Unique Positive Equilibrium Point Research Articles

The Spatial Pattern Dynamics of Reaction-Diffusion Instability in Tumor Cell–Immune Cell System

In this paper, spatial patterns of a diffusive Tumor cell and immune cell model with sigmoid ratio-dependent functional response are investigated. The asymptotic stability behavior of the corresponding nonspatial model around the unique positive interior equilibrium point in homogeneous steady state is obtained. We have obtained the optimal condition under which the system loses stability and a Turing pattern occurs. Numerical simulations have been carried out in order to show the significant role of reaction-diffusion coefficients and other important parameters of the system. Various contour figures of spatial patterns through Turing instability are portrayed and analyzed in order to substantiate the applicability of the present model. The paper ends with an extended discussion of biological implications of the immune system.

Global asymptotical stability in a rational difference equation

In this paper we prove a global attractivity result for the unique positive equilibrium point of a difference equation, which improves and generalizes some known ones in the existing literature. Especially, our results completely solve an open problem and some conjectures proposed in [1, 2, 3, 4].

Global stability analysis of an unemployment model with distributed delay

A mathematical model with distributed time delay describing the labour market is investigated, focusing on the asymptotic stability of the unique positive equilibrium point. The positivity and boundedness of the solutions are proved and the local stability analysis reveals that the positive equilibrium point is asymptotically stable, regardless of the distributed time delay considered in the model. Moreover, the construction of a suitable Lyapunov function leads to global asymptotic stability results. Numerical simulations are presented with the aim of substantiating the theoretical statements.

Dynamic behavior of the p53-Mdm2 core module under the action of drug Nutlin and dual delays.

Nutlin is a family of p53-targeting drugs. It is able to bind to Mdm2, thereby accelerate the accumulation of p53 that is a prominent tumor suppressor. An integrated module of the Nutlin PBK and p53 pathway is composed of positive feedback mediated by Mdm2 mRNA as well as the drug Nutlin and negative feedback mediated by Mdm2 protein. The main research content of our paper is how the time delay of protein synthesis, response time delay of Nutlin drug, the degradation rate of Mdm2, the degradation rate of p53 depended on Mdm2 and the actual dose of Nutlin in the cell affect the oscillatory behavior caused by Hopf bifurcation in the integrated network system of Nutlin PBK and p53 pathways. The stability of the unique positive equilibrium point and the existence of Hopf bifurcation are studied by taking the time delays as the bifurcation parameters and applying bifurcation theory. Based on the normal form theory and central manifold theorem, explicit criteria to determine the Hopf bifurcation direction and stability of the bifurcated periodic solution are established. It is found that the time delays and key parameters in the integrated network system of Nutlin PBK and p53 pathways play an important role in the amplitude and period of p53 oscillation according to the results from the numerical simulation and theoretical calculation. These results may provide us with a better understanding of the biological functions of the p53 pathway and some clues for cancer treatment.

Pattern dynamics of a reaction-diffusion predator-prey system with both refuge and harvesting

We are concerned with a reaction-diffusion predator–prey model under homogeneous Neumann boundary condition incorporating prey refuge (proportion of both the species) and harvesting of prey species in this contribution. Criteria for asymptotic stability (local and global) and bifurcation of the subsequent temporal model system are thoroughly analyzed around the unique positive interior equilibrium point. For partial differential equation (PDE), the conditions of diffusion-driven instability and the Turing bifurcation region in two-parameter space are investigated. The results around the unique interior feasible equilibrium point specify that the effect of refuge and harvesting cooperation is an important part of the control of spatial pattern formation of the species. A series of computer simulations reveal that the typical dynamics of population density variation are the formation of isolated groups within the Turing space, that is, spots, stripe-spot mixtures, labyrinthine, holes, stripe-hole mixtures and stripes replication. Finally, we discuss spatiotemporal dynamics of the system for a number of different momentous parameters via numerical simulations.

Lyapunov Function for Two-species Mutualism Model with Constant Harvesting

In this paper, the researcher proposes a simple mathematical model consisting of mutualistic interactions among two-species with constant harvesting. Mutualism is one kind of interaction that ends up being a win-win situation for both species involved. The interacting species benefit from this interaction and ultimately are better adapted for continuous existence. The harvesting function is implemented to describe the rate of removal of the species. This paper aims to investigate the global stability of the unique positive equilibrium point of the model. The global stability of the model is studied by using Lyapunov function method. By constructing a suitable Lyapunov function, it has been proven that the unique positive equilibrium point is globally asymptotically stable in a nonlinear system. Finally, numerical simulation is shown to illustrate theoretical results and to simulate the trajectories around the stable equilibrium point. From the numerical analysis, it is observed that both the species persist.

