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In this paper, a discrete predator-prey model incorporating herd behaviour and square root response function is deduced from its continuous version by the semi-discretization method. Firstly, the existence and local stability of fixed points of the system are studied by applying a key lemma. Secondly, by employing the centre manifold theorem and bifurcation theory, the conditions for the occurrences of the transcritical bifurcation and Neimark-Sacker bifurcation are obtained. Not only that but the direction and stability conditions of the bifurcated closed orbits are also clearly shown. Finally, numerical simulations are also given to confirm the existence of Neimark-Sacker bifurcation. [ABSTRACT FROM AUTHOR]

This article presents an analysis and control approach for a non-smooth air-gap permanent magnet synchronous motor (NSAG-PMSM) in the absence of external disturbances. The analytical study of NSAG-PMSM shows the existence of equilibrium points. Based on the Routh–Hurwitz criterion, the stability of the equilibrium points reveals the existence of transcritical bifurcation. NSAG-PMSM exhibits various dynamical behaviours, such as bistable chaos, periodic spiking oscillations, chaotic spiking characteristics, coexistence between periodic and chaotic behaviours and periodic evolution towards monostable chaos as system parameters change. The research uses microcontroller implementation to validate the dynamical characteristics observed during the numerical simulations of the NSAG-PMSM. The study of NSAG-PMSM proposes a strategy to mitigate chaos and stabilise the system using two simple controllers, with a comparative study presented using peak overshoot and settling time diagrams. By combining these different aspects, this article significantly contributes to the understanding of the operation in NSAG-PMSM, highlighting specific aspects related to the application of microcontroller techniques in the field of electrical engineering and solutions to chaos control. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

A four-compartmental fractional-order epidemic model has been investigated to understand the transmission mechanism of infectious diseases with the population's memory effect. The existence and uniqueness criterion of the model solution of the proposed fractional-order model is verified. Utilizing the next-generation matrix method, a threshold quantity called, the basic reproduction number ( R 0 ) is obtained. The model possesses two equilibrium points, infection-free and endemic. The asymptotic stability (local and global) of the proposed system at the equilibrium points has been analyzed thoroughly. It is observed that the total number of infections during the disease is influenced by the fractional-order of the model which represents the population's memory. A transcritical bifurcation is exhibited around the infection-free equilibrium point when the basic reproduction number crosses unity. Additionally, a fractional-order optimal control problem has been studied by considering two disease interventions: media awareness and treatment. The policy containing infectious disease spread has been determined based on a cost-effectiveness analysis. Sensitivity indices are computed to determine which parameters significantly impact R 0 and hence may used in controlling the disease. Some numerical simulations have been performed to verify analytical results by using MATLAB2022a. [ABSTRACT FROM AUTHOR]

In this paper, a discrete predator-prey model incorporating herd behaviour and square root response function is deduced from its continuous version by the semi-discretization method. Firstly, the existence and local stability of fixed points of the system are studied by applying a key lemma. Secondly, by employing the centre manifold theorem and bifurcation theory, the conditions for the occurrences of the transcritical bifurcation and Neimark-Sacker bifurcation are obtained. Not only that but the direction and stability conditions of the bifurcated closed orbits are also clearly shown. Finally, numerical simulations are also given to confirm the existence of Neimark-Sacker bifurcation.

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 R0{R}_{0}. We observe that our model exhibits transcritical forward bifurcation at R0=1{R}_{0}=1. We also performed the sensitivity analysis to determine the sensitivity of parameters of the effective reproduction number R0{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.

The global crisis of the COVID-19 pandemic has highlighted the need for mathematical models to inform public health strategies. The present study introduces a novel six-compartment epidemiological model that uniquely incorporates a higher isolation rate for unreported symptomatic cases of COVID-19 compared to reported cases, aiming to enhance prediction accuracy and address the challenge of initial underreporting. Additionally, we employ optimal control theory to assess the cost-effectiveness of interventions and adapt these strategies to specific epidemiological scenarios, such as varying transmission rates and the presence of asymptomatic carriers. By applying this model to COVID-19 data from India (30 January 2020 to 24 November 2020), chosen to capture the initial outbreak and subsequent waves, we calculate a basic reproduction number of 2.147, indicating the high transmissibility of the virus during this period in India. A sensitivity analysis reveals the critical impact of detection rates and isolation measures on disease progression, showing the robustness of our model in estimating the basic reproduction number. Through optimal control simulations, we demonstrate that increasing isolation rates for unreported cases and enhancing detection reduces the spread of COVID-19. Furthermore, our cost-effectiveness analysis establishes that a combined strategy of isolation and treatment is both more effective and economically viable. This research offers novel insights into the efficacy of non-pharmaceutical interventions, providing a tool for strategizing public health interventions and advancing our understanding of infectious disease dynamics. [ABSTRACT FROM AUTHOR]

