5. Hopf bifurcation and its nature
Set\(f\left(k_{2}\right)=a_{1}\left(k_{2}\right)a_{2}\left(k_{2}\right)-a_{3}\left(k_{2}\right).\)
Theorem 4 . If there exists\(k_{2}=k_{2}^{*}\) such that\(\left(i\right)\ a_{2}\left(k_{2}^{*}\right)>0,\ \left(\text{ii}\right)f\left(k_{2}^{*}\right)=0,\ \left(\text{iii}\right)\ f^{/}\left(k_{2}^{*}\right)>0\)then the positive equilibrium point \(E^{*}\) is unstable if \(k_{2}<k_{2}^{*}\) but is stable for \(k_{2}>k_{2}^{*}\)and a Hopf bifurcation of periodic solution occurs at\(k_{2}=k_{2}^{*}.\)
Proof . Proceeding along the lines in [21], we note that\(f(k_{2})\) is monotonic increasing function in the neighbourhood of\(k_{2}=k_{2}^{*}.\) As\(a_{2}\left(k_{2}\right)>0,\ f\left(k_{2}\right)<0\)for\(\ k_{2}<k_{2}^{*}\ \mathrm{\text{thus}}\ E^{*}\) becomes unstable. Again, it is obvious that \(,\ f\left(k_{2}\right)>0\)for\(\ k_{2}>k_{2}^{*}\) and hence \(E^{*}\) is stable. Therefore, Hopf bifurcation follows from a result in [15].
Similarly, bifurcation phenomenon can be shown by considering\(k_{1}\)as a bifurcation parameter.
5.1. Stability of the limit cycle
Stability of the limit cycle can be derived by calculating the coefficient of curvature of the limit cycle [26]. Detail calculation can be found in [17]. Then the coefficient of curvature of limit cycle of system (2) is
\(\sigma_{1}^{0}\) \(=\frac{1}{16}(\alpha-\beta).\)
Thus we observed that the coefficient of curvature\(\sigma_{1}^{0}<0\ \mathrm{\text{\ if\ }}\alpha<\ \beta\) in that case the limit cycle of system (2) will be stable. From above analysis one can conclude that the intraspecific competition rate between the prey species plays a vital role for determining the nature of the limit cycle emerging through a Hopf bifurcation.
In the following table, we summarise the stability criteria of different equilibria of system (2).
Table 1 . Dynamics of system (2). LAS= Locally asymptotically stable, GAS= Globally asymptotically stable.