In this paper we investigate the existence of positive solutions of the $q$ -difference equation $-D_q^2u(t)=a(t)\,f(u(t))$ with some boundary conditions by applying a fixed point theorem in cones.

In this paper, a fractional-order model of palm trees, the lesser date moth and the predator is presented. Existence conditions of the local asymptotic stability of the equilibrium points of the fractional system are analyzed. We prove that the positive equilibrium point is globally stable also. The numerical simulations come to illustrate the dynamical behaviors of the model such as bifurcation and chaos phenomenon, and the numerical simulations confirm the validity of our theoretical results.

In this paper, a fractional-order eco-epidemiological model with two disease strains in the predator population incorporating harvesting is formulated and analyzed. The model assumes that the population is divided into a prey population, a susceptible predator population, a predator population infected by the first disease, and a predator population infected by the second disease. A mathematical analysis and numerical simulations are performed to explain the dynamics and properties of the proposed fractional-order eco-epidemiological model. The positivity, boundedness, existence, and uniqueness of the solutions are examined. The basic reproduction number and some sufficient conditions for the existence of four equilibrium points are obtained. In addition, some sufficient conditions are proposed to ensure the local and global asymptotic stability of the equilibrium points. Theoretical results are illustrated through numerical simulations, which also highlight the effect of the fractional order.

In this paper, the dynamical behavior of a discrete SIR epidemic model with fractional-order with non-monotonic incidence rate is discussed. The sufficient conditions of the locally asymptotic stability and bifurcation analysis of the equilibrium points are also discussed. The numerical simulations come to illustrate the dynamical behaviors of the model such as flip bifurcation, Hopf bifurcation and chaos phenomenon. The results of numerical simulation verify our theoretical results.

In this paper, a deterministic prey–predator model is proposed and analyzed. The interaction between three predators and a single prey was investigated. The impact of harvesting on the three predators was studied, and we concluded that the dynamics of the population can be controlled by harvesting. Some sufficient conditions were obtained to ensure the local and global stability of equilibrium points. The transcritical bifurcation was investigated using Sotomayor’s theorem. We performed a stochastic extension of the deterministic model to study the fluctuation environmental factors. The existence of a unique global positive solution for the stochastic model was investigated. The exponential–mean–squared stability of the resulting stochastic differential equation model was examined, and it was found to be dependent on the harvesting effort. Theoretical results are illustrated using numerical simulations.