American Institute of Mathematical Sciences

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doi: 10.3934/dcdsb.2021206
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Predator-prey interactions under fear effect and multiple foraging strategies

Susmita Halder 1, , Joydeb Bhattacharyya 2,, and  Samares Pal 1,

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

Received  February 2021 Revised  May 2021 Early access August 2021

Fund Project: 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.

Keywords: Fear effect, foraging, harvesting, transcritical bifurcation, saddle-node bifurcation, Hopf bifurcation.
Mathematics Subject Classification: Primary: 92B05, 92D40; Secondary: 92D25.
Citation: Susmita Halder, Joydeb Bhattacharyya, Samares Pal. Predator-prey interactions under fear effect and multiple foraging strategies. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021206
References:
[1]

M. A. Aziz-Alaoui and M. Daher Okiye, Boundedness and global stability for a predator-prey model with modified leslie-gower and holling-type ii schemes, Appl. Math. Lett., 16 (2003), 1069-1075.  doi: 10.1016/S0893-9659(03)90096-6.  Google Scholar

[2]

G. Barabás and G. Meszéna, When the exception becomes the rule: the disappearance of limiting similarity in the Lotka–Volterra model, J. Theoret. Biol., 258 (2009), 89-94.  doi: 10.1016/j.jtbi.2008.12.033.  Google Scholar

[3]

S. Creel and D. Christianson, Relationships between direct predation and risk effects, Trends in Ecology & Evolution, 23 (2008), 194-201.  doi: 10.1016/j.tree.2007.12.004.  Google Scholar

[4]

W. Cresswell, Predation in bird populations, Journal of Ornithology, 152 (2011), 251-263.  doi: 10.1007/s10336-010-0638-1.  Google Scholar

[5]

Y.-J. Gong and J.-C. Huang, Bogdanov-Takens bifurcation in a Leslie-Gower predator-prey model with prey harvesting, Acta Math. Appl. Sin. Engl. Ser., 30 (2014), 239-244.  doi: 10.1007/s10255-014-0279-x.  Google Scholar

[6]

E. González-Olivares, J. Mena-Lorca, A. Rojas-Palma and J. D. Flores, Dynamical complexities in the Leslie–Gower predator–prey model as consequences of the Allee effect on prey, Appl. Math. Model., 35 (2011), 366-381.  doi: 10.1016/j.apm.2010.07.001.  Google Scholar

[7]

R. P. Gupta, M. Banerjee and P. Chandra, Bifurcation analysis and control of Leslie–Gower predator–prey model with Michaelis–Menten type prey-harvesting, Differ. Equ. Dyn. Syst., 20 (2012), 339-366.  doi: 10.1007/s12591-012-0142-6.  Google Scholar

[8]

R. P. Gupta and P. Chandra, Bifurcation analysis of modified Leslie–Gower predator–prey model with Michaelis–Menten type prey harvesting, J. Math. Anal. Appl., 398 (2013), 278-295.  doi: 10.1016/j.jmaa.2012.08.057.  Google Scholar

[9]

S. Halder, J. Bhattacharyya and S. Pal, Comparative studies on a predator–prey model subjected to fear and Allee effect with type Ⅰ and type Ⅱ foraging, J. Appl. Math. Comput., 62 (2020), 93-118.  doi: 10.1007/s12190-019-01275-w.  Google Scholar

[10]

G. W. Harrison, Global stability of predator-prey interactions, J. Math. Biol., 8 (1979), 159-171.  doi: 10.1007/BF00279719.  Google Scholar

[11]

C. S. Holling, The components of predation as revealed by a study of small-mammal predation of the European pine sawfly1, The Canadian Entomologist, 91 (1959), 293-320.   Google Scholar

[12]

C. S. Holling, Some characteristics of simple types of predation and parasitism, Canadian Entomologist, 91 (1959), 385-398.  doi: 10.4039/Ent91385-7.  Google Scholar

[13]

V. Křivan, On the Gause predator–prey model with a refuge: A fresh look at the history, J. Theoret. Biol., 274 (2011), 67-73.  doi: 10.1016/j.jtbi.2011.01.016.  Google Scholar

[14]

P. H. Leslie and J. C. Gower, The properties of a stochastic model for the predator-prey type of interaction between two species, Biometrika, 47 (1960), 219-234.  doi: 10.1093/biomet/47.3-4.219.  Google Scholar

