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).