On global stability of the intra-host dynamics of malaria and the immune system

In this paper we consider an intra-host model for the dynamics of malaria. The model describes the dynamics of the blood stage malaria parasites and their interaction with host cells, in particular red blood cells (RBC) and immune effectors. We establish the equilibrium points of the system and analyze their stability using the theory of competitive systems, compound matrices and stability of periodic orbits. We established that the disease-free equilibrium is globally stable if and only if the basic reproduction number satisfies R₀?1 and the parasite will be cleared out of the host. If R₀>1, a unique endemic equilibrium is globally stable and the parasites persist at the endemic steady state. In the presence of the immune response, the numerical analysis of the model shows that the endemic equilibrium is unstable.     

The Stability of Self-Organized Rule-Following Work Teams

Self-organized rule-following systems are increasingly relevant objects of study in organization theory due to such systems&2018; capacity to maintain control while enabling decentralization of authority. This paper proposes a network model for such systems and examines the stability of the networks&2018; repetitive behavior. The networks examined are Ashby nets, a fundamental class of binary systems: connected aggregates of nodes that individually compute an interaction rule, a binary function of their three inputs. The nodes, which we interpret as workers in a work team, have two network inputs and one self-input. All workers in a given team follow the same interaction rule.We operationalize the notion of stability of the team&2018;s work routine and determine stability under small perturbations for all possible rules these teams can follow. To study the organizational concomitants of stability, we characterize the rules by their memory, fluency, homogeneity, and autonomy. We relate these measures to work routine stability, and find that stability in ten member teams is enhanced by rules that have low memory, high homogeneity, and low autonomy.     

Landau Damping to Partially Locked States in the Kuramoto Model

In the Kuramoto model of globally coupled oscillators, partially locked states (PLS) are stationary solutions that capture the emergence of partial synchrony when the interaction strength increases. While PLS have long been considered, existing results on their stability are limited to neutral stability of the linearized dynamics in strong topology or to specific invariant subspaces (obtained via the so‐called Ott‐Antonsen (OA) ansatz) with specific frequency distributions for the oscillators. In the mean‐field limit, the Kuramoto model shows various ingredients of the Landau damping mechanism in the Vlasov equation. This analogy has been a source of inspiration for stability proofs of regular Kuramoto equilibria. In addition, the major mathematical issue with PLS asymptotic stability is that these states consist of heterogeneous and singular measures. Here we establish an explicit criterion for their spectral stability and prove their local asymptotic stability in weak topology for a large class of analytic frequency marginals. The proof strongly relies on a suitable functional space that contains (Fourier transforms of) singular measures, and for which the linearized dynamics is well under control. For illustration, the stability criterion is evaluated in some standard examples. We confirm in particular that no loss of generality results in assuming the OA ansatz. To the best of our knowledge, our result provides the first proof of Landau damping to heterogeneous and irregular equilibria in the absence of dissipation. © 2018 Wiley Periodicals, Inc.     

Bottom up and top down effect on toxin producing phytoplankton and its consequence on the formation of plankton bloom

We consider a plankton-nutrient interaction model consisting of phytoplankton, zooplankton and dissolved limiting nutrient with general nutrient uptake functions and instantaneous nutrient recycling. In this model, it is assumed that phytoplankton releases toxic chemical for self defense against their predators. The model system is studied analytically and the threshold values for the existence and stability of various steady states are worked out. It is observed that if the maximal zooplankton conversion rate crosses a certain critical value, the system enters into Hopf bifurcation. Finally it is observed that to control the planktonic bloom and to maintain stability around the coexistence equilibrium we have to control the nutrient input rate specially caused by artificial eutrophication. In case if it is not possible to control the nutrient input rate, one could use toxic phytoplankton to prevent the recurrence bloom.     

The effect of crack faces contact interaction on the critical strain in composites under compressive loading

Aviation and aerospace structural components made of composite laminates due to their internal structure and manufacturing methods often contain a number of inter- and intra-component defects which size, dispersion and interaction alter significantly the critical compression strain level [1]. The current study investigates the effect of the cracks interaction and crack faces contact interaction on the critical strain in laminar transversally isotropic material (cross-ply) compressed in a static manner along interlaminar defects. The frictionless Hertzian contact and the shear and extensional mode of stability loss are considered for the interacting crack faces. The statement of the problem is based on the most accurate approach, the model of piecewise-homogenous medium and the 3-D stability theory [2]. The moment of stability loss in the microstructure of material is treated as the onset of the fracture process. The complex non-classical fracture mechanics problem is solved utilizing the finite elements analysis. The results are obtained for the typical dispositions of cracks. It was found that the crack faces contact interaction alter significantly the critical strain level of the composite. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability of stationary states of non-local equations with singular interaction potentials

We study the large-time behaviour of a non-local evolution equation for the density of particles or individuals subject to an external and an interaction potential. In particular, we consider interaction potentials which are singular in the sense that their first derivative is discontinuous at the origin.For locally attractive singular interaction potentials we prove under a linear stability condition local non-linear stability of stationary states consisting of a finite sum of Dirac masses. For singular repulsive interaction potentials we show the stability of stationary states of uniformly bounded solutions under a convexity condition.Finally, we present numerical simulations to illustrate our results.     

