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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]

Abstract: This paper deals with the qualitative analysis of the travelling waves solutions of a reaction diffusion model that refers to the competition between the predator and prey with modified Leslie–Gower and Holling type II schemes. The well posedeness of the problem is proved. We establish sufficient conditions for the asymptotic stability of the unique nontrivial positive steady state of the model by analyzing roots of the forth degree exponential polynomial characteristic equation. We also prove the existence of a Hopf bifurcation which leads to periodic oscillating travelling waves by considering the diffusion coefficient as a parameter of bifurcation. Numerical simulations are given to illustrate the analytical study. [Copyright &y& Elsevier]

Abstract: The model analyzed in this paper is based on the model set forth by [M.A. Aziz-Alaoui, M. Daher Okiye, Boundedness and global stability for a predator–prey model with modified Leslie–Gower and Holling-type II schemes, Appl. Math. Lett. 16 (2003) 1069–1075, A.F. Nindjin, M.A. Aziz-Alaoui, M. Cadivel, Analysis of a a predator–prey model with modified Leslie–Gower and Holling-type II schemes with time delay, Nonlinear Anal. Real World Appl., in Press.] with time delay, which describes the competition between predator and prey. This model incorporates a modified version of Leslie–Gower functional response as well as that of the Holling-type II. In this paper, we consider the model with one delay and a unique non-trivial equilibrium and the three others are trivial. Their dynamics are studied in terms of the local stability and of the description of the Hopf bifurcation at for small and large delays and at the third trivial equilibrium that is proven to exist as the delay (taken as a parameter of bifurcation) crosses some critical values. We illustrate these results by numerical simulations. [Copyright &y& Elsevier]

Abstract: In this paper, we consider an ordinary differential equations (ODE) for tumor–immune system with positive immune response and a unique nontrivial positive equilibrium. Its dynamics are studied in terms of the local stability and of the description of the Hopf bifurcation which is proven to exist as the parameter of the normal rate of the flow of adult ECs into the tumor site crosses some critical values. We illustrate these results numerically. [Copyright &y& Elsevier]