In this paper, we study the dynamics behaviour of a stratum of plant–herbivore which is modelled through the following F(x, y)=(f(x, y), g(x, y)) two-dimensional map with four parameters defined by where x≥0, y≥0, and the real parameters a, b, r, k are all positive. We will focus on the case a≠b. We study the stability of fixed points and do the analysis of the period-doubling and the Neimark–Sacker bifurcations in a standard way.

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In this paper, we propose an optimal control problem for an HIV infection model with cellular and humoral immune responses, logistic growth of uninfected cells, cell-to-cell spread, saturated infection, and cure rate. The model describes the interaction between uninfected cells, infected cells, free viruses, and cellular and humoral immune responses. We use two control functions in our model to show the effectiveness of drug therapy on inhibiting virus production and preventing new infections. We apply Pontryagin maximum principle to study these two control functions. Next, we simulate the role of optimal therapy in the control of the infection by numerical simulations and AMPL software.

a b s t r a c tIn this work we consider the number of limit cycles that can bifurcate from periodic orbits located inside a double cuspidal loop of the quintic Hamiltonian vector fieldwhere 0 <| ε | 1 and α, β, γ are real constants. Using Picard-Fuchs equations for related abelian integrals, asymptotic expansion of these integrals about critical level curves of H, and some geometric properties of the curves defined by ratios of two especial integrals, we show that the least upper bound for the number of limit cycles appeared in this bifurcation is two.