Conditions For Local Asymptotic Stability Research Articles

Transient Dynamics of a Food Chain Model with Vigilance in the Biparameter Spaces

This article focuses on the asymptotic and transient behavior of a food chain model with vigilance in the biparameter spaces. Firstly, the local asymptotic stability conditions of the positive equilibrium point are obtained. Then, the transient dynamical behavior of the positive equilibrium point is analyzed based on eight parameters. Using numerical simulation, the stable regions of the model with single and double parameters are plotted, as well as the value changes of resilience, reactivity, and amplification envelope within the stable regions. The results show that the coexistence equilibrium point is always reactive, and the corresponding parameter ranges for system stability are large, but the resilience values are small. The vigilance parameter of the middle- and lower-level organisms has the greatest impact on the system’s resilience and reactivity, and the lowest vigilance corresponds to the maximum resilience and reactivity. Our results provide important references for effective management and protection of wild animal populations by the potential for transient amplification, disturbance growth time, and amplitude.

Evolutionary dynamics on a regular networked structured and unstructured multi‐population

AbstractIn this paper, we study collective decision‐making in a multi‐population framework, where groups of individuals represent whole populations that interact by means of a regular network. Each group consists of a number of players and every player can choose between two options. A group is characterized by three variables, the first two denoting the fractions of individuals committed to each option and the last one representing the fraction of players not committed to either option. The state of every population is influenced by the state of neighboring groups. The contribution of this work is the following. First, we find explicit expressions for all the equilibrium points of the proposed system, and show that these represent equilibrium points where populations reach consensus, namely, where all populations have the same states. We also derive a sufficient condition for local asymptotic stability as well as exponential asymptotic stability. Then, we study a structured model where every population is now assumed to represent a structured complex network. We conclude the paper by providing a set of simulations that corroborate the theoretical findings.

Ultimate Dynamics Analysis of the 5D Structural Leukemia Model and Partitioning of the Parameter Space

In this paper, we study the global dynamics of the 5D structural leukemia model with 14 parameters as developed by Clapp et al. [2015]. This model describes the interaction between leukemic cell populations and the immune system. Our analysis is based on the localization method of compact invariant sets. We develop this method by introducing the notion of a partitioning of the parameter space and the notion of a localization set corresponding to this partitioning as its parameters change. Further, we obtain ultimate upper and lower bounds for all variables of a state vector without imposing additional restrictions. Local asymptotic stability conditions with respect to the leukemia-free equilibrium point (EP) are given. We deduce formulas describing inner EPs expressed in terms of positive roots of one 7D equation. Based on this equation, it is shown that the number of inner EPs cannot exceed 3 and one case of a global bifurcation of EPs is detected. Next, we prove the existence of the attracting set. Further, in two theorems, the global eradication/extinction leukemia theorems are described. The impact of using parametrically variable localization sets for a qualitative analysis of the ultimate behavior of leukemic cell populations is discussed.

Memory effect on prey–predator dynamics: Exploring the role of fear effect, additional food and anti-predator behaviour of prey

Memory over past events has great importance on the survival of species in an ecological system. It is helpful in searching favourite food, escaping from predator attacks, group defence, use of environmental protection in the future time for the species. To explore the role of different ecological effects like fear effect, additional food and anti-predator behaviour of prey in the presence memory on an ecological system, in this paper we have proposed a two-dimensional prey–predator model and formulated mathematically using a system of two fractional order differential equations. Here the Caputo sense fractional derivative is used to construct the fractional order differential equations. The existence–uniqueness, non-negativity and boundedness of system solutions have been established. The detailed mathematical and graphical analysis for the feasibility of different equilibrium points and their analytical conditions for local asymptotic stability has been studied. The sufficient parametric conditions for the global stability of non-trivial equilibrium points are investigated. The Hopf bifurcation analysis of the system around the coexistence equilibrium point is derived in terms of system parameters. Using numerical simulations we have established that the memory effect can stabilize the system from an unstable periodic situation and makes the above-considered ecological effects more positively effective.

