Bogdanov Takens Research Articles

Spatiotemporal and Trade-Off Dynamics in Prey–Predator Model with Domed Functional Response and Fear Effect

In the ecological scenario, predators often risk their lives pursuing dangerous prey, potentially reducing their chances of survival due to injuries. Prey, on the other hand, try to strike a balance between reproduction rates and safety. In our study, we introduce a two-dimensional prey–predator model inspired by Tostowaryk’s work, specifically focusing on the domed-shaped functional response observed in interactions between pentatomid predators and neo-diprionid sawfly larvae. To account for the varying effectiveness of larval group defense, we incorporate a new component into the response equation. Our investigation delves into predator trade-off dynamics by adjusting the predator’s mortality rate to reflect losses incurred during encounters with dangerous prey and prey’s trade-off between safety and reproduction rate incorporating this domed-shaped functional response. Our model demonstrates bistability and undergoes various bifurcations, including transcritical, saddle-node, Hopf, Bogdanov–Takens, and Homoclinic bifurcations. Critical parameters impact both predator and prey populations, potentially leading to predator extinction if losses due to dangerous prey encounters become excessive, highlighting the risks predators face for their survival. Furthermore, the efficacy of group defense mechanisms can further endanger predators. Expanding our analysis to a spatially extended model under different perturbations, we explore Turing instability to explain the relationship between diffusion and encounter parameters through both stationary and dynamic pattern formation. Sensitivity to initial conditions uncovers spatiotemporal chaos. These findings provide valuable insights into comprehending the intricate dynamics of prey–predator interactions within ecological systems.

Study of a cannibalistic prey–predator model with Allee effect in prey under the presence of diffusion

In this study, we have investigated the temporal and spatio-temporal behavior of a prey–predator model with weak Allee effect in prey and the quality of being cannibalistic in a specialist predator. The parameters responsible for the Allee effect and cannibalism impact both the existence and stability of coexistence steady states of the temporal system. The temporal system exhibits various kinds of local bifurcations such as saddle–node, Hopf, Generalized Hopf (Bautin), Bogdanov–Takens, and global bifurcation like homoclinic, saddle–node bifurcation of limit cycles. For the model with self-diffusion, we establish the non-negativity and prior bounds of the solution. Subsequently, we derive the theoretical conditions in which self-diffusion leads to the destabilization of the interior equilibrium. Additionally, we explore the conditions under which cross-diffusion induces the Turing-instability where self-diffusion fails to do so. Further, we present different kinds of stationary and dynamic patterns on varying the values of diffusion coefficients to depict the spatio-temporal model’s rich dynamics. It has been found that the addition of self and cross-diffusion in a prey–predator model with the Allee effect in prey and cannibalistic predator play essential roles in comprehending the pattern formation of a distributed population model. It is expected that the comprehensive mathematical analysis and extensive numerical simulations used in investigating the global dynamics of the proposed model can facilitate researchers in studying the temporal and spatial aspects of prey–predator models in more significant detail.

Hopf and Bogdanov–Takens Bifurcations of a Delayed Bazykin Model

Hopf and Bogdanov–Takens Bifurcations of a Delayed Bazykin Model

BIFURCATION ANALYSIS OF AN SIR MODEL WITH SATURATED INCIDENCE RATE AND STRONG ALLEE EFFECT

In this paper, rich dynamics and complex bifurcations of an SIR epidemic model with saturated incidence rate and strong Allee effect are investigated. First, the existence of disease-free and endemic equilibria is explored, and we prove that the system has at most three positive equilibria, which exhibit different types such as hyperbolic saddle and node, degenerate unstable saddle (node) of codimension 2, degenerate saddle-node of codimension 3 at disease-free equilibria, and cusp, focus, and elliptic types Bogdanov–Takens singularities of codimension 3 at endemic equilibria. Second, bifurcation analysis at these equilibria are investigated, and it is found that the system undergoes a series of bifurcations, including transcritical, saddle node, Hopf, degenerate Hopf, homoclinic, cusp type Bogdanov–Takens of codimensional 2, and focus and elliptic type Bogdanov–Takens bifurcation of codimension 3 which are composed of some bifurcations with lower codimension. The system shows very rich dynamics such as the coexistence of multiple periodic orbits and homoclinic loops. Finally, numerical simulations are conducted on the theoretical results.

