4. Existence of equilibria and Stability analysis
Evidently, system (2) has six non-negative equilibrium points.
- The population free equilibrium point \(E_{0}=(0,0,0)\).
- The second prey and predator free equilibrium point\(E_{1}=(r_{1},0,0)\).
- The first prey and predator free equilibrium point\(E_{2}=(0,r_{2},0)\).
- If\(\alpha<\frac{r_{1}}{r_{2}}<\ \frac{1}{\beta}\ \mathrm{\text{or\ }}\alpha>\frac{r_{1}}{r_{2}}>\ \frac{1}{\beta}\ \)then there exists unique predator free equilibrium point\(E_{12}=({\overset{\overline{}}{x}}_{1},{\overset{\overline{}}{x}}_{2},0)\)where\({\overset{\overline{}}{x}}_{1}=\frac{r_{1}-r_{2}\alpha}{1-\alpha\beta},\ {\overset{\overline{}}{x}}_{2}=\frac{r_{2}-r_{1}\beta}{1-\alpha\beta}\).
- If \(d<r_{1}c_{1}\varepsilon\) then there exists unique second prey
free equilibrium point \(E_{13}=({\hat{x}}_{1},0,\hat{y})\) where\({\hat{x}}_{1}=\frac{h\hat{y}+d}{c_{1}\varepsilon}\) and\(\hat{y}\) is the positive root of the equation
\(\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }k_{1}(h+c_{1}\varepsilon^{2})y^{2}+\left(h+dk_{1}+c_{1}\varepsilon^{2}\right)y+d-r_{1}c_{1}\varepsilon=0\).
If \(d<r_{2}c_{2}\mu\) then there exists unique first prey free
equilibrium point \(E_{23}=(0,{\tilde{x}}_{2},\tilde{y})\) where\({\tilde{x}}_{2}=\frac{h\tilde{y}+d}{c_{2}\mu}\) and\(\tilde{y}\) is the positive root of the equation
\(\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }k_{2}(h+c_{2}\mu^{2})y^{2}+\left(h+dk_{2}+c_{2}\mu^{2}\right)y+d-r_{2}c_{2}\mu=0\).
\(E_{0}\) is always unstable.
\(E_{1}\) is stable if\(r_{2}<\ \beta r_{1}\ \mathrm{\text{and\ }}d>\ c_{1}\varepsilon r_{1}.\)
\(E_{2}\) is stable if\(r_{1}<\ \alpha r_{2}\ \mathrm{\text{and\ }}d>\ c_{2}\mu r_{2}.\)
\(E_{12}\) is stable if\(\alpha\beta<1\ \mathrm{\text{and\ }}d>\ c_{1}\varepsilon{\overset{\overline{}}{x}}_{1}+c_{2}\mu{\overset{\overline{}}{x}}_{2}.\)
\(E_{13}\ \mathrm{\text{is\ stable\ if\ }}\frac{r_{2}}{1+k_{2}\hat{y}}<\ \beta{\hat{x}}_{1}+\mu\hat{y}\)and\(E_{23}\ \mathrm{\text{is\ stable\ if\ }}\frac{r_{1}}{1+k_{1}\tilde{y}}<\ \alpha{\tilde{x}}_{2}+\varepsilon\tilde{y}.\)
To find the existence condition of positive equilibrium point we first
show uniform persistence of system (2) and then application of a result
in [10] ensures the existence.