Supporting Tet A. Stabilit Analsis of Sstem In this Appendi, we stud the stabilit of the equilibria of sstem. If we redefine the sstem as, T when -, then there are at most three equilibria: E,, E κ -,, κ κ E,, where E eists if >. To determine the properties of these three equilibria and the topological structures, set,. ecall, we have d dt d dt κ κ., [A] otice that E,, E κ -, are still equilibria of sstem A, and the third equilibrium E becomes κ E,,. We first discuss the stabilit of E and E. Proposition A. The semitrivial equilibrium E κ -, is stable if < and unstable if >.
Proof. The variational matri of sstem A at E takes the form J. The eigenvalues are < λ and λ. Thus, E is stable if < and unstable if >. Proposition A. The positive equilibrium, E is locall asmptoticall stable if >. Proof. The variational matri of sstem A at E takes the form J. We then have
det > J and. < J tr Therefore, E is locall asmptoticall stable., E is a degenerate equilibrium of sstem A. To determine its propert, rescale the age variable b dt da, sstem A becomes
d dt d dt 3 κ κ X Y, φ,,, ψ,, [A] where X, and Y, are homogeneous polnomials in and of degree, φ, and ψ, are polnomials in and of higher degrees. Let r cosθ, r sinθ, θ π /. Define G θ cosθ Y cosθ,sinθ sinθ X cosθ,sinθ. The characteristic equation is as follows: G θ r cosθ sinθ cosθ sinθ. [A3] We can see that either G θ has a finite number of real roots θ k,,..., n or k G θ. B the results in Zhang et al., no orbit of sstem A can tend to, spirall. If G θ, then it is singular. If Gθ is not identicall zero, there are at most 5 directions θ θ i along which an orbit of sstem A ma approach the origin; these directions θ θ i are given b solutions of the characteristic Eq. A3. If the orbit of sstem A tends to the origin as a sequence, { t n } tends to or along a direction θ θ, then the direction is called a characteristic direction. The orbits of i sstem A, which approach the origin along characteristic directions, divide a neighborhood of the origin into a finite number of open regions, called sectors. or an analtic sstem, there are three tpes of sectors: hperbolic, parabolic, and elliptic [see Perko ].
Solving for θ in Eq. A3 for θ π /, we obtain three simple roots: θ, θ π /, θ arctan. 3 Proposition A3. There eist ε > and r > such that there is a unique orbit of sstem A in {, θ : < r < r, θ θ < ε } 3 r that tends to the trivial equilibrium E, along θ θ 3 as t. Proof. Making the Briot-Bouquet transformation, z, ds dt, we transform sstem A into the following sstem: d z ds κ dz z z. ds, [A4] The Briot-Bouquet transformation maps the first, second, third, and fourth quadrants in the, plane into the first, third, second, and fourth quadrants in the, z plane, respectivel. The inverse Briot-Bouquet transformation maps the z ais in the, z plane to the point, in the, plane, and maps the orbits in the left of the z ais in the, z plane to orbits in the left of the ais in the, plane with reversed directions. Thus, we consider onl equilibria of sstem A4 in the z ais. There are two equilibria:, and, /. Linear analsis shows that, is an unstable node, and, / is an unstable saddle. Thus, there is a unique separatri in the interior of the first quadrant of sstem A4, which tends to, / as t.
B the inverse Briot-Bouquet transformation, there eist ε > and r >, such that there is a unique orbit of sstem A in {, θ : < r < r, θ θ < ε }, along θ θ 3 as t. 3 r that tends to Proposition A4. Sstem A does not have limit ccles. Proof. Denote the functions on the right-hand side in sstem A b f, and g,. Choose a Dulac function D, and notice that, >. We have D, f, D, g, <. This implies that sstem A does not have a limit ccle. Combining Propositions A-A4 and using Poincaré-Bendison s Theorem [see Perko ], we have the following theorem about the dnamical behavior of solutions to sstems A and thus sstem. Theorem A5. or the original sstem, the topological structure of the trivial equilibrium E in the interior of the first quadrant consists of a parabolic sector and a hperbolic sector; the semitrivial equilibrium E is stable if < and unstable if >; the positive equilibrium E is a global attractor if >. B. Stead States for Sstem 3 To find the stead states of sstem 3, consider
λ νs η[ Φ i i Φ i Φ ]S, d da i a µ ai a, d da i a µ ai a. [B] rom the first equation in B, we have λ S ν ηφ, Φ [ Φ i Φ i Φ i ]. [B] rom the second and third equations in B, we have i a ep i a ep µ a ˆ dˆ a i ep µ a ˆ dˆ a i ep µ a ˆ dˆ a η Φ i i Φ µ a ˆ dˆ a η i Φ [ ] [ ] Using the definitions of T, and T, we have T i η ai ai da ep η[ i T i ]S T µ a ˆ dˆ a S [B3] and i η ai da ep µ a ˆ dˆ a S η[ i ]S [B4] T rom B, we have Φ λ ν S /η S. B3 and B4 impl that i i η ΦS. We then have the following claim:
Claim B. If i i, then i i λ S. µ ow B4 implies that S η T if i. [B5] Substituting B5 into B3, we have i [ i T i, which implies that T ]/T i i T T T if i. [B6] We now prove the second claim. Claim B. A necessar condition for i > is T >. T Because T i i T λ ν S i T T T, T we have i λ ν ηt T T, T T [B7] and
i λ ν ηt T T T. T T [B8] Claim B3. A necessar condition for i > and i > is λ > ν/ηt. or the eistence of stead states, we have three cases. a If λη/ν mat, T <, then i. B3 thus implies i. The equilibrium is given b S λ / µ, I, I. b If λη/ν mat, T λη/νt >, then i > and i. B3 and B4 give the stead-state values S /ηt, I λ ν ep ηt µ a ˆ dˆ a, I. c If λη/ν mat, T λη/νt >, then i > and i >. B5, B7, and B8 ield the following stead-state values: S / ηt I, I ν λ ηt ν λ ηt T T T T T T T T T ep ep µ ˆ ˆ a da. µ ˆ ˆ a da,. Zhang, Z., Ding, T., Huang, W. & Dong, Z. 99 Qualitative Theor of Differential Equations Transl. Math. Monogr., Am. Math. Soc., Providence, I.
. Perko, L. 996 Differential Equations and Dnamical Sstems Springer, ew York.