Bifurcations and global dynamics of a predator–prey mite model of Leslie type

AbstractIn this paper, we study a predator–prey mite model of Leslie type with generalized Holling IV functional response. The model is shown to have very rich bifurcation dynamics, including subcritical and supercritical Hopf bifurcations, degenerate Hopf bifurcation, focus‐type and cusp‐type degenerate Bogdanov–Takens bifurcations of codimension 3, originating from a nilpotent focus or cusp of codimension 3 that acts as the organizing center for the bifurcation set. Coexistence of multiple steady states, multiple limit cycles, and homoclinic cycles is also found. Interestingly, the coexistence of two limit cycles is guaranteed by investigating generalized Hopf bifurcation and degenerate homoclinic bifurcation, and we also find that two generalized Hopf bifurcation points are connected by a saddle‐node bifurcation curve of limit cycles, which indicates the existence of global regime for two limit cycles. Our work extends some results in the literature.

Bifurcation Analysis of a Predator–Prey Model with Alternative Prey and Prey Refuges

In this paper, we study the codimensions of Hopf bifurcation and Bogdanov–Takens bifurcation of a predator–prey model with alternative prey and prey refuges, which was proposed by Chen et al. [ 2023 ]. The results show that the predator–prey model can undergo a supercritical Hopf bifurcation or a Bogdanov–Takens bifurcation of codimension two under certain parameter conditions. It means that there are some predator–prey models with an alternative prey and prey refuges which have a limit cycle or a homoclinic loop. Moreover, it is also shown that the codimension of Hopf bifurcation is at most one and codimension of Bogdanov–Takens bifurcation is at most two.

Dynamic Relationship Between Informal Sector and Unemployment: A Mathematical Model

Shortage of formal jobs, lack of skills in workforce and increasing human population proliferate the informal sector. This sector provides an opportunity to unskilled workers to gain skills along with earnings. In this paper, a deterministic nonlinear mathematical model is developed to study the effects of informal skill learning and job generation on unemployment. For the formulated system, feasibility of equilibria and their stability properties are discussed. A pertinent quantity ([Formula: see text]), known as the reproduction number, is calculated and it is shown that the formulated system undergoes transcritical, saddle-node, Hopf and Bogdanov–Takens bifurcations on the variation of [Formula: see text]. The analytically obtained results are validated through numerical simulations. The results obtained from this study indicate that a substantial rate of job generation by self-employed individuals has a stabilizing effect on the system. Moreover, self-employment along with informal skill acquisition through engaging in informal work proves to be an effective measure in curbing the issue of unemployment in society.

Impact of Fear and Group Defense on the Dynamics of a Predator–Prey System

To reduce the chance of predation, many prey species adopt group defense mechanisms. While it is commonly believed that such defense mechanisms lead to positive feedback on prey density, a closer observation reveals that it may impact the growth rate of species. This is because individuals invest more time and effort in defense rather than reproductive activities. In this study, we delve into a predator–prey system where predator-induced fear influences the birth rate of prey, and the prey species exhibit group defense mechanism. We adopt a nonmonotonic functional response to govern the predator–prey interaction, which effectively captures the group defense mechanism. We present a detailed mathematical analysis, encompassing the determination of feasible equilibria and their stability conditions. Through the analytical approach, we demonstrate the occurrence of Hopf and Bogdanov–Takens (BT) bifurcations. We observe two distinct types of bistabilities in the system: one between interior and predator-free equilibria, and another between limit cycle and predator-free equilibrium. Our findings reveal that the parameters associated with group defense and predator-induced fear play significant roles in the survival and extinction of populations.

