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In this paper, we propose a reaction–diffusion mathematical model augmented with self/cross-diffusion in 2D domain which describes the oncolytic virotherapy treatment of a tumor with its growth following the logistic law. The tumor cells are divided into uninfected and infected cells and the virus transmission is supposed to be in a direct mode (from cell to cell). In the absence of cross-diffusion, we establish well posedness of the problem, non-negativity and boundedness of solutions, nonexistence of positive solutions, local and global stability of the nontrivial steady-state and the nonoccurrence of Turing instability. In the presence of cross-diffusion, we prove the occurrence of Turing instability by using the cross-diffusion coefficient of infected cells as a parameter. To have an idea about different patterns, we derive the corresponding amplitude equation by using the nonlinear analysis theory. In the end, we perform some numerical simulations to illustrate the obtained theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we propose a delayed mathematical model describing oncolytic virotherapy treatment of a tumour that proliferates according to the logistic growth function, incorporating viral lytic cycle. The tumour population cells are divided into uninfected and infected cell sub-populations and the virus spreading is supposed to be in a direct mode (i.e. from cell to cell). Depending on the time delay, we analyze the positivity and boundedness of solutions and the stability of tumour, infected and uninfected free equilibria (TFE, IFE, UFE) and uninfected–infected equilibrium (UIE) is established. We prove that, delay can lead to "Jeff's phenomenon" observed in a laboratory which causes oscillations in tumour size whose phase and period change over time. With nonlinear dependence of UIE equilibrium on time delay, we develop a more general algorithm determining the stability/instability of the oscillating periodic solutions bifurcating from the UIE equilibrium. Finally, we present numerical simulations illustrating our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the mathematical analysis of a mathematical model describing the virotherapy treatment of a cancer with logistic growth and the effect of viral cycle presented by a time delay. The cancer population size is divided into uninfected and infected compartments. Depending on time delay, we prove the positivity and boundedness and the stability of equilibria. We give conditions on which the viral cycle leads to “Jeff’s phenomenon” observed in laboratory and causes oscillations in cancer size via Hopf bifurcation theory. We establish an algorithm that determines the bifurcation elements via center manifold and normal form theories. We give conditions which lead to a supercritical or subcritical bifurcation. We end with numerical simulations illustrating our theoretical results. [ABSTRACT FROM AUTHOR]

In this work, we develop a mathematical model to study the effect of drug on the development of cancer including the quiescent compartment. The model is governed by a system of delay differential equations where the delay represents the time that the cancer cell take to proliferate. Our analytical study of the stability shows that by considering the time delay as a parameter of bifurcation, it is possible to have stability switch and oscillations through a Hopf bifurcation. Moreover, by introducing the drug intervention term, the critical delay value increases. This indicates that the system can tolerate a longer delay before oscillations start. In the end, we present some numerical simulations illustrating our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate theoretically and numerically a 2-D spatio-temporal dynamics of a predator-prey mathematical model which incorporates the Holling type II and a modified Leslie–Gower functional response and logistic growth of the prey. This system is modeled by a reaction diffusion equations defined on a disc domain { ( x , y ) ∈ R 2 / x 2 + y 2 < R 2 } with Dirichlet initial conditions and Neumann boundary conditions. We study the local and global stability of the positive equilibrium point. We show that the diffusion can induce instability of the uniform equilibrium point which is stable with respect to a constant perturbation as shown by Turing in 1950s and derive the conditions for Hopf and Turing bifurcation in the spatial domain. Numerical results are given in order to illustrate how biological processes affect spatiotemporal pattern formation in a spatial domain. We perform the computations and generalize, on a circular domain, the results presented in Camara and Aziz-Alaoui [6]. [ABSTRACT FROM AUTHOR]

The Spatio-temporal pattern induced by self-diffusion of a predator-prey model with Boiling type III functional response that incorporates the hunting cooperation between predators has been investigated in this paper. For the local model without structure, the stability of non-negative equilibria with or without collaborative hunting in predators is studied. For the Spatio-temporal model, we analyze the effect of hunting cooperation term on diffusion-driven Turing instability of the homogeneous positive equilibria. To get an idea about patterns formation near the Turing bifurcation, we derive and give a detailed study of the amplitude equation using the multiple-scale analysis. Our result shows that hunting cooperation plays a crucial role in determining the stability and the Turing bifurcation of the model, which is in sharp contrast to the case without cooperation in hunting. Furthermore, some numerical simulations are illustrated to visualize the complex dynamic behavior of the model. [ABSTRACT FROM AUTHOR]