Positive Equilibrium State Research Articles

Multiple spreading phenomena for a free boundary problem of a reaction–diffusion equation with a certain class of bistable nonlinearity

This paper deals with a free boundary problem for diffusion equation with a certain class of bistable nonlinearity which allows two positive stable equilibrium states as an ODE model. This problem models the invasion of a biological species and the free boundary represents the spreading front of its habitat. Our main interest is to study large-time behaviors of solutions for the free boundary problem. We will completely classify asymptotic behaviors of solutions and, in particular, observe two different types of spreading phenomena corresponding to two positive stable equilibrium states. Moreover, it will be proved that, if the free boundary expands to infinity, an asymptotic speed of the moving free boundary for large time can be uniquely determined from the related semi-wave problem.

A Free Boundary Problem for the Predator–Prey Model with Double Free Boundaries

In this paper we investigate a free boundary problem for a predator-prey model with double free boundaries in one space dimension. This system models the expanding of an invasive or new predator species in which the free boundaries represent expanding fronts of the predator species and are described by Stefan-like condition. We prove a spreading-vanishing dichotomy for this model, namely the predator species either successfully spreads to infinity as $t\to \infty$ at both fronts and survives in the new environment, or it fails to establish and dies out in the long run while the prey species stabilizes at a positive equilibrium state. The long time behavior of solution and criteria for spreading and vanishing are also obtained.

A free boundary problem of the ratio-dependent prey-predator model

This paper deals with a free boundary problem for the ratio-dependent prey–predator model over a one-dimensional habitat, in which the free boundary represents the spreading front and is caused only by predator. In this problem, it is assumed that the species can only invade further into the new environment from the right end of the initial region, and the spreading front expands at a speed that is proportional to the predator’s population gradient at the front. The main objective is to realize the dynamics/variations of the prey, predator, and the free boundary. We prove a spreading–vanishing dichotomy for this model, namely the predator species either successfully spreads to infinity as at the front and survives in the new environment, or it fails to establish and dies out in the long run while the prey species stabilizes at a positive equilibrium state. The long-time behavior of solution and criteria for spreading and vanishing are obtained.

Age-dependency in host-vector models: The global analysis

In this paper, we introduce and analyze two structured models for the transmission of a vector-borne infectious disease. The first of these models assumes that the level of contagiousness and the rate of removal (recovery) of infected hosts depends on the infection age. In the second model the hosts population is structured with respect to the physical age of the hosts, and the susceptibility of the hosts is assumed to be age-dependent. For these models, the threshold parameter for the existence of a positive (endemic) equilibrium state is determined, and the global asymptotic stability of the equilibrium states are established by the Lyapunov’s direct method.

The influence of time-dependent delay on behavior of stochastic population model with the Allee effect

This paper presents the analysis of behavior of stochastic time-dependent delay population model with the Allee effect. We prove the existence-and-uniqueness of positive solution of considered model. Then, we find the sufficient conditions under which the population will become extinct. We also show that if the initial population size exceeds environmental carrying capacity and time delay is sufficiently long, considered population is non-persistent in mean. The sufficient conditions for asymptotical mean square stability and stability in probability of the positive equilibrium states of the model, in terms of Lyapunov functional method, are obtained. Finally, as an illustration, we apply our mathematical results and predict time which a population of the African wild dog Lycaon pictus needs to reach it is equilibrium states, and also confirm that population of brown tree snake Boiga irregularis is non-persistent in mean if the initial population size is greater than carrying capacity and time delay is long enough.

Stability of a positive equilibrium state for a stochastically perturbed mathematical model of glassy-winged sharpshooter population.

The known nonlinear mathematical model of the Glassy-winged Sharpshooter is considered. It is assumed that this model is influenced by stochastic perturbations of the white noise type and these perturbations are directly proportional to the deviation of the system state from the positive equilibrium point. A necessary and sufficient condition for asymptotic mean square stability of the equilibrium point of the linear part of the considered stochastic differential equation is obtained. This condition is at the same time a sufficient one for stability in probability of the equilibrium point of the initial nonlinear equation. Numerical calculations and figures illustrate the obtained results.

Asymptotics of the solutions of the generalized Hutchinson equation

The problem of the behavior of solutions of the Hutchinson equation and its generalizations is considered. Results regarding the estimation of the domain of the global stability of the positive equilibrium state in the parameter space are obtained. The problems of the existence, stability, and asymptotics of a slowly oscillating periodic solution are approached in the basic propositions. The problem of the dynamic properties of the system of ordinary differential equations that describes the well-known Belousov-Zhabotinsky reaction is considered as an application of the newly developed asymptotic methods.

