Most research works on discrete epidemic models concern the definition of the basic reproductive number, the global stability of the disease free equilibrium, the persistence of diseases, the existence and local stability of endemic equilibria, the existence of flip bifurcation and Hopf bifurcation.
Motivating by the above papers, we study now the boundedness, the persistence, the existence of unbounded solutions, the attractivity, the stability of the positive solutions, and the two-period solutions of the difference equation (1.4).
Theorem 3.1 and Theorem 4.1 indicate that the distance θ between impulse points, the values of the impulse coefficients h i k ( i = 1, 2, …, n, k ∈ Z ), and the number A of the impulse points in each interval of length 1 are harmful for the uniform persistence and existence of a unique globally asymptotically stable positive almost periodic solution for the model.
The main results obtained in this paper are completely new and the method used in this paper provides a possible method to study the uniform persistence and existence of a unique globally asymptotically stable positive almost periodic solution of the models with impulsive perturbations in biological populations.
Probably the most accepted mechanism of persistence is the existence of viral reservoirs in latent and stably infected CD4+ T cells [ 8– 10].
In Section 4, we present the disease persistence and existence of the endemic equilibrium.
The phylogenetic history of these extended haplotypes was examined using a haplotype network [20], since network approaches account for the persistence of ancestral haplotypes, the existence of multiple descendant haplotypes, recombination, and low levels of sequence variation [21].
Network approaches have several advantages over traditional phylogenetic methods such as trees, since they account for the persistence of ancestral haplotypes, the existence of multiple descendant haplotypes, recombination and low levels of sequence variation [21].
Let (sigma _{1}=0.01), (sigma _{2}=0.01), (sigma _{3}=0.01), the conditions of stochastic persistence and the existence of stationary distribution (see Theorem 4(2) and Theorem 5) are satisfied, and the population densities fluctuate around the deterministic steady state ((x_{1}^,y^{2}^,y^)), respectively, as is shown in Fig. 1.
During the last decade, the dynamic behaviors of the mutualism model have been extensively investigated [1 8] and many excellent results were obtained, which concern the persistence, existence of a positive periodic solution, and stability of the system.
