Transcritical Bifurcation Research Articles

Dynamics of a discrete Rosenzweig–MacArthur predator–prey model with piecewise-constant arguments

This paper investigates the dynamics of a new discrete form of the classical continuous Rosenzweig–MacArthur predator–prey model and the biological implications of the dynamics. The discretization method used here is to modify the continuous model to another with piecewise-constant arguments and then to integrate the modified model, which is quite different from the traditional Euler discretization method. First, the existence and local stability of fixed points have been thoroughly discussed. Then, all codimension-1 bifurcations have been studied, including the transcritical bifurcations at the trivial fixed point and the boundary fixed point, and the Neimark–Sacker bifurcation at the unique positive fixed point. Furthermore, it is proved that there are no codimension-2 bifurcations. Finally, the control of the bifurcations is studied. Regarding the biological significance, we have considered the following four aspects: When no harvesting effort is applied to both predators and prey, it is observed that providing more resources to the prey species leads to predator extinction, causing the ecosystem to lose stability. This implies that the paradox of enrichment occurs. When predators are harvested, the system exhibits the hydra effect, i.e. increasing the harvesting effort on predators will lead to an increase in the mean population density of predators. When the prey is harvested, numerical examples indicate that increasing the harvesting effort initially reduces prey density. Upon reaching the bifurcation point, prey density stabilizes while the predator density continues to decline. When both prey and predators are harvested, the biological significance is similar to the previous case. The outcomes of this paper have significant theoretical meaning in the study of population ecology. By MATLAB, the bifurcation diagrams, maximal Lyapunov exponent diagrams, phase portraits and mean population density curves are plotted. Numerical simulations not only show the correctness of the theoretical findings but also reveal many new and interesting dynamics.

Dynamical transition in controllable quantum neural networks with large depth

Understanding the training dynamics of quantum neural networks is a fundamental task in quantum information science with wide impact in physics, chemistry and machine learning. In this work, we show that the late-time training dynamics of quantum neural networks with a quadratic loss function can be described by the generalized Lotka-Volterra equations, leading to a transcritical bifurcation transition in the dynamics. When the targeted value of loss function crosses the minimum achievable value from above to below, the dynamics evolve from a frozen-kernel dynamics to a frozen-error dynamics, showing a duality between the quantum neural tangent kernel and the total error. In both regions, the convergence towards the fixed point is exponential, while at the critical point becomes polynomial. We provide a non-perturbative analytical theory to explain the transition via a restricted Haar ensemble at late time, when the output state approaches the steady state. Via mapping the Hessian to an effective Hamiltonian, we also identify a linearly vanishing gap at the transition point. Compared with the linear loss function, we show that a quadratic loss function within the frozen-error dynamics enables a speedup in the training convergence. The theory findings are verified experimentally on IBM quantum devices.

Bifurcations in a Model of Criminal Organizations and a Corrupt Judiciary

Let a population be composed of members of a criminal organization and judges of the judicial system, in which the judges can be co-opted by this organization. In this article, a model written as a set of four nonlinear differential equations is proposed to investigate this population dynamics. The impact of the rate constants related to judges’ co-optation and ex-convicts’ recidivism on the population composition is explicitly examined. This analysis reveals that the proposed model can experience backward and transcritical bifurcations. Also, if all ex-convicts relapse, organized crime cannot be eradicated even in the absence of corrupt judges. The results analytically derived here are illustrated by numerical simulations and discussed from a crime-control perspective.

