Table 2 Equilibria and their stability of model (2), where \(r_1=b_1-d_1\), \(r_2=b_2-d_2\), \(\alpha _1=r_1p/k_1-\varepsilon a\), \(\alpha _2=r_2q/k_2+a\), \(S_1^*=\frac{1}{\beta _1}+\frac{\varepsilon aN_2^*}{\beta _1}\) and \(I_1^*=N_1^*-\frac{1}{\beta _1}-\frac{\varepsilon aN_2^*}{\beta _1}\) (see Supplementary analysis of the model with disease).
From: Concurrent dilution and amplification effects in an intraguild predation eco-epidemiological model
Equilibrium | Existence | Stability condition |
|---|---|---|
\(E_0\left( 0,0,0,0\right)\) | Always | Always unstable |
\(E_1\left( k_1,0,0,0\right)\) | Always | \(r_2<\alpha _2k_1\) and \(R_0^{(1,1)}<1\) |
\(E_2\left( 0,0,k_2,0\right)\) | Always | \(r_1<\alpha _1k_2\) and \(R_0^{(1,2)}<1\) |
\(E_3\left( \frac{b_1}{\beta _1},k_1-\frac{b_1}{\beta _1},0,0\right)\) | \(R_{0}^{(1,1)}>1\) | \(r_2<\alpha _2k_1\) and \(R_0^{(1,1)}>1\) |
\(E_4\left( 0,0,\frac{b_2}{\beta _2},k_2-\frac{b_2}{\beta _2}\right)\) | \(R_{0}^{(1,2)}>1\) | \(r_1<\alpha _1k_2\) and \(R_0^{(1,2)}>1\) |
\(E_5\left( N_1^*,0, N_2^*,0\right)\) | \(\alpha _1k_2<r_1\), \(\alpha _2k_1<r_2\) | \(R_{0}^{(2,1)}<1\) and \(R_{0}^{(2,2)}<1\) |
and \(\alpha _1\alpha _2<\left( \frac{r_1}{k_1}\right) \left( \frac{r_2}{k_2}\right)\) | ||
\(E_6\left( S_1^*, I_1^*, N_2^*,0\right)\) | \(R_0^{(2,1)}>1\) | \(R_{0}^{(2,1)}>1\) and \(R_{0}^{(2,2)}<1\) |
\(E_7\left( N_1^*,0,\frac{b_2}{\beta _2},N_2^*-\frac{b_2}{\beta _2}\right)\) | \(R_0^{(2,2)}>1\) | \(R_{0}^{(2,1)}<1\) and \(R_{0}^{(2,2)}>1\) |
\(E^*\left( S_1^*, I_1^*, \frac{b_2}{\beta _2},N_2^*-\frac{b_2}{\beta _2}\right)\) | \(R_0^{(2,1)}>1\) and \(R_0^{(2,2)}>1\) | \(R_{0}^{(2,1)}>1\) and \(R_{0}^{(2,2)}>1\) |