This paper is the first to propose an allelopathic phytoplankton competition ODE model influenced by the fear effect based on natural biological phenomena. It is shown that the interplay of this fear effect and the allelopathic term cause rich dynamics in the proposed competition model, such as global stability, transcritical bifurcation, pitchfork bifurcation, and saddle-node bifurcation. We also consider the spatially explicit version of the model and prove analogous results. Numerical simulations verify the feasibility of the theoretical analysis. The results demonstrate that the primary cause of the extinction of non-toxic species is the fear of toxic species compared to toxins. Allelopathy only affects the density of non-toxic species. The discussion guides the conservation of species and the maintenance of biodiversity. [ABSTRACT FROM AUTHOR]

The commensal symbiosis system with the Allee effect and single feedback control is proposed and analyzed in this paper. The stability analysis of all possible equilibrium points is discussed, and the sufficient conditions for global stability of the interior equilibrium points are obtained. The occurrence of transcritical bifurcation and saddle-node bifurcation around the equilibrium points is investigated. Finally, the main results of the model are illustrated by numerical simulations. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. 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.

The transition between strong and weak Allee effects in prey provides a simple regime shift in ecology. In this article, we study a discrete predator-prey system with Holling type II functional response and Allee effect. First, the number of fixed points of the system, local stability, and global stability is discussed. The population changes of predator and prey under strong or weak Allee effects are proved using the nullclines and direction field, respectively. Second, using the bifurcation theory, the bifurcation conditions for the system to undergo transcritical bifurcation and Neimark-Sacker bifurcation at the equilibrium point are obtained. Finally, the dynamic behavior of the system is analyzed by numerical simulation of bifurcation diagram, phase diagram, and maximum Lyapunov exponent diagram. The results show that the system will produce complex dynamic phenomena such as periodic state, quasi-periodic state, and chaos.

Capturing complicated dynamics and understanding the underlying controlling ecological processes is one of the major ecological issues. The Allee effect is an essential component in ecology, and considering it can have a substantial impact on system dynamics. In the present investigation, we analysed a prey-predator scenario in which the predator is a generalist since it feeds on prey populations and the Allee phenomenon impacts the prey population's growth. The influence of the Allee effect on the changing nature of the system is investigated. The stability and boundedness of the model's equilibria are extensively investigated. We found that including the Allee effect enhances the system's local and global behaviours through a detailed bifurcation analysis. The chaotic nature of the system is strongly impacted by the Allee effect, particularly once a specific threshold value is reached. In the study of bifurcation analysis, we looked into bifurcations such the presence of transcritical bifurcation and Hopf-bifurcation to chaos. We added stochastic perturbation to this problem by including random fluctuations in the sensitive parameters. Finally, we analysed the system's mean-square stochastic stability towards the internal equilibrium. As a result, it is discovered that the Allee effect and stochastic perturbation considerably influence the behaviour of the prey-predator system.

The rarity of species increases its market price, consequently leading to the overexploitation of the species and even the extinction of the species. We study how the harvest intensity and the additive Allee effect impact on the Gordon–Schaefer model. In addition, by Sotomayor's theorem and Poincaré–Andronov theorem, we prove the existence of Hopf bifurcation, saddle-node bifurcation and transcritical bifurcation, respectively. Finally, we illustrate our results by numerical simulations. We find that both the cost per unit of harvest and the additive Allee effect have a significant impact on human exploitation of the population. As the additive Allee effect reduces to the weak Allee effect, the lower harvest cost encourages humans to increase the exploitation of species. This threshold is a switch that controls the strong Allee effect. If it exceeds its threshold, then the motivation of humans to exploit the species increases. [ABSTRACT FROM AUTHOR]

Zika is a mosquito-transmitted viral disease which may spread directly by the vector or sexual transmission. Zika virus may persist in semen and urine for a long time after disappearance from the blood, those persons are known as the convalescent humans. It can also be transmitted vertically among mosquitoes. In this paper, we have considered an eight-compartment Zika model to study the effect of all the said aspects on the virus dynamics in deterministic as well as stochastic environment. In analytic part, we have computed basic reproduction number and discussed stability of different equilibria. We have shown the proposed model undergoes through transcritical bifurcation when the reproduction number is unity and validate the model with real infection data of Dominican Republic in 2016. To study the model in stochastic environments, the additive noise is taken into consideration which is formulated considering the standard Brownian white noises proportional to each class. We have obtained the condition for disease extinction and persistence in mean. All theoretical findings are justified by numerical simulations. Lastly, the paper is ended with some conclusions. [ABSTRACT FROM AUTHOR]