Stability analysis in COVID-19 within-host model with immune response

The 2019 coronavirus disease (COVID-19) is now a global pandemic. Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2) is the causative pathogen of COVID-19. Here, we study an in-host model that highlights the effector T cell response to SARS-CoV-2. The stability of a unique positive equilibrium point, with viral load V*, suggests that the virus may replicate fast enough to overcome T cell response and cause infection. This overcoming is the bifurcation point, near which the orders of magnitude for V* can be sensitive to numerical changes in the parameter values. Our work offers a mathematical insight into how SARS-CoV-2 causes the disease.

Combating unemployment through skill development

In this paper, we propose and analyze a nonlinear mathematical model to study the effect of skill development on unemployment. We assume that government promulgates different levels of skill development programs for unemployed persons through which two different categories of skilled persons, namely, the low-skilled and the highly-skilled persons, are coming out and the highly-skilled persons are able to create vacancies. The model is studied using stability theory of nonlinear differential equations. We find analytically that there exists a unique positive equilibrium point of the proposed model system under some conditions. Also, the resulting equilibrium is locally as well as globally stable under certain conditions. The effective use of implemented policies to control unemployment by providing skills to unemployed persons and the new vacancies created by highly-skilled persons are identified by using optimal control analysis. Finally, numerical simulation is carried out to support analytical findings.

Dynamical Analysis of a Beddington–DeAngelis Interacting Species System with Prey Harvesting

A reaction–diffusion interacting species system with Beddington–DeAngelis functional response that has been proposed in the environment of mathematical ecology, which provides the rise to spatial pattern formation, is investigated and associated with the models of deterministic dynamics. The dynamical behaviour of a generalist predator–prey system with linear harvesting of each species and predator-dependent functional response is fully analyzed. Conditions of stability behaviour of the interior equilibrium point are established properly. Furthermore, we have recognized that the unique positive equilibrium point of the system is globally stable via appropriate Lyapunov function structure, which signifies that appropriate harvesting has no impact on the persistence property of the harvesting system. Also, we establish the conditions for the existence of bifurcation phenomena including a saddle-node bifurcation and a Hopf bifurcation. Subsequently, complete analysis regarding the impact of harvesting is carried out, and an interesting decision is that under some appropriate constraints, harvesting has immense impact on the final size of the interacting species. In addition, in accordance with Turing’s ideas on morphogenesis , our analysis shows that harvesting effort in a reaction–diffusion predator–prey system plays a vital function for geological conservation of interacting species. Finally, we discuss sufficient conditions for the existence of bionomic equilibrium point and the optimal harvesting policy attained by using the Pontryagin maximal principle.

Global dynamics of neoclassical growth model with multiple pairs of variable delays

Taking into account the effects of multiple pairs of variable delays, this paper deals with the global dynamics for a class of differential neoclassical growth models. We aim to obtain significant insights into better understanding of how the multiple pairs of variable delays essentially affect the stability and attractiveness of the unique positive equilibrium point. First of all, we prove that every solution of the IVP (initial value problem) with respect to the addressed system exists globally and is positive and bounded above. Secondly, with the help of the methods of fluctuation lemma and analytical techniques, two delay-independent criteria and one delay-dependent criterion on the attractivity of the unique positive equilibrium point are established, which improve and complement some published results. Lastly, two examples with the numerical simulation are arranged to illustrate the effectiveness and feasibility of the obtained theoretical results.

On the global asymptotic stability of a predator–prey model with Crowley–Martin function and stage structure for prey

In this work, we present a mathematically rigorous analysis for the local and global asymptotic stability of a predator–prey model with Crowley–Martin function and stage structure for prey, which was proposed in a previous work [Applied Mathematics and Computation 232(2014) 810–819]. In the previous work, the local and global stability of the unique positive equilibrium point of this model were established under some sufficient conditions. Our objective is to re-establish the stability of this equilibrium point without these sufficient conditions. For this purpose, we propose a novel approach to show that the sufficient conditions proposed in the previous work are completely freed. More clearly, we use a well-known stability theorem of continuous-time nonlinear cascade systems and Lyapunov stability theorem to prove that the unique positive equilibrium point is asymptotically stable if it exists. Furthermore, the global stability of the boundary equilibrium point is also established based on the approach. Finally, some numerical simulations are performed to confirm the validity of the constructed theoretical results. The results indicate that there is a good agreement between the numerical simulations and theoretical results.

