Sentence examples for condition for permanence from inspiring English sources

The condition for permanence is also obtained.

The condition for permanence is obtained.

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}}.

In [1], Liao et al. obtained sufficient conditions for permanence of the system (1.4).

We build some new results about the sufficient conditions for permanence, extinction, and balancing survival.

For system (1.2) in the n-dimensional competitive and prey-predator cases, in [3], Hofbauer et al. obtained conditions for permanence of the system (1.2).

Also, ∫ − τ i j 0 V i d θ = ∫ − τ i 0 W i d θ = ∫ − η i 0 U i d θ = 1, i, j = 1, 2, …, N. Very recently, Chen and Xie [7] obtained a set of sufficient conditions for permanence of the system above.

Under the condition ((mathrm{H}_{1})), for any time delay τ, ((mathrm{H}_{3})) is the necessary and sufficient condition for the permanence of (1.2).

In [1], a sufficient and necessary condition for the permanence of system (1) is established by the theorem, which Theorem 5 in this study.

In fact, we show the local stability of the prey-free periodic solution under some conditions and give a sufficient condition for the permanence of the system (1.3) by applying the Floquet theory.