One of the most important agricultural activities worldwide, coffee cultivation, is severely affected by the Coffee Berry Borer (CBB), Hypothenemus hampei, considered the primary coffee pest. The CBB is a tiny beetle that diminishes the quantity and quality of coffee beans by penetrating them to feed on the endosperm and deposit its eggs, continuing its life cycle. One strategy to combat CBBs is using biological control agents, such as certain species of ants. Here, a mathematical model (consisting of a system of nonlinear ordinary differential equations) is formulated to describe the prey–predator interaction between CBBs and an unspecified species of ants. From this mathematical perspective, the model allows us to determine conditions for the existence and stability of extinction, persistence or co-existence equilibria. Transitions among those equilibrium states are investigated through the maximum per capita consumption rate of the predator as a bifurcation parameter, allowing us to determine the existence of transcritical and saddle-node bifurcations. Phase portraits of the system are presented for different values of bifurcation parameter, to illustrate stability outcomes and the occurrence of bifurcations. It is concluded that an increase in bifurcation parameters significantly reduces the CBB population, suggesting that ant predation is an effective control strategy, at least theoretically. [ABSTRACT FROM AUTHOR]

This paper studies the dynamics of interacting Tilapia fish and Pelican bird population in the Salton Sea. We assume that the diseases spread in Tilapia fish follows the Holling type II response function, and the interaction between Tilapia and Pelican follows the Beddington–DeAngelis response function. The dynamics of diffusive and delayed system are discussed separately. Analytically, all the feasible equilibria and their stability are discussed. The criterion for Turing instability is derived. Based on the normal form theory and center manifold arguments, the existence of stability criterion and the direction of Hopf bifurcation are obtained. Numerical simulation shows the occurrence Hopf bifurcation, double Hopf bifurcation and transcritical bifurcation scenarios. The snap shot shows the spot, spot-strip mix patterns in the whole domain. Further, the stability switching phenomena is observed in the delayed system. Our comprehensive study highlights the effect of different parameters, multiple time delay and extinction in Pelican populations. [ABSTRACT FROM AUTHOR]

In this work, the global behavior of a discrete population model { xn+1 = αxne-yn + β, yn+1 = αxn(1-e-yn), n = 0,1,2,..., is considered, where α ∈ (0, 1), β ∈ (0, +∞), and the initial value (x0, y0) ∈ [0, ∞)×[0, ∞). To illustrate the dynamics behavior of this model, the boundedness, periodic character, local stability, bifurcation, and the global asymptotic stability of the solutions are investigated. [ABSTRACT FROM AUTHOR]

In this paper, we couple population dynamics and evolutionary game theory to establish a prey–predator system in which individuals in the predator population need to choose between group hunting strategies and isolated hunting strategies. This system includes two types of delay: fitness delay and hunting delay. In the absence of delays, we discuss the stability of boundary and interior equilibria. In addition, the condition that the non-delayed system undergoes transcritical bifurcation is obtained. For the delayed system, we explore the stability of the interior equilibrium and obtain the conditions for the existence of Hopf bifurcation. The conditions for determining the direction and stability of the Hopf bifurcation and the periodic variation in the periodic solution are introduced by using the normal form theory and center manifold theory. Finally, we simulate non-delayed and delayed systems. The results indicate that when the availability of prey is high, the isolated hunting strategy is the dominant strategy. When the availability of prey is low, mixed strategies appear and the proportion of the group hunting strategy increases as the availability of prey decreases. Furthermore, large delays lead to the disappearance of the mixed hunting strategy and its replacement by pure hunting strategies. [ABSTRACT FROM AUTHOR]

Dengue fever is currently one of the most serious mosquito‐borne infectious disease in the world. How to effectively prevent the outbreak of dengue fever has become a matter of significant public health concern. In this work, the cytoplasmic incompatibility induced by Wolbachia is assumed to be complete. Based on this assumption, we establish an extended discrete Ricker model with overlapping generations to investigate the suppression of mosquito population in the wild by adopting two different release strategies: the constant release strategy and the proportional release strategy. We prove the nonnegativity, boundedness, and stability of equilibrium points and finally find the release threshold, denoted as r1∗$$ {r}_1^{\ast } $$ and k1∗$$ {k}_1^{\ast } $$, for the successful suppression in these two release strategies. In addition, we demonstrate that the model that adopts the constant release strategy, respectively, has a saddle node bifurcation when r=r1∗$$ r={r}_1^{\ast } $$ and a stable period‐doubling bifurcation when r=r2∗$$ r={r}_2^{\ast } $$. While in the case of proportional release strategy, the model exhibits a transcritical bifurcation when k=k1∗$$ k={k}_1^{\ast } $$ and a stable period‐doubling bifurcation when k=k2∗$$ k={k}_2^{\ast } $$. Finally, we substantiate the conclusions through numerical simulations. [ABSTRACT FROM AUTHOR]