Allee effects in a Ricker-type predator–prey system

We study a discrete host–parasitoid system where the host population follows the classical Ricker functional form and is also subject to Allee effects. We determine basins of attraction of the local attractors of the single population model when the host intrinsic growth rate is not large. In this situation, existence and local stability of the interior steady states for the host–parasitoid interaction are completely analysed. If the host's intrinsic growth rate is large, then the interaction may support multiple interior steady states. Linear stability of these steady states is provided.     

Survival of three species in a nonautonomous Lotka–Volterra system

In Ahmad and Stamova (2004) [1], the author considers a competitive Lotka–Volterra system of three species with constant interaction coefficients. In this paper, we study a nonautonomous Lotka–Volterra model with one predator and two preys. The explorations involve the persistence, extinction and global asymptotic stability of a positive solution.     

Nonconstant steady states and pattern formations of generalized 1D cross-diffusion systems with prey-taxis

Cross-diffusion effects and tactic interactions are the processes that preys move away from the highest density of predators preferentially, or vice versa. It is renowned that these effects have played significant roles in ecology and biology, which are also essential to the maintenance of diversity of species. To simulate the stability of systems and illustrate their spatial distributions, we consider positive nonconstant steady states of a generalized cross-diffusion model with prey-taxis and general functional responses in one dimension. By applying linear stability theory, we analyze the stability of the interior equilibrium and show that even in the case of negative cross-diffusion rate, which appeared in many models, the corresponding cross-diffusion model has opportunity to achieve its stability. Meanwhile, in addition to the cross-diffusion effect, tactic interactions can also destabilize the homogeneity of predator–prey systems if the tactic interaction coefficient is negative. Otherwise, taxis effects can stabilize the homogeneity.     

Bifurcation analysis of a delayed mathematical model for tumor growth

In this study, we present a modified mathematical model of tumor growth by introducing discrete time delay in interaction terms. The model describes the interaction between tumor cells, healthy tissue cells (host cells) and immune effector cells. The goal of this study is to obtain a better compatibility with reality for which we introduced the discrete time delay in the interaction between tumor cells and host cells. We investigate the local stability of the non-negative equilibria and the existence of Hopf-bifurcation by considering the discrete time delay as a bifurcation parameter. We estimate the length of delay to preserve the stability of bifurcating periodic solutions, which gives an idea about the mode of action for controlling oscillations in the tumor growth. Numerical simulations of the model confirm the analytical findings.     

On the stability of evolutionary dynamics in games with incomplete information

In an interaction it is possible that one agent has features it is aware of but the opponent is not. These features (e.g. cost, valuation or fighting ability) are referred to as the agent’s type. The paper compares two models of evolution in symmetric situations of this kind. In one model the type of an agent is fixed and evolution works on strategies of types. In the other model every agent adopts with fixed probabilities both types, and type-contingent strategies are exposed to evolution. It is shown that the dynamic stability properties of equilibria may differ even when there are only two types and two strategies. However, in this case the dynamic stability properties are generically the same when the payoff of a player does not depend directly on the type of the opponent. Examples illustrating these results are provided.     

Real and financial interacting markets: A behavioral macro-model

In the present paper we propose a model in which the real side of the economy, described via a Keynesian good market approach, interacts with the stock market with heterogeneous speculators, i.e., optimistic and pessimistic fundamentalists, that respectively overestimate and underestimate the reference value due to a belief bias. Agents may switch between optimism and pessimism according to which behavior is more profitable. To the best of our knowledge, this is the first contribution considering both real and financial interacting markets and an evolutionary selection process for which an analytical study is performed. Indeed, employing analytical and numerical tools, we detect the mechanisms and the channels through which the stability of the isolated real and financial sectors leads to instability for the two interacting markets. In order to perform such analysis, we introduce the “interaction degree approach”, which allows us to study the complete three-dimensional system by decomposing it into two subsystems, i.e., the isolated financial and real markets, easier to analyze, that are then linked through a parameter describing the interaction degree between the two markets. We derive the stability conditions both for the isolated markets and for the whole system with interacting markets. Next, we show how to apply the interaction degree approach to our model. Among the various scenarios we are led to analyze, the most interesting one is that in which the isolated markets are stable, but their interaction is destabilizing. We choose such setting to give an economic interpretation of the model and to explain the rationale for the emergence of boom and bust cycles. Finally, we add stochastic noises to the optimists and pessimists demands and show how the model is able to reproduce the stylized facts for the real output data in the US.     

