Global dynamics of a compartmental model which describes virus propagation in vivo is studied using the direct Lyapunov method, where the incidence rate of the infection and the removal rate of the virus are assumed to be nonlinear. In the case where the functional quotient between the force of infection and the removal rate of the virus is a nonincreasing function of the virus concentration, the existence of a threshold parameter, i.e., the basic reproduction number or basic reproductive ratio, is established and the global stability of the equilibria is discussed. Moreover, in the absence of the above-mentioned monotonicity property, estimations for the sizes of the domains of attraction are given. Biological significance of the results and possible extensions of the model are also discussed.

From a practical point of view, the most efficient strategy for pest control is to combine an array of techniques to control the wide variety of potential pests that may threaten crops in an approach known as integrated pest management (IPM). In this paper, we propose a predator-prey (pest) model of IPM in which pests are impulsively controlled by means of spraying pesticides (the chemical control) and releasing natural predators (the biological control). It is assumed that the biological and chemical control are used with the same periodicity, but not simultaneously. The functional response of the predator is allowed to be predator-dependent, in the form of a Beddington-DeAngelis functional response, rather than to have a perhaps more classical prey-only dependence. The local and global stability of the pest-eradication periodic solution, as well as the permanence of the system, are obtained under integral conditions which are shown to have biological significance. In a certain limiting case, it is shown that a nontrivial periodic solution emerges via a supercritical bifurcation. Finally, our findings are confirmed by means of numerical simulations.

The global properties of a predator-prey model with nonlinear functional response and stage structure for the predator are studied using Lyapunov functions and LaSalle's invariance principle. It is found that, under hypotheses which ensure the uniform persistence of the system and the existence of a unique positive steady state, a feasible a priori lower bound condition on the abundance of the prey population ensures the global asymptotic stability of the positive steady state. A condition which leads to the extinction of the predators is indicated. We also obtain results on the existence and stability of periodic solutions. In particular, when (4.2) fails to hold and the unique positive steady state E * becomes unstable, the coexistence of prey and predator populations is ensured for initial populations not on the one-dimensional stable manifold of E * , albeit with fluctuating population sizes.

In this paper, we consider an integrated pest management model which is impulsively controlled by means of biological and chemical controls. These controls are assumed to act in a periodic fashion, a nonlinear incidence rate being used to account for the dynamics of the disease caused by the application of the biological control. The Floquet theory for impulsive ordinary differential equations is employed to obtain a condition in terms of an inequality involving the total action of the nonlinear force of infection in a period, under which the susceptible pest-eradication solution is globally asymptotically stable. If the opposite inequality is satisfied, then it is shown that the system under consideration becomes uniformly persistent. A biological interpretation of the persistence condition is also provided.

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This paper deals with an impulsive predator-prey model with Beddington–DeAngelis functional response and time delay, in which the evolution of the predators takes them through two stages, juvenile and mature. It is assumed that only mature predators are able to hunt for prey and reproduce and the time delay is understood as being the time spent by the juvenile predators from birth to maturity. It is first seen that the dynamics of the model can be completely determined through the use of a reduced system consisting of the equations for prey and mature predators, respectively. Using the discrete dynamical system determined by the stroboscopic map, one first determines the mature predator-free periodic solution of the reduced system. By means of comparison techniques, one then deduces sufficient criteria for the global stability of the mature predator-free periodic solution and for the permanence of the reduced system, which yield similar properties for the initial system. As a result, it is observed that time delay and pulses have a crucial effect upon the dynamics of our model.

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