[15]

M. Liu and K. Wang, Dynamics of a Leslie–Gower Holling-type ii predator–prey system with Lévy jumps, Nonlinear Anal., 85 (2013), 204-213.  doi: 10.1016/j.na.2013.02.018.  Google Scholar

[16]

K. J. MacLeod, C. J. Krebs, R. Boonstra and M. J. Sheriff, Fear and lethality in snowshoe hares: The deadly effects of non-consumptive predation risk, Oikos, 127 (2018), 375-380.  doi: 10.1111/oik.04890.  Google Scholar

[17]

P. Mishra, S. N. Raw and B. Tiwari, Study of a Leslie–Gower predator-prey model with prey defense and mutual interference of predators, Chaos Solitons Fractals, 120 (2019), 1-16.  doi: 10.1016/j.chaos.2019.01.012.  Google Scholar

[18]

A. Oaten and W. W. Murdoch, Functional response and stability in predator-prey systems, The American Naturalist, 109 (1975), 289-298.   Google Scholar

[19]

L. Perko, Differential Equations and Dynamical Systems, vol. 7, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4684-0392-3.  Google Scholar

[20]

E. L. Preisser and D. I. Bolnick, The many faces of fear: Comparing the pathways and impacts of nonconsumptive predator effects on prey populations, PloS One, 3 (2008), e2465. doi: 10.1371/journal.pone.0002465.  Google Scholar

[21]

H. Seno, A discrete prey–predator model preserving the dynamics of a structurally unstable Lotka–Volterra model, J. Difference Equ. Appl., 13 (2007), 1155-1170.  doi: 10.1080/10236190701464996.  Google Scholar

[22]

M. K. Singh, B. S. Bhadauria and B. K. Singh, Bifurcation analysis of modified leslie-gower predator-prey model with double allee effect, Ain Shams Engineering Journal, 9 (2018), 1263-1277.  doi: 10.1016/j.asej.2016.07.007.  Google Scholar

[23]

E. van Leeuwen, V. A. A. Jansen and P. W. Bright, How population dynamics shape the functional response in a one-predator–two-prey system, Ecology, 88 (2007), 1571-1581.  doi: 10.1890/06-1335.  Google Scholar

[24]

J. Wang, Y. Cai, S. Fu and W. Wang, The effect of the fear factor on the dynamics of a predator-prey model incorporating the prey refuge, Chaos, 29 (2019), 083109, 10 pp. doi: 10.1063/1.5111121.  Google Scholar

[25]

X. Wang, L. Zanette and X. Zou, Modelling the fear effect in predator–prey interactions, J. Math. Biol., 73 (2016), 1179-1204.  doi: 10.1007/s00285-016-0989-1.  Google Scholar

[26]

X. Wang and X. Zou, Modeling the fear effect in predator–prey interactions with adaptive avoidance of predators, Bull. Math. Biol., 79 (2017), 1325-1359.  doi: 10.1007/s11538-017-0287-0.  Google Scholar

[27]

Y. Xia and S. Yuan, Survival analysis of a stochastic predator–prey model with prey refuge and fear effect, J. Biol. Dyn., 14 (2020), 871-892.  doi: 10.1080/17513758.2020.1853832.  Google Scholar

[28]

Z. Xiao, Z. Li et al., Stability analysis of a mutual interference predator-prey model with the fear effect, Journal of Applied Science and Engineering, 22 (2019), 205–211. Google Scholar

[29]

Z. Zhang, R. K. Upadhyay and J. Datta, Bifurcation analysis of a modified Leslie–Gower model with Holling type-Ⅳ functional response and nonlinear prey harvesting, Adv. Difference Equ., 2018 (2018), Paper No. 127, 21 pp. doi: 10.1186/s13662-018-1581-3.  Google Scholar

[30]

Z.-Z. Zhang and H.-Z. Yang, Hopf bifurcation in a delayed predator-prey system with modified Leslie-Gower and Holling type Ⅲ schemes, Acta Automat. Sinica, 39 (2013), 610-616.  doi: 10.3724/SP.J.1004.2013.00610.  Google Scholar

[31]

