Boundary Equilibrium Points Research Articles

Stability analysis of solutions of certain May's host-parasitoid model by using KAM theory

<abstract><p>We use the Kolmogorov-Arnold-Moser (KAM) theory to investigate the stability of solutions of a system of difference equations, a certain class of a generalized May's host-parasitoid model. We show the existence of the extinction, interior, and boundary equilibrium points and examine their stability. When the rate of increase of hosts is less than one, the zero equilibrium is globally asymptotically stable, which means that both populations are extinct. We thoroughly describe the dynamics of 1:1 non-isolated resonance fixed points and have used the KAM theory to determine the stability of interior equilibrium point. Also, we have conducted several numerical simulations to support our findings by using the software package Mathematica.</p></abstract>

A double time-delay Holling Ⅱ predation model with weak Allee effect and age-structure

<abstract><p>A double-time-delay Holling Ⅱ predator model with weak Allee effect and age structure was studied in this paper. First, the model was converted into an abstract Cauchy problem. We also discussed the well-posedness of the model and the existence of the equilibrium solution. We analyzed the global stability of boundary equilibrium points, the local stability of positive equilibrium points, and the conditions of the Hopf bifurcation for the system. The conclusion was verified by numerical simulation.</p></abstract>

The Impact of Predator-dependent Prey Refuge on the Dynamics of a Leslie-Gower Predator-prey Model

In this paper, we propose a new Leslie-Gower predator-prey model with predator-dependent prey refuge. Firstly, we obtain the positivity and boundedness of the system solution. Secondly, we prove that the origin is unstable using blow-up method, analyze the existence and local stability of the boundary equilibrium point and positive equilibrium point, and prove that the unique positive equilibrium point of the system is globally asymptotically stable by constructing a suitable Dulac function. Finally, mathematic analysis and numerical simulation show that: (1) when the strength of the predator-dependent prey refuge k = 0 , the dynamics of the predator-prey system without predator-dependent prey refuge are consistent with the results obtained from the traditional Leslie-Gower predator-prey system; (2) when k tends to positive infinity, the predator-dependent refuge lead to prey population densities fall somewhere between without prey refuge and with proportional refuge. However, the predator densities within this new form of the predator-dependent prey refuge is greater than the densities of predators without prey refuge and with proportional refuge; (3) increasing the strength k of the predator-dependent prey refuge can increase the densities of predator and prey populations respectively.

On the undesired equilibria induced by control barrier function based quadratic programs

In this paper, we analyze the system behavior for general nonlinear control-affine systems when a control barrier function-induced quadratic program-based controller is employed for feedback. In particular, we characterize the existence and locations of possible equilibrium points of the closed-loop system and also provide analytical results on how design parameters affect them. Based on this analysis, a simple modification on the existing quadratic program-based controller is provided, which, without any assumptions other than those taken in the original program, inherits the safety set forward invariance property, and further guarantees the complete elimination of undesired equilibrium points in the interior of the safety set as well as one type of boundary equilibrium points, and local asymptotic stability of the origin. Numerical examples are given alongside the theoretical discussions.

Lyapunov functions for studying global asymptotic stability of two rumor spreading models

In a previous work (2018, Commun. Theor. Phys. 70, 795–802), a new compartment model for the spreading of rumors was introduced and analyzed. However, only the local asymptotic stability of this model was discussed. In the present work, we first provide a rigorous mathematical analysis for the global asymptotic stability (GAS) of the above-mentioned rumor spreading model. By constructing suitable Lyapunov candidate functions, we obtain the GAS of a rumor-free (boundary) equilibrium point and a unique rumor-spreading (positive) equilibrium point. After that, we utilize the approach based on the Lyapunov candidate functions to study the GAS of another rumor spreading model with control strategies, which was proposed in (2022, Physica A 606, 128157). As an important consequence, the GAS of the rumor spreading model with control strategies is determined fully without resorting to technical hypotheses used in the benchmark work. Lastly, the theoretical findings are supported by a set of illustrative numerical examples. The obtained results not only improve the ones constructed in the two above-mentioned benchmark papers but also can be extended to study the global dynamics of other rumor propagation models in the context of both integer-order and fractional-order derivatives.

