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In recent years, the stability of the predator–prey model subject to the Allee effect has become an interesting issue. This study investigates the effect of Allee effect on the stability of a discrete-time predator–prey model with Holling type-II functional response. Using equilibrium analysis, stability analysis and bifurcation theory, the mathematical characteristics of the proposed model are examined. Model experiences flip bifurcation and Neimark–Sacker bifurcation based on the center manifold theorem and bifurcation theory. Our analytical results are demonstrated by numerical simulations.

The extinction and the persistence of the population of the harmful sheep blowfly (Lucilia cuprina) are discussed in this paper through a stochastic mathematical model. Using appropriate Lyapunov functionals, the extinction of these flies depends on the time to oviposition and the time between generations. The oviposition period must be sufficiently short. Another species of blowfly, called Lucilia sericata, also attacks sheep in many areas. Both blowflies are ectoparasites of warm-blooded vertebrates, particularly domestic sheep. These two blowflies are related to sharing the same mitochondrial DNA sequences, although the two species are distinct. A sufficiently small egg-laying delay then leads to the rapid extinction of both blowfly populations.

In this paper, a new mathematical model that describes the dynamics of the within-host COVID-19 epidemic is formulated. We show the stochastic dynamics of Target-Latent-Infected-Virus free within the human body with discrete delay and noise. Positivity and uniqueness of the solutions are established. Our study shows the extinction and persistence of the disease inside the human body through the stability analysis of the disease-free equilibrium $$E_0$$ E 0 and the endemic equilibrium $$E^*$$ E ∗ , respectively. Moreover, we show the impact of delay tactics and noise on the extinction of the disease. The most interesting result is even if the deterministic system is inevitably pandemic at a specific point, extinction will become possible in the stochastic version of our model.

A discrete-time predator–prey model is investigated in this paper. In considered model, the population is assumed to follow the model suggested by Ricker 1954. Existence and stability of equilibria are studied. Numerical simulations reveal that, depending on the parameters, the system has complicated and rich dynamics and can exhibit complex patterns. Also the bifurcation diagrams are presented.

We are concerned about the stochastic nonlinear delay differential equation. The stochasticity arises from the white Gaussian noise, which is the time derivative of the standard Brownian motion. The main objective of this paper is to introduce a new technique using the Lyapunov functional for the study of stability of the zero solution of the stochastic delay differential system. Constructing a new appropriate deterministic system in the neighborhood of the origin is an effective way to investigate the necessary and sufficient conditions of stability in the sense of the mean square. Nicholson’s blowflies equation is one of the major problems in ecology; necessary conditions for the possible extinction of the Nicholson’s blowflies population are investigated. We support our theoretical results by providing areas of stability and some numerical simulations of the solution of the system using the Euler–Maruyama scheme, which is mean square stable Maruyama (Rendiconti del Circolo Matematico di Palermo 4(1):48, 1955), Cao et al. (Appl Math Comput 159(1):127–135, 2004).

This manuscript is involved in the study of stability of the solutions of functional differential equations (FDEs) with random coefficients and/or stochastic terms. We focus on the study of different types of stability of random/stochastic functional systems, specifically, stochastic delay differential equations (SDDEs). Introducing appropriate Lyapunov functionals enables us to investigate the necessary conditions for stochastic stability, asymptotic stochastic stability, asymptotic mean square stability, mean square exponential stability, global exponential mean square stability, and practical uniform exponential stability. Some examples with numerical simulations are presented to strengthen the theoretical results. Using our theoretical study, important aspects of epidemiological and ecological mathematical models can be revealed. In ecology, the dynamics of Nicholson’s blowflies equation is studied. Conditions of stochastic stability and stochastic global exponential stability of the equilibrium point at which the blowflies become extinct are investigated. In finance, the dynamics of the Black–Scholes market model driven by a Brownian motion with random variable coefficients and time delay is also studied.