Seasonal refuge patterns of phytoplankton trigger irregular bloom events in a contaminated environment

Several experimental evidences and field data documented that zooplankton may alter its behavioral response in the presence of toxic phytoplankton, reducing its consumption to the point of starvation. This paper is devoted to the mathematical study of such interactions of toxic phytoplankton with grazer zooplankton. The non-toxic phytoplankton is assumed to adopt a density-dependent refuge strategy to avoid over-predation by zooplankton. Both groups of phytoplankton are assumed to suffer direct harm from anthropogenic toxicants, while zooplankton is affected indirectly by ingesting contaminated phytoplankton. We calibrate the proposed model with the field data from Talsari and Digha Mohana, India, and estimate some crucial model parameters consistent with the behavior of the observed data. Our results demonstrate that zooplankton grazing on toxic phytoplankton plays a key role in the emergence or mitigation of plankton blooms. We also highlight the system’s potential to exhibit multiple stable configurations under the same ecological conditions. The plankton system experiences significant regime shifts, which are explored through various bifurcation scenarios, such as transcritical and saddle-node bifurcations. These shifts are influenced by changes in refuge capacity, species growth rates, and environmental carrying capacity. Furthermore, we incorporate environmental variations due to seasonal periodic or almost periodic changes, allowing the refuge parameter to be time-dependent. We observe that the forced system exhibits double periodic solutions. Moreover, stronger seasonal variations in the refuge pattern lead to irregular chaotic blooms. In conclusion, the results offer valuable insights into the sustainability of biodiversity, potentially shedding light on the origin of diverse plankton bloom phenomena.

Bifurcation in a single-species logistic model with addition Allee effect and fear effect-type feedback control

In this paper, a single-species logistic model with both fear effect-type feedback control and additive Allee effect is developed and investigated using the new coronavirus as a feedback control variable. When the system introduces additive Allee effect and fear effect-type feedback control, more complicated dynamical behavior is obtained. The system can undergo transcritical bifurcation, saddle-node bifurcation, degenerate Hopf bifurcation and Bogdanov–Takens bifurcation. By numerical simulations, the system exhibits homoclinic bifurcation and saddle-node bifurcation of limit cycles as parameters are altered. Remarkably, it is the first time that two limit cycles have been discovered in a single-species logistic model with the Allee effect. Further, stronger Allee effect or stronger fear effect can lead to the extinction of the species population.

DYNAMICAL BEHAVIOR IN THE COMPETITIVE MODEL INCORPORATING THE FEAR EFFECT OF PREY DUE TO ALLELOPATHY WITH SHARED BIOTIC RESOURCES

This research develops a mathematical model of a natural phenomenon, namely sea snails that can release toxins (allelopathy) so that non-toxic sea snails become afraid. In addition, toxic and non-toxic sea snails share biotic resources. Based on the existing phenomenon, the model of fear effect caused by allelopathy in the competitive interaction model with shared biotic resources will be studied. In this system, three equilibrium points are obtained: extinction point of prey, extinction point of predator, and coexisting point under certain conditions. Analysis of local stability at equilibrium points by linearization shows that all equilibrium points are asymptotically stable with certain conditions. Numerical simulations at the equilibrium point show the same results as the analysis results. Then, numerical continuity was carried out by selecting variation of the fear effect parameter for Numerical continuity results show that changes in these parameters affect the population of toxic and non-toxic species, marked by the emergence of Transcritical bifurcations, Bifurcation occurs at the first Saddle-Node bifurcation at and the second Saddle-Node bifurcation at

Self-financing model for cabbage crops with pest management.

Smallholder farmers rely on their farm earnings to cover operating costs and generate income. That is not an easy task because of the pests, which reduce yields and generate plant protection costs. The farm yield and plant protection depend on the budget capacity of the farmer. In this work, we want to explore conditions for a sustainable and self-financing cabbage farm. We propose then a non-linear mathematical model for cabbage crops by considering the current account of the plantation as a dynamic variable. We assume that this variable increases due to the sale of cabbages, and provides for the seedling purchase, the plant protection costs, and the grower's income. In the first part, we analyze the model without pest management. We determine how the budget must be spent and we show the existence of a double transcritical bifurcation. We quantify the seasonal yield and income, and estimate the damage due to pest herbivory. In the second part, we analyze a slightly simplified version of our model and obtain the existence of a backward bifurcation. Furthermore, we show that botanical pesticides can be used to prevent pest spread with relatively low plant protection costs.

