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# Predator-prey interactions under fear effect and multiple foraging strategies

• * Corresponding author: Joydeb Bhattacharyya

SH is supported by CSIR, Govt. of India grant (09/106(0161)/2017-EMR-I). JB is supported by SERB, Govt. of India grant (TAR/2018/000283). SP is supported by WBSCST, Govt. of India grant (ST/P/S & T/16G-22/2018)

• We propose and analyze the effects of a generalist predator-driven fear effect on a prey population by considering a modified Leslie-Gower predator-prey model. We assume that the prey population suffers from reduced fecundity due to the fear of predators. We investigate the predator-prey dynamics by incorporating linear, Holling type Ⅱ and Holling type Ⅲ foraging strategies of the generalist predator. As a control strategy, we have considered density-dependent harvesting of the organisms in the system. We show that the systems with linear and Holling type Ⅲ foraging exhibit transcritical bifurcation, whereas the system with Holling type Ⅱ foraging has a much more complex dynamics with transcritical, saddle-node, and Hopf bifurcations. It is observed that the prey population in the system with Holling type Ⅲ foraging of the predator gets severely affected by the predation-driven fear effect in comparison with the same with linear and Holling type Ⅱ foraging rates of the predator. Our model simulation results show that an increase in the harvesting rate of the predator is a viable strategy in recovering the prey population.

Mathematics Subject Classification: Primary: 92B05, 92D40; Secondary: 92D25.

 Citation:

• Figure 1.  Mutual position of prey-nullclines (red) and predator-nullclines (blue) of the system (3) due to the changes in $h_1$ and $h_2$, other parameters are taken from Table 2. The system is LAS at $(a)$ $E^*$ ($E_0$, $E_1$ and $E_2$ are unstable), $(b)$ $E_2$ ($E_0$ and $E_1$ are unstable; $E^*$ does not exist), $(c)$ $E_1$ ($E_0$ is unstable; $E^*$ and $E_2$ do not exist) and $(d)$ $E_0$ ($E^*$, $E_1$ and $E_2$ do not exist)

Figure 2.  One-parameter bifurcation plots of the system $(4)$ due to the changes in $(a)$ $h_1$ where $h_2<r$, $(b)$ $h_1$ where $h_2>r$, $(c)$ $h_2$ where $h_1<h_1^*$, and $(d)$ $h_2$ where $h_1>h_1^*$

Figure 3.  Two-parameter bifurcation plots with the bifurcation parameters $(a)$ $h_1$ and $h_2$, $(b)$ $h_1$ and $\beta$, $(c)$ $h_2$ and $\beta$, $(d)$ $\beta$ and $\eta_1$, $(e)$ $h_1$ and $\eta_1$, $(f)$ $h_2$ and $\eta_1$. All other parameters are taken from Table 2. The coloured bars represent the prey population density

Figure 4.  Mutual position of prey-nullclines (red) and predator-nullclines (blue) of the system (4) due to the changes in $h_1$ and $h_2$, all other parameters are taken from Table 2. $(a)$ The system is LAS at the unique interior equilibrium $E^*_1$ ($E_0$, $E_1$ and $E_2$ are unstable). $(b)$ The system has bistability at $E_2$ and $E^*_1$ ($E_0$, $E_1$ and $E^*_2$ are unstable). The system is LAS at $(c)$ $E_2$ ($E_0$ and $E_1$ are unstable; $E^*_i$ does not exist), $(d)$ $E_2$ ($E_0$ is unstable; $E^*_i$ and $E_1$ do not exist), $(e)$ $E_1$ ($E_0$ is unstable; $E^*_i$ and $E_2$ do not exist) and $(f)$ $E_0$ ($E^*_i$, $E_1$ and $E_2$ do not exist) $(i = 1,2)$

Figure 5.  One-parameter bifurcation plots of the system (4) due to the changes in $h_1$ $(a)$ for $h_2<r$ and $\eta_1 = 0.05$, where a transcritical bifurcation occurs at $h_1^{**} = 0.6585$; $(b)$ for $h_2<r$ and $\eta_1 = 0.125$, a transcritical and a saddle-node bifurcation occur at $h_1^{**} = 0.1475$ and $h_{1cr}^- = 0.3685$ respectively. One-parameter bifurcation plots of the system (4) due to the changes in $h_2$ $(c)$ for $h_1<h_1^*$ and $\eta_1 = 0.05$, where a transcritical bifurcation occurs at $h_2^{**} = 0.3495$; $(d)$ for $h_1<h_1^*$ and $\eta_1 = 0.125$, a transcritical and a saddle-node bifurcation occur at $h_2^{**} = 0.74$ and $h_{2sn} = 0.64$ respectively

Figure 6.  Two-parameter bifurcation plots with $h_1$ and $h_2$ as bifurcation parameters where $(a)$ $\eta_1 = 0.05$ and $(c)$ $\eta_1 = 0.25$, where $f_{SN} = 0$ is a saddle-node bifurcation curve, $h_i = h_i^{**}$ $(i = 1,2)$ and $h_2 = r$ are transcritical bifurcation curves