Analysis of periodic solutions in an eco-epidemiological model with saturation incidence and latency delay

In the present work, a mathematical model of predator–prey ecological interaction with infected prey is investigated. A saturation incidence function is used to model the behavioral change of the susceptible individuals when their number increases or due to the crowding effect of the infected individuals [V. Capasso, G. Serio, A generalization of the Kermack–McKendrick deterministic epidemic model, Math. Biosci. 42 (1978) 41–61]. Stability criteria for the infection-free and the endemic equilibria are deduced in terms of system parameters. The basic model is then modified to incorporate a time delay, describing a latency period. Stability and bifurcation analysis of the resulting delay differential equation model is carried out and ranges of the delay inducing stability and as well as instability for the system are found. Finally, a stability analysis of the bifurcating solutions is performed and the criteria for subcritical and supercritical Hopf bifurcation derived. The existence of a delay interval that preserves the stability of periodic orbits is demonstrated. The analysis emphasizes the importance of differential predation and a latency period in controlling disease dynamics.     

Modeling the depletion of dissolved oxygen due to algal bloom in a lake by taking Holling type-III interaction

In this paper a non-linear mathematical model for depletion of dissolved oxygen due to algal bloom in a lake is proposed and analyzed. The model is formulated by considering four variables namely, cumulative concentration of nutrients, density of algal population, density of detritus and concentration of dissolved oxygen. In the modeling process it is assumed that nutrients are continuously coming with a constant rate to the lake through water runoff from agricultural fields and domestic drainage. The Holling type-III interaction between nutrients and algal population is considered. Equilibrium values have been obtained and their stability analysis has also been performed. Numerical simulations are carried out to explain the mathematical results.     

Bubble dynamics: Long-time behavior and interaction in viscous liquids

A mathematical two-phase model is used to numerically investigate physical and rheological effects on small, individual bubbles in high-viscosity liquids under pressure impact. It is found out that bubbles remain stable over time at high viscosity and surface tension. The steady case is considered and connected to the stability behavior of the bubble. An upper bound for the bubble radius is derived and the new equilibrium state of the bubble can be predicted by means of stability theorems of differential equations. Finally, the interaction of a limited number of well separated bubbles in an Hele-Shaw flow is mathematically analyzed to visualize and physically interpret their trajectories. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the stability of evolutionary dynamics in games with incomplete information

In an interaction it is possible that one agent has features it is aware of but the opponent is not. These features (e.g. cost, valuation or fighting ability) are referred to as the agent’s type. The paper compares two models of evolution in symmetric situations of this kind. In one model the type of an agent is fixed and evolution works on strategies of types. In the other model every agent adopts with fixed probabilities both types, and type-contingent strategies are exposed to evolution. It is shown that the dynamic stability properties of equilibria may differ even when there are only two types and two strategies. However, in this case the dynamic stability properties are generically the same when the payoff of a player does not depend directly on the type of the opponent. Examples illustrating these results are provided.     

一类三种群生态系统平衡点稳定性的研究

从具有捕食被捕食关系的三种群之间相互作用的数学模型出发 ,讨论了模型平衡点的稳定性     

A stability and numerical study of the solutions of a Timoshenko system with distributed delay

In this work, we analyze the stability of the semigroup associated with a Timoshenko beam model with distributed delay in the rotation angle equation. We show that the type of stability resulting from the semigroup is directly related to some model coefficients, which constitute the velocities of the system's component equations. In the case of stability of the polynomial type, we prove that rate obtained is optimal. We conclude the work performing a numerical study of the solutions and their energies, associated to discrete system.     

Stability analysis of a virus infection model with humoral immunity response and two time delays

In this paper, we investigate the dynamical properties for a model of delay differential equations, which describes a virus‐immune interaction in vivo. By analyzing corresponding characteristic equations, the local stability of the equilibria for infection‐free, antibody‐free, and antibody response and the existence of Hopf bifurcation with antibody response delay as a bifurcation parameter at the antibody‐activated infection equilibrium are established, respectively. Global stability of the equilibria for infection‐free, antibody‐free, and antibody response, respectively, also are established by applying the Lyapunov functionals method. The numerical simulations are performed in order to illustrate the dynamical behavior of the model. Copyright © 2016 John Wiley & Sons, Ltd.     

Asymptotic stability of traveling waves in a discrete convolution model for phase transitions

In a recent paper [P. Bates, A. Chmaj, A discrete convolution model for phase transition, Arch. Rational Mech. Anal. 150 (1999) 281-305], a discrete convolution model for Ising-like phase transition has been derived, and the existence, uniqueness of traveling waves and stability of stationary solution have been studied. This nonlocal model describes l₂-gradient flow for a Helmholts free energy functional with general range interaction. In this paper, by using the comparison principle and the squeezing technique, we prove that the traveling wavefronts with nonzero speed is globally asymptotic stable with phase shift.