Global Dynamics of an Oligopoly Game Model with Nonlinear Costs and Strategic Delegation

A dynamic oligopoly game model with nonlinear cost and strategic delegation is built on the basis of isoelastic demand in this paper. And the dynamic characteristics of this game model are investigated. The local stability of the boundary equilibrium points is analyzed by means of the stability theory and Jacobian matrix, and the stability region of the Nash equilibrium point is obtained by Jury criterion. It is concluded that the system may lose stability through Flip bifurcation and Neimark–Sacker bifurcation. And the effects of speed of adjustment, price elasticity, profit weight coefficient and marginal cost on the system stability are discussed through numerical simulation. After that, the coexistence of attractors is analyzed through the basin of attraction, where multiple stability always means path dependence, implying that the long-term behavior of enterprises is strongly affected by historical contingency. In other words, a small perturbation of the initial conditions will have a significant impact on the system. In addition, the global dynamical behavior of the system is analyzed by using the critical curves, the basin of attraction, absorbing areas and a noninvertible map, revealing that three global bifurcations, the first two of which are caused by the interconversion of simply-connected and multiply-connected regions in the basin of attraction, and the third global bifurcation, that is, the final bifurcation is caused by the contact between attractors and the boundary of the basin of attraction.

Dynamical properties of single species stage structured model with Michaelis-Menten type harvesting on adult population and linear harvesting on juvenile population

A very common and effective way for investigating future demographics is the study of stage structured models. The focus of this article is to propose a modified model to study the impact of population harvesting on their juvenile and adult stages, and analyze the dynamical properties from both qualitative and numerical perspective. It studies single species stage structured model with linear harvesting on juvenile group and Michaelis-Menten type harvesting on adult group. We exploit general ideas in mathematical modeling process to study the dynamical properties and their biological, ecological, and economic implications. It discusses that bi-stability phenomena may exist, global asymptotic stability at boundary equilibrium points and internal equilibrium points are investigated from construction of suitable Lyapunov and Dulac functions. It has been observed that a suitable linear harvesting on juvenile population can feasibly be carry out along with Michaelis-Menten type harvesting on adult population without endangering extinction of any group of population.

A Filippov type predator prey system with selective harvesting of predator

In this paper, piece-wise smooth predator prey Filippov system to describe the role of refuge to prey and additional food to predators is explored. Logistic growth of prey and Holling type II predation is considered. Selective proportional harvesting of predators is included. Refuge is provided to a fraction of preys when their density is below a threshold value. This reduces the food availability for predators. To compensate this, additional food is provided to predators. The dynamics on the boundary is computed using the Filippov regularization method. The stability and existence of different equilibrium states of the independent smooth subsystems are analyzed. The role of quality of additional food with respect to prey in the characterization of the equilibrium points as regular or virtual equilibrium points for the Filippov system is investigated. It is verified that if only refuge is provided to prey with no additional food to predators then both the interior equilibrium state will never be regular simultaneously for the Filippov system. It is observed from the sliding mode dynamics that there is no escaping region. It is proved that both refuge to prey and additional food to predators enhances the sliding region on the boundary. The analysis of boundary equilibrium points, tangent points and pseudoequilibrium points on the boundary is carried out. The role of effort on harvesting on the expansion or shrinking of the limit cycle is elaborated numerically. Boundary bifurcation for the interior equilibrium state of the subsystems is demonstrated. Numerical simulation is used to elaborate the dynamics of Filippov system.

Dynamical behaviours of discrete amensalism system with fear effects on first species.

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.

A Fractional-Order SIR-C Cyber Rumor Propagation Prediction Model with a Clarification Mechanism

As communication continues to develop, the high freedom and low cost of the communication network environment also make rumors spread more rapidly. If rumors are not clarified and controlled in time, it is very easy to trigger mass panic and undermine social stability. Therefore, it is important to establish an efficient model for rumor propagation. In this paper, the impact of rumor clarifiers on the spread of rumors is considered and fractional order differentiation is introduced to solve the problem that traditional models do not take into account the “anomalous propagation” characteristics of information. A fractional-order Susceptible-Infected-Removal-Clarify (SIR-C) rumor propagation prediction model featuring the clarification mechanism is proposed. The existence and asymptotic stability conditions of the rumor-free equilibrium point (RFEP) E0; the boundary equilibrium points (BEPs) E1 and E2 are also given. Finally, the stability conditions and practical cases are verified by numerical simulations. The experimental results confirm the analysis of the theoretical study and the model fits well with the real-world case data with just minor deviations. As a result, the model can play a positive and effective role in rumor propagation prediction.

