# K br we obtain br K br Furthermore br K

K

we obtain

K

Furthermore,

K

K

K

K

when αβ ≤ 1. If αβ > 1, we use simple algebraic computations to obtain

Clearly, the sign of D is determined by

D

Since αβ > 1, the preceding arguments for Theorem 3.1 implies that X2∗ is the smaller positive root of (A.4). Thus, 0 < X2∗ < −λ2/(2λ1 ). It follows from (B.3) that D1 > 0. Then the Routh–Hurwitz criteria means that the eigenvalues of J have negative real parts. Therefore the equilibrium point E2 is locally asymptotically stable. By similar discussions to above, one can verify that the determinant of the Jacobian matrix at E1 is positive and the determinant of the Jacobian matrix at E2 is negative. Therefore, we conclude that E1 = (X11, X21 ) is asymptotically stable and E2 = (X12, X22 ) is a saddle. The proof of Theorem 3.2 is thus completed.

Appendix C. Proof of Theorem 3.3

K

r

K

K

r

K

It follows from Dulac’s criterion that there is no closed orbit in (3.1). Therefore, by Poincaré–Bendixson Theorem and the local Atezolizumab of the equilibria imply that E1 is globally stable if (3.4) is satisfied and E2 is globally asymptotically stable under (3.2). The proof is completed.

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