**Proof.**

	- (a) When *T*(3) <sup>3</sup> <sup>&</sup>gt; 0, *<sup>D</sup>*(3) <sup>3</sup> > 0 hold, Equation (22) has no positive root, the equilibrium *E*<sup>3</sup> is locally asymptotically stable for any *τ* 0;
	- (b) When *D*(3) <sup>3</sup> < 0 holds, Equation (22) has one positive root *z*2, and *Sign*(Re( *<sup>d</sup><sup>λ</sup> <sup>d</sup><sup>τ</sup>* )−<sup>1</sup> *<sup>τ</sup>*=*τ*(*j*) 2 ) = *Sign*(*h* (*z*2)) > 0, and thus, all the roots of Equation (22) have negative real parts for *<sup>τ</sup>* <sup>∈</sup> [0, *<sup>τ</sup>*(0) <sup>2</sup> ), and Equation (22) has at least one pair of roots with positive real part when *τ* > *τ*(0) <sup>2</sup> ;
	- (c) If *T*(3) <sup>3</sup> <sup>&</sup>lt; 0, *<sup>D</sup>*(3) <sup>3</sup> > 0 hold, *h*(*z*) = 0 has two positive roots *z*<sup>3</sup> and *z*4, and *Sign*(Re( *<sup>d</sup><sup>λ</sup> <sup>d</sup><sup>τ</sup>* )−<sup>1</sup> *<sup>τ</sup>*=*τ*(*j*) 3 ) = *Sign*(*h* (*z*3)) < 0 and *Sign*(Re( *<sup>d</sup><sup>λ</sup> <sup>d</sup><sup>τ</sup>* )−<sup>1</sup> *<sup>τ</sup>*=*τ*(*j*) 4 ) = *Sign*(*h* (*z*4)) > 0, thus, there exists *m* ∈ *N*, such that all the roots of Equation (15) have negative real parts when*<sup>τ</sup>* <sup>∈</sup> [0, *<sup>τ</sup>*(0) <sup>4</sup> ) <sup>∪</sup> '*<sup>m</sup> l*=1 (*τ*(*l*−1) <sup>3</sup> , *<sup>τ</sup>*(*l*) <sup>4</sup> ), and Equation (15) has at least one root with a positive real part when *<sup>τ</sup>* <sup>∈</sup> *<sup>m</sup>*'−<sup>1</sup> *l*=0 (*τ*(*l*) <sup>4</sup> , *<sup>τ</sup>*(*l*) <sup>3</sup> ) <sup>∪</sup> (*τ*(*m*) <sup>4</sup> , +∞), and the conclusion is immediate.