Complex Dynamic Behaviors of a Modified Discrete Leslie–Gower Predator–Prey System with Fear Effect on Prey Species

A discrete modified Leslie–Gower prey-predator model considering the effect of fear on prey species is proposed and studied in this paper. First, we discuss the existence of equilibria and the local stability of the model. Second, we use the iterative method and comparison principle to obtain the set of conditions which ensures the global attractivity of positive equilibrium point. The results show that prey and predator can coexist stably when the intrinsic growth rates of both prey and predator are maintained within a certain range. Then, we study the global attractivity of the boundary equilibrium point. Our results suggest that when the intrinsic rate of prey is small enough or the fear factor is large enough, the prey will tend to go extinct, while the predator can survive stably due to the availability of other food sources. Subsequently, we discuss flip bifurcation, transcritical bifurcation at the equilibria of the system, by using the center manifold theorem and bifurcation theory. We find that system changes from chaotic state to four-period orbit, two-period orbit, stable state, and finally prey species will be driven to extinction, while predator species survive in a stable state for enough large birth rate of prey species with the increasing of fear effect. Finally, we verify the feasibility of the main results by numerical simulations, and discuss the influence of the fear effect. The results show that the fear effect within a certain range can enhance the stability of the system.

Asymptotic behavior and extinction of a stochastic predator–prey model with Holling type II functional response and disease in the prey

In this paper, we formulate and investigate a stochastic one‐prey and two‐predator model with Holling II functional response and disease in the prey, in which the predators only feed on infected prey. The existence and uniqueness of global positive solution is proved by using conventional methods. The corresponding deterministic model has a disease‐free equilibrium point if the basic reproduction number , and it has three boundary equilibrium points and one positive equilibrium point if . For the stochastic model, we investigate the asymptotic behavior around all of the five equilibrium points and prove that there is a unique ergodic stationary distribution under certain conditions. Moreover, we obtain the condition on which the population of the infected prey and the two predators will die out in the time mean sense. Finally, numerical simulations are conducted to illustrate our analysis results.

Stability and Hopf bifurcation of a diffusive plankton model with time-delay and mixed nonlinear functional responses

In this paper, we deal with a plankton reaction–diffusion model with time-delay and two different functional responses. Firstly, we consider the global stability of boundary equilibrium point. Secondly, we investigate the existence, uniqueness and stability of internal equilibrium point without time-delay. Then, we analyze the existence of Hopf bifurcation emitting from internal equilibrium point and give some characteristics on Hopf branch in detail. A new finding is presented, specifically, we find that there exist two critical values which have important effects on the occurrence of Hopf bifurcation. Finally, a few numerical examples are presented to check and illustrate the theoretical analysis, some simulation graphs, including the spatiotemporal graphs, trajectory graphs and phase portraits are depicted graphically.

"A dynamic Cournot mixed oligopoly model with time delay for competitors"

"This paper deals with the analysis of a discrete-time Cournot game with time delay, where the interactions of one public firm and n private firms on the market are considered. The production of the public firm is adjusted based on the past production levels of the private firms. At the same time, the productions of the private firms are updated with respect to the past production of the public firm. Two equilibrium points are determined for the discrete-time nonlinear mathematical model. The analysis of the stability reveals that the boundary equilibrium point is a saddle point, while the positive one, under some conditions, is asymptotically stable for any time delays. Numerical simulations illustrate the complex dynamic behaviour of the system."

Dynamics Analysis of Predator-Prey Model with Double Allee Effects and Holling Type II Functional Response

In this article, a predator prey model with double Allee effects and Holling type II functional response is discussed. Strong and weak Allee effects were analyzed separately. The dynamic behavior of the model is analyzed by determining the equilibrium point and stability around the equilibrium point. From the analysis result, it is obtained that the trivial equilibrium point is locally asymptotically stable for the case of the strong Allee effect and the saddle unstable for the case of the weak Allee effect, while the boundary and coexistence equilibrium points are locally asymptotically stable if it satisfies several parameter conditions. Numerical simulations are carried out around the coexistence equilibrium point. The simulation results show that the Allee effect threshold affects prey population growth when experiencing a strong Allee effect. The growth of the prey population also depends on the initial conditions of the prey and predator population density. Furthermore, when the prey population experiences a weak Allee effect, there is no threshold must be exceeded for the population to survive so that for each initial condition however, the population will not experience extinction.