Bifurcation Analysis of Bogdanov–Takens Bifurcations in Delay Differential Equations

Bifurcation Analysis of Bogdanov–Takens Bifurcations in Delay Differential Equations

Interplay between Normal Forms and Center Manifold Reduction for Homoclinic Predictors near Bogdanov–Takens Bifurcation

Interplay between Normal Forms and Center Manifold Reduction for Homoclinic Predictors near Bogdanov–Takens Bifurcation

Bifurcation analysis of predator–prey model with Cosner type functional response and combined harvesting

In this paper, we consider a predator–prey model with Cosner type functional response and combined harvesting. First, we explore the existence and stability of the equilibria. Then using the center manifold theorem and normal form theory, we investigate codimension one and codimension two bifurcations of the model. The analysis shows that the system has a variety of bifurcation phenomena including transcritical bifurcation, saddle‐node bifurcation, Hopf bifurcation, Bogdanov–Takens bifurcation and homoclinic bifurcation. Our findings indicate that the dynamics with harvesting are significantly richer than the system without harvesting. Finally, numerical simulations are provided to support the analytical results.

Modeling the impact of hospital beds and vaccination on the dynamics of an infectious disease

The unprecedented scale and rapidity of dissemination of re-emerging and emerging infectious diseases impose new challenges for regulators and health authorities. To curb the dispersal of such diseases, proper management of healthcare facilities and vaccines are core drivers. In the present work, we assess the unified impact of healthcare facilities and vaccination on the control of an infectious disease by formulating a mathematical model. To formulate the model for any region, we consider four classes of human population; namely, susceptible, infected, hospitalized, and vaccinated. It is assumed that the increment in number of beds in hospitals is continuously made in proportion to the number of infected individuals. To ensure the occurrence of transcritical, saddle–node and Hopf bifurcations, the conditions are derived. The normal form is obtained to show the existence of Bogdanov–Takens bifurcation. To validate the analytically obtained results, we have conducted some numerical simulations. These results will be useful to public health authorities for planning appropriate health care resources and vaccination programs to diminish prevalence of infectious diseases.

The impact of social media advertisements and treatments on the dynamics of infectious diseases with optimal control strategies

The dissemination of public health information through television and social media posts is essential for informing the public about the transmission of contagious diseases, which is crucial in preventing the spread of various infectious diseases. In this paper, we propose a non-linear mathematical model to assess the effect of advertisements through social media in creating awareness and limiting treatment on spreading infectious diseases. These initiatives may alter population behaviour and divide the susceptible population into subgroups. In addition, to comprehend these dynamics better, we use half-saturation constant rates for media coverage and treatment. The model’s well-posedness and feasibility are evaluated. The possible biological equilibrium points are calculated. Local and global stability are carried out. The objective of our study is to produce the model’s bifurcation. Transcritical, Saddle–node, Hopf bifurcation of codimension 1 and Cusp, Generalized-Hopf (Bautin), and Bogdanov–Takens (BT) bifurcation of codimension 2 are studied for this purpose. Due to the limited medical resources and supply efficiency, the model exhibits backward bifurcation, resulting in bistability. Moreover, the occurrence condition for stability and direction of Hopf bifurcation is discussed. This model study demonstrates that the system is significantly influenced by the pace with which awareness programmes are implemented and that raising this value above a threshold may result in continuous oscillation. Sensitivity analysis employs the normalized forward sensitivity index of the basic reproduction number to provide a comprehensive understanding of the effect of various parameters on accelerating and limiting disease spread. Further, the minimum possible cost is determined by formulating an optimal control system based on sensitivity analysis and applying Pontryagin’s maximum principle. Methods of cost-effectiveness, such as ACER and ICER, are used to determine the most cost-effective control intervention strategy among all the strategies. Numerical simulations have been done to support all theoretical findings.

