Table 4 Local stability analysis of different scenarios at each equilibrium point
Scenarios | Constraints | Points | det(\({\varvec{J}}\)) | tr(\({\varvec{J}}\)) | Results |
|---|---|---|---|---|---|
Scenario 1 | \(\left\{\begin{array}{c}{\theta }_{i}{R}_{i}-{\beta }_{i}{R}_{j}<{\alpha }_{i}{R}_{i}\\ {\theta }_{j}{R}_{j}-{\beta }_{j}{R}_{i}<{\alpha }_{j}{R}_{j}\end{array}\right.\) | \({E}_{1}\) | \(+\) | \(-\) | ESS |
\({E}_{2}\) | \(-\) | Uncertain | Saddle point | ||
\({E}_{3}\) | \(-\) | Uncertain | Saddle point | ||
\({E}_{4}\) | \(+\) | \(+\) | Unstable | ||
\({E}_{5}\) | \(-\) | 0 | Saddle point | ||
Scenario 2 | \(\left\{\begin{array}{c}{\theta }_{i}{R}_{i}-{\beta }_{i}{R}_{j}>{\alpha }_{i}{R}_{i}\\ {\theta }_{j}{R}_{j}-{\beta }_{j}{R}_{i}>{\alpha }_{j}{R}_{j}\end{array}\right.\) | \({E}_{1}\) | \(+\) | \(-\) | ESS |
\({E}_{2}\) | \(+\) | \(+\) | Unstable | ||
\({E}_{3}\) | \(+\) | \(+\) | Unstable | ||
\({E}_{4}\) | \(+\) | \(-\) | ESS | ||
\({E}_{5}\) | \(-\) | 0 | Saddle point |