Nonlinear dynamics of endothelial cells

We propose a mathematical model that governs endothelial cell pattern formation on a biogel surface. The model accounts for diffusion and chemotactic motion of the cells, diffusion of the growth factor and effective biochemical reactions. The model admits a basic steady state that corresponds to a spatially uniform distribution of both the cells and the growth factor. We perform a weakly nonlinear stability analysis of the basic state in order to determine whether spatially nonuniform steady patterns can appear in the system when the basic state becomes unstable. The main results can be summarized as follows. No steady patterns can bifurcate from the basic state if the rate of decay of the growth factor is small. Increasing the rate of decay of the growth factor allows one to observe steady patterns, provided that diffusion of the growth factor is sufficiently slow. Specifically, the work focuses on the occurrence of hexagons and stripes. Most often hexagons are observed. In order for stripes to occur, the chemotactic sensitivity of the endothelial cells and/or their biochemical activity have to be reduced.     

Asymptotic behavior and stability switch for a mature–immature model of cell differentiation

We consider a nonlinear age-structured model, inspired by hematopoiesis modelling, describing the dynamics of a cell population divided into mature and immature cells. Immature cells, that can be either proliferating or non-proliferating, differentiate in mature cells, that in turn control the immature cell population through a negative feedback. We reduce the system to two delay differential equations, and we investigate the asymptotic stability of the trivial and the positive steady states. By constructing a Lyapunov function, the trivial steady state is proven to be globally asymptotically stable when it is the only equilibrium of the system. The asymptotic stability of the positive steady state is related to a delay-dependent characteristic equation. Existence of a Hopf bifurcation and stability switch for the positive steady state is established. Numerical simulations illustrate the stability results.     

Boundedness and Lyapunov function for a nonlinear system of hematopoietic stem cell dynamics

We investigate a system of nonlinear differential equations with distributed delays, arising from a model of hematopoietic stem cell dynamics. We state uniqueness of a global solution under a classical Lipschitz condition. Sufficient conditions for the global stability of the population are obtained, through the analysis of the asymptotic behavior of the trivial steady state and using a Lyapunov function. Finally, we give sufficient conditions for the unbounded proliferation of a given cell generation.     

Stability Analysis of a Diffusive Predator-prey Model with Hunting Cooperation

In this paper, we are concerned with the dynamics of a diffusive predator-prey model that incorporates the functional response concerning hunting cooperation. First, we investigate the stability of the semi-trivial steady state. Then, we investigate the influence of the diffusive rates on the stability of the positive constant steady state. It is shown that there exists diffusion-driven Turing instability when the diffusive rate of the predator is smaller than the critical value, which is dependent on the diffusive rate of the prey, and the semi-trivial steady state and the positive constant steady state are both locally asymptotically stable when the diffusive rate of the predator is larger than the critical value. Finally, the nonexistence of nonconstant steady states is discussed.     

A microbial continuous culture system with diffusion and diversified growth

A reaction-diffusion model is presented to describe the microbial continuous culture with diversified growth. The existence of nonnegative solutions and attractors for the system is obtained, the stability of steady states and the steady state bifurcation are studied under three growth conditions. In the case of no growth inhibition or only product inhibition, the system admits one positive constant steady state which is stable; in the case of growth inhibition only by substrate, the system can have two positive constant steady states, explicit conditions of the stability and the steady state bifurcation are also determined. In addition, numerical simulations are given to exhibit the theoretical results.     

On the stability of hematopoietic model with feedback control

We propose and analyze a mathematical model of the production and regulation of blood cell population in the bone marrow (hematopoiesis). This model includes the primitive hematopoietic stem cells (PHSC), the three lineages of their progenitors and the corresponding mature blood cells (red blood cells, white cells and platelets). The resulting mathematical model is a nonlinear system of differential equations with several delays corresponding to the cell cycle durations for each type of cells. We investigate the local asymptotic stability of the trivial steady state by analyzing the roots of the characteristic equation. We also prove by a Lyapunov function the global asymptotic stability of this steady state. This situation illustrates the extinction of the cell population in some pathological cases.     

