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This paper investigates the asymptotic analysis of spatially heterogeneous viral transmission, incorporating cell‐to‐cell transmission, virus nonlocal dispersal, and intracellular delay. Due to the noncompactness of the semiflow, we used the Kuratowski measure of noncompactness to demonstrate the existence of a global compact attractor. This noncompactness issue generates difficulties in calculating the basic reproduction number R0$$ {R}_0 $$, which is the principal eigenvalue of the next‐generation operator. The threshold role of this number is determined, where we derived two different cases: (i) the global stability of the virus‐free steady state, which is globally stable for R0<1$$ {R}_0<1 $$ by the Lyapunov direct method, and (ii) the global stability of the virus steady state for R0>1$$ {R}_0>1 $$. Indeed, the second case is demonstrated through several steps that include uniform persistence, the existence of a virus steady state, and the global stability of the virus steady state. The results are supported by different graphical representations with proper biological justifications. [ABSTRACT FROM AUTHOR]

This research is concerned to study the global threshold dynamics of a hybrid viral infection model, with the assumption that all parameters are space dependent. The considered hybrid model contains three principal equations, the first is an ordinary differential equation that represents the uninfected cells, an age-structured equation modeled by the transport equation for infected-cells, and a diffusive equation that studies the evolution of the virus. The challenging mathematical aspect of this research lies in the fact that the model is partially degenerate and the solution map is not compact. Also, understanding the global threshold dynamics in terms of this partial degeneracy, mostly with age-structured equation and nonlocal diffusion is the ultimate challenging point of this research. The existence and the uniqueness of a global solution are proved. Further, the existence of a global compact attractor is shown. Moreover, the basic reproduction number $ \mathfrak{R}_0 $ is identified for the model with its threshold role: If $ \mathfrak{R}_0<1 $, the infection-free-steady state is globally asymptotically stable; for $ \mathfrak{R}_0>1 $, the solution of the studied model is uniformly persistent and the pathogen persists in a unique positive steady state, which it is shown using the Lyapunov approach. The results are supported by some graphical representations for illustrating the finding. [ABSTRACT FROM AUTHOR]

A chemostat is a laboratory device (of the bioreactor type) in which organisms (bacteria, phytoplankton) develop in a controlled manner. This paper studies the asymptotic properties of a chemostat model with generalized interference function and Poisson noise. Due to the complexity of abrupt and erratic fluctuations, we consider the effect of the second order Itô-Lévy processes. The dynamics of our perturbed system are determined by the value of the threshold parameter C 0 ⋆ . If C 0 ⋆ is strictly positive, the stationarity and ergodicity properties of our model are verified (practical scenario). If C 0 ⋆ is strictly negative, the considered and modeled microorganism will disappear in an exponential manner. This research provides a comprehensive overview of the chemostat interaction under general assumptions that can be applied to various models in biology and ecology. In order to verify the reliability of our results, we probe the case of industrial waste-water treatment. It is concluded that higher order jumps possess a negative influence on the long-term behavior of microorganisms in the sense that they lead to complete extinction. [ABSTRACT FROM AUTHOR]

We aim in this research to determine the global stability of equilibrium states for an SIR epidemic model with nonlocal diffusion and nonlinear incidence function in a heterogeneous environment. For achieving this result, we consider that the model parameters are Lipschitz continuous functions. At the infection free‐equilibrium, the eigenvalue problem has a principal simple eigenvalue λ0$$ {\lambda}_0 $$ corresponding to strictly positive eigenfunction which is expressed as λ0=R0−1$$ {\lambda}_0={R}_0-1 $$. By a Lyapunov function approach, we show the global stability of drug‐free equilibrium for R0<1$$ {R}_0<1 $$. For R0>1$$ {R}_0>1 $$, and using persistence theory for dynamical systems, we show that the epidemic will persist and the unique endemic equilibrium state is globally stable by constructing a proper Lyapunov function. [ABSTRACT FROM AUTHOR]

We are dealing with the classical Landau–Lifshitz equation with an unusual exchange field expressed in terms of a p -Laplacian operator for p greater than 2. We obtain weak solutions to the model by using penalization, compactness arguments, and monotonicity method. [ABSTRACT FROM AUTHOR]