This study investigates the dynamics of predator–prey interactions with non-overlapping generations under the influence of fear effects, a crucial factor in ecological research. We propose a novel discrete-time model that addresses limitations of previous models by explicitly incorporating fear. Our primary question is: How does fear influence the stability of predator–prey populations and the potential for chaotic dynamics? We analyze the model to identify biologically relevant equilibria (fixed points) and determine the conditions for their stability. Bifurcation analysis reveals how changes in fear levels and predation rates can lead to population crashes (transcritical bifurcation) and complex population fluctuations (period-doubling and Neimark–Sacker bifurcations). Furthermore, we explore the potential for controlling chaotic behavior using established methods. Finally, two-parameter analysis employing Lyapunov exponents, spectrum, and Kaplan–Yorke dimension quantifies the chaotic dynamics of the proposed system across a range of fear and predation levels. Numerical simulations support the theoretical findings. This study offers valuable insights into the impact of fear on predator–prey dynamics and paves the way for further exploration of chaos control in ecological models. [ABSTRACT FROM AUTHOR]

The emergence of Lorentzian soliton is studied in an ultra-relativistic degenerate dense plasma by using the bifurcation technique. Based on the quantum hydrodynamic model for electrostatic excitations, the stability of possible equilibrium points is investigated which shows the occurrence of transcritical bifurcation. At the half-stable critical point, the plasma system supports a new localized structure with different tails from the regular solitons. According to the analytical solutions and phase portrait analysis, we find the effects of critical value of plasma density and Mach number on the appearance of Lorentzian solitons. Numerical simulations are performed to further verify the existence of the chaotic motions in the perturbed plasma system. The Lorentzian solitons at the bifurcation point undergo the onset of oscillatory instability, and the route from the stationary structure to the chaos state proceeds through quasi-periodic dynamics. The work presented here is related to electrostatic waves in dense astrophysical environment such as white dwarfs, neutron stars, and the core of massive planets. [ABSTRACT FROM AUTHOR]

COVID-19 is a respiratory disease triggered by an RNA virus inclined to mutations. Since December 2020, variants of COVID-19 (especially Delta and Omicron) continuously appeared with different characteristics that influenced death and transmissibility emerged around the world. To address the novel dynamics of the disease, we propose and analyze a dynamical model of two strains, namely native and mutant, transmission dynamics with mutation and imperfect vaccination. It is also assumed that the recuperated individuals from the native strain can be infected with mutant strain through the direct contact with individual or contaminated surfaces or aerosols. We compute the basic reproduction number, R 0 , which is the maximum of the basic reproduction numbers of native and mutant strains. We prove the nonexistence of backward bifurcation using the center manifold theory, and global stability of disease-free equilibrium when R 0 < 1 , that is, vaccine is effective enough to eliminate the native and mutant strains even if it cannot provide full protection. Hopf bifurcation appears when the endemic equilibrium loses its stability. An intermediate mutation rate ν 1 leads to oscillations. When ν 1 increases over a threshold, the system regains its stability and exhibits an interesting dynamics called endemic bubble. An analytical expression for vaccine-induced herd immunity is derived. The epidemiological implication of the herd immunity threshold is that the disease may effectively be eradicated if the minimum herd immunity threshold is attained in the community. Furthermore, the model is parameterized using the Indian data of the cumulative number of confirmed cases and deaths of COVID-19 from March 1 to September 27 in 2021, using MCMC method. The cumulative cases and deaths can be reduced by increasing the vaccine efficacies to both native and mutant strains. We observe that by considering the vaccine efficacy against native strain as 90%, both cumulative cases and deaths would be reduced by 0.40%. It is concluded that increasing immunity against mutant strain is more influential than the vaccine efficacy against it in controlling the total cases. Our study demonstrates that the COVID-19 pandemic may be worse due to the occurrence of oscillations for certain mutation rates (i.e., outbreaks will occur repeatedly) but better due to stability at a lower infection level with a larger mutation rate. We perform sensitivity analysis using the Latin Hypercube Sampling methodology and partial rank correlation coefficients to illustrate the impact of parameters on the basic reproduction number, the number of cumulative cases and deaths, which ultimately sheds light on disease mitigation. [ABSTRACT FROM AUTHOR]

In general, there is an imperative to amalgamate timely interventions and comprehensive measures for the efficacious control of infectious diseases. The deployment of such measures is intricately tied to the system's state and its transmission rate, presenting formidable challenges for stability and bifurcation analyses. In our pursuit of devising qualitative techniques for infectious disease analysis, we introduced a model that incorporates state-dependent transmission interventions. Through the introduction of state-dependent control, characterized by a non-linear action threshold contingent upon the combination of susceptible population density and its rate of change, we employ analytical methods to scrutinize various facets of the model. This encompasses addressing the existence, stability, and bifurcation phenomena concerning disease-free periodic solutions (DFPS). The analysis of the established Poincaré map leads us to the conclusion that DFPS indeed exists and maintains stability under specific conditions. Significantly, we have formulated a distinctive single-parameter family of discrete mappings, leveraging the bifurcation theorems of discrete maps to dissect the transcritical bifurcations around DFPS with respect to parameters such as $ ET $ and $ \eta_{1} $. Under particular conditions, these phenomena may give rise to effects like backward bifurcation and bistability. Through the analytical methodologies developed in this study, our objective is to unveil a more comprehensive understanding of infectious disease models and their potential relevance across diverse domains.