Travelling waves of a delayed SIRS epidemic model with spatial diffusion

This paper is concerned with the existence of travelling waves to an SIRS epidemic model with bilinear incidence rate, spatial diffusion and time delay. By analysing the corresponding characteristic equations, the local stability of a disease-free steady state and an endemic steady state to this system under homogeneous Neumann boundary conditions is discussed. By using the cross iteration method and the Schauder’s fixed point theorem, we reduce the existence of travelling waves to the existence of a pair of upper-lower solutions. By constructing a pair of upper-lower solutions, we derive the existence of a travelling wave solution connecting the disease-free steady state and the endemic steady state. Numerical simulations are carried out to illustrate the main results.     

Global asymptotic stability for an age-structured model of hematopoietic stem cell dynamics

We investigate a system of two nonlinear age-structured partial differential equations describing the dynamics of proliferating and quiescent hematopoietic stem cell (HSC) populations. The method of characteristics reduces the age-structured model to a system of coupled delay differential and renewal difference equations with continuous time and distributed delay. By constructing a Lyapunov–Krasovskii functional, we give a necessary and sufficient condition for the global asymptotic stability of the trivial steady state, which describes the population dying out. We also give sufficient conditions for the existence of unbounded solutions, which describe the uncontrolled proliferation of HSC population. This study may be helpful in understanding the behavior of hematopoietic cells in some hematological disorders.     

Properties of stability and Hopf bifurcation for a HIV infection model with time delay

In this paper, we consider the classical mathematical model with saturation response of the infection rate and time delay. By stability analysis we obtain sufficient conditions for the global stability of the infection-free steady state and the permanence of the infected steady state. Numerical simulations are carried out to explain the mathematical conclusions.     

VOLE POPULATION DYNAMICS UNDER THE INFLUENCE OF SPECIALIST AND GENERALIST PREDATION

Abstract To understand the impact of predation by different types of predators on the vole population dynamics, we formulate a three differential equation model describing the population dynamics of voles, the “specialist predator” and the “generalist predator.” First we perform a local stability study of the different steady states of the basic model and deduce that the predation rates of the “specialist” as well as the “generalist” predator are the main parameters controlling the existence/extinction criteria of the concerned populations. Next we analyze the model from a thermodynamic perspective and study the thermodynamic stability of the different equilibria. Finally using stochastic driving forces, we incorporate the exogenous factor of environmental forcing and investigate the stochastic stability of the system. We compare the stability criteria of the different steady states under deterministic, thermodynamic and stochastic situations. The analysis reveals that when the “specialist” and the “generalist” predator are modeled separately, the system exhibits rich dynamics and the predation rates of both types of predators play a major role in controlling vole oscillation and/or stability. These findings are also seen to resemble closely with the observed behavior of voles in the natural setting. Numerical simulations are carried out to illustrate analytical findings.     

Stability of a stochastic SEIS model with saturation incidence and latent period

In this paper, a stochastic SEIS epidemic model with a saturation incidence rate and a time delay describing the latent period of the disease is investigated. The model inherits the endemic steady state from its corresponding deterministic counterpart. We first show the existence and uniqueness of the global positive solution of the model. Then, by constructing Lyapunov functionals, we derive sufficient conditions ensuring the stochastic stability of the endemic steady state. Numerical simulations are carried out to confirm our analytical results. Furthermore, our simulation results shows that the existence of noise and delay may cause the endemic steady state to be unstable.     

On the dynamics of a predator–prey model with nonconstant death rate and diffusion

The main goal of this paper is to describe the global dynamic of a predator–prey model with nonconstant death rate and diffusion. We obtain necessary and sufficient conditions under which the system is dissipative and permanent. We study the global stability of the nontrivial equilibrium, when it is unique. Finally, we show that there are no nontrivial steady state solutions for certain parameter configuration.     