Fractional Order Turbidostat Model with the Discrete Delay of Digestion

In this paper, we consider the turbidostat system with fractional delayed differential equations. Stability of the unique positive equilibrium point of the systems with and without delay are investigated. We prove the existence of Hopf bifurcation for delayed case. Then, the influence of fractional order and time delay parameter on the system is examined through numerical simulations.

Oscillatory behavior of p53-Mdm2 system driven by transcriptional and translational time delays

Biological experiments clarify that p53-Mdm2 module is the core of tumor network and p53 oscillation plays an important role in determining the tumor cell fate. In this paper, we investigate the effect of time delay on the oscillatory behavior induced by Hopf bifurcation in p53-Mdm2 system. First, the stability of the unique positive equilibrium point and the existence of Hopf bifurcation are investigated by using the time delay as the bifurcation parameter and by applying the bifurcation theory. Second, the explicit criteria determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are developed based on the normal form theory and the center manifold theorem. In addition, the combination of numerical simulation results and theoretical calculation results indicates that time delays in p53-Mdm2 system are critical for p53 oscillations. The results may help us to better understand the biological functions of p53 pathway and provide clues for treatment of cancer.

Bifurcations and chaos control in a discrete-time biological model

In this paper, bifurcations and chaos control in a discrete-time Lotka–Volterra predator–prey model have been studied in quadrant-[Formula: see text]. It is shown that for all parametric values, model has boundary equilibria: [Formula: see text], and the unique positive equilibrium point: [Formula: see text] if [Formula: see text]. By Linearization method, we explored the local dynamics along with different topological classifications about equilibria. We also explored the boundedness of positive solution, global dynamics, and existence of prime-period and periodic points of the model. It is explored that flip bifurcation occurs about boundary equilibria: [Formula: see text], and also there exists a flip bifurcation when parameters of the discrete-time model vary in a small neighborhood of [Formula: see text]. Further, it is also explored that about [Formula: see text] the model undergoes a N–S bifurcation, and meanwhile a stable close invariant curves appears. From the perspective of biology, these curves imply that between predator and prey populations, there exist periodic or quasi-periodic oscillations. Some simulations are presented to illustrate not only main results but also reveals the complex dynamics such as the orbits of period-2,3,13,15,17 and 23. The Maximum Lyapunov exponents as well as fractal dimension are computed numerically to justify the chaotic behaviors in the model. Finally, feedback control method is applied to stabilize chaos existing in the model.

Neimark‐Sacker bifurcation and hybrid control in a discrete‐time Lotka‐Volterra model

We explore the local dynamics, N‐S bifurcation, and hybrid control in a discrete‐time Lotka‐Volterra predator‐prey model in . It is shown that parametric values, model has two boundary equilibria: and , and a unique positive equilibrium point: if . We explored the local dynamics along with different topological classifications about equilibria: , , and of the model. It is proved that model cannot undergo any bifurcation about and but it undergoes an N‐S bifurcation when parameters vary in a small neighborhood of by using a center manifold theorem and bifurcation theory and meanwhile, invariant close curves appears. The appearance of these curves implies that there exist a periodic or quasiperiodic oscillations between predator and prey populations. Further, theoretical results are verified numerically. Finally, the hybrid control strategy is applied to control N‐S bifurcation in the discrete‐time model.

Supercritical Neimark–Sacker Bifurcation and Hybrid Control in a Discrete-Time Glycolytic Oscillator Model

We study the local dynamical properties, Neimark–Sacker bifurcation, and hybrid control in a glycolytic oscillator model in the interior of ℝ+2. It is proved that, for all parametric values, Pxy+α/β+α2,α is the unique positive equilibrium point of the glycolytic oscillator model. Further local dynamical properties along with different topological classifications about the equilibrium Pxy+α/β+α2,α have been investigated by employing the method of linearization. Existence of prime period and periodic points of the model under consideration are also investigated. It is proved that, about the fixed point Pxy+α/β+α2,α, the discrete-time glycolytic oscillator model undergoes no bifurcation, except Neimark–Sacker bifurcation. A further hybrid control strategy is applied to control Neimark–Sacker bifurcation in the discrete-time model. Finally, theoretical results are verified numerically.