MODELING THE TRANSMISSION PHENOMENA OF COVID-19 INFECTION WITH THE EFFECT OF VACCINATION VIA NONINTEGER DERIVATIVE UNDER REAL STATISTIC

The infection of coronavirus (COVID-19) is a dangerous and life-threatening disease which spread to almost all parts of the globe. We present a mathematical model for the transmission of COVID-19 with vaccination effects. The basic properties of fractional calculus are presented for the inspection of the model. We calculate the equilibria of the model and determined the reproduction number [Formula: see text]. Local asymptotic stability conditions for the disease-free are obtained which determines the conditions to stabilize the exponential spread of the disease. The nonlinear least-square procedure is utilized to parameterize the model from actual cases reported in Pakistan. By fixed point theory, we prove the existence of a unique solution. We also present numerical results to simulate virus transmission and compare the results with those of the Caputo derivative. We study the solution pathways of the COVID-19 system to provide effective control policies for the infection. Significant changes have been noticed by lowering the order of fractional derivative.

Dynamic Analysis of a Delayed Fractional Infectious Disease Model with Saturated Incidence

This paper addresses a new fractional order infectious disease model with saturated incidence and time delay. In the new model, the isolated population and the asymptomatic infected population are considered. The invariant region and positive analysis of the solution of the model are established. Next, the basic reproduction number is obtained by the next-generation matrix method, and the sufficient conditions for local asymptotic stability for arbitrary time delays are proposed. Finally, the correctness of the theoretical results is verified by some numerical simulations.

Stability and Bifurcation Analysis of Fractional-Order Delayed Prey–Predator System and the Effect of Diffusion

Most biological systems have long-range temporal memory. Such systems can be modeled using fractional-order differential equations. The combination of fractional-order derivative and time delay provides the system more consistency with the reality of the interactions and higher degree of freedom. A fractional-order delayed prey–predator system with the fear effect has been proposed in this work. The time delay is considered in the cost of fear; therefore, there are no dynamical changes observed in the system due to time delay in the absence of fear. The existence and uniqueness of the solutions of the proposed system are studied along with non-negativity and boundedness. The existence of biologically relevant equilibria is discussed, and the conditions for local asymptotic stability are derived. Hopf bifurcation occurs in the system with respect to delay parameter. Further, a spatially extended system is proposed and analyzed. Hopf bifurcation also occurs in the extended system due to the delay parameter. Numerical examples are provided in support of analytical findings. Fractional-order derivative improves the stability and damps the oscillatory behaviors of the solutions of the system. Bistability behavior of the system admits stable dynamics by decreasing the fractional-order. Also, chaotic behavior is destroyed by decreasing fractional-order.

Stability Analysis of Systems with Recurrent Neural Network Controllers

The increasing utilization of recurrent neural networks (RNNs) in feedback control and the complexity of RNNs motivate us to provide a method for verifying the stability of RNN-controlled systems. To circumvent the nonlinearities in RNNs, we describe them using local integral quadratic constraints. A sufficient condition based on linear matrix inequalities is achieved for local asymptotic stability of feedback interconnections between a linear time-invariant plant and an RNN controller. This condition is employed as a constraint to construct semidefinite programming for finding a large ellipsoidal inner approximation to the region of attraction. A bisection algorithm is proposed to enlarge the inner approximation further. A necessary condition for local asymptotic stability is derived from linearization, which can be exploited to determine initial values for the bisection algorithm. Illustrative examples complete the presentation and show the improvement by our work.

Spatiotemporal Evolution Characteristics of Time‐Delay Ecological Competition Systems with Food‐Limited and Diffusion

In this paper, we put forward a time‐delay ecological competition system with food restriction and diffusion terms under Neumann boundary conditions. For the case without delay, the conditions for local asymptotic stability and Turing instability are constructed. For the case with delay, the existence of Hopf bifurcation is demonstrated by analyzing the root distribution of the corresponding characteristic equations. Furthermore, by using the normal form theory and the center manifold reduction of partial functional differential equations, explicit formulas are obtained to determine the direction of bifurcations and the stability of bifurcating periodic solutions. Finally, some simulation examples are provided to substantiate our analysis.

DEVELOPMENT AND ANALYSIS OF A SEIR MODEL FOR COVID-19 EPIDEMIC WITH VACCINATION AND NONSINGULAR KERNEL

We propose a new model to investigate the recent coronavirus (COVID-19) epidemic. The fundamental reproductive number is determined using the next-generation matrix, and the model’s soundness is tested using stability theory. The local asymptotic stability conditions for the disease-free equilibrium are determined. The model is extended to the fractional domain, and its properties are investigated using the Caputo–Fabrizio (CFC) robust nonsingular and nonlocal fractional operator. We focused on numerous exciting elements of the expanded model, such as solution positivity. In numerical aspects, different values of numerous parameters are employed to view certain physical behavior of the model.