Hopf bifurcation in a delayed reaction–diffusion–advection population model

In this paper, we investigate a reaction–diffusion–advection model with time delay effect. The stability/instability of the spatially nonhomogeneous positive steady state and the associated Hopf bifurcation are investigated when the given parameter of the model is near the principle eigenvalue of an elliptic operator. Our results imply that time delay can make the spatially nonhomogeneous positive steady state unstable for a reaction–diffusion–advection model, and the model can exhibit oscillatory pattern through Hopf bifurcation. The effect of advection on Hopf bifurcation values is also considered, and our results suggest that Hopf bifurcation is more likely to occur when the advection rate increases.     

Stability analysis of diffusive predator-prey model with modified Leslie-Gower and Holling-type III schemes

The stability of a diffusive predator-prey model with modified Leslie-Gower and Holling-type III schemes is investigated. A threshold property of the local stability is obtained for a boundary steady state, and sufficient conditions of local stability and un-stability for the positive steady state are also obtained. Furthermore, the global asymptotic stability of these two steady states are discussed. Our results reveal the dynamics of this model system.     

A note on a diffusive predator-prey model and its steady-state system

In this note, a diffusive predator-prey model subject to the homogeneous Neumann bound- ary condition is investigated and some qualitative analysis of solutions to this reaction-diffusion system and its corresponding steady-state problem is presented. In particular, by use of a Lyapunov function, the global stability of the constant positive steady state is discussed. For the associated steady state problem, a priori estimates for positive steady states are derived and some non-existence results for non-constant positive steady states are also established when one of the diffusion rates is large enough. Consequently, our results extend and complement the existing ones on this model.     

Stability Analysis of Diffusive Predator-Prey Model with Modified Leslie-Gower and Holling-Type II Schemes

The stability of a diffusive predator-prey model with modified Leslie-Gower and Holling-type II schemes is investigated. A threshold property of the local stability is obtained for a boundary steady state, and sufficient conditions of local stability and un-stability for the positive steady state are also obtained. Furthermore, the global asymptotic stability of these two steady states are discussed. Our results reveal the dynamics of this model system.     

Analysis of stability and Hopf bifurcation for HIV-1 dynamics with PI and three intracellular delays

Mathematical model for the effects of protease inhibitor on the dynamics of HIV-1 infection model with three delays is proposed and analyzed. Some analytical results on the global stability of viral free steady state and infected steady state are obtained. The stability/instability of the positive steady state and associated Hopf bifurcation are investigated by analyzing the characteristic equations.     

Finding all steady state solutions of chemical kinetic models

Ordinary differential equations are used frequently by theoreticians to model kinetic process in chemistry and biology. These systems can have stable and unstable steady states and oscillations. This paper presents an algorithm to find all steady state solutions to a restricted class of ODE models, for which the right-hand sides are linear combinations of rational functions of variables and parameters. The algorithm converts the steady state equations into a system of polynomial equations and uses a globally convergent homotopy method to find all the roots of the system of polynomials. All steady state solutions of the original ODEs are guaranteed to be present as roots of the polynomial equations. The conversion may generate some spurious roots that do not correspond to steady state solutions. The stability properties of the steady states are not revealed. This paper explains the algorithms used and gives results for a cell cycle modeling problem.     

Steady state analysis of a nonlinear renewal equation

In the analysis of a nonlinear renewal equation it is natural to anticipate the existence of nonzero steady states and deal with the question of their stability. Sufficient conditions for existence and uniqueness for these steady states are given. The study of the linearized version of the renewal equation around the steady state helps to a great extent to have insight into some complicated dynamics of the full problem. At this stage the first eigenvalue of the steady state plays a vital role. The characteristic equation, a functional equation whose roots are the eigenvalues, is derived. We give various structures showing that the steady state may be stable or unstable (though the fertility rate is decreasing with competition). A similar study is carried out on a nonlinear model motivated by neuroscience in which the total population is conserved.