Stability and bifurcations for a 3D Filippov SEIS model with limited medical resources

In this paper, a three-dimensional (3D) Filippov SEIS epidemic model is proposed to characterize the impact of limited medical resources on disease transmission with discontinuous treatment functions. Qualitative analysis of non-smooth dynamical behaviors are performed on two subsystems and sliding modes. Criteria on the stability of various kinds of feasible equilibria and bifurcations, e.g., saddle-node bifurcation, transcritical bifurcation, and boundary equilibrium bifurcation, are established. The theoretical results are illustrated by numerical simulation, from which we find there could exist bistable phenomena, e.g., endemic and pseudo-equilibria, endemic equilibria of the two subsystems, or endemic and disease-free equilibria, even the basic reproduction numbers of two subsystems are less than 1. The disease spread is dependent both on the limited medical resources and latent compartment, which are more beneficial to effective disease control than planar Filippov and smooth models.

Stability and bifurcation in a two-patch commensal symbiosis model with nonlinear dispersal and additive Allee effect

In this paper, a two-patch model with additive Allee effect, nonlinear dispersal and commensalism is proposed and studied. The stability of equilibria and the existence of saddle-node bifurcation, transcritical bifurcation are discussed. Through qualitative analysis of the model, we know that the persistence and the extinction of population are influenced by the Allee effect, dispersal and commensalism. Combining with numerical simulation, the study shows that the total population density will increase when the Allee effect constant [Formula: see text] increases or [Formula: see text] decreases. In addition to suppress the Allee effect, nonlinear dispersal and commensalism are crucial to the survival of the species in the two patches.

Some Bifurcations of Codimensions 1 and 2 in a Discrete Predator–Prey Model with Non-Linear Harvesting

This paper delves into the dynamics of a discrete-time predator–prey system. Initially, it presents the existence and stability conditions of the fixed points. Subsequently, by employing the center manifold theorem and bifurcation theory, the conditions for the occurrence of four types of codimension 1 bifurcations (transcritical bifurcation, fold bifurcation, flip bifurcation, and Neimark–Sacker bifurcation) are examined. Then, through several variable substitutions and the introduction of new parameters, the conditions for the existence of codimension 2 bifurcations (fold–flip bifurcation, 1:2 and 1:4 strong resonances) are derived. Finally, some numerical analyses of two-parameter planes are provided. The two-parameter plane plots showcase interesting dynamical behaviors of the discrete system as the integral step size and other parameters vary. These results unveil much richer dynamics of the discrete-time model in comparison to the continuous model.

Analysis of stability and bifurcation and design of optimal control for phytoplankton–zooplankton–fish model with additional food and fish harvesting

ABSTRACT This paper investigates the dynamics of a four-dimensional plankton-fish system that contains three communities of plankton: non-toxic phytoplankton, toxic-producing phytoplankton, and zooplankton. The zooplankton consumes both non-toxic and toxic-producing phytoplankton. First, the positivity and boundedness of the solutions in the plankton-fish model are evaluated. Then, we derive the conditions for the existence of ecologically possible equilibria and their local and global asymptotic nature. Furthermore, we evaluate the criteria for the existence of Hopf-bifurcation and transcritical bifurcation. By applying Pontryagin’s maximum principle, we investigate the optimal control of fish harvesting, where fishing effort is taken as a control parameter. By using phase-space diagrams, time responses, bifurcation diagrams, and numerical analyses validate the theoretical results. The numerical results including maximum Lyapunov exponent verify chaotic dynamics of the system. We notice that a suitable choice of harvesting, phytoplankton growth rate, and additional food parameters can control the chaotic behavior of the presented model. The proposed novel mathematical model gives deeper insights to theoretical ecologists on how toxic-producing phytoplankton affects zooplankton and fish populations in aquatic systems. Furthermore, the proposed model is more appropriate for fish harvesting to maximize profit as well as species conversion in the presence of toxic phytoplankton.