Amensalism, a rare yet impactful symbiotic relationship in ecological systems, is the focus of this study. We examine a discrete-time amensalism system by incorporating the fear effect on the first species. We identify the plausible equilibrium points and analyze their local stability conditions. The global attractivity of the positive equilibrium, $ E^* $, and the boundary equilibrium, $ E_1 $, are analyzed by exploring threshold conditions linked to the level of fear. Additionally, we analyze transcritical bifurcations and flip bifurcations exhibited by the boundary equilibrium points analytically. Considering some biologically feasible parameter values, we conduct extensive numerical simulations. From numerical simulations, it is observed that the level of fear has a stabilizing effect on the system dynamics when it increases. It eventually accelerates the extinction process for the first species as the level of fear continues to increase. These findings highlight the complex interplay between external factors and intrinsic system dynamics, enriching potential mechanisms for driving species changes and extinction events.

In this paper, we investigate the impact of functional shifts in a time-discrete cross-catalytic system. We use the hypercycle model considering that one of the species shifts from a cooperator to a degrader. At the bifurcation caused by this functional shift, an invariant curve collapses to a point P while, simultaneously, two fixed points collide with P in a transcritical bifurcation. Moreover, all points of a line containing P become fixed points at the bifurcation and only at the bifurcation in a degenerate scenario. We provide a complete analytical description of this degenerate bifurcation. As a result of our study, we prove the existence of the invariant curve arising from the transition to cooperation. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Agroforestry is a land-use system based on the simultaneous cultivation of crops and trees. The abundance and diversity of species living in agroforestry is usually higher than in monoculture. Hence, agroforestry may naturally control agricultural pests, by conserving their natural enemies. In monoculture, pest control requires the application of synthetic pesticides, which can cause harmful effects. In agroforestry, however, crops and trees compete for resources, which may reduce the crop production, as compared to monoculture. Here, a mathematical model is proposed to analyze this scenario. The model is written as a set of three nonlinear differential equations, in which the variables are the densities of crops, pests, and trees. It is analytically shown that transcritical and Hopf bifurcations can occur. Thus, by varying the parameter values, the pest can either be eradicated or endemically persist at constant level or endemically persist in oscillatory regime. For the Hopf bifurcation, the first Lyapunov exponent is computed. The analytical results are illustrated by computer simulations and discussed from an agroecological perspective. [ABSTRACT FROM AUTHOR]

In this study, we examine the plant–herbivore discrete model of apple twig borer and grape vine interaction, with a particular emphasis on the extended weak-predator response to Holling type-II response. We explore the dynamical and qualitative analysis of this model and investigate the conditions for stability and bifurcation. Our study demonstrates the presence of the Neimark–Sacker bifurcation at the interior equilibrium and the transcritical bifurcation at the trivial equilibrium, both of which have biological feasibility. To avoid unpredictable outcomes due to bifurcation, we employ chaos control methods. Furthermore, to support our theoretical and mathematical findings, we develop numerical simulation techniques with examples. In summary, our research enhances the comprehension of the dynamics pertaining to interactions between plants and herbivores in the context of discrete-time population models. [ABSTRACT FROM AUTHOR]

This paper discusses the stability analysis of the model for the spread of tuberculosis and the effects of treatment. It is shown the analyze the dynamic behavior of the model to investigate the local stability properties of the model equilibrium point. The Routh-Hurwitz criterion is used to analyze local stability at the disease-free equilibrium point, while the Transcritical Bifurcation theorem is used to investigate the local stability properties of the endemic equilibrium point. The discussion results show that the equilibrium point's stability properties depend on the value of the basic reproduction number, which is calculated based on the Next Generation Matrix (NGM). When the basic reproduction number value is less than one, then the disease-free equilibrium point is locally asymptotically stable, whereas if it is more than one, then the endemic equilibrium point is locally asymptotically stable. Numerical simulations are included to explain the dynamic behavior of disease spread and to understand the effectiveness of tuberculosis treatment in a given population. The simulation results show that treatment in the infected individual phase is known to be more effective than treatment in latent individuals. [ABSTRACT FROM AUTHOR]

