Theory and Modern Applications
Table 1 Global dynamics of system (1) with (2) summarized from Sects. 3, 4, 5 and 6
From: Modeling plant virus propagation with Filippov control
Case | Condition | Equilibrium | Main result |
|---|---|---|---|
\(Y_{c}< Y_{3}^{*}< Y_{2}^{*}< Y_{1}^{*}\) | \(I_{c}< g_{4}< g_{5}\) | \(E_{1}^{V}\), \(E_{2}^{V}\), \(E_{3}^{R}\) | (I) |
\(Y_{c}< Y_{3}^{*}< Y_{2}^{*}< Y_{1}^{*}\) | \(g_{4}< I_{c}< g_{5}\), \(I_{c}<\min \{I_{3}^{*}, g_{5}\}\) | \(E_{1}^{V}\), \(E_{2}^{V}\), \(E_{3}^{R}\) | (I) |
\(Y_{c}< Y_{3}^{*}< Y_{2}^{*}< Y_{1}^{*}\) | \(g_{4}< I_{c}< g_{5} \), \(I_{3}^{*}< I_{c}< g_{5}\) | \(E_{1}^{V}\), \(E_{2}^{V}\), \(E_{3}^{V}\), \(E_{p2}\) | (II) |
\(Y_{c}< Y_{3}^{*}< Y_{2}^{*}< Y_{1}^{*}\) | \(g_{4}< g_{5}< I_{c}\), \(I_{c}<\max \{I_{3}^{*}, g_{5}\}\) | \(E_{1}^{V}\), \(E_{2}^{V}\), \(E_{2}^{R}\) | (I) |
\(Y_{c}< Y_{3}^{*}< Y_{2}^{*}< Y_{1}^{*}\) | \(g_{4}< g_{5}< I_{c}\), \(\max \{I_{3}^{*}, g_{5}\}< I_{c}< I_{1}^{*}\) | \(E_{1}^{V}\), \(E_{2}^{V}\), \(E_{3}^{V} E_{p2}\) | (II) |
\(Y_{c}< Y_{3}^{*}< Y_{2}^{*}< Y_{1}^{*}\) | \(g_{4}< g_{5}< I_{c}\), \(I_{c}>I_{1}^{*}\) | \(E_{1}^{R}\), \(E_{2}^{V}\), \(E_{3}^{V}\) | (III) |
\(Y_{3}^{*}< Y_{c}< Y_{2}^{*}< Y_{1}^{*}\) | \(I_{c}< g_{4}< g_{5}\) | \(E_{1}^{V}\), \(E_{2}^{V}\), \(E_{3}^{V}\), \(E_{p3}\) | (I) |
\(Y_{3}^{*}< Y_{c}< Y_{2}^{*}< Y_{1}^{*}\) | \(g_{4}< I_{c}< g_{5}\), \(I_{c}< g_{3}\) | \(E_{1}^{V}\), \(E_{2}^{V}\), \(E_{3}^{V}\), \(E_{c}\) | (II) |
\(Y_{3}^{*}< Y_{c}< Y_{2}^{*}< Y_{1}^{*}\) | \(g_{4}< I_{c}< g_{4}\), \(I_{c}>g_{3}\) | \(E_{1}^{V}\), \(E_{2}^{V}\), \(E_{3}^{V}\), \(E_{p2}\) | (II) |
\(Y_{3}^{*}< Y_{c}< Y_{2}^{*}< Y_{1}^{*}\) | \(g_{4}< g_{5}< I_{c}\), \(I_{c}< I_{1}^{*}\) | \(E_{1}^{V}\), \(E_{2}^{V}\), \(E_{3}^{V}\), \(E_{p2}\) | (II) |
\(Y_{3}^{*}< Y_{c}< Y_{2}^{*}< Y_{1}^{*}\) | \(g_{4}< g_{5}< I_{c}\), \(I_{c}>I_{1}^{*}\) | \(E_{1}^{R}\), \(E_{2}^{V}\), \(E_{3}^{V}\) | (III) |