Figure 7.  Two-parameter bifurcation plots with the bifurcation parameters $(a)$ $h_1$ and $\beta$, $(b)$ $h_2$ and $\beta$, $(c)$ $h_1$, and $\eta_1$, $(d)$ $h_2$ and $\eta_1$; other parameter values are taken from Table 2

Figure 8.  $(a)$ For $\alpha = 0.4993$, the phase space shows the existence of stable (in green) and unstable (in red) manifolds of the system (4). The unstable limit cycle around $E^*_1$ is represented in blue. The curves representing the changes in $(b)$ ${\rm{Tr}}(J^*_1)$, ${\rm{Det}}(J^*_1)$ and $(c)$ $\frac{d}{d\alpha}{\rm{Tr}}(J^*_1)$ due to the changes in $\alpha$ verify the occurrence of a Hopf bifurcation of the system (4) at $\alpha = 0.4993$

Figure 9.  Mutual position of prey-nullclines (red) and predator-nullclines (blue) of the system (5) due to the changes in $h_1$ and $h_2$, other parameters are taken from Table $1$. The system is LAS at $(a)$ $E_*$ ($E_0$, $E_1$ and $E_2$ are unstable), $(b)$ $E_2$ ($E_0$ and $E_1$ are unstable; $E_*$ does not exist), $(c)$ $E_1$ ($E_0$ is unstable; $E_*$ and $E_2$ do not exist) and $(d)$ $E_0$ ($E_*$, $E_1$ and $E_2$ do not exist)

Figure 10.  One-parameter bifurcation plots of the system (5) due to the changes in $(a)$ $h_1$ where $h_2<r$, $(b)$ $h_2$ where $h_1<h_1^{\#}$. $(c)$ A two-parameter bifurcation plot with $h_1$ and $h_2$ as bifurcation parameters, where $h_1 = 1$, $h_1 = h_1^{\#}$ and $h_2 = r$ are transcritical bifurcation curves. All other parameters are taken from Table 2

Figure 11.  Two-parameter bifurcation plots of the system (5) with the bifurcation parameters $(a)$ $h_1$ and $\beta$, $(b)$ $h_2$ and $\beta$, $(c)$ $h_1$ and $\eta_1$, $(d)$ $h_2$ and $\eta_1$, where $h_1 = h_1^{\#}$, $h_2 = h_2^{\#}$, and $h_2 = r$ are transcritical bifurcation curves

Figure 12.  Local sensitivity of the prey response for $(a)$ linear, $(b)$ Holling type Ⅱ, and $(c)$ Holling type Ⅲ foraging of the predator. For comparison, model simulations before parameter manipulations, are shown in black line. The prey density from simulations where particular parameter values were increased by $10\%$ are shown in red lines, as are the prey density from simulations where particular parameter values were decreased by $10\%$ in blue lines

Figure 13.  One-parameter bifurcation plots due to the changes in $\beta$, where $h_1<1$ and $h_2<r$ for $(a)$ linear, $(b)$ Holling type Ⅱ, and $(c)$ Holling type Ⅲ foraging rates

Figure 14.  One-parameter bifurcation plots due to the changes in $\eta_1$, where $h_1<1$ and $h_2<r$ for $(a)$ linear, $(b)$ Holling type Ⅱ, and $(c)$ Holling type Ⅲ foraging rates

Figure 15.  Local sensitivity of the predator response for $(a)$ linear, $(b)$ Holling type Ⅱ, and $(c)$ Holling type Ⅲ foraging of the predator. For comparison, model simulations before parameter manipulations, are shown in black line. The predator density from simulations where particular parameter values were increased by $10\%$ are shown in red lines, as are the predator density from simulations where particular parameter values were decreased by $10\%$ in blue lines

