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\noindent where, $A=(1+\gamma\varphi)\beta\epsilon-\alpha(\beta\delta_1-\delta\chi)$ and $B=\alpha N\left(\beta-\delta\right)+\beta\,\gamma$,
such that $\omega^2=A^2-4\beta B=\left( \left( 1-\gamma \right) \beta-\alpha\, \left( \beta-\delta \right) \right) ^{2}-4\,\alpha\,\beta\, \left( N+\gamma \right)
\left( \beta-\delta \right)$.
\noindent The roots $x_i^*$, $i=1,2$ will be distinct and positive when $A>0$ and $\omega^2>0$ which gives $(1+\gamma)\beta>(\alpha-\delta)$ and $\left( \alpha-\delta-\beta(1-\gamma) \right) ^{2}>4\,\beta\, \left( \alpha\phi-\delta \right) \left( N+\gamma \epsilon\right)$ where $\alpha>\delta$.
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