RETRACTED ARTICLE:Existence of global solutions for the\t\t\tsemilinear nonlocal fractional Cauchy problem of the Schr\xf6dinger equation

This paper is concerned with the global existence of solutions for the\t\t\t\tsemilinear nonlocal fractional Cauchy problem of the Schrödinger equation. Firstly,\t\t\t\tbased on the Schrödinger approximation technique and the theory of a family of\t\t\t\tpotential wells, the authors obtain the invariant sets and vacuum isolating of\t\t\t\tglobal solutions including the critical case. Then, the global existence of\t\t\t\tsolutions and the stability of equilibrium points are discussed. Finally, the global\t\t\t\tasymptotic stability of the unique positive equilibrium point of the system is\t\t\t\tproved by applying the Leray–Schauder alternative fixed point theorem.

Global existence and asymptotic stability in a predator–prey chemotaxis model

In this paper, we consider the global behavior of the fully parabolic predator–prey chemotaxis model u1t=d1Δu1+χ∇⋅(u1∇v)+μ1u1(1−u1−e1u2),x∈Ω,t>0,u2t=d2Δu2−ξ∇⋅(u2∇v)+μ2u2(1+e2u1−u2),x∈Ω,t>0,vt=d3Δv+αu1+βu2−γv,x∈Ω,t>0,∂u1∂ν=∂u2∂ν=∂v∂ν=0,x∈∂Ω,t>0,u1(x,0)=u1,0(x),u2(x,0)=u2,0(x),v(x,0)=v0(x),x∈Ωin a smooth bounded domain Ω⊂Rn, where d1,d2,d3,χ,ξ,μ1,μ2,e1,e2,β,γ are positive constants, α∈R. It is proved that if n≤2 and the parameters μ1, μ2, e1, e2 satisfy some suitable conditions, then for all appropriate regular nonnegative initial data, the model has a unique global classical solution (u1,u2,v). Furthermore, the following criteria on the global asymptotic stability of the equilibria to the model are given by constructing Lyapunov functions. (i) If e1<1 and both μ1χ2 and μ2ξ2 are sufficiently large, then the solution (u1,u2,v) satisfying u1,0,u2,0≥(⁄≡)0 converges to a unique positive equilibrium point of the model. (ii) If e1≥1 and μ2ξ2 is sufficiently large, then the solution (u1,u2,v) with u2,0≥(⁄≡)0 converges to the semi-trivial equilibrium point (0,1,βγ).In particular, the respective convergence rates are at least exponential if e1≠1, and algebraic if e1=1.

Global stability in a modified Leslie-Gower type predation model assuming mutual interference among generalist predators.

In the ecological literature, mutual interference (self-interference) or competition among predators (CAP) to effect the harvesting of their prey has been modeled through different mathematical formulations. In this work, the dynamical properties of a Leslie-Gower type predation model is analyzed, incorporating one of these forms, which is described by the function $g\left(y\right) =y^{\beta }$, with $0<\beta <1$. This function $g$ is not differentiable for $y=0$, and neither the Jacobian matrix of the system is not defined in the equilibrium points over the horizontal axis ($x-axis$). To determine the nature of these points, we had to use a non-standard methodology. Previously, we have shown the fundamental properties of the Leslie-Gower type model with generalist predators, to carry out an adequate comparative analysis with the model where the competition among predators (CAP) is incorporated. The main obtained outcomes in both systems are: (i) The unique positive equilibrium point, when exists, is globally asymptotically stable (g.a.s), which is proven using a suitable Lyapunov function. (ii) There not exist periodic orbits, which was proved constructing an adequate Dulac function.

Global boundedness and stability for a chemotaxis model of Boló's concentric sclerosis.

Baló's concentric sclerosis (BCS) is considered a variant of inflammatory demyelinating disease closely related to multiple sclerosis characterized by a discrete concentrically layered lesion in the cerebal white matter. Khonsari and Calvez (Plos ONE. 2(2007)) proposed a parabolic-elliptic-ODE chemotaxis model for BCS which describes the evolution of the densities of activated macrophages, cytokine and apoptotic oligodendrocytes. Because "classically activated" M1 microglia can produce cytotoxicity, we introduce a linear production term from the activated microglia in the ODE for pro-inflammatory cytotoxic. For the new BCS chemotaxis model, we first investigate the uniform boundedness and global existence of classical solutions, and then get a range of the chemosensitive rate χ where the unique positive equilibrium point is exponentially asymptotically stable.