Optimization of SOX2 Expression for Enhanced Glioblastoma Stem Cell Virotherapy

The Zika virus has been shown to infect glioblastoma stem cells via the membrane receptor αvβ5, which is activated by the stem-specific transcription factor SOX2. Since the expression level of SOX2 is an important predictive marker for successful virotherapy, it is important to understand the fundamental mechanisms of the role of SOX2 in the dynamics of cancer stem cells and Zika viruses. In this paper, we develop a mathematical ODE model to investigate the effects of SOX2 expression levels on Zika virotherapy against glioblastoma stem cells. Our study aimed to identify the conditions under which SOX2 expression level, viral infection, and replication can reduce or eradicate the glioblastoma stem cells. Analytic work on the existence and stability conditions of equilibrium points with respect to the basic reproduction number are provided. Numerical results were in good agreement with analytic solutions. Our results show that critical threshold levels of both SOX2 and viral replication, which change the stability of equilibrium points through population dynamics such as transcritical and Hopf bifurcations, were observed. These critical thresholds provide the optimal conditions for SOX2 expression levels and viral bursting sizes to enhance therapeutic efficacy of Zika virotherapy against glioblastoma stem cells. This study provides critical insights into optimizing Zika virus-based treatment for glioblastoma by highlighting the essential role of SOX2 in viral infection and replication.

Dynamics of a SVEIR Epidemic Model with a Delay in Diagnosis in a Changing Environment

A SVEIR epidemic model with a delay in diagnosis is studied in a constant and variable environment. The mathematical analysis shows that the dynamics of the model in the constant environment are completely determined by the magnitude of the delay-induced reproduction number . We established that if < 1, the disease-free equilibrium is globally asymptotically stable, and when > 1 the endemic equilibrium is globally asymptotically stable. In the variable environment, the model undergoes a transcritical bifurcation for = 1 leading to changes in the stability of the equilibrium points. The analytical effect of the delays in epidemic diagnosis is investigated. A minimum diagnosis rate αmin has been determined to face or control the disease effectively. Finally, numerical illustrations were presented to support the theoretical results.

Dynamical and optimal control analysis of lymphatic filariasis and buruli ulcer co-infection

This study delves into the dynamics of lymphatic filariasis and buruli ulcer coinfection, two overlooked yet impactful tropical diseases. With lymphatic filariasis, commonly referred to as elephantiasis, and buruli ulcer, a chronic affliction caused by mycobacterium ulcerans, both posing significant health challenges, understanding their interaction is crucial. Utilizing a mathematical model, this research aims to analyze the dynamics of this coinfection, elucidating its complexities. The study establishes the local asymptotic stability of the disease-free equilibrium and calculates the basic reproduction number using the next generation matrix. It uncovers transcritical and backward bifurcation phenomena within the model. Additionally, the integration of time-dependent controls enables the exploration of optimal disease management strategies. Numerical simulations highlight the efficacy of employing a comprehensive approach, utilizing all available controls simultaneously, as the most effective strategy for disease control. These findings underscore the importance of integrated interventions in combating lymphatic filariasis and buruli ulcer coinfection, offering valuable insights for public health policymakers and practitioners.

Impact of awareness and time-delayed saturated treatment on the transmission of infectious diseases

We have proposed a mathematical model with saturated incidence and treatment rates along with awareness and delay in treatment. We analyze the model and find the equilibrium points and their stability. We also find the basic reproduction number R0 to understand the disease dynamics. For R0=1, we get transcritical backward bifurcation which means that the disease persists in the population even if R0<1. We performed the sensitivity analysis and found that treatment along with awareness plays a significant role in controlling infectious disease. We deduce that awareness about the disease affects the transmission rate of infection. As people become aware of aspects of the infectious disease, they amend their behavior so that they fend themselves from catching the disease. We have introduced a time lag in treatment and found the threshold value of the time delay. It is observed that when the value of the time delay crosses the threshold value, we get a Hopf bifurcation i.e. endemic steady state becomes unstable above the threshold value and it may become difficult to control the disease beyond the threshold value of delay in treatment.