Eradication Conditions of Infected Cell Populations in the 7-Order HIV Model with Viral Mutations and Related Results

In this paper, we study possibilities of eradication of populations at an early stage of a patient’s infection in the framework of the seven-order Stengel model with 11 model parameters and four treatment parameters describing the interactions of wild-type and mutant HIV particles with various immune cells. We compute ultimate upper bounds for all model variables that define a polytope containing the attracting set. The theoretical possibility of eradicating HIV-infected populations has been investigated in the case of a therapy aimed only at eliminating wild-type HIV particles. Eradication conditions are expressed via algebraic inequalities imposed on parameters. Under these conditions, the concentrations of wild-type HIV particles, mutant HIV particles, and infected cells asymptotically tend to zero with increasing time. Our study covers the scope of acceptable therapies with constant concentrations and values of model parameters where eradication of infected particles/cells populations is observed. Sets of parameter values for which Stengel performed his research do not satisfy our local asymptotic stability conditions. Therefore, our exploration develops the Stengel results where he investigated using the optimal control theory and numerical dynamics of his model and came to a negative health prognosis for a patient. The biological interpretation of these results is that after a sufficiently long time, the concentrations of wild-type and mutant HIV particles, as well as infected cells will be maintained at a sufficiently low level, which means that the viral load and the concentration of infected cells will be minimized. Thus, our study theoretically confirms the possibility of efficient treatment beginning at the earliest stage of infection. Our approach is based on a combination of the localization method of compact invariant sets and the LaSalle theorem.

A nullcline approach to global stability in discrete-time predator–prey models

In this paper, we consider a two-dimensional discrete-time predator–prey model that was recently developed in Ackleh et al. [Persistence and stability analysis of discrete-time predator-prey models: A study of population and evolutionary dynamics, J. Differ. Equ. Appl. 25 (2019), pp. 1568–1603]. Utilizing a novel approach that is based on nullcline analysis, we derive conditions for the global stability of the interior equilibrium. This result significantly expands the parameter ranges under which global stability was shown to hold in Ackleh et al. [Long-term dynamics of discrete-time predator–prey models: Stability of equilibria, cycles and chaos, J. Differ. Equ. Appl. 26 (2020), pp. 693–726] using Lyapunov functions. We then extend these global stability results to a predator–prey model with evolution in the prey to obtain sharper conditions on the persistence of the system and to establish global-stability results for the interior equilibrium of this three-dimensional model. Numerical results corroborating these theoretical findings and demonstrating a relationship between the conditions for local asymptotic stability and global asymptotic stability are also presented.

Bifurcation analysis of a discrete-time four-dimensional cubic autocatalator chemical reaction model with coupling through uncatalysed reactant

In this manuscript, we discuss a four-dimensional cubic autocatalator chemical reaction model in continuous form. We investigate the existence of one and only positive fixed point and then we have obtained some parametric conditions for local stability of continuous system by using Routh-Hurwitz stability criteria. Moreover, we discretize the four-dimensional continuous cubic autocatalator chemical reaction model by using Euler’s forward method and then by using a nonstandard difference scheme we obtained a consistent discrete-time counterpart of four-dimensional cubic autocatalator chemical reaction model. Parametric conditions for local asymptotic stability of one and only positive fixed point of obtained system are also discussed. It is shown that the obtained system experiences the Neimark-Sacker bifurcation at one and only positive fixed point by using a general standard for Neimark-Sacker bifurcation. The discrete-time counterpart of genuine four-dimensional system displays chaotic dynamics at different standards of bifurcation parameter. Furthermore, the control of Neimark-Sacker bifurcation and chaos is also deliberated by using a generalized hybrid control scheme, which is based on parameter perturbation and feedback control. Finally, some numerical examples are given to strengthen our theoretical results.

English

Coral reefs can undergo relatively rapid changes in the dominant biota, a phenomenon referred to as phase shift. Degradation of coral reefs is often associated with changes in community structure towards a macroalgae-dominated reef ecosystem due to the reduction in herbivory caused by overfishing. We investigate the coral-macroalgal phase shift due to the effects of harvesting of herbivorous reef fish by means of a continuous time model in the food chain. Conditions for local asymptotic stability of steady states are derived. We have shown that under certain conditions the system is uniformly persistent in presence of all the organisms. Moreover, it is shown that the system undergoes a Hopf bifurcation when the carrying capacity of macroalgae crosses certain critical value. Computer simulations have been carried out to illustrate different analytical results.

