria allow an analytical solution with explicit expressions which allow for a full geometrical interpretation of the occurring bifurcations related to stationary dynamics. A Bogdanov-Takens point is the starting point in the parameter space where oscillatory endemic dynamics occurs including a homoclinic connection. These findings bring additional insights on biological mechanisms able to generate rich and complicated dynamical behavior in simple epidemic models that are, so far, largely unexplored.

IMPACT OF HUNTING COOPERATION AND FEAR EFFECT IN A GENERALIST PREDATOR–PREY MODEL

Predator foraging facilitation or hunting cooperation and the antipredator behavior of prey are essential mechanisms in evolutionary biology and ecology and may strongly influence the predator–prey dynamics. In a real-world scenario, this behavioral tendency is well documented, but less is known about how it could affect the dynamics between predator and prey. Here, we investigate the impact of the fear of predator on prey and the hunting cooperation in predator on the predator–prey dynamics, where the predator is assumed to be of generalist type. We observe that without fear, even with the high level of hunting cooperation, both populations may coexist, though the increasing level of hunting cooperation reduces the prey density at coexistence equilibrium. Moreover, increasing level of fear also destabilizes the system with and without hunting cooperation. Further, in the presence of hunting cooperation and fear effect, the model shows three different types of bistability phenomena: bistability between two coexisting equilibria, bistability between coexisting equilibria and prey-free equilibrium, and bistability between stable limit cycle and coexisting equilibria. In addition, saddle-node, Hopf, transcritical bifurcation of codimension one, Bautin (generalized Hopf), Bogdanov–Takens, and cusp bifurcation of codimension two are observed.

Resilience assessment in complex natural systems.

Ecological resilience is the capability of an ecosystem to maintain the same structure and function and avoid crossing catastrophic tipping points (i.e. undergoing irreversible regime shifts). While fundamental for management, concrete ways to estimate and interpret resilience in real ecosystems are still lacking. Here, we develop an empirical approach to estimate resilience based on the stochastic cusp model derived from catastrophe theory. The cusp model models tipping points derived from a cusp bifurcation. We extend cusp in order to identify the presence of stable and unstable states in complex natural systems. Our Cusp Resilience Assessment (CUSPRA) has three characteristics: (i) it provides estimates on how likely a system is to cross a tipping point (in the form of a cusp bifurcation) characterized by hysteresis, (ii) it assesses resilience in relation to multiple external drivers and (iii) it produces straightforward results for ecosystem-based management. We validate our approach using simulated data and demonstrate its application using empirical time series of an Atlantic cod population and marine ecosystems in the North Sea and the Mediterranean Sea. We show that Cusp Resilience Assessment is a powerful method to empirically estimate resilience in support of a sustainable management of our constantly adapting ecosystems under global climate change.

Hierarchical Symmetry-Breaking Model for Stem Cell Differentiation

Waddington envisioned stem cell differentiation as a marble rolling down a hill, passing through hierarchically branched valleys representing the cell’s temporal state. The terminal valleys at the bottom of the hill indicate the possible committed cells of the multicellular organism. Although originally proposed as a metaphor, Waddington’s hypothesis establishes the fundamental principles for characterizing the differentiation process as a dynamic system: the generated equilibrium points must exhibit hierarchical branching, robustness to perturbations (homeorhesis), and produce the appropriate number of cells for each cell type. This article aims to capture these characteristics using a mathematical model based on two fundamental hypotheses. First, it is assumed that the gene regulatory network consists of hierarchically coupled subnetworks of genes (modules), each modeled as a dynamical system exhibiting supercritical pitchfork or cusp bifurcation. Second, the gene modules are spatiotemporally regulated by feedback mechanisms originating from epigenetic factors. Analytical and numerical results show that the proposed model exhibits self-organized multistability with hierarchical branching. Moreover, these branches of equilibrium points are robust to perturbations, and the number of different cells produced can be determined by the system parameters.