Tuberculosis (TB) remains a significant global health concern, necessitating effective control strategies. This article presents a mathematical model to evaluate the comparative effectiveness of medical mask usage and case detection in TB control. The model is constructed as a system of ordinary differential equations and incorporates crucial aspects of TB dynamics, including slow–fast progression, medical mask use, case detection, treatment interventions and differentiation between symptomatic and asymptomatic cases. A key objective of TB control is to ensure that the reproduction number, [Formula: see text], remains below unity to achieve TB elimination or persistence if [Formula: see text] exceeds 1. Our mathematical analysis reveals the presence of a transcritical bifurcation when the [Formula: see text] signifies a critical juncture in TB control strategies. These results confirm that the effectiveness of case detection in diminishing the endemic population of symptomatic individuals within a TB-endemic equilibrium depends on exceeding a critical threshold value. Furthermore, our model is calibrated using TB yearly case incidence data per 100 000 population from Indonesia, India, Lesotho and Angola. We employed the bootstrap resampling residual approach to assess the uncertainty inherent in our parameter estimates which provides a comprehensive distribution of the parameter values. Despite a declining trend in new incidence, these four countries exhibit a reproduction number greater than 1, indicating persistent TB cases in the presence of ongoing TB control programmes. We employ the partial rank correlation coefficient in conjunction with the Latin hypercube sampling method to conduct a global sensitivity analysis of the [Formula: see text] parameter for each fitted parameter in every country. We find that the medical mask use is more sensitive to reduce [Formula: see text] compared with the case detection implementation. To further gain insight into the necessary control strategy, we formulated an optimal control and studied the cost-effectiveness analysis of our model to investigate the impact of case detection and medical mask use as control measures in TB spread. Cost-effectiveness analysis demonstrates that combining these interventions emerges as the most cost-effective strategy for TB control. Our findings highlight the critical importance of medical masks and their efficacy coupled with case detection in shaping TB control dynamics, elucidating the primary parameter of concern for managing the control reproduction number. We envisage our findings to have implications and be vital for TB control if implemented by policymakers and healthcare practitioners involved in TB control efforts.

This research focuses on the interaction between the grape borer and grapevine using a discrete-time plant-herbivore model with Allee's effect. We specifically investigate a model that incorporates a strong predator functional response to better understand the system's qualitative behavior at positive equilibrium points. In the present study, we explore the topological classifications at fixed points, stability analysis, Neimark-Sacker, Transcritical bifurcation and State feedback control in the two-dimensional discrete-time plant-herbivore model. It is proved that for all involved parameters ς1,ϱ1,γ1 and ϒ1, discrete-time plant-herbivore model has boundary and interior fixed points: c1=(0,0), c2=(ς1−1ϱ1,0) and c3=(ϒ1(1−γ1)2γ1−1,γ1(2ς1+ϱ1ϒ1−2)−ϱ1ϒ1+1−ς12γ1−1) respectively. Then by linear stability theory, local dynamics with different topological classifications are investigated at fixed points: c1=(0,0), c2=(ς1−1ϱ1,0) and c3=(ϒ1(1−γ1)2γ1−1,γ1(2ς1+ϱ1ϒ1−2)−ϱ1ϒ1+1−ς12γ1−1). Our investigation uncovers that the boundary equilibrium c2=(ς1−1ϱ1,0) experiences a transcritical bifurcation, whereas the unique positive steady-state c3=(ϒ1(1−γ1)2γ1−1,γ1(2ς1+ϱ1ϒ1−2)−ϱ1ϒ1+1−ς12γ1−1) of the discrete-time plant-herbivore model undergoes a Neimark-Sacker bifurcation. To address the periodic fluctuations in grapevine population density and other unpredictable behaviors observed in the model, we propose implementing state feedback chaos control. To support our theoretical findings, we provide comprehensive numerical simulations, phase portraits, dynamics diagrams, and a graph of the maximum Lyapunov exponent. These visual representations enhance the clarity of our research outcomes and further validate the effectiveness of the chaos control approach.

The four-dimensional food-web system consisting of two prey species for a generalist middle predator and a top predator is proposed and investigated. The middle predator is predating over both the prey species with a modified Holling type-II functional response. The food-web model is effectively formulated, exhibits bounded behavior, and displays dissipative dynamics. The proposed model’s essential dynamical features are studied regarding local stability. We investigated the four species’ survival and established their persistence criteria. In the proposed model, a transcritical bifurcation occurs at the axial equilibrium point. The numerical simulations reveal the persistence of a chaotic attractor or stable focus. The conclusion is that increasing the food available to the middle predator may make it possible to manage and mitigate the chaos within the food chain.

In this work, the dynamics of a food chain model with disease in the predator and the Allee effect in the prey have been investigated. The model also incorporates a Holling type-III functional response, accounting for both disease transmission and predation. The existence of equilibria and their stability in the model have also been investigated. The primary objective of this research is to examine the effects of the Allee parameter. Hopf bifurcations are explored about the interior and disease-free equilibrium point, where the Allee is taken as a bifurcation point. In numerical simulation, phase portraits have been used to look into the existence of equilibrium points and their stability. The bifurcation diagrams that have been drawn clearly demonstrate the presence of significant local bifurcations, including Hopf, transcritical, and saddle-node bifurcations. Through the phase portrait, limit cycle, and time series, the stability and oscillatory behaviour of the equilibrium point of the model are investigated. The numerical simulation has been done using MATLAB and Matcont.

