6. Numerical simulations
In this section, we carry out numerical simulations of our system (2). First we investigate the effect of fear on the dynamics of system (2). So it is reasonable to study the system (2) without fear effect (i.e.,\(k_{i}=0,\ i=1,\ 2\)). We choose the other parametric value as
\(r_{1}=3.125,\ r_{2}=2,\ \alpha=1.4,\ \beta=1,\ \varepsilon=1,\ \mu=0.01,\ c_{1}=1,\ c_{2}=1,\ d=1,\ h=0.01.\)(4)
The numerical outputs are depicted in Fig. 1.
Fig. 1a shows the phase diagram of system (2) for\(k_{i}=0,\ i=1,\ 2\). In absence of fear, the system (2) has a unique coexistence equilibrium point \(E^{*}=(0.8211,\ 1.0539,\ 0.9203)\) which is unstable in nature and surrounded by a limit cycle.
Fig. 1b,\(k_{1}=0.1,\ k_{2}=0.01\), the dynamics remains the same as in Fig. 1a and the coexistence equilibrium point\(E^{*}=(1.0530,\ 0.8535,\ 0.7660)\). Here the value of \(x_{2}^{*},\ y^{*}\ \mathrm{\text{of\ }}E^{*}\)decreases while the value of \(x_{1}^{*}\ \mathrm{\text{of\ }}E^{*}\)increases.
Fig. 1c, \(k_{1}=0.1,\ k_{2}=0.08\), system (2) has a unique co-existence equilibrium point \(E^{*}=(0.9973,\ 0.8918,\ 0.6794)\)which is stable in nature. In this case, the increase amount of fear on second prey stabilize the system and reduces the predator density.
Fig. 1d, \(k_{1}=1,\ k_{2}=0.08\), system (2) has a unique coexistence equilibrium point \(E^{*}=(0.9973,\ 0.8918,\ 0.6794)\)which is unstable due to the increase amount of fear on first prey species.
It is to be noted that the increase amount of intraspecific competition within the predator
population can induces stability of the system. Taking the value of parameter \(h=0.5\) and all other parameters are same as in Fig. 1d, we observe the stable coexistence equilibrium point\(E^{*}=(1.1077,\ 0.8532,\ 0.2327)\)(see Fig. 1e). Thus the Fig. 1d and 1e indicates that the fear factor and intraspecific competition acts in opposite way concerning the stability of the system.
Lastly, we consider the following set of parameters\(k_{1}=0.6,\ \ k_{2}=0.01,\)\(r_{1}=12,\ r_{2}=2,\ \alpha=5,\ \beta=1,\ \varepsilon=1,\ \mu=0.01,\ c_{1}=1,\ c_{2}=1,\ d=1,\ h=0.001.\)In this case a chaotic type attractor arises enclosing the coexistence equilibrium point \(E^{*}=(4.6856,\ 0.5102,\ 0.5397)\)(see Fig. 2).