This paper investigates the dynamics behavior of the mathematical model of Human Immunodeficiency Virus (HIV) influenced by stochastic perturbations. This paper is involved in exploring the disease-free equilibrium’s stochastic global exponential stability with a basic reproductive number $$R_{0} < 1$$ . Necessary and sufficient conditions for stochastic global exponential stability of the disease-free equilibrium $$E^{0}$$ of the nonlinear HIV stochastic system are derived. This can be accomplished by the exponential analysis of the global mean square stability of trivial equilibrium of the corresponding linear system. Finally, we provide areas of stability of $$E^{0}$$ and numerical simulations to confirm the analytical results by using the fundamental Euler–Maruyama (EM) algorithm.

We deal with the form of the solutions for the following systems of rational difference equations , , with nonzero real numbers initial conditions. Also we investigate some properties of the obtained solutions and present some numerical examples.

We study some qualitative properties of the solutions of a system of difference equations, which describes an economic model. The study of the local stability of the equilibrium points is carried out. We give some important results of the invariant and the boundedness of the solutions to the considered system. The global convergence of the solutions is presented and investigated.

We obtain in this paper the solutions of the following recursive sequences 𝑥 𝑛 + 1 = 𝑥 𝑛 𝑥 𝑛 − 3 / 𝑥 𝑛 − 2 ( ± 1 ± 𝑥 𝑛 𝑥 𝑛 − 3 ) , 𝑛 = 0 , 1 , … , where the initial conditions are arbitrary real numbers and we study the behaviors of the solutions and we obtained the equilibrium points of the considered equations. Some qualitative behavior of the solutions such as the boundedness, the global stability, and the periodicity character of the solutions in each case have been studied. We presented some numerical examples by giving some numerical values for the initial values and the coefficients of each case. Some figures have been given to explain the behavior of the obtained solutions in the case of numerical examples by using the mathematical program Mathematica to confirm the obtained results.

A discrete-time prey–predator model with Holling type II is investigated. For this model, the existence and stability of three fixed points are analyzed. The bifurcation diagrams, phase portraits and Lyapunov exponents are obtained for different parameters of the model. The fractal dimension of a strange attractor of the model was also calculated. Numerical simulations show that the discrete model exhibits rich dynamics compared with the continuous model, which means that the present model is a chaotic, and complex one.

We study the asymptotic behavior of some higher order extension forms of some known population models. We establish that every solution of the considered difference equations is bounded from above and bounded from below by positive constants. We also provide sufficient conditions for the global asymptotically stable of all solutions of that higher order difference equations.

The objective of this paper is to investigate some qualitative behavior of the solutions of an economic model. This is accomplished by studying a higher order difference equation where we establish some results about the boundedness, the periodicity, and the global attractivity of the solutions of this higher order difference equation and then apply the obtained results to give a complete description of global stability and the periodic character of the solutions of the model.

Consider the system of neutral delay differential equations d d t ( x (t)+p x (t−τ))+Q(t) x (t−σ)= 0 , where p∈ R ,τ∈(0,∞),σ∈[0,∞] and Q is continuous matrix, and the system d d t ( x (t)+B x (t−τ))+Q(t) x (t−σ)= 0 , where B is a matrix. We obtain the sufficient condition for the existence of certain types of solutions of the above equation to be ∫ ∞ ||Q(s)|| d s for p≠−1 .

We establish a global convergence result and then apply it to show that under appropriate hypotheses every positive solution of the difference equation x n + 1 = ∑ i = 0 m A i x n − 2 i , n = 0, 1,…, converges to a period two solution.

In this paper, we build an economic model of a non-linear system of difference equations and present a qualitative study for the obtained model, where a mathematical model of a bounded rationality multiple game with an exponential demand function will be introduced, and then we obtain the equilibrium points of the model and classify if they are locally stable or not. Also, we investigate the boundedness and global convergence of solutions for the obtained system.

We investigate some qualitative behavior of the solutions of the difference equation x n + = ax n - bx n /( cx x - dx n -1 ), n = 0,1,..., where the initial conditions x -1 , x 0 are arbitrary real numbers and a , b , c , d are positive constants.

In this paper, we deal with the existence of solutions and the periodicity character of the following systems of rational difference equations with order three: where the initial conditions , , , , and are nonzero real numbers. MSC:39A10.

In this article we study the difference equation where the initial conditions x -r , x -r+1, x -r+2,..., x 0 are arbitrary positive real numbers, r = max{l, k, p, q} is nonnegative integer and a, b, c are positive constants: Also, we study some special cases of this equation.