A free boundary problem of a predator–prey model with higher dimension and heterogeneous environment

This paper is concerned with a free boundary problem for a prey–predator model in higher space dimensions and heterogeneous environment. Such a model may be used to describe the spreading of an invasive or new predator species in which the free boundary represents the spreading front of the predator species and is described by Stefan-like condition. For simplicity, we assume that the environment and solutions are radially symmetric. We prove a spreading–vanishing dichotomy for this model, namely the predator species either successfully spreads to infinity as t→∞ and survives in the new environment, or it fails to establish and dies out in the long run while the prey species stabilizes at a positive equilibrium state. The criteria for spreading and vanishing are given.

On the Dynamic Regulation of a Non Linear Model Fish Population

In this work we deal with the stability property of a discrete-time nonlinear model of a harvested fish population with the Beverthon-Holt recruitment relationship. We use the method of Lyapunov functions to show that the positive equilibrium state of the stage structured model fish corresponding to a constant fishing intensity is globally and asymptotically stable. Since the fishing intensity is often variable, we also propose how a variable fishing intensity may be controlled through feedback in order to stabilize the fish population state around the positive equilibrium state.

On the stability of non-autonomous perturbed Lotka–Volterra models

The paper is devoted to an extended Lotka–Volterra system of differential equations of predator–prey model. The extension is proposed with perturbation terms, which are null for the positive equilibrium state. In the original Lotka–Volterra system, the equilibrium state is not asymptotically stable due to the fact that perturbations are periodic in time. The aim of the paper is to characterize a form of perturbation terms guaranteeing the asymptotic stability or instability of equilibrium state. The reason of the proposed model is that for large time scale, the Lotka–Volterra model is too simple to be realistic. In the paper, the non-autonomous perturbations do not change the equilibrium state but introduce functions of time as well as for additional perturbed terms as for the main part of the equations modified from Lotka–Volterra model. Theorems are proposed in a renormalized form of the differential equations for time and the two variables. The key point of the paper comes from the use of a Liapunov function introduced in Section 2 which allows to obtain conditions for the asymptotic stability (Section 3) and instability (Section 4) by using a Cetaiev instability theorem following conditions on the renormalized coefficients in time of System (6). An appendix recalls the main results of the Liapunov Direct Method for non-autonomous binary systems of ordinary differential equations.

Temephos spraying and thermal fogging efficacy on Aedes aegypti in homogeneous urban residences

An optimal control model of Aedes aegypti population dynamics concerning classification of indoor-outdoor life cycles is taken into account in this paper. A dengue epidemics measure, the basic mosquito offspring, is obtained from the well-known next-generation matrix. The number is used to analyse the stability of the mosquito-free equilibrium. This mosquito-free equilibrium describes a steady-state condition in the mosquito population dynamics where there are no mosquitoes. Comprehensive analysis on the existence and stability of the positive non-trivial equilibrium state is shown as well. Further work deals with designation of the control measures and numerical implementation of the optimal control model using various scenarios. We highlight those measures expressed as the semi-discrete mass profile of the Temephos spraying and the thermal fogging.

Computing weakly reversible linearly conjugate chemical reaction networks with minimal deficiency

Mass-action kinetics is frequently used in systems biology to model the behavior of interacting chemical species. Many important dynamical properties are known to hold for such systems if their underlying networks are weakly reversible and have a low deficiency. In particular, the Deficiency Zero and Deficiency One Theorems guarantee strong regularity with regards to the number and stability of positive equilibrium states. It is also known that chemical reaction networks with distinct reaction structure can admit mass-action systems with the same qualitative dynamics. The theory of linear conjugacy encapsulates the cases where this relationship is captured by a linear transformation. In this paper, we propose a mixed-integer linear programming algorithm capable of determining the minimal deficiency weakly reversible reaction network which admits a mass-action system which is linearly conjugate to a given reaction network.

The Stefan problem for the Fisher–KPP equation

We study the Fisher–KPP equation with a free boundary governed by a one-phase Stefan condition. Such a problem arises in the modeling of the propagation of a new or invasive species, with the free boundary representing the propagation front. In one space dimension this problem was investigated in Du and Lin (2010) [11], and the radially symmetric case in higher space dimensions was studied in Du and Guo (2011) [10]. In both cases a spreading-vanishing dichotomy was established, namely the species either successfully spreads to all the new environment and stabilizes at a positive equilibrium state, or fails to establish and dies out in the long run; moreover, in the case of spreading, the asymptotic spreading speed was determined. In this paper, we consider the non-radially symmetric case. In such a situation, similar to the classical Stefan problem, smooth solutions need not exist even if the initial data are smooth. We thus introduce and study the “weak solution” for a class of free boundary problems that include the Fisher–KPP as a special case. We establish the existence and uniqueness of the weak solution, and through suitable comparison arguments, we extend some of the results obtained earlier in Du and Lin (2010) [11] and Du and Guo (2011) [10] to this general case. We also show that the classical Aronson–Weinberger result on the spreading speed obtained through the traveling wave solution approach is a limiting case of our free boundary problem here.