Y. Zhu and K. Wang, Existence and global attractivity of positive periodic solutions for a predator–prey model with modified Leslie–Gower Holling-type Ⅱ schemes, J. Math. Anal. Appl., 384 (2011), 400-408.  doi: 10.1016/j.jmaa.2011.05.081.  Google Scholar

show all references

References:
[1]

M. A. Aziz-Alaoui and M. Daher Okiye, Boundedness and global stability for a predator-prey model with modified leslie-gower and holling-type ii schemes, Appl. Math. Lett., 16 (2003), 1069-1075.  doi: 10.1016/S0893-9659(03)90096-6.  Google Scholar

[2]

G. Barabás and G. Meszéna, When the exception becomes the rule: the disappearance of limiting similarity in the Lotka–Volterra model, J. Theoret. Biol., 258 (2009), 89-94.  doi: 10.1016/j.jtbi.2008.12.033.  Google Scholar

[3]

S. Creel and D. Christianson, Relationships between direct predation and risk effects, Trends in Ecology & Evolution, 23 (2008), 194-201.  doi: 10.1016/j.tree.2007.12.004.  Google Scholar

[4]

W. Cresswell, Predation in bird populations, Journal of Ornithology, 152 (2011), 251-263.  doi: 10.1007/s10336-010-0638-1.  Google Scholar

[5]

Y.-J. Gong and J.-C. Huang, Bogdanov-Takens bifurcation in a Leslie-Gower predator-prey model with prey harvesting, Acta Math. Appl. Sin. Engl. Ser., 30 (2014), 239-244.  doi: 10.1007/s10255-014-0279-x.  Google Scholar

[6]

E. González-Olivares, J. Mena-Lorca, A. Rojas-Palma and J. D. Flores, Dynamical complexities in the Leslie–Gower predator–prey model as consequences of the Allee effect on prey, Appl. Math. Model., 35 (2011), 366-381.  doi: 10.1016/j.apm.2010.07.001.  Google Scholar

[7]

R. P. Gupta, M. Banerjee and P. Chandra, Bifurcation analysis and control of Leslie–Gower predator–prey model with Michaelis–Menten type prey-harvesting, Differ. Equ. Dyn. Syst., 20 (2012), 339-366.  doi: 10.1007/s12591-012-0142-6.  Google Scholar

[8]

R. P. Gupta and P. Chandra, Bifurcation analysis of modified Leslie–Gower predator–prey model with Michaelis–Menten type prey harvesting, J. Math. Anal. Appl., 398 (2013), 278-295.  doi: 10.1016/j.jmaa.2012.08.057.  Google Scholar

[9]

S. Halder, J. Bhattacharyya and S. Pal, Comparative studies on a predator–prey model subjected to fear and Allee effect with type Ⅰ and type Ⅱ foraging, J. Appl. Math. Comput., 62 (2020), 93-118.  doi: 10.1007/s12190-019-01275-w.  Google Scholar

[10]

G. W. Harrison, Global stability of predator-prey interactions, J. Math. Biol., 8 (1979), 159-171.  doi: 10.1007/BF00279719.  Google Scholar

[11]

C. S. Holling, The components of predation as revealed by a study of small-mammal predation of the European pine sawfly1, The Canadian Entomologist, 91 (1959), 293-320.   Google Scholar

[12]

C. S. Holling, Some characteristics of simple types of predation and parasitism, Canadian Entomologist, 91 (1959), 385-398.  doi: 10.4039/Ent91385-7.  Google Scholar

[13]

V. Křivan, On the Gause predator–prey model with a refuge: A fresh look at the history, J. Theoret. Biol., 274 (2011), 67-73.  doi: 10.1016/j.jtbi.2011.01.016.  Google Scholar

[14]

P. H. Leslie and J. C. Gower, The properties of a stochastic model for the predator-prey type of interaction between two species, Biometrika, 47 (1960), 219-234.  doi: 10.1093/biomet/47.3-4.219.  Google Scholar

[15]

M. Liu and K. Wang, Dynamics of a Leslie–Gower Holling-type ii predator–prey system with Lévy jumps, Nonlinear Anal., 85 (2013), 204-213.  doi: 10.1016/j.na.2013.02.018.  Google Scholar

[16]

K. J. MacLeod, C. J. Krebs, R. Boonstra and M. J. Sheriff, Fear and lethality in snowshoe hares: The deadly effects of non-consumptive predation risk, Oikos, 127 (2018), 375-380.  doi: 10.1111/oik.04890.  Google Scholar