In this research, we are interested in discussing the evolution of the COVID-19 infection cases and predicting the spread of COVID-19 disease in Algeria and India. To this aim, we will approximate the transmission rate in terms of the measures taken by the governments. The least square method is used with an accuracy of 95% for fitting the artificial solution with the real data declared by WHO for the purpose of approximating the density of asymptomatic individuals for COVID-19 disease. As a result, we obtained the different values of the basic reproduction number (BRN) corresponding to each measure taken by the governments. Moreover, we estimate the number of asymptomatic infected persons at the epidemic peak for each country. Further, we will determine the needed ICU beds (intense medical carte beds) and regular treatment beds. Also, we provide the outcome of governmental strategies in reducing the spread of disease. Combining all these components, we offer some suggestions about the necessity of using the recently discovered vaccines as Pfizer/Bioentec and Moderna for limiting the spread of the COVID-19 disease in the studied countries. [ABSTRACT FROM AUTHOR]

In this work, we propose an age‐structured prey–predator model with infection developed in the prey population. Our main goal is to distinguish the effect of the predator maturation measured by the age on the interaction between the predator and the prey, even on the spread of the infectious disease. However, it is obtained that the minimal maturation duration can affect the behavior of the solution, where it can lead to the existence of periodic solutions generated by the presence of Hopf bifurcation for three different equilibrium states. The obtained mathematical results are checked numerically using graphical illustrations. [ABSTRACT FROM AUTHOR]

We investigate through this research the numerical inversion technique for the Laplace transforms cooperated by the integration Boubaker polynomials operational matrix. The efficiency of the presented approach is demonstrated by solving some differential equations. Also, this technique is combined with the standard Laplace Homotopy Perturbation Method. The numerical results highlight that there is a very good agreement between the estimated solutions with exact solutions. [ABSTRACT FROM AUTHOR]

In this research, we investigate the influence of predator harvesting on the predator–prey interaction in the presence of prey social behavior using a reaction–diffusion system subject to the Neumann boundary conditions. It has been proved that the investigated model can undergo Hopf, Turing–Hopf bifurcation, which indicates the possibility of having a homogenous/nonhomogeneous periodic solution under some conditions on the model parameters. The stability of these periodic solutions is studied using the normal form on the center of the manifold theory. The obtained mathematical results are checked numerically. [ABSTRACT FROM AUTHOR]

We are concerned with a reaction-diffusion predator–prey model under homogeneous Neumann boundary condition incorporating prey refuge (proportion of both the species) and harvesting of prey species in this contribution. Criteria for asymptotic stability (local and global) and bifurcation of the subsequent temporal model system are thoroughly analyzed around the unique positive interior equilibrium point. For partial differential equation (PDE), the conditions of diffusion-driven instability and the Turing bifurcation region in two-parameter space are investigated. The results around the unique interior feasible equilibrium point specify that the effect of refuge and harvesting cooperation is an important part of the control of spatial pattern formation of the species. A series of computer simulations reveal that the typical dynamics of population density variation are the formation of isolated groups within the Turing space, that is, spots, stripe-spot mixtures, labyrinthine, holes, stripe-hole mixtures and stripes replication. Finally, we discuss spatiotemporal dynamics of the system for a number of different momentous parameters via numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, we investigate with a time fractional‐order derivative in a three‐species predator‐prey model with the presence of prey social behavior. A new approximation for predator‐prey interaction in the presence of prey social behavior has been considered. For the model analysis, the study has been divided into two principal parts. First of all, we study the local stability of the equilibria and the existence of Hopf bifurcation. Then, for the numerical analysis, the Caputo fractional derivative operator is utilized to approximate the numerical solution of the model. An excellent agreement is seen between the numerical results and the theoretical predictions. [ABSTRACT FROM AUTHOR]

This paper deals with a very general case of predator-prey model called Gauss type model, in the presence of cross-diffusion, where the existence of global solution has been shown and some priori bound of solution. The dynamic near Turing-Hopf bifurcation has been studied, where the existence of Hopf and Turing-Hopf bifurcation has been shown, and for the stability of the periodic solution the normal form on the center of the manifold of Turing-Hopf bifurcation has been calculated. Further, an application of the considered model has been applied in treated models and non treated ones, also an extend numerical simulation has been used to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]