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In a region under consideration, let at any instant (t >0), x and y represent the prey and predator population densities, respectively. The rate of change of each model species density at time t is made on the following assumptions:

1.

Prey population grow logistically in the absence of predator with birth rate r, which is affected by the fear ((f_1)) of predator (when predators are around).

2.

There is a reduction in the rate of prey density change due to three types of death, namely, natural death with the rate (d_1), fear related death5 with the level of fear (f_2) and over crowding death with the rate (d_2).

3.

Also, the rate of change of prey density decreases due to predation of predator population following a predator-dependent functional response describing both predatory and prey schooling behaviors10. Response function is expressed in functional form describing as (zeta (x, y)=frac{cxy}{1+chxy}), where c denotes the rate of consumption and h represents handling time of predator for one prey.

4.

Predator population survive in the system by consuming prey population only. They grow with conversion efficiency (c_1) of prey biomass into predator biomass.

5.

Predator population harvested from the system which reduces its rate of density. We consider a nonlinear harvesting term (Michaelis-Menten type) given by, (H(y)=dfrac{qEy}{p_1E+p_2y}). Here, parameters q and E, respectively, represent the catchability rate and harvesting effort. It is easy to observe that (Hrightarrow frac{q}{p_1}y) as (Erightarrow infty) for a fixed value of y. Also, (Hrightarrow frac{q}{p_2}E) as (yrightarrow infty) for a fixed value of E. Therefore, at higher effort levels, (p_1) is proportional to the stock level-catch rate ratio and at higher levels of stock, (p_2) is proportional to the effort level-catch rate ratio.

6.

Lastly, we assume that the predator population experience natural as well as over crowding related death with the rates (d_3) and (d_4), respectively.

Keeping all these above assumptions in mind, we formulate the following prey–predator model:$$begin{aligned} frac{dx}{dt}= & {} frac{rx}{1+f_1y}-(1+f_2y)d_1x-d_2x^2-frac{cxy^2}{1+chxy}nonumber ,\ frac{dy}{dt}= & {} frac{c_1cxy^2}{1+chxy}-d_3y-d_4y^2-frac{qEy}{p_1E+p_2y}. end{aligned}$$
(1)
System (1) is to be analyzed with the initial conditions (x(0),y(0) >0). All the model parameters are assumed to be positive constants and their hypothetical values that we used for numerical calculations are as follows:$$begin{aligned}{} & {} r=3.1, f_1=1, f_2=0.4, d_1=0.1, d_2=0.08, c=0.11, h=0.1, c_1=0.5, d_3=0.1,nonumber \{} & {} d_4=0.06, q=0.65, E=0.5, p_1=0.5, p_2=0.65. end{aligned}$$
(2)
In Table 1, we have provided system’s equilibria, sufficient conditions of their existence and stability. Mathematically, it is difficult to determine the existence of coexistence (interior) equilibrium point(s) by the given nullclines. So, we visualize it numerically (see Fig. 1). It is apparent from the figure that on increasing the value of E, number of coexistence equilibrium points reduces and after a certain range there is no coexistence equilibrium point.Table 1 Sufficient conditions for the existence and stability of different equilibrium points of system (1).Full size tableFigure 1Nullclines for different values of E. Other parameters are same as in (2).Full size image
Transcritical bifurcationFrom Table 1, it is clear that the equilibrium (E_0) is stable if (rr^{TB}=d_1=0.1)) then (E_0) becomes unstable and the equilibrium point (E_1) exists and becomes stable.Figure 2Transcritical bifurcation with respect to r. Rest of the parameters are same as in (2).Full size imageHopf bifurcationOne of the most common dynamics in interacting population dynamics is oscillating behavior, which implies that there is a Hopf bifurcation. By local changes in equilibrium properties, Hopf bifurcation describes when a periodic solution appears or disappears. In this section, we study the Hopf bifurcation through the coexistence equilibrium (E^*) with respect to the model parameter E. Discussion for the existence of Hopf bifurcation is as follows:As it is easy to follow, we verify Hopf bifurcation numerically. We have considered the parameters value same as in (2) except (c=0.1) and E. At (E=E^{[HB]}=0.1196559641), the trace of the Jacobian matrix at (E^*(2.618402886, 2.352228027)) is zero and determinant, (Det(J_{E^*})=0.4474794791 >0). The value of (dfrac{d(Tr(J_{E^*}))}{dE}Big |_{E=E^{[HB]}}=-0.02965188514ne 0). Therefore, the transversality condition for Hopf bifurcation is also satisfied at (E=E^{[HB]}). Thus, these results confirm that the system (1) experiences a Hopf bifurcation2 around (E^*(2.618402886, 2.352228027)).Moreover, we obtain Lyapunov number (L_1=-0.04728284756pi More

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