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Highlights • We study the impact of herd shape on the the prey and predator equilibrium densities. • In the absence of the time delay we studied the local stability and bifurcation. • The effect of the time delay on the stability of the interior equilibrium where Hopf bifurcation is obtained. Abstract In this paper, a delayed predator-prey model with the presence of a social behavior for the prey population has been investigated. A new functional response is obtained. We studied the effect of the herd shape for the prey population on the prey and predator equilibrium densities. The analysis of the system has been also established where the boundedness, stability, Hopf bifurcation, stability of Hopf bifurcation are obtained. [ABSTRACT FROM AUTHOR]

• The cross-diffusive predator-prey model with the inclusion of prey refuge is formulated. • A brief re-examination of different dynamical properties corresponding to its temporal counterpart is performed. • The stability analyses of the cross-diffusive system along with Turing space are carried out successfully. • Special emphasis is paid on the impact of prey refuge in the cross-diffusive system to observe how the growth of the interacting species gets substantially perturbed. • The emergence of spatial patterns based on numerical simulations is pointed out. • Conclusions are drawn from several numerical outcomes of the proposed model. The present study appertains to a reaction-diffusion system embracing two-dimensional continuous Beddington-DeAngelis predator-prey model incorporating intra-specific competition among predators and prey refuge in proportion to both the species as well. The existence of all conceivable ecologically significant equilibria is explored and consequently the diffusion-driven instability around the coexistence equilibrium position is reviewed. Numerical simulation with zero-flux boundary conditions discloses that the system under consideration experiences the occurrence of diffusion-driven instability. The dynamical system in Turing space emerges further to get influenced by prey refuge while it unveils diffusion controlled spatiotemporal pattern formation namely, growth of spots, stripe-spot mixtures, stripes, labyrinthine, stripe-hole mixtures and holes reproduction. The quantitative analysis reveals that the interaction of both self- and cross-diffusion plays a significant role on the pattern formation of the present system in a way to enrich the pattern at a greater height. [ABSTRACT FROM AUTHOR]

In nature, the consumers (predator) use many strategies for hunting the resources (prey). The hunting cooperation is an effective strategy for the consumers for giving more accuracy to the hunting. This strategy works on perturbing the resources for facilitating the hunting. In this paper, we are concerned to analyze a superdiffusive resource-consumer system with hunting cooperation functional response. The presence of supperdiffusion represents the effect of the fear of the resources and the organized hunting strategy of the consumer. In fact, the effect of the superdiffusion is successfully established, where it is shown that the superdiffusion undergoes a complex patterns. Also, it noticed that the investigated system has a complex patters as Hopf bifurcation (HB), Turing bifurcation (TB), and Turing-Hopf bifurcation (THB). Furthermore, the super-diffusion can influence the stability of some equilibria. The obtained results are tested numerically. [ABSTRACT FROM AUTHOR]

In this research, we are interested in predicting the epidemic peak outbreak of the Coronavirus in South Africa, Turkey, and Brazil. Until now, there is no known safe treatment, hence the immunity system of the individual has a crucial role in recovering from this contagious disease. In general, the aged individuals probably have the highest rate of mortality due to COVID-19. It is well known that this immunity system can be affected by the age of the individual, so it is wise to consider an age-structured SEIR system to model Coronavirus transmission. For the COVID-19 epidemic, the individuals in the incubation stage are capable of infecting the susceptible individuals. All the mentioned points are regarded in building the responsible predictive mathematical model. The investigated model allows us to predict the spread of COID-19 in South Africa, Turkey, and Brazil. The epidemic peak outbreak in these countries is considered, and the estimated time of the end of infection is regarded by the help of some numerical simulations. Further, the influence of the isolation of the infected persons on the spread of COVID-19 disease is investigated. [ABSTRACT FROM AUTHOR]

This manuscript aims to consider a variety of fractional predator-prey models in the presence of an infection developed in the predator population. The crowding behavior plays an essential role in some species surviving, which is a useful strategy for defending the inside group prey. The purpose of considering the fractional-order-derivative is to study the memory effects on the mutual interactions, which has been confirmed to be an intrinsic feature of a dynamic biological system. From the perspective of mathematical results, the local behavior of the equilibrium points, and the existence of Hopf bifurcation are obtained. Besides, the influence of some crucial parameters as memory rate, herd shape, the infection rate in determining the asymptotic behavior of prey and predator are investigated. Further, an efficient numerical technique has been employed to illustrate some illustrative representations for numerical approximation for the solutions. This iterative scheme has been designed using the fundamental theorem of calculus in the fractional sense, and linear polynomial interpolation. [ABSTRACT FROM AUTHOR]

• A diffusive mussel-algae model with time fractional-order derivative is proposed. • The existence of Turing instability, Hopf bifurcation, Turing-Hopf bifurcation are achieved in the presence of time-fractional derivative. • The influence of the time fractional derivative on the spatiotemporal dynamics is investigated. • The obtained mathematical results are tested numerically. In this paper, we consider a time fractional-order derivative for a diffusive mussel–algae model. The existence of pattern formation was the subject of interest of many previous research works in the case of the diffusive mussel–algae model. Examples include the Turing instability, Hopf bifurcation, Turing-Hopf bifurcation, and others. The presence of the time–fractional–order derivative never been investigated in this model. Next to it ecological relevant, it can generate some important patterns. One of these patterns is produced by the presence of the Turing-Hopf bifurcation. Therefore, our main interest is to analyze the effect of the time fractional–order derivative on the spatiotemporal behavior of the solution, which never been achieved for the mussel-algae model. Besides, Turing–Hopf was studied exclusively on the classical reaction-diffusion systems, where it was also considered for the diffusive mussel-algae model. Thus, our paper puts the fist steps on proving the existence of this type of codimension bifurcation on the diffusive systems with time fractional–order–derivative systems. Further, a suitable numerical simulations are used for confirming the theoretical obtained results. [ABSTRACT FROM AUTHOR]

• A predator-prey model with prey protection zone and predator harvesting is investigated. • Existence, positivity, uniqueness of solution is shown. • The existence and uniqueness of the positive homogeneous steady state is proved. • Global stability of the boundary equilibrium steady state is investigated under some value of model parameters. • The existence of Hopf, Turing, Turing-Hopf bifurcations is investigated. • The stability of the periodic solutions are studied using the normal form. • Some numerical simulations are used for discussing the impact of the predator harvesting on the spatiotemporal behavior of solution. In this paper, a diffusive predator-prey model subject to the zero flux boundary conditions is considered, in which the prey population exhibits social behavior and the harvesting functional of the predator population is assumed to be considered in a quadratic form. The existence of a positive solution and its bounders is investigated. The global stability of the semi trivial constant equilibrium state is established. Concerning the non trivial equilibrium state, the local stability, Hopf bifurcation, diffusion driven instability, Turing-Hopf bifurcation are investigated. The direction and the stability of Hopf bifurcation relying on the system parameters is derived. Some numerical simulations are used to extend the analytical results and show the occurrence of the homogeneous and non homogeneous periodic solutions. Further the effect of the rivalry rate on the dynamical behavior of the studied species. [ABSTRACT FROM AUTHOR]