Bifurcation Analysis of a New Aquatic Ecological Model with Aggregation Effect and Harvesting

In this paper, we investigated the dynamics of the interaction between Microcystis aeruginosa and filter-feeding fish in a new aquatic ecological model and considered the effects of aggregation and harvesting and focused on studying the critical threshold conditions through the analysis of saddle-node bifurcation, Hopf bifurcation, and Bogdanov–Takens bifurcation. We also conducted numerical simulations to illustrate our findings and provided biological interpretations. The results obtained indicate that the aggregation effect or harvesting can disrupt the coexistence of Microcystis aeruginosa and filter-feeding fish. The filter-feeding fish population may go extinct while the Microcystis aeruginosa population could survive. We identified the importance of finding an appropriate timing for harvesting Microcystis aeruginosa in order to promote the growth of the filter-feeding fish population. This optimal timing may be influenced by the carrying capacity of Microcystis aeruginosa. Taken together, our study sheds light on the dynamics of Microcystis aeruginosa and filter-feeding fish in an aquatic ecosystem, highlighting the critical role of aggregation, harvesting, and timing in determining the coexistence and survival of these species.

Dynamical Analysis of a Predator–Prey Model with Additive Allee Effect and Migration

In this paper, a predator–prey model in which the prey has the additive Allee effect and the predator has artificially controlled migration is proposed. When the system introduces additive Allee effect and artificially controlled migration, more complicated dynamical behavior is obtained. The system can undergo saddle-node bifurcation, transcritical bifurcation, pitchfork bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation. Two limit cycles are found and discussed. The influence of the additive Allee effect and artificially controlled migration on the dynamics of the system is also presented. In detail, when the Allee effect is large, the prey will become extinct. When the artificially controlled migration rate is larger, the intensity of the prey (pest) will be smaller and the intensity of the predator will be larger. This indicates that artificially controlled migration can be effectively used to control the pest.

Simple Time-Periodic Delay Can Support Complex Dynamics

In this paper, we investigate the complex dynamics of a mapping derived from a differential equation with simple time-periodic delay. Firstly, we calculate the truncated normal form of 1:1 resonance of the mapping at a degenerate fixed point and obtain an approximating system of the mapping by using Picard iteration. By analyzing the approximate system, we find that the mapping will undergo a 1:1 resonance at the degenerate fixed point. Secondly, the qualitative property and the stability of the degenerate fixed point are determined, which provide a new view to understand the dynamic of differential equation with simple time-periodic delay. However, the approximate system does not have the versal unfolding of the Bogdanov–Takens singularity of codimension 2. These phenomena show that simple time-periodic delay can support complex dynamics. Finally, a numerical simulation is carried out to verify the analytic results.

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.

Bogdanov–Takens Bifurcation of Codimension 3 in the Gierer–Meinhardt Model

Bogdanov–Takens Bifurcation of Codimension 3 in the Gierer–Meinhardt Model

Influence of asymmetrical factors on aircraft ground steering stability

An aircraft taxiing on the ground is affected by several nonlinear forces so that the directional instability may occur during the steering process, and the asymmetric factors could exacerbate this phenomenon. A nonlinear asymmetrical aircraft ground taxiing dynamic model is established in MATLAB/Simulink in this study. Then the numerical continuation method based on the bifurcation theory is used to study the influence of the asymmetrical aircraft structure on the taxiing directional stability by taking the distance between the eccentric mass and the aircraft symmetry plane as the bifurcation parameter. In addition, a dual-parameter bifurcation is conducted to study the influence of the nose-wheel steering angle and the asymmetrical mass eccentric distance on the directional stability. Moreover, the effect of the asymmetrical thrust on the taxiing steering bifurcation characteristics is investigated, and the instability mechanism is revealed by analyzing the aircraft taxiing motion states under five working conditions. Then the dual-parameter bifurcation curves are obtained by introducing the nose wheel steering angle as the additional bifurcation parameter. The Bogdanov-Takens (BT) bifurcation and the Generalized Hopf (GH) bifurcation in the nonlinear ground turning dynamic system are also discussed. The results show that the asymmetrical factors can significantly affect the aircraft ground taxiing directional stability.

Bogdanov–Takens Bifurcation of Codimensions 3 and 4 in a Holling and Leslie type Predator–Prey System with Strong Allee Effect

Bogdanov–Takens Bifurcation of Codimensions 3 and 4 in a Holling and Leslie type Predator–Prey System with Strong Allee Effect