Sentence examples for condition of permanence from inspiring English sources

In the Appendix, the local and global stability of equilibria for system (2) are discussed, the condition of permanence is also derived, we can compare these results with stochastic system (5), it shows that the environmental random perturbation plays an important role, it can not be neglected.

In [4, 5, 10, 11], the comparison method was used, and some sufficient conditions of permanence of biological systems were established.

However, in the cooperative case, he suggested that the conditions of permanence may fail based on the results obtained in [6].

In contrast to conditions of permanence for the deterministic system in Additional file 1, it shows that environmental stochastic perturbation can reduce the size of population to a certain extent.

We obtain the conditions of permanence, global attractivity and uniqueness of positive almost periodic solutions of the system by using the Ascoli theorem, Lebesgue dominated convergence theorem, Lyapunov functions and comparison theorem.

Furthermore, sufficient conditions of permanence for deterministic system (2) are derived in Additional file 1, which can give us a contrast between stochastic system (5) and its corresponding deterministic system (2).

By means of the comparison theorem, Ascoli theorem and Lebesgue dominated convergence theorem, we establish the sufficient conditions of permanence and investigate the existence of a unique almost periodic solution.

By means of the M-matrix analysis and Lyapunov functions, we study stochastically ultimate boundedness and stochastic permanence, and the sufficient condition of stochastic permanence is given in Section 3.

We also obtain the condition of the permanence of system (2.2).

The condition for permanence of system (1.4) is (rtheta-lnxi>0), where xi=frac{alpha+beta M}{alpha-gamma}=frac{alpha a+beta r}{a alpha-gamma)}>0.

The condition for permanence of system (1.4) is (rtheta-lnxi>0), that is, begin{gathered} rtheta>lnfrac{alpha a+beta r}{a alpha-gamma)}, e^{r}{a alpha-gammaha a+beta r}{a(alpha-gamma)}, a(alpha-gamma)>frac{alpha a+be^{rtheta}>frac{alphalpha+betaa>fr}{a alpha-gamma r}{a alpha-gammaend{gathered} then we ha alpha-gammapha alpha-gammaa+beta r}{ae^{rtheta}}.