\(Y_{3}^{*}< Y_{2}^{*}< Y_{c}< Y_{1}^{*}\) | \(I_{c}< g_{4}< g_{5}\), \(I_{c}< I_{2}^{*}\) | \(E_{1}^{V}\), \(E_{2}^{R}\), \(E_{3}^{V}\) | (I) |
\(Y_{3}^{*}< Y_{2}^{*}< Y_{c}< Y_{1}^{*}\) | \(I_{c}< g_{4}< g_{5}\), \(I_{c}< I_{2}^{*}\) | \(E_{1}^{V}\), \(E_{2}^{V}\), \(E_{3}^{V}\), \(E_{p1}\) | (II) |
\(Y_{3}^{*}< Y_{2}^{*}< Y_{c}< Y_{1}^{*}\) | \(g_{4}< I_{c}< g_{5}\), \(I_{c}< g_{1}\) | \(E_{1}^{V}\), \(E_{2}^{V}\), \(E_{3}^{V}\), \(E_{p1}\) | (II) |
\(Y_{3}^{*}< Y_{2}^{*}< Y_{c}< Y_{1}^{*}\) | \(g_{4}< I_{c}< g_{5}\), \(g_{1}< I_{c}< g_{3}\) | \(E_{1}^{V}\), \(E_{2}^{V}\), \(E_{3}^{V}\), \(E_{c}\) | (II) |
\(Y_{3}^{*}< Y_{2}^{*}< Y_{c}< Y_{1}^{*}\) | \(g_{4}< I_{c}< g_{5}\), \(I_{c}>g_{3}\) | \(E_{1}^{V}\), \(E_{2}^{V}\), \(E_{3}^{V}\), \(E_{p2}\) | (II) |
\(Y_{3}^{*}< Y_{2}^{*}< Y_{c}< Y_{1}^{*}\) | \(g_{4}< g_{5}< I_{c}\), \(I_{c}< I_{1}^{*}\) | \(E_{1}^{V}\), \(E_{2}^{V}\), \(E_{3}^{V}\), \(E_{p2}\) | (II) |
\(Y_{3}^{*}< Y_{2}^{*}< Y_{c}< Y_{1}^{*}\) | \(g_{4}< g_{5}< I_{c}\), \(I_{c}>I_{1}^{*}\) | \(E_{1}^{R}\), \(E_{2}^{V}\), \(E_{3}^{V}\) | (III) |
\(Y_{3}^{*}< Y_{2}^{*}< Y_{1}^{*}< Y_{c}\) | \(I_{c}< g_{4}< g_{5}\), \(I_{c}< I_{2}^{*}\) | \(E_{1}^{V}\), \(E_{2}^{R}\), \(E_{3}^{V}\) | (I) |
\(Y_{3}^{*}< Y_{2}^{*}< Y_{1}^{*}< Y_{c}\) | \(I_{c}< g_{4}< g_{5}\), \(I_{2}^{*}< I_{c}<\min \{I_{1}^{*}, g_{4}\}\) | \(E_{1}^{V}\), \(E_{2}^{V}\), \(E_{3}^{V}\), \(E_{p1}\) | (II) |
\(Y_{3}^{*}< Y_{2}^{*}< Y_{1}^{*}< Y_{c}\) | \(I_{c}< g_{4}< g_{5}\), \(I_{1}^{*}< I_{c}< g_{4}\) | \(E_{1}^{R}\), \(E_{2}^{V}\), \(E_{3}^{V}\) | (III) |
\(Y_{3}^{*}< Y_{2}^{*}< Y_{1}^{*}< Y_{c}\) | \(g_{4}< I_{c}< g_{5}\), \(g_{4}< I_{c}<\min \{I_{1}^{*}, g_{5}\}\) | \(E_{1}^{V}\), \(E_{2}^{V}\), \(E_{3}^{V}\), \(E_{p1}\) | (II) |
\(Y_{3}^{*}< Y_{2}^{*}< Y_{1}^{*}< Y_{c}\) | \(g_{4}< I_{c}< g_{5}\), \(\max \{I_{1}^{*}, g_{4}\}< I_{c}< g_{5}\) | \(E_{1}^{R}\), \(E_{2}^{V}\), \(E_{3}^{V}\) | (III) |
\(Y_{3}^{*}< Y_{2}^{*}< Y_{1}^{*}< Y_{c}\) | \(g_{4}< g_{5}< I_{c}\) | \(E_{1}^{R}\), \(E_{2}^{V}\), \(E_{3}^{V}\) | (III) |