Lemma 5 ��All solutions (x1(t), x2(t), x3(t)) of (1) www.selleckchem.com/products/ABT-888.html with initial value (x10, x20, x30) +3 are bounded.Proof ��Let (x1(t), x2(t), x3(t)) be a solution of (1) with a positive initial value (x10, x20, x30) and t��0.(8)Then, if t �� nT, t �� (n + l ??letV(t)=a31a13x(t)+a32a23x2(t)+x3(t), 1)T, and t > 0, we obtain t?t��nT,?+x2(a20a32a23+a30?a32a22a23x2)?a33x32,t��(n+l?1)T,??a30x3?a33x32.(9)ThendVdt+a30V(t)=x1(a10a31a13+a30?a31a11a13x1)?thatdVdt=a31a13x1(a10?a11x1)+a32a23x2(a20?a22x2)>0.(10)As the right-hand side of (10) is bounded from above denoted by D, it follows t��(n+l?1)T,??t��nT,??t?thatdVdt+a30V(t)��D,>0(11)together withV((n+l?1)T+)��(1?��)V((n+l?1)T),V(nT+)��(1?��)V(nT).

(12)By Lemma 3, it follows thatV(t)��V(0+)(��0<(n+l?1)T0(14)and since the limit of the right-hand side of (14) for t �� �� isV(t)��Da30<��,(15)it easily follows that V(t) is bounded in its domain. Consequently, (x1(t), x2(t), x3(t)) are bounded by a constant ��D/a30�� for sufficiently lager t.3. Stability of the Giant Panda-Free Periodic SolutionsFirst, we will give the basic properties of the following differential equations considering the absence of the giant panda.When the giant panda x3(t) is eradicated, it is easy to see that the equations in (1) decouple, and then we consider the properties of the t=nT,x2(0+)=x20.

(17)Lemma?t��nT,��x2(t)=?��x2(t),?t=(n+l?1)T,x1(0+)=x10,(16)dx2dt=x2(a20?a22x2),?t��(n+l?1)T,��x1(t)=?��x1(t),?subsystems:dx1dt=x1(a10?a11x1), 6 (see [14]) ��Suppose that ln (1 ? ��) + a10T > 0, then the system (16) has a periodic solution x1*(t) with this notation, and the following properties lim?t����|x1(t)?x1?(t)|=0(18)for?��0Tx1?(t)dt=1a11[ln??(1?��)+a10T],(ii)?are satisfied:(i) all solutions x1(t) of (16) starting with strictly positive x10.Similarly, we have the following Lemma 7.Lemma 7 ��Suppose that ln (1 ? ��) + a20T > 0, then the system (17) has a periodic solution x2*(t) with this notation, and the following properties are lim?t����|x2(t)?x2?(t)|=0(19)for?��0Tx1?(t)dt=1a22[ln??(1?��)+a20T],(iv)?satisfied:(iii) all solutions x2(t) of (17) starting with strictly positive x20.It follows from Lemmas 6 and 7 that the system (1) has a giant panda-free periodic solution (x1*(t), x2*(t), 0).

Now, we study the local stability of the giant panda-free periodic solution (x1*(t), x2*(t), 0) by means of the Floquent theory. (We can see Anacetrapib the details from Page 26 to 35 of [13].)Theorem 8 ��Suppose that ln (1 ? ��) + a10T > 0 and ln (1 ? ��) + a20T > 0 ??and?a30T+a31a11(ln?(1?��)+a10T)+a32a22(ln?(1?��)+a20T)<0(20)hold, and then the giant panda-free periodic solution (x1*(t), x2*(t), 0) is locally stable.Proof ��The local stability of the periodic solution (x1*(t), x2*(t), 0) may be determined by considering the behavior of small-amplitude perturbations of the solution. x2(t)=v(t)+x2?,x3(t)=w(t).