A Lyapunov Function and Global Stability for a Class of Predator‐Prey Models

We construct a new Lyapunov function for a class of predation models. Global stability of the positive equilibrium states of these systems can be established when the Lyapunov function is used.

Bifurcation analysis in a diffusive ‘food-limited’ model with time delay

The dynamics of a diffusive ‘food-limited’ population model with delay and Dirichlet boundary condition are investigated. The occurrence of steady state bifurcation with changes of parameter is proved by applying phase plane ideas. The existence of Hopf bifurcation at the positive steady state with the changes of specific parameter is obtained, and the phenomenon that the unstable positive equilibrium state without dispersion may become stable with dispersion under certain conditions, which is found by analysing the distribution of the eigenvalues. By the theory of normal form and centre manifold, an explicit algorithm for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are derived. Examples of numerical simulation are carried out to support the analytic results.

Spreading-Vanishing Dichotomy in the Diffusive Logistic Model with a Free Boundary

In this paper we investigate a diffusive logistic model with a free boundary in one space dimension. We aim to use the dynamics of such a problem to describe the spreading of a new or invasive species, with the free boundary representing the expanding front. We prove a spreading-vanishing dichotomy for this model, namely the species either successfully spreads to all the new environment and stabilizes at a positive equilibrium state, or it fails to establish and dies out in the long run. Sharp criteria for spreading and vanishing are given. Moreover, we show that when spreading occurs, for large time, the expanding front moves at a constant speed. This spreading speed is uniquely determined by an elliptic problem induced from the original model.

The chaos and optimal control of cancer model with complete unknown parameters

In this paper, we study the chaos and optimal control of cancer model with completely unknown parameters. The stability analysis of the biologically feasible steady-states of this model will be discussed. It is proved that the system appears to exhibit periodic and quasi-periodic limit cycles and chaotic attractors for some ranges of the system parameters. The necessary optimal controllers input for the asymptotic stability of some positive equilibrium states are derived. Numerical analysis and extensive numerical examples of the uncontrolled and controlled systems were carried out for various parameters values and different initial densities.

Bifurcation analysis in a delayed diffusive Nicholson’s blowflies equation

The dynamics of a diffusive Nicholson’s blowflies equation with a finite delay and Dirichlet boundary condition have been investigated in this paper. The occurrence of steady state bifurcation with the changes of parameter is proved by applying phase plane ideas. The existence of Hopf bifurcation at the positive equilibrium with the changes of specify parameters is obtained, and the phenomenon that the unstable positive equilibrium state without dispersion may become stable with dispersion under certain conditions is found by analyzing the distribution of the eigenvalues. By the theory of normal form and center manifold, an explicit algorithm for determining the direction of the Hopf bifurcation and stability of the bifurcating periodic solutions are derived.

HOPF BIFURCATION ANALYSIS OF DIFFUSIVE BASS MODEL WITH DELAY UNDER "NEGATIVE-WORD-OF-MOUTH"

The original Bass product growth model has been modified in this paper by incorporating a diffusion term and a delay. The imitation parameter β is assumed to be negative which represents the effect of "negative-word-of-mouth" (NWOM). The occurrence of Hopf bifurcation at the positive equilibrium of the remaining market growth u* was proved as β passes through a critical value. The analysis of the distribution of eigenvalues led to the following conclusion. The unstable positive equilibrium state at zero diffusion (d = 0) under NWOM becomes stable when d > 0 for a wide range of β. This observation suggests that when there is a critical NWOM, favorable conditions for the future stable state of u* could be achieved by promoting the product in space (new geographical areas).

Chaos and optimal control of cancer self-remission and tumor system steady states

This paper is devoted to study the problem of optimal control of cancer self-remission and tumor unstable steady-states. The stability analysis of the biologically feasible equilibrium states is presented using a local stability approach. The system appears exhibit a chaotic behavior for some ranges of the system parameters. The necessary optimal control inputs for the asymptotic stability of the positive equilibrium states and minimizes the require performance measure are obtained as nonlinear function of the system densities. Analysis and extensive numerical examples of the uncontrolled and controlled systems were carried out for various parameters values and different initial densities.