Table 3 The expression of the determinant and trace at five equilibrium points
Equilibrium points | det(\({\varvec{J}}\)) | tr(\({\varvec{J}}\)) |
|---|---|---|
\({E}_{1}(\mathrm{0,0})\) | \(({\alpha }_{i}{R}_{i})\times ({\alpha }_{j}{R}_{j})\) | \(-({\alpha }_{i}{R}_{i}+{\alpha }_{j}{R}_{j})\) |
\({E}_{2}(\mathrm{1,0})\) | \({\alpha }_{i}{R}_{i}\times ({\theta }_{j}{R}_{j}-{\beta }_{j}{R}_{i}-{\alpha }_{j}{R}_{j})\) | \({\alpha }_{i}{R}_{i}+({\theta }_{j}{R}_{j}-{\beta }_{j}{R}_{i}-{\alpha }_{j}{R}_{j})\) |
\({E}_{3}(\mathrm{0,1})\) | \({\alpha }_{j}{R}_{j}\times ({\theta }_{i}{R}_{i}-{\beta }_{i}{R}_{j}-{\alpha }_{i}{R}_{i})\) | \({\alpha }_{j}{R}_{j}+({\theta }_{i}{R}_{i}-{\beta }_{i}{R}_{j}-{\alpha }_{i}{R}_{i})\) |
\({E}_{4}(\mathrm{1,1})\) | \(\left({\theta }_{i}{R}_{i}-{\beta }_{i}{R}_{j}-{\alpha }_{i}{R}_{i}\right)\times\) \(({\theta }_{j}{R}_{j}-{\beta }_{j}{R}_{i}-{\alpha }_{j}{R}_{j})\) | \(-[({\theta }_{i}{R}_{i}-{\beta }_{i}{R}_{j}-{\alpha }_{i}{R}_{i})+\) \(({\theta }_{j}{R}_{j}-{\beta }_{j}{R}_{i}-{\alpha }_{j}{R}_{j})]\) |
\({E}_{5}({x}^{*},{y}^{*})\) | \(-\left({\alpha }_{i}{R}_{i}\right)\times \left({\alpha }_{j}{R}_{j}\right)\) \(\times (1-{x}^{*})\times (1-{y}^{*})\) | 0 |