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.

Dynamics and bifurcations in a model of chronic myeloid leukemia with optimal immune response windows.

Chronic Myeloid Leukemia is a blood cancer for which standard therapy with Tyrosine-Kinase Inhibitors is successful in the majority of patients. After discontinuation of treatment half of the well-responding patients either present undetectable levels of tumor cells for a long time or exhibit sustained fluctuations of tumor load oscillating at very low levels. Motivated by the consequent question of whether the observed kinetics reflect periodic oscillations emerging from tumor-immune interactions, in this work, we analyze a system of ordinary differential equations describing the immune response to CML where both the functional response against leukemia and the immune recruitment exhibit optimal activation windows. Besides investigating the stability of the equilibrium points, we provide rigorous proofs that the model exhibits at least two types of bifurcations: a transcritical bifurcation around the tumor-free equilibrium point and a Hopf bifurcation around a biologically plausible equilibrium point, providing an affirmative answer to our initial question. Focusing our attention on the Hopf bifurcation, we examine the emergence of limit cycles and analyze their stability through the calculation of Lyapunov coefficients. Then we illustrate our theoretical results with numerical simulations based on clinically relevant parameters. Besides the mathematical interest, our results suggest that the fluctuating levels of low tumor load observed in CML patients may be a consequence of periodic orbits arising from predator-prey-like interactions.

Analyzing Bifurcations and Optimal Control Strategies in SIRS Epidemic Models: Insights from Theory and COVID-19 Data

This study investigates the dynamic behavior of an SIRS epidemic model in discrete time, focusing primarily on mathematical analysis. We identify two equilibrium points, disease-free and endemic, with our main focus on the stability of the endemic state. Using data from the US Department of Health and optimizing the SIRS model, we estimate model parameters and analyze two types of bifurcations: Flip and Transcritical. Bifurcation diagrams and curves are presented, employing the Carcasses method. for the Flip bifurcation and an implicit function approach for the Transcritical bifurcation. Finally, we apply constrained optimal control to the infection and recruitment rates in the discrete SIRS model. Pontryagin’s maximum principle is employed to determine the optimal controls. Utilizing COVID-19 data from the USA, we showcase the effectiveness of the proposed control strategy in mitigating the pandemic’s spread.

Effect of awareness and saturated treatment on the transmission of infectious diseases

Abstract In this article, we study the role of awareness and its impact on the control of infectious diseases. We analyze a susceptible-infected-recovered model with a media awareness compartment. We find the effective reproduction number R 0 {R}_{0} . We observe that our model exhibits transcritical forward bifurcation at R 0 = 1 {R}_{0}=1 . We also performed the sensitivity analysis to determine the sensitivity of parameters of the effective reproduction number R 0 {R}_{0} . In addition, we study the corresponding optimal control problem by considering control in media awareness and treatment. Our studies conclude that we can reduce the rate of spread of infection in the population by increasing the treatment rate along with media awareness.

Research on Stability and Bifurcation for Two-Dimensional Two-Parameter Squared Discrete Dynamical Systems

This study investigates a class of two-dimensional, two-parameter squared discrete dynamical systems. It determines the conditions for local stability at the fixed points for these proposed systems. Theoretical and numerical analyses are conducted to examine the bifurcation behavior of the proposed systems. Conditions for the existence of Naimark–Sacker bifurcation, transcritical bifurcation, and flip bifurcation are derived using center manifold theorem and bifurcation theory. Results of the theoretical analyses are validated by numerical simulation studies. Numerical simulations also reveal the complex bifurcation behaviors exhibited by the proposed systems and their advantage in image encryption.