Stability and Bifurcation Analysis of Hassell–Varley Prey–Predator System with Fear Effect

In this work, we study a prey–predator system with Hassell–Varley functional response, which is a predator-dependent functional response. Our goal is to study the dynamics of the system in the presence of fear of predation risk. Hassell–Varley functional response is used when the predator population forms a fixed number of tight groups like for foraging purposes. Those predators can be terrestrial and aquatic. We assume that the birth rate of prey population reduces due to fear of the predator. Here, we have discussed the existence of biologically relevant equilibria and derived the condition for local asymptotic stability. Further, permanence and persistence are discussed to understand the dynamics of the system. The system undergoes Hopf bifurcation for both: the fear parameter f and interference coefficient $$\gamma $$ . Stability and the direction of Hopf bifurcation are also discussed. We observe that limit cycle oscillation can be prevented by increasing the cost of fear of the predator population. Numerical simulation is performed to substantiate our analytical findings.

Modeling the effect of temperature variability on malaria control strategies

In this study, a non-autonomous (temperature dependent) and autonomous (temperature independent) models for the transmission dynamics of malaria in a population are designed and rigorously analysed. The models are used to assess the impact of temperature changes on various control strategies. The autonomous model is shown to exhibit the phenomenon of backward bifurcation, where an asymptotically-stable disease-free equilibrium (DFE) co-exists with an asymptotically-stable endemic equilibrium when the associated reproduction number is less than unity. This phenomenon is shown to arise due to the presence of imperfect vaccines and disease-induced mortality rate. Threshold quantities (such as the basic offspring number, vaccination and host type reproduction numbers) and their interpretations for the models are presented. Conditions for local asymptotic stability of the disease-free solutions are computed. Sensitivity analysis using temperature data obtained from Kwazulu Natal Province of South Africa [K. Okuneye and A.B. Gumel. Mathematical Biosciences 287 (2017) 72–92] is used to assess the parameters that have the most influence on malaria transmission. The effect of various control strategies (bed nets, adulticides and vaccination) were assessed via numerical simulations.

Bifurcation analysis and chaos control in a discrete‐time plant quality and larch budmoth interaction model with Ricker equation

We investigate the dynamics of two‐dimensional discrete‐time model of leaf quality and larch budmoth interaction with Ricker equation. More precisely, the qualitative behavior of larch budmoth model is discussed in which the effect of food source upon the moth population is through intrinsic growth rate. We find the parametric conditions for local asymptotic stability of the unique positive fixed point. It is also proved that under certain parametric conditions, the system undergoes period‐doubling bifurcation with the help of center manifold theory. The parametric conditions for existence and direction of Neimark‐Sacker bifurcation at positive fixed point is investigated with the help of standard mathematical techniques of bifurcation theory. The chaos control in the system is discussed through implementation of hybrid control methodology. Finally, numerical simulations are provided to illustrate theoretical results. These results of numerical simulations demonstrate chaotic long‐term behavior over a broad range of parameters. The computation of the maximum Lyapunov exponents confirms the presence of chaotic behavior in the system.

Some Qualitative Behavior of Solutions of General Class of Difference Equations

In this work, we consider the general class of difference equations (covered many equations that have been studied by other authors or that have never been studied before), as a means of establishing general theorems, for the asymptotic behavior of its solutions. Namely, we state new necessary and sufficient conditions for local asymptotic stability of these equations. In addition, we study the periodic solution with period two and three. Our results essentially extend and improve the earlier ones.

Global stabilization of nonlinear systems via hybrid implementation of dynamic continuous-time local controllers

Given a continuous-time system and a dynamic control law such that the closed-loop system satisfies standard Lyapunov conditions for local asymptotic stability, we propose a hybrid implementation of the continuous-time control law. We demonstrate that subject to certain “relaxed” conditions, the hybrid implementation yields global asymptotic stability properties. These conditions can be further specialized to yield local/regional asymptotic stability with an enlarged basin of attraction with respect to the original control law. Two illustrative numerical examples are provided to demonstrate the main results.

A study of stability and bifurcation analysis in discrete-time predator–prey system involving the Allee effect

This paper deals with a discrete-time predator–prey system which is subject to an Allee effect on prey. We investigate the existence and uniqueness and find parametric conditions for local asymptotic stability of fixed points of the discrete dynamic system. Moreover, using bifurcation theory, it is shown that the system undergoes Neimark–Sacker bifurcation in a small neighborhood of the unique positive fixed point and an invariant circle will appear. Then the direction of bifurcation is given. Furthermore, numerical analysis is provided to illustrate the theoretical discussions with the help of Matlab packages. Thus, the main theoretical results are supported with numerical simulations.