In this paper, we revisit a discrete prey-predator model with the Allee effect in prey to find its more complex dynamical properties. After pointing out and correcting those known errors for the local stability of the unique positive fixed point $ E_*, $ unlike previous studies in which the author only considered the codim 1 Neimark-Sacker bifurcation at the fixed point $ E_*, $ we focus on deriving many new bifurcation results, namely, the codim 1 transcritical bifurcation at the trivial fixed point $ E_1, $ the codim 1 transcritical and period-doubling bifurcations at the boundary fixed point $ E_2, $ the codim 1 period-doubling bifurcation and the codim 2 1:2 resonance bifurcation at the positive fixed point $ E_* $. The obtained theoretical results are also further illustrated via numerical simulations. Some new dynamics are numerically found. Our new results clearly demonstrate that the occurrence of 1:2 resonance bifurcation confirms that this system is strongly unstable, indicating that the predator and the prey will increase rapidly and breakout suddenly.

The classic SIR model is often used to evaluate the effectiveness of controlling infectious diseases. Moreover, when adopting strategies such as isolation and vaccination based on changes in the size of susceptible populations and other states, it is necessary to develop a non-smooth SIR infectious disease model. To do this, we first add a non-linear term to the classical SIR model to describe the impact of limited medical resources or treatment capacity on infectious disease transmission, and then involve the state-dependent impulsive feedback control, which is determined by the convex combinations of the size of the susceptible population and its growth rates, into the model. Further, the analytical methods have been developed to address the existence of non-trivial periodic solutions, the existence and stability of a disease-free periodic solution (DFPS) and its bifurcation. Based on the properties of the established Poincaré map, we conclude that DFPS exists, which is stable under certain conditions. In particular, we show that the non-trivial order-1 periodic solutions may exist and a non-trivial order-$ k $ ($ k\geq 1 $) periodic solution in some special cases may not exist. Moreover, the transcritical bifurcations around the DFPS with respect to the parameters $ p $ and $ AT $ have been investigated by employing the bifurcation theorems of discrete maps.

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.

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.

In predator–prey system, the impact of predation fear on the prey population has received significant attention from researchers recently. A predator–prey community with infected predator in presence of predation fear has not yet been studied. We model a predator–prey system with disease transmission in predator and predator-induced fear in prey. We consider two crucial mechanisms where fear of prey can be detrimental to the prey due to low reproduction, and disease in predator can be beneficial to the prey due to low predation risk. In this study, we investigate the simultaneous effect of disease transmission and the cost of predation fear on the system dynamics. We study the dynamics of such system around the equilibrium points by stability and bifurcation analysis. Analytical results establish the occurrence of transcritical and Hopf bifurcation into the system dynamics. Numerical results reveal that high strength of predation fear and disease transmissions can stabilize the system by excluding oscillations. We show that the increased strength of disease transmission control or extinction of the predator yet permits the prey species to recuperate. The study indicates that infectious diseases may act as a biological control to control undesirable species. In the presence of fear, the half-saturation constant plays a vital role in shrinking or expanding the oscillatory region. Also, we identify a scenario in which the disease transmission rate produces a bubbling effect around an endemic equilibrium. We validate our analytical findings through numerical simulations. [ABSTRACT FROM AUTHOR]

We develop a deterministic predator–prey compartmental model to investigate the impact of their velocities on their interactions. Prey hides in a refuge area and comes out of this area when predation pressure declines. To avoid predation, prey can limit their velocity. For antipredator behaviour, we examined that prey mortality increases when either predator or prey velocity increases while raising antipredator behaviour increases prey density. We proved that predator free equilibrium is globally asymptotically stable and co-existing equilibrium will be globally stable under certain conditions. We find that transcritical bifurcations occur at predator-free equilibrium at the certain value of the death rate of the predator. [ABSTRACT FROM AUTHOR]

The dynamics of prey–predator system, when one or both the species are harvested non-linearly, has become a topic of intense study because of its wide applications in biological control and species conservation. In this paper we have discuss different bifurcation analysis of a two dimensional prey–predator model with Beddington–DeAngelis type functional response in the presence of prey refuge and non-linear harvesting of both species. We have studied the positivity and boundedness of the model system. All the biologically feasible equilibrium points are investigated and their local stability is analyzed in terms of model parameters. The global stability of coexistence equilibrium point has been discussed. Depending on the prey harvesting effort ( E 1 ) and degree of competition among the boats, fishermen and other technology ( l 1 ) used for prey harvesting, the number of axial and interior equilibrium points may change. The system experiences different type of co-dimension one bifurcations such as transcritical, Hopf, saddle-node bifurcation and co-dimension two Bogdanov–Takens bifurcation. The parameter values at the Bogdanov–Takens bifurcation point are highly sensitive in the sense that the nature of coexistence equilibrium point changes dramatically in the neighbourhood of this point. The feasible region of the bifurcation diagram in the l 1 - E 1 parametric plane divides into nine distinct sub-regions depending on the number and nature of equilibrium points. We carried out some numerical simulations using the Maple and MATLAB software to justify our theoretical findings and finally some conclusions are given. [ABSTRACT FROM AUTHOR]

