In this paper, we consider a fractional-order eco-epidemic model based on the Rosenzweig–MacArthur predator–prey model. The model is derived by assuming that the prey may be infected by a disease. In order to take the memory effect into account, we apply two fractional differential operators, namely the Caputo fractional derivative (operator with power-law kernel) and the Atangana–Baleanu fractional derivative in the Caputo (ABC) sense (operator with Mittag–Leffler kernel). We take the same order of the fractional derivative in all equations for both senses to maintain the symmetry aspect. The existence and uniqueness of solutions of both eco-epidemic models (i.e., in the Caputo sense and in ABC sense) are established. Both models have the same equilibrium points, namely the trivial (origin) equilibrium point, the extinction of infected prey and predator point, the infected prey free point, the predator-free point and the co-existence point. For a model in the Caputo sense, we also show the non-negativity and boundedness of solution, perform the local and global stability analysis and establish the conditions for the existence of Hopf bifurcation. It is found that the trivial equilibrium point is a saddle point while other equilibrium points are conditionally asymptotically stable. The numerical simulations show that the solutions of the model in the Caputo sense strongly agree with analytical results. Furthermore, it is indicated numerically that the model in the ABC sense has quite similar dynamics as the model in the Caputo sense. The essential difference between the two models is the convergence rate to reach the stable equilibrium point. When a Hopf bifurcation occurs, the bifurcation points and the diameter of the limit cycles of both models are different. Moreover, we also observe a bistability phenomenon which disappears via Hopf bifurcation.

We consider a mathematical model to describe the interaction between predator and prey that includes predator cannibalism and refuge. We aim to study the dynamics and its long-term behavior of the proposed model, as well as to discuss the effects of crucial parameters associated with the model. We first show the boundedness and positivity of the solution of the model. Then, we study the existence and stability of all possible equilibrium points. The local stability of the model around each equilibrium point is studied via the linearized system, while the global stability is performed by defining a Lyapunov function. The model has four equilibrium points. It is found that the equilibrium point representing the extinction of both prey and predator populations is always unstable, while the other equilibrium points are conditionally stable. In addition, there is forward bifurcation phenomena that occur under certain condition. To support our analytical findings, we perform some numerical simulations.

The harvesting management is developed to protect the biological resources from over-exploitation such as harvesting and trapping. In this article, we consider a predator–prey interaction that follows the fractional-order Rosenzweig–MacArthur model where the predator is harvested obeying a threshold harvesting policy (THP). The THP is applied to maintain the existence of the population in the prey–predator mechanism. We first consider the Rosenzweig–MacArthur model using the Caputo fractional-order derivative (that is, the operator with the power-law kernel) and perform some dynamical analysis such as the existence and uniqueness, non-negativity, boundedness, local stability, global stability, and the existence of Hopf bifurcation. We then reconsider the same model involving the Atangana–Baleanu fractional derivative with the Mittag–Leffler kernel in the Caputo sense (ABC). The existence and uniqueness of the solution of the model with ABC operator are established. We also explore the dynamics of the model with both fractional derivative operators numerically and confirm the theoretical findings. In particular, it is shown that models with both Caputo operator and ABC operator undergo a Hopf bifurcation that can be controlled by the conversion rate of consumed prey into the predator birth rate or by the order of fractional derivative. However, the bifurcation point of the model with the Caputo operator is different from that of the model with the ABC operator.

We study the dynamics of a modified Leslie–Gower one prey–two predators model with competition between predator populations. The model describes complex dynamics in the permanence, global stability and bifurcation. It is shown that there are eight possible equilibrium states. Two equilibrium states, i.e., the extinction of all of the species state and the extinction of both predators state are always unstable, while the other equilibrium states are conditionally locally and globally asymptotically stable. We also analyzed numerically the effect of competition between predators. Our numerical simulations showed that the competition rate of the second-predator may induce the transcritical bifurcation, the saddle-node bifurcation as well as the bi-stability phenomenon. Such numerical results are consistent with the analytical results.

This paper discusses a dynamical analysis on a model that governs the growth of tumour cell under a therapy by using oncolytic viruses, on the standard incident rate. The model is a modification of the similar one by replacing the bilinear incident rate with the standard one. The conducted dynamical analysis consists of the determination of equilibrium points and their existence conditions, followed by local as well as global stability analysis of the equilibrium points. The analytical result shows that there are two equilibrium points, namely uninfected and the endemic point, which needs a condition to exist. Stability analysis shows that there is a dimensionless basic reproduction number that marks the existence as well as the stability of equilibrium points. When basic reproduction number is less than one, there is only the uninfected equilibrium, which is global asymptotically stable. On the other hands, both of equilibrium points exist when the basic reproduction number is more than one, but the uninfected point is not stable anymore, while the endemic one is local asymptotically stable under a condition. Some numerical simulations are performed to illustrate the analytical result. Numerically, it can also be demonstrated that there is a set of parameters which indicates that tumour can be fully removed.

This research concern with dynamical analysis of a SVLIT (Susceptible Vaccination Latent Infective Treatment) model. It represents the spread of tuberculosis with vaccination and saturated incident rate. The incident rate is considered because of the barriers effect due to changes in susceptible individuals behavior. This model has two equilibrium points, namely disease-free equilibrium point which always exists and an endemic equilibrium point that exists under some certain conditions. The local stability of the equilibrium points is investigated by using Routh-Hurwitz criteria. The method of next generation matrix is applied to determine the basic reproduction number R ₀. It can be shown numerically that disease-free equilibrium point is local asymptotically stable when R ₀ < 1, while the endemic equilibrium point exist and local asymptotically stable when R ₀ > 1. Numerical simulations are given to illustrate the theoretical results.