Mechanisms of stable species coexistence in food chain systems: Strength of odor disturbance and group defense

Odor disturbance and group defense generally play an important role in ecosystem stability. By incorporating odor disturbances and group defenses into the traditional three-species food chain system, where the preys are disturbed by the odor of predators, and group together to defend. We proved the boundedness of solutions of the system, and discussed the existence of equilibrium points. By Lyapunov’s first method and Routh-Hurwitz criterion, the sufficient conditions for stability of equilibrium points are obtained. It is found that, with the increasing of group defense, the density of the preys increases, while the odor disturbance of predators decreases the density of preys. Interestingly, the group defense induces a transition from a boundary equilibrium point to a positive equilibrium one, or from no Hopf-bifurcation to a Hopf-bifurcation. Furthermore, we demonstrated that the system undergoes a saddle-node bifurcation at mc=0.3362, below which the system has a stable node and a saddle point, suggesting that the less odor disturbance avail system stability; while above which the system has no equilibrium point, indicating that the greater odor disturbance causes the collapse of the system. To reveal the underlying mechanisms of the interplay of odor disturbance and group defense, the bifurcation diagrams of the preys with two factors are given respectively.

Mutual interference and its effects on searching efficiency between predator–prey interaction with bifurcation analysis and chaos control

In this paper, we study the dynamic of the predator–prey model based on mutual interference and its effects on searching efficiency. The parametric conditions, existence, and stability for trivial and boundary equilibrium points are studied. Also, it has shown that by applying the center manifold theorem and bifurcation theory, system undergoes Neimark–Sacker bifurcation across the neighborhood of a positive fixed point. Moreover, due to the bifurcation and chaos which objectively exist in a system, three chaos control strategies are designed and used. Moreover, to validate our theoretical and analytical discussions, numerical simulations are applied to show complex and chaotic behavior. Finally, theoretical discussions are validated with experimental field data.

Evaluation of the Number of Visits to Chinese Medical Institutions Using a Logistic Differential Equation Model

In this study, we established a two-dimensional logistic differential equation model to study the number of visits in Chinese PHCIs and hospitals based on the behavior of patients. We determine the model's equilibrium points and analyze their stability and then use China medical services data to fit the unknown parameters of the model. Finally, the sensitivity of model parameters is evaluated to determine the parameters that are susceptible to influence the system. The results indicate that the system corresponds to the zero-equilibrium point, the boundary equilibrium point, and the positive equilibrium point under different parameter conditions. We found that, in order to substantially increase visits to PHCIs, efforts should be made to improve PHCI comprehensive capacity and maximum service capacity.

Sliding Dynamics and Bifurcations in the Extended Nonsmooth Filippov Ecosystem

We propose a nonsmooth Filippov refuge ecosystem with a piecewise saturating response function and analyze its dynamics. We first investigate some key elements to our model which include the sliding segment, the sliding mode dynamics and the existence of equilibria which are classified into regular/virtual equilibrium, pseudo-equilibrium, boundary equilibrium and tangent point. In particular, we consider how the existence of the regular equilibrium and the pseudo-equilibrium are related. Then we study the stability of the standard periodic solution (limit cycle), the sliding periodic solutions (grazing or touching cycle) and the dynamics of the pseudo equilibrium, using quantitative analysis techniques related to nonsmooth Filippov systems. Furthermore, as the threshold value is varied, the model exhibits several complex bifurcations which are classified into equilibria, sliding mode, local sliding (boundary node and focus) and global bifurcations (grazing or touching). In conclusion, we discuss the importance of the refuge strategy in a biological setting.

Stability of May’s Host–Parasitoid model with variable stocking upon parasitoids

Using the Kolmogorov–Arnold–Mozer (KAM) theory, we investigate the stability of May’s host–parasitoid model’s solutions with proportional stocking upon the parasitoid population. We show the existence of the extinction, boundary, and interior equilibrium points. When the host population’s intrinsic growth rate and the releasement coefficient are less than one, both populations are extinct. There are an infinite number of boundary equilibrium points, which are nonhyperbolic and stable. Under certain conditions, there appear 1:1 nonisolated resonance fixed points for which we thoroughly described dynamics. Regarding the interior equilibrium point, we use the KAM theory to prove its stability. We give a biological meaning of obtained results. Using the software package Mathematica, we produce numerical simulations to support our findings.