International Journal of Biomathematics

We formulate general plant–herbivore interaction models with monotone plant growth functions (rates). We study the impact of monotone plant growth functions in general plant–herbivore models on their dynamics. Our study shows that all monotone plant growth models generate a unique interior equilibrium and they are uniform persistent under certain range of parameters values. However, if the attacking rate of herbivore is too small or the quantity of plant is not enough, then herbivore goes extinct. Moreover, these models lead to noise sensitive bursting which can be identified as a dynamical mechanism for almost periodic outbreaks of the herbivore infestation. Montone and non-monotone plant growth models are contrasted with respect to bistability and crises of chaotic attractors.

The plateau pika is a keystone species of Qinghai–Tibet plateau, but its overabundance aggravates the degradation of alpine meadow. Grazing is the most convenient manner to utilize alpine meadow. Grazing would change vegetation condition, that is, change the habitat of plateau pika and so lead to variation of plateau pika population. Based on ecological characteristics of plateau pika and alpine meadow, a cellular-automata model is established to investigate the influence of grazing on dynamics of plateau pika population. Vegetation shortens with the increase of grazing intensity. When grazing intensity is light, the height of vegetation under summer grazing, continuous grazing, rotational grazing and winter grazing decrease in turn. The ACC (average carrying capacity of plateau pika) is higher on degraded meadow and is lower on undegraded meadow. On undegraded meadow grazing affects the value of ACC, whereas, on degraded meadow grazing has slight effect on it. On undegraded meadow, plateau pika occupies all cells speedly, the amount of damaged cells and the average amount of live holes in occupied cells decrease or hold the line on temporal dimension. On degraded meadow, the dispersal of plateau pika is restrained, the amount of damaged cells and the average amount of live holes in occupied cells increase on temporal dimension.

In this paper, a stage-structured predator–prey model is proposed and analyzed with density-dependent maturation delay. We studied the dynamics of our model analytically and obtained conditions which influence the positivity and boundedness of all populations. Criteria for the existence of a non-trivial equilibrium and conditions for the uniqueness of this equilibrium are given. A linearized analysis on the equilibria, which is algebraically very complicated in the case of non-trivial equilibrium, is carried out. We proved that the system is globally asymptotically stable in the situation when non-trivial equilibrium does not exist. To accomplish our all analytical findings and to investigate the effect of density-dependent maturation delay on the system behavior, we presented a numerical simulation. It is concluded that variations in parameter, which we introduce in the system to observe the effect of density-dependent maturation delay, produces significant quantitative changes in system behavior and also qualitative changes in the behavior of immature predator population.

The stability and almost periodic solution for generalized logistic almost periodic system with infinite and discrete delays is considered. Some sufficient conditions for the boundedness of the system are obtained to guarantee that the system is globally asymptotically stable. We also show that the almost periodic system has a unique globally asymptotically stable strictly positive almost periodic solution by using the almost periodic functional Hull theory and new computational techniques. Furthermore, some recent results are improved, and an open question is answered.

An age-structured epidemiological model for the disease transmission dynamics of TB is studied. We show that the infection-free steady state is locally and globally asymptotically stable if the basic reproductive number is below one, and in this case, the disease always dies out. We prove that the endemic steady state exists when the basic reproductive number is above one. In addition, the endemic steady state is globally asymptotically stable if the basic reproductive number is above one and death rate due to TB is zero.

The paper deals with a problem related to the flow of blood in vessels. It is known that physiological pumps produce flow by alternate contraction and expansion of the vessel. When muscles start squeezing the vessel wall, the valve at the upstream end gets closed and that at the downstream end becomes open. As a result of this, blood is pumped out in the downstream direction. During systole, when the blood in the left ventricle is forced into the aorta, the mitral valve becomes closed, while the atrioventricular valve becomes open. In a situation such as this, the left ventricle takes the form of a vessel with one end closed. In order to study such a situation, a model has been developed here that consists of a fluid-filled long cylindrical pipe with one end closed by a compliant membrane (that prevents the axial motion of the fluid, leaving radial motion completely unrestricted). An approximate solution of the Navier–Stokes equation for the flow of a second-order fluid is presented. The derived analytical expressions are computed numerically. The computed values indicate that if the Reynolds number of the flow is low, the effect of fluid viscoelasticity appears in higher-order terms.

Peristaltic transport of an incompressible couple stress fluid in a two-dimensional uniform channel with wall effects and in the presence of magnetic field has been studied. Using long wavelength approximation with low Reynolds number and dynamic boundary conditions, analytical expressions have been derived for velocity and stream function and the effects of pertinent parameters on these flow variables have been studied. The numerical results show that the time average velocity decreases with viscous damping force. Further, it has been observed that trapping occurs and the size of the trapped bolus decreases with Hartman number.