Depopulation of birds has always been an effective method not only to control the transmission of avian influenza in bird populations but also to eliminate influenza viruses. We introduce a Filippov avian-only model with culling of susceptible and/or infected birds. For each susceptible threshold level [Formula: see text], we derive the phase portrait for the dynamical system as we vary the infected threshold level [Formula: see text], focusing on the existence of endemic states; the endemic states are represented by real equilibria, pseudoequilibria and pseudo-attractors. We show generically that all solutions of this model will approach one of the endemic states. Our results suggest that the spread of avian influenza in bird populations is tolerable if the trajectories converge to the equilibrium point that lies in the region below the threshold level [Formula: see text] or if they converge to one of the pseudoequilibria or a pseudo-attractor on the surface of discontinuity. However, we have to cull birds whenever the solution of this model converges to an equilibrium point that lies in the region above the threshold level [Formula: see text] in order to control the outbreak. Hence a good threshold policy is required to combat bird flu successfully and to prevent overkilling birds.
A canonical/lognormal model for human demography is established, specifying the net maternity function and the age distribution for mothers of new-borns using a single macroscopic parameter vector of dimension five. The age distribution of mothers is canonical, while the net maternity function normalizes to a lognormal density. Comparison of an actual population with the model serves to identify anomalies in the population which may be indicative of phase transitions or influences from levels outside the demographic. Tracking the time development of the parameter vector may be used to predict the future state of a population, or to interpolate for data missing from the record. In accordance with classical theoretical considerations of Backman, Prigogine, et al., it emerges that the logarithm of a mother's age is the most fundamental time variable for demographic purposes.
A characteristic of ecosystems is the existence of manifold of independencies which are highly complex. Various mathematical models have made considerable contributions in gaining a better understanding of the predator-prey interactions. The main components of any predator-prey models are, firstly, how the different population classes grow and secondly, how the prey and predator interacts. In this paper, the two populations' growth rates obey the logistic law and the carrying capacity of the predator depends on the available number of prey are considered. Our aim is to clarify the relationship between models and Holling types functional and numerical responses in order to gain insights into predator interferences and to answer an important question how competition is carried out. We consider a predator-prey model and a two-predator one-prey model to explain the idea. The novel approach is explained for the mechanism measurement of predator interference through depending on numerical response. Our approach gives good correspondence between an important real data and computer simulations.