Table 4 Equilibrium point stability analysis.

O(0, 0)

\(\frac{1}{9}(2\theta R - 3{P_{1}})(2\theta F - 3({P_{2}} - rW + {T_{2}} + W)) > 0\)

\(\frac{2}{3}\theta (F + R) - {P_{1}} - {P_{2}} + (r - 1)W - {T_{2}} < 0\)

A(0, 1)

\(\frac{1}{9}(3{P_{1}} - 2\theta R)(2\theta F + 3{I_{2}} + 3((r - 1)W - {T_{2}} + {T_{\textrm{3}}})) > 0\)

\({I_2} + {P_1} + \frac{2}{3}\theta \left( {F - R} \right) - {T_2} + {T_3} + \left( { - 1 + r} \right) W < 0\)

B(1, 0)

\(- \frac{1}{9}(3{I_{1}} + 2\theta R)(2\theta F - 3({P_{2}} - rW + {T_{2}} + W)) > 0\)

\(\frac{2}{3}\theta (R - F) + {I_{1}} + {P_{2}} - rW + {T_{2}} + W < 0\)

C(1, 1)

\(\frac{1}{9}\left( {3{I_1} + 2\theta R} \right) \left( {3{I_2} + 2\theta F + 3\left( { - {T_2} + {T_3} + \left( { - 1 + r} \right) W} \right) } \right) > 0\)

\(- {I_1} - {I_2} - \frac{2}{3}\theta \left( {F + R} \right) + {T_2} - {T_3} + W - rW < 0\)