We propose and study a class of discrete-time commensalism systems with additive Allee effects on the host species. First, the single species with additive Allee effects is analyzed for existence and stability, then the existence of fixed points of discrete systems is given, and the local stability of fixed points is given by characteristic root analysis. Second, we used the center manifold theorem and bifurcation theory to study the bifurcation of a codimension of one of the system at non-hyperbolic fixed points, including flip, transcritical, pitchfork, and fold bifurcations. Furthermore, this paper used the hybrid chaos method to control the chaos that occurs in the flip bifurcation of the system. Finally, the analysis conclusions were verified by numerical simulations. Compared with the continuous system, the similarities are that both species' densities decrease with increasing Allee values under the weak Allee effect and that the host species hastens extinction under the strong Allee effect. Further, when the birth rate of the benefited species is low and the time is large enough, the benefited species will be locally asymptotically stabilized. Thus, our new finding is that both strong and weak Allee effects contribute to the stability of the benefited species under certain conditions. [ABSTRACT FROM AUTHOR]

This paper investigates multiple bifurcations analyses and strong resonances of the Bazykin–Berezovskaya predator–prey model in depth using analytical and numerical bifurcation analysis. The stability conditions of fixed points, codim-1 and codim-2 bifurcations to include multiple and generic bifurcations are studied. This model exhibits transcritical, flip, Neimark–Sacker, and 1 : 2 , 1 : 3 , 1 : 4 strong resonances. The normal form coefficients and their scenarios for each bifurcation are examined by using the normal form theorem and bifurcation theory. For each bifurcation, various types of critical states are calculated, such as potential transformations between the one-parameter bifurcation point and different bifurcation points obtained from the two-parameter bifurcation point. To validate our analytical findings, the bifurcation curves of fixed points are determined by using MatcontM. [ABSTRACT FROM AUTHOR]

In this paper, we revisit a discrete prey-predator model with the Allee effect in prey to find its more complex dynamical properties. After pointing out and correcting those known errors for the local stability of the unique positive fixed point E ∗ , unlike previous studies in which the author only considered the codim 1 Neimark-Sacker bifurcation at the fixed point E ∗ , we focus on deriving many new bifurcation results, namely, the codim 1 transcritical bifurcation at the trivial fixed point E 1 , the codim 1 transcritical and period-doubling bifurcations at the boundary fixed point E 2 , the codim 1 period-doubling bifurcation and the codim 2 1:2 resonance bifurcation at the positive fixed point E ∗ . The obtained theoretical results are also further illustrated via numerical simulations. Some new dynamics are numerically found. Our new results clearly demonstrate that the occurrence of 1:2 resonance bifurcation confirms that this system is strongly unstable, indicating that the predator and the prey will increase rapidly and breakout suddenly. [ABSTRACT FROM AUTHOR]

In this paper, we utilize the semi-discretization method to construct a discrete model from a continuous predator-prey model with herd behaviour and group defense. Specifically, some new results for the transcritical bifurcation, the period-doubling bifurcation, and the Neimark-Sacker bifurcation are derived by using the center manifold theorem and bifurcation theory. Novelty includes a smooth transition from individual behaviour (low number of prey) to herd behaviour (large number of prey). Our results not only formulate simpler forms for the existence conditions of these bifurcations, but also clearly present the conditions for the direction and stability of the bifurcated closed orbits. Numerical simulations are also given to illustrate the existence of the derived Neimark-Sacker bifurcation.

One of the most important agricultural activities worldwide, coffee cultivation, is severely affected by the Coffee Berry Borer (CBB), Hypothenemus hampei, considered the primary coffee pest. The CBB is a tiny beetle that diminishes the quantity and quality of coffee beans by penetrating them to feed on the endosperm and deposit its eggs, continuing its life cycle. One strategy to combat CBBs is using biological control agents, such as certain species of ants. Here, a mathematical model (consisting of a system of nonlinear ordinary differential equations) is formulated to describe the prey–predator interaction between CBBs and an unspecified species of ants. From this mathematical perspective, the model allows us to determine conditions for the existence and stability of extinction, persistence or co-existence equilibria. Transitions among those equilibrium states are investigated through the maximum per capita consumption rate of the predator as a bifurcation parameter, allowing us to determine the existence of transcritical and saddle-node bifurcations. Phase portraits of the system are presented for different values of bifurcation parameter, to illustrate stability outcomes and the occurrence of bifurcations. It is concluded that an increase in bifurcation parameters significantly reduces the CBB population, suggesting that ant predation is an effective control strategy, at least theoretically.