Predator-prey interactions under fear effect and multiple foraging strategies

Susmita Halder; Joydeb Bhattacharyya; Samares Pal

doi:10.3934/dcdsb.2021206

Article Contents

2022, Volume 27, Issue 7: 3779-3810. Doi: 10.3934/dcdsb.2021206

This issue Previous Article Well-posedness of the initial-boundary value problem for the fourth-order nonlinear Schrödinger equation Next Article Stabilization for hybrid stochastic differential equations driven by Lévy noise via periodically intermittent control

Predator-prey interactions under fear effect and multiple foraging strategies

1.
Department of Mathematics, University of Kalyani, Nadia, West Bengal-741235, India
2.
Department of Mathematics, Karimpur Pannadevi College, Nadia, West Bengal-741152, India

^* Corresponding author: Joydeb Bhattacharyya
^* Corresponding author: Joydeb Bhattacharyya

Received: January 31, 2021

Revised: April 30, 2021

Published: July 2022

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)

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Abstract

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.

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

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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)

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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^* $

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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

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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) $

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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

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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

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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

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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 $

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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)

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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

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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

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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

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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

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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

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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

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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 $y(0)\geq 0$.

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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

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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_2<r $	$ h_{1}>h_{1}^* $ and $ h_{2}<r $
$ E^* $	$ h_1<\min\{1,h_{1}^*\} $ and $ h_2<r $	$ h_1<\min\{1,h_{1}^*\} $ and $ h_2<r $

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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_2<r $	$ h_{1}>h_{1}^{**} $ and $ h_{2}<r $
$ E^*_i $	$ h_1<\min\{1,h_{1}^{**}\} $ and $ h_2<r $	$ \mbox{Tr}(J^_i)<0 $ and $ \mbox{Det}(J^_i)>0 $

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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_2<r $	$ h_{1}>h_1^{\#} $ and $ h_{2}<r $
$ E_* $	$ h_1<\min\{1,h_1^{\#}\} $ and $ h_2<r $	$ \mbox{Tr}(J_)<0 $ and $ \mbox{Det}(J_)>0 $

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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
Bifurcation parameter	Threshold	Bifurcation	Threshold	Bifurcation	Threshold	Bifurcation
$h_1$ ($h_2 < r$)	$h_1^*=0.8971$	TB	$h_1^{**}=0.1475$ $h_{1sn}^-=0.3685$	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	$h_2^{**}=0.74$ $h_{2sn}=0.64$	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

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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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