Nonlinear dynamic modelling and analysis of a rotating composite thin-walled beam considering ice coating

For rotating structures operating in severe weather conditions, ice coating phenomenon is an established issue that impairs the reliability and effectiveness of the structures by changing the dynamic behaviors. To explore the influence mechanisms of ice coating on the dynamic behaviors of rotating structures, this work presents an analytical dynamic model for nonlinear forced vibration of a composite thin-walled beam under both rotating and ice coating conditions. The ice coating is considered as C-shaped configuration which can affect the system parameters of the beam, especially aerodynamic performance. With effect of out-of-plane warping taken into account, the nonlinear dynamic equation of the beam is established using Hamilton's principle. The Galerkin discrete method together with the incremental harmonic balance method (IHB) as well as pseudo arclength continuation technique are adopted to obtain the periodic response of the nonlinear system. Then, a detailed parametric study on natural frequency, frequency-response and bifurcation behaviors is numerically conducted. Special attention is paid to the effects of ice coating, rotating angular velocity, inflow velocity and design parameters on the linear and nonlinear dynamic behaviors of the beam. Results reveal a prominent change in natural frequency, vibration amplitude and cusp bifurcation of multivalued region under the combined effects of ice coating and rotating motion. Such ice-induced changes exhibit the requirement of vibration control and optimization design consideration for rotating structures operating in icing coating conditions.

Effect of motor suspension parameters on bifurcations for a nonlinear bogie system

This paper aims to investigate the effect of motor suspension parameters, specifically damping and stiffness, on the bifurcations of a bogie system. A motor bogie model with a nonlinear smooth equivalent conicity function is established. The study includes qualitative analyses of the stability and the Hopf bifurcation of the equilibrium, with the running speed as the single parameter. Furthermore, the generalised Hopf bifurcation and Hopf-Hopf bifurcation of the equilibrium, based on two motor suspension parameters, are also analysed. The paper delves into the codimension-1 and codimension-2 bifurcations of the limit cycles generated by the Hopf bifurcation, including Neimark–Sacker bifurcation, fold bifurcation, cusp bifurcation, 1:3 resonance, and 1:4 resonance. Analytical investigations reveal that the cusp bifurcation alters the type of fold bifurcation (subcritical or supercritical), and the fold bifurcation influences the stability and bifurcation direction of the limit cycle. Additionally, resonance occurs between the motor and the bogie frame. Since the subcritical (supercritical) Neimark–Sacker bifurcation produces an unstable (a stable) torus, the resonance points associated with the subcritical Neimark–Sacker bifurcation will lead to the instability of the motor bogie. The bifurcation analysis on the motor suspension parameters in this paper offers a theoretical reference for enhancing the stability of the motor bogie.

Instability in optical injection locking semiconductors lasers using multiparametric bifurcation analysis.

In this paper, we investigate bifurcations of equilibria and transients by using modified rate equations of semiconductor lasers (SCLs) subjected to optical injection. An analytical study is performed to demonstrate some two-parameter bifurcations, inter alia, Bogdanov-Takens and Gavrilov-Guckenheimer bifurcations. A detailed numerical study based on the multiparametric bifurcation method and using 3D-plots and projections reveal a rich locking dynamics of SCLs. In this way, a so-called zero frequency detuning well is highlighted in the vicinity of a Hopf bifurcation confining minimal states of the larger Lyapunov exponent in injection locking curves. Three-parameter bifurcation curves mainly underscore cusp bifurcation and resizing of its multi-equilibrium region by the specific control parameter defined in this model. The bursting phenomenon observed in the transient regime is discussed by using various numerical approaches wherefrom another quantifying method tapping into two-parameter bifurcation analysis is proposed. Thereafter, metastable chaos dynamics supported by spiraling relaxation oscillations is also investigated as well as planar saddle-node bifurcations with three homoclinic orbits for high positive and negative detunings. At last, zero α-factor effects contribute to drastically shrink the unlocking region of SCLs, twofold increase in Hopf bifurcation along with evidencing of complex chaotic sine-shaped and folded torus-shaped attractors.

Bifurcation analysis in a predator–prey model with strong Allee effect on prey and density‐dependent mortality of predator

The influences of Allee effect and density‐dependent mortality on population growth are of great significance in ecology. In this paper, we first consider a Gause‐type predator–prey model with simplified Holling type IV functional response, strong Allee effect on prey, and density‐dependent mortality of predator. It is shown that the system exhibits rich and complex dynamics like bistability, local and global bifurcations such as transcritical bifurcation, saddle‐node bifurcation, Hopf bifurcation, cusp bifurcation, Bogdanov–Takens bifurcation, Bautin bifurcation, homoclinic bifurcation, saddle‐node bifurcation of limit cycle, and heteroclinic bifurcation. Next, we separately explore the influences of Allee effect and density‐dependent mortality on the dynamics of the same model. The results show that the strong Allee effect induces the occurrence of heteroclinic bifurcation and the reduction of the number of limit cycles, while the density‐dependent mortality stabilizes the system. Since the analytical expressions of the interior equilibria are difficult to derive, the results are verified numerically.