Table 1.  Non-dimensionalized system

 $\delta$ $\delta=1$ (Linear) $\delta=2$ (Holling type-Ⅱ) $\delta=3$ (Holling type-Ⅲ) Transformation $X=Kx$, $Y= \frac{R_{1}}{A_{1}}y$, $T= \frac{1}{R_{1}}t$, $r=\frac{R_{2}}{R_{1}}$, $\beta=\frac{BR_{1}}{A_{1}}$, $h_{1}=\frac{H_{1}}{R_{1}}$, $h_{2}=\frac{H_{2}}{R_{1}}$, $\alpha=\frac{A_{2}}{A_{1}K}$, $\eta_{1}=\frac{\eta}{K}$ $X=Kx$, $Y= \frac{R_{1}K}{A_{1}}y$, $T= \frac{1}{R_{1}}t$, $r=\frac{R_{2}}{R_{1}}$, $\beta=\frac{BR_{1}K}{A_{1}}$, $h_{1}=\frac{H_{1}}{R_{1}}$, $h_{2}=\frac{H_{2}}{R_{1}}$, $\alpha=\frac{A_{2}}{A_{1}}$, $\eta_{1}=\frac{\eta}{K}$, $b=\frac{B_{1}}{K}.$ $X=Kx$, $Y= \frac{R_{1}K}{A_{1}}y$, $T= \frac{1}{R_{1}}t$, $r=\frac{R_{2}}{R_{1}}$, $\beta=\frac{BR_{1}K}{A_{1}}$, $h_{1}=\frac{H_{1}}{R_{1}}$, $h_{2}=\frac{H_{2}}{R_{1}}$, $\alpha=\frac{A_{2}R_{1}}{A_{1}}$, $\eta_{1}=\frac{\eta}{K}$, $b_{1}=\frac{B_{1}}{K^2}.$ Non-dimensional system $\begin{array}{ll}\frac{dx}{dt}=\frac{x(1-x)}{1+\beta y}- xyc_\delta(x)-h_{1}x\equiv f^\delta_1 \; \;\;\;\;\;\;\;\;\;(2)\end{array}$ $\frac{dy}{dt}=y\left(r-\frac{\alpha y}{x+\eta_{1}}\right)-h_{2}y\equiv f^\delta_2,$ where $c_{\delta}(x)=\left\{\begin{array}{ll} 1, &{ \textrm{if }}\;\delta=1\\ \frac{1}{b+x}, &{ \textrm{if }}\;\delta=2\\ \frac{x}{b_1+x^2}, &{ \textrm{if }}\;\delta=3, \end{array}\right.$ $x(0)\geq 0$ and $y(0)\geq 0$.

Table 2.  Tables of parameter values

 (a) Original parameters Parameter Description Value $R_1$ Intrinsic growth rate of prey 0.03 $R_2$ Intrinsic growth rate of predator 0.03 $B$ The level of fear 4 $A_1$ Consumption rate of predator 0.5 $A_2$ Intraspecific competition of predator 0.5 $K$ Carrying capacity of prey 2 $\eta$ Alternative prey density 0.25 $B_1$ Half saturation coefficient 0.1 $H_1$ Harvesting rate of prey 0.01 $H_2$ Harvesting rate of predator 0.02 (b) Non-dimensional parameters Parameter Value $r$ 1 $\alpha$ 0.03 $\beta$ 0.48 $\eta_1$ 0.125 $b$ 0.025 $h_1$ 0.333 $h_2$ 0.667

Table 3.  Existence and local stability of equilibria of system (3)

 Equilibria Sufficient condition for existence Local asymptotic stability $E_0$ Always $h_1>1$ and $h_2>r$ $E_1$ $h_1<1$ $h_{1}<1$ and $h_{2}>r$ $E_2$ $h_2h_{1}^*$ and $h_{2} Table 4. Existence and local stability of equilibria of system (4)  Equilibria Sufficient condition for existence Local asymptotic stability$ E_0 $Always$ h_1>1 $and$ h_2>r  E_1  h_1<1  h_{1}<1 $and$ h_{2}>r  E_2  h_2h_{1}^{**} $and$ h_{2}0 $Table 5. Existence and local stability of equilibria of system (5)  Equilibria Sufficient condition for existence Local asymptotic stability$ E_0 $Always$ h_1>1 $and$ h_2>r  E_1  h_1<1  h_{1}<1 $and$ h_{2}>r  E_2  h_2h_1^{\#} $and$ h_{2}0 $Table 6. Comparison of the critical threshold values for transcritical bifurcation (TB) and saddle-node bifurcation (SNB) of the three systems  Bifurcation parameter Linear Holling type Ⅱ Holling type Ⅲ Threshold Bifurcation Threshold Bifurcation Threshold Bifurcation$h_1$($h_2 < r$)$h_1^*=0.8971$TB$h_1^{**}=0.1475h_{1sn}^-=0.3685$TB SNB$h_1^{\#}=0.79$TB$h_1$($h_2>r$)$h_1^*=1$TB$h_1^{**}=1$TB$h_1^{\#}=1$TB$h_2$($h_1 < 1$)$h_2^*=1$TB$h_2^{**}=0.74h_{2sn}=0.64$TB SNB$h_2^{\#}=1$TB$\beta$($h_1 < 1$&$h_2 < r$)$\beta^*=16.8$TB$\beta^{**}=16\beta_{sn}=18.51$TB SNB$\beta^{\#}=1.5$TB$\eta_1$($h_1 < 1$&$h_2 < r$)$\eta_1^*=0.825$TB$\eta_1^{**}=0.2\eta_{1sn}=0.441$TB SNB$\eta_1^{\#}=0.3721$TB Table 7. Bifurcation parameters with different foraging types and corresponding basins of attraction at$ E^* $ Parameters Largest basin of recovery Smallest basin of recovery$ h_1 $&$ h_2 $Linear Holling type Ⅱ$ h_1 $&$ \beta $Linear Holling type Ⅲ$ h_2 $&$ \beta $Holling type Ⅲ Holling type Ⅱ$ h_1 $&$ \eta_1 $Linear Holling type Ⅱ$ h_2 $&$ \eta_1 \$ Linear Holling type Ⅱ
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Tables(7)