stability

Stability analysis and Hopf bifurcation of a delayed prey–predator model with Hattaf–Yousfi functional response and Allee effect

The Allee effect is an important phenomena in the context of ecology characterized by a correlation between population density and the mean individual fitness of a population.  In this work, we examine the influences of Allee effect on the dynamics of a delayed prey–predator model with Hattaf–Yousfi functional response.  We first prove that the proposed model with Allee effect is mathematically and ecologically well-posed.  Moreover, we study the stability of equilibriums and discuss the local existence of Hopf bifurcation.

Fractional derivative model for tumor cells and immune system competition

Modeling a dynamics of complex biologic disease such as cancer still present a complex dealing.  So, we try in our case to study it by considering the system of normal cells, tumor cells and immune response as mathematical variables structured in fractional-order derivatives equations which express the dynamics of cancer's evolution under immunity of the body.  We will analyze the stability of the formulated system at different equilibrium points.  Numerical simulations are carried out to get more helpful and specific outcome about the variations of the cancer's dynamics.

Dynamical behavior of predator–prey model with non-smooth prey harvesting

The objective of the current paper is to investigate the dynamics of a new predator–prey model, where the prey species obeys the law of logistic growth and is subjected to a non-smooth switched harvest: when the density of the prey is below a switched value, the harvest has a linear rate.  Otherwise, the harvesting rate is constant.  The equilibria of the proposed system are described, and the boundedness of its solutions is examined.  We discuss the existence of periodic solutions; we show the appearance of two limit cycles, an unstable inner limit cycle and a stable o

A continuous SIR mathematical model of the spread of infectious illnesses that takes human immunity into account

A mathematical model of infectious disease contagion that accounts for population stratification based on immunity criteria is proposed.  Our goal is to demonstrate the effectiveness of this idea in preventing different epidemics and to lessen the significant financial and human costs these diseases cause.  We determined the fundamental reproduction rate, and with the help of this rate, we were able to examine the stability of the free equilibrium point and then proposed two control measures.  The Pontryagin's maximum principle is used to describe the optimal controls,

Optimal control of tritrophic reaction–diffusion system with a spatiotemporal model

In this paper, we propose a new model of spatio-temporal dynamics concerning the tritrophic reaction-diffusion system by introducing Phytoplankton and Zooplankton.  We recall that the phytoplankton and zooplankton species are the basis of the marine food chain.  There is prey in each marine tritrophic system.  The main objective of this work is to control this species's biomass to ensure the system's sustainability.  To achieve this, we determine an optimal control that minimizes the biomass of super predators.

Dynamical analysis of an HCV model with cell-to-cell transmission and cure rate in the presence of adaptive immunity

In this paper, we will study mathematically and numerically the dynamics of the hepatitis C virus disease with the consideration of two fundamental modes of transmission of the infection, namely virus-to-cell and cell-to-cell.  In our model, we will take into account the role of cure rate of the infected cells and the effect of the adaptive immunity.  The model consists of five nonlinear differential equations, describing the interaction between the uninfected cells, the infected cells, the hepatitis C virions and the adaptive immunity.  This immunity will be represented by the humoral and

Mathematical modeling and analysis of Phytoplankton–Zooplankton–Nanoparticle dynamics

In this paper, we investigate the population dynamics of phytoplankton–zooplankton–nanoparticle model with diffusion and density dependent death rate of predator.  The functional response of predator in this model is considered as Beddington–DeAngelis type.  The stability analysis of the equilibrium points is observed by applying the Routh–Hurwitz criterion.  Numerical simulations are given to illustrate the theoretical results.

On the maximal output set of fractional-order discrete-time linear systems

In this paper, we consider a linear discrete-time fractional-order system defined by \[\Delta ^{\alpha }x_ {k+1}=Ax_k+B u_k, \quad k \geq 0, \quad x_{0} \in \mathbb{R}^{n};\] \[y_{k}=Cx_k, \quad k \geq 0,\] where $A$, $B$ and $C$ are appropriate matrices, $x_{0}$ is the initial state, $\alpha$ is the order of the derivative, $y_k$ is the signal output and $u_k=K x_k$ is feedback control.  By defining the fractional derivative in the Grunwald–Letnikov sense, we investigate the characterization of the maximal output set, $\Gamma(\Omega)=\lbrace x_{0} \in \mathbb{R}^{n}/y_

State regulation of stability and effectiveness of money systems

The problems of stability providing by the way of using criterion of effectiveness of money systems models in the transitional and modern economys are researched. The methodological approaches with the increasing of effectiveness and stability of money systems through the possibility of achievement of certain economic and social priorities are considered.