[17]

P. Mishra, S. N. Raw and B. Tiwari, Study of a Leslie–Gower predator-prey model with prey defense and mutual interference of predators, Chaos Solitons Fractals, 120 (2019), 1-16.  doi: 10.1016/j.chaos.2019.01.012.  Google Scholar

[18]

A. Oaten and W. W. Murdoch, Functional response and stability in predator-prey systems, The American Naturalist, 109 (1975), 289-298.   Google Scholar

[19]

L. Perko, Differential Equations and Dynamical Systems, vol. 7, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4684-0392-3.  Google Scholar

[20]

E. L. Preisser and D. I. Bolnick, The many faces of fear: Comparing the pathways and impacts of nonconsumptive predator effects on prey populations, PloS One, 3 (2008), e2465. doi: 10.1371/journal.pone.0002465.  Google Scholar

[21]

H. Seno, A discrete prey–predator model preserving the dynamics of a structurally unstable Lotka–Volterra model, J. Difference Equ. Appl., 13 (2007), 1155-1170.  doi: 10.1080/10236190701464996.  Google Scholar

[22]

M. K. Singh, B. S. Bhadauria and B. K. Singh, Bifurcation analysis of modified leslie-gower predator-prey model with double allee effect, Ain Shams Engineering Journal, 9 (2018), 1263-1277.  doi: 10.1016/j.asej.2016.07.007.  Google Scholar

[23]

E. van Leeuwen, V. A. A. Jansen and P. W. Bright, How population dynamics shape the functional response in a one-predator–two-prey system, Ecology, 88 (2007), 1571-1581.  doi: 10.1890/06-1335.  Google Scholar

[24]

J. Wang, Y. Cai, S. Fu and W. Wang, The effect of the fear factor on the dynamics of a predator-prey model incorporating the prey refuge, Chaos, 29 (2019), 083109, 10 pp. doi: 10.1063/1.5111121.  Google Scholar

[25]

X. Wang, L. Zanette and X. Zou, Modelling the fear effect in predator–prey interactions, J. Math. Biol., 73 (2016), 1179-1204.  doi: 10.1007/s00285-016-0989-1.  Google Scholar

[26]

X. Wang and X. Zou, Modeling the fear effect in predator–prey interactions with adaptive avoidance of predators, Bull. Math. Biol., 79 (2017), 1325-1359.  doi: 10.1007/s11538-017-0287-0.  Google Scholar

[27]

Y. Xia and S. Yuan, Survival analysis of a stochastic predator–prey model with prey refuge and fear effect, J. Biol. Dyn., 14 (2020), 871-892.  doi: 10.1080/17513758.2020.1853832.  Google Scholar

[28]

Z. Xiao, Z. Li et al., Stability analysis of a mutual interference predator-prey model with the fear effect, Journal of Applied Science and Engineering, 22 (2019), 205–211. Google Scholar

[29]

Z. Zhang, R. K. Upadhyay and J. Datta, Bifurcation analysis of a modified Leslie–Gower model with Holling type-Ⅳ functional response and nonlinear prey harvesting, Adv. Difference Equ., 2018 (2018), Paper No. 127, 21 pp. doi: 10.1186/s13662-018-1581-3.  Google Scholar

[30]

Z.-Z. Zhang and H.-Z. Yang, Hopf bifurcation in a delayed predator-prey system with modified Leslie-Gower and Holling type Ⅲ schemes, Acta Automat. Sinica, 39 (2013), 610-616.  doi: 10.3724/SP.J.1004.2013.00610.  Google Scholar

[31]

Y. Zhu and K. Wang, Existence and global attractivity of positive periodic solutions for a predator–prey model with modified Leslie–Gower Holling-type Ⅱ schemes, J. Math. Anal. Appl., 384 (2011), 400-408.  doi: 10.1016/j.jmaa.2011.05.081.  Google Scholar

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 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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Table 2. The coloured bars represent the prey population density">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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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 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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Table 2">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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Table 2">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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$ \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
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(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_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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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 $
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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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 $
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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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 $
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 $h_1^{**}=0.1475$
$h_{1sn}^-=0.3685$		 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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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 $h_1^{**}=0.1475$
$h_{1sn}^-=0.3685$		 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
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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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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