Table 2 Equilibrium points and stability conditions for the four-actor symbiotic evolution model in DIES.

E₁(0,0,0,0)

All positive

Not a stable point

E₂(N_1, 0,0,0)

All negative

\((1+\alpha _{21})(1+\alpha _{31})(1+\alpha _{41})<0\)

E₃(0, N₂,0,0)

All negative

\((1+\alpha _{12})(1+\alpha _{32})(1+\alpha _{42})<0\)

E₄(0,0, N₃,0)

All negative

\((1+\alpha _{13})(1+\alpha _{23})(1+\alpha _{43})<0\)

E₅(0,0,0, N₄)

All negative

\((1+\alpha _{14})(1+\alpha _{24})(1+\alpha _{34})<0\)

\(E_6=(\frac{{N_1(1 - \alpha _{12})}}{{1 - \alpha _{12}\alpha _{21}}},\frac{{N_2(1 - \alpha _{21})}}{{1 - \alpha _{12}\alpha _{21}}},0,0)\)

There is a positive

Not a stable point

\(E_7=(\frac{{N_1(1 - \alpha _{13})}}{{1 - \alpha 13\alpha 31}},0,\frac{{N_3(1 - \alpha _{31})}}{{1 - \alpha _{13}\alpha _{31}}},0)\)

There is a positive

Not a stable point

\(E_8=(\frac{{N_1(1+\alpha _{14})}}{{1 - \alpha _{14}\alpha _{41}}},0,0,\frac{{N_4(1+\alpha _{41})}}{{1 - \alpha _{14}\alpha _{41}}})\)

There is a positive

Not a stable point

\(E_9=(0,\frac{{N_2(1+\alpha _{23})}}{{1 - \alpha _{23}\alpha _{32}}},\frac{{N_3(1+\alpha _{32})}}{{1 - \alpha _{23}\alpha _{32}}},0)\)

There is a positive

Not a stable point

\(E_{10}=(0,\frac{{N_2(1+\alpha _{24})}}{{1 - \alpha _{24}\alpha _{42}}},0,\frac{{N_4(1+\alpha _{42})}}{{1 - \alpha _{24}\alpha _{42}}})\)

There is a positive

Not a stable point

\(E_{11}=(0,0,\frac{{N_3(1+\alpha _{34})}}{{1 - \alpha _{34}\alpha _{43}}},\frac{{N_4(1+\alpha _{43})}}{{1 - \alpha _{34}\alpha _{43}}})\)

There is a positive

Not a stable point

E₁₂

All negative

\(\begin{gathered}\frac{\alpha_{23}\alpha_{34} + \alpha_{23} + \alpha_{24}\alpha_{43} + \alpha_{24} - \alpha_{34}\alpha_{43} + 1}{\alpha_{23}\alpha_{32} + \alpha_{23}\alpha_{34}\alpha_{42} + \alpha_{24}\alpha_{32}\alpha_{43} + \alpha_{24}\alpha_{42} + \alpha_{34}\alpha_{43} - 1} < 0 \hfill \\\frac{\alpha_{24}\alpha_{32} - \alpha_{24}\alpha_{42} + \alpha_{32} + \alpha_{34}\alpha_{42} + \alpha_{34} + 1}{\alpha_{23}\alpha_{32} + \alpha_{23}\alpha_{34}\alpha_{42} + \alpha_{24}\alpha_{32}\alpha_{43} + \alpha_{24}\alpha_{42} + \alpha_{34}\alpha_{43} - 1} < 0 \hfill \\\frac{\alpha_{23}\alpha_{32} - \alpha_{23}\alpha_{42} - \alpha_{23}\alpha_{43} - \alpha_{42} - \alpha_{43} - 1}{\alpha_{23}\alpha_{32} + \alpha_{23}\alpha_{34}\alpha_{42} + \alpha_{24}\alpha_{32}\alpha_{43} + \alpha_{24}\alpha_{42} + \alpha_{34}\alpha_{43} - 1} > 0 \hfill \\\end{gathered}\)

E₁₃

All negative

\(\begin{gathered} \frac{{{\alpha _{13}}{\alpha _{34}}+{\alpha _{13}}+{\alpha _{14}}{\alpha _{43}}+{\alpha _{14}} - {\alpha _{34}}{\alpha _{43}}+1}}{{{\alpha _{13}}{\alpha _{31}}+{\alpha _{13}}{\alpha _{34}}{\alpha _{41}}+{\alpha _{14}}{\alpha _{31}}{\alpha _{43}}+{\alpha _{14}}{\alpha _{41}}+{\alpha _{34}}{\alpha _{43}} - 1}}<0 \hfill \\ \frac{{{\alpha _{14}}{\alpha _{31}} - {\alpha _{14}}{\alpha _{41}}+{\alpha _{31}}+{\alpha _{34}}{\alpha _{41}}+{\alpha _{34}}+1}}{{{\alpha _{13}}{\alpha _{31}}+{\alpha _{13}}{\alpha _{34}}{\alpha _{41}}+{\alpha _{14}}{\alpha _{31}}{\alpha _{43}}+{\alpha _{14}}{\alpha _{41}}+{\alpha _{34}}{\alpha _{43}} - 1}}<0 \hfill \\ \frac{{{\alpha _{13}}{\alpha _{31}} - {\alpha _{13}}{\alpha _{41}} - {\alpha _{31}}{\alpha _{43}} - {\alpha _{41}} - {\alpha _{43}} - 1}}{{{\alpha _{13}}{\alpha _{31}}+{\alpha _{13}}{\alpha _{34}}{\alpha _{41}}+{\alpha _{14}}{\alpha _{31}}{\alpha _{43}}+{\alpha _{14}}{\alpha _{41}}+{\alpha _{34}}{\alpha _{43}} - 1}}>0 \hfill \\ \end{gathered}\)

E₁₄

All negative

\(\begin{gathered} \frac{{{\alpha _{12}}{\alpha _{24}}+{\alpha _{12}}+{\alpha _{14}}{\alpha _{42}}+{\alpha _{14}} - {\alpha _{24}}{\alpha _{42}}+1}}{{{\alpha _{12}}{\alpha _{21}}+{\alpha _{12}}{\alpha _{24}}{\alpha _{41}}+{\alpha _{14}}{\alpha _{21}}{\alpha _{42}}+{\alpha _{14}}{\alpha _{41}}+{\alpha _{24}}{\alpha _{42}} - 1}}<0 \hfill \\ \frac{{{\alpha _{14}}{\alpha _{21}} - {\alpha _{14}}{\alpha _{41}}+{\alpha _{21}}+{\alpha _{24}}{\alpha _{41}}+{\alpha _{24}}+1}}{{{\alpha _{12}}{\alpha _{21}}+{\alpha _{12}}{\alpha _{24}}{\alpha _{41}}+{\alpha _{14}}{\alpha _{21}}{\alpha _{42}}+{\alpha _{14}}{\alpha _{41}}+{\alpha _{24}}{\alpha _{42}} - 1}}<0 \hfill \\ \frac{{{\alpha _{12}}{\alpha _{21}} - {\alpha _{12}}{\alpha _{41}} - {\alpha _{21}}{\alpha _{42}} - {\alpha _{41}} - {\alpha _{42}} - 1}}{{{\alpha _{12}}{\alpha _{21}}+{\alpha _{12}}{\alpha _{24}}{\alpha _{41}}+{\alpha _{14}}{\alpha _{21}}{\alpha _{42}}+{\alpha _{14}}{\alpha _{41}}+{\alpha _{24}}{\alpha _{42}} - 1}}>0 \hfill \\ \end{gathered}\)

E₁₆

All negative

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