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doi: 10.3934/dcdsb.2022017
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The role of vigilance on a discrete-time predator-prey model

1. 

Department of Mathematics and Statistics, American University of Sharjah, P. O. Box 26666, Sharjah, UAE

2. 

Department of Mathematics, Visva-Bharati, Santiniketan 731235, India

3. 

Agricultural and Ecological Research Unit, Indian Statistical Institute, 203, B. T. Road, Kolkata 700108, India

*Corresponding author: Nikhil Pal

Received  April 2021 Revised  September 2021 Early access February 2022

Fund Project: The first author is supported by AUS grant FRG19-S-S141

The change of behaviors of prey in the form of vigilance significantly affects the dynamics of a predator-prey system. In this paper, we consider a discrete-time predator-prey model, where the vigilance of prey acts as a trade-off between the safety and growth rate of the prey. Mathematical properties such as stability, permanence, both flip and Neimark-Sacker bifurcations of the model are investigated. Numerical simulations are carried out to illustrate the analytical findings and to explore the impact of prey vigilance on the dynamics of the system.

Citation: Ziyad AlSharawi, Nikhil Pal, Joydev Chattopadhyay. The role of vigilance on a discrete-time predator-prey model. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022017
References:
[1]

A. S. AcklehM. I. HossainA. Veprauskas and A. Zhang, Persistence and stability analysis of discrete-time predator–prey models: A study of population and evolutionary dynamics, J. Difference Equ. Appl., 25 (2019), 1568-1603.  doi: 10.1080/10236198.2019.1669579.

[2]

Z. AlSharawiS. PalN. Pal and J. Chattopadhyay, A discrete-time model with non-monotonic functional response and strong Allee effect in prey, J. Difference Equ. Appl., 26 (2020), 404-431.  doi: 10.1080/10236198.2020.1739276.

[3]

J. S. BrownJ. W. Laundré and M. Gurung, The ecology of fear: Optimal foraging, game theory, and trophic interactions, Journal of Mammalogy, 80 (1999), 385-399. 

[4]

S. CreelP. Schuette and D. Christianson, Effects of predation risk on group size, vigilance, and foraging behavior in an African ungulate community, Behavioral Ecology, 25 (2014), 773-784.  doi: 10.1093/beheco/aru050.

[5]

J. K. Hale, Dissipation and attractors, In International Conference on Differential Equations, (Berlin, 1999), World Sci. Publ., River Edge, NJ, 1, 2 (2000), 622–637.

[6]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, volume 25 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1988. doi: 10.1090/surv/025.

[7]

J. Hofbauer and J. W-H So, Uniform persistence and repellors for maps, Proc. Amer. Math. Soc., 107 (1989), 1137-1142.  doi: 10.1090/S0002-9939-1989-0984816-4.

[8]

J. HuangS. LiuS. Ruan and D. Xiao, Bifurcations in a discrete predator–prey model with nonmonotonic functional response, J. Math. Anal. Appl., 464 (2018), 201-230.  doi: 10.1016/j.jmaa.2018.03.074.

[9]

T. KimbrellR. D. Holt and P. Lundberg, The influence of vigilance on intraguild predation, J. Theoret. Biol., 249 (2007), 218-234.  doi: 10.1016/j.jtbi.2007.07.031.

[10]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 3$^rd$ edition, Applied Mathematical Sciences, 112. Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-3978-7.

[11]

S. L. Lima and L. M. Dill, Behavioral decisions made under the risk of predation: A review and prospectus, Canadian Journal of Zoology, 68 (1990), 619-640.  doi: 10.1139/z90-092.

[12]

M. A. MaloneA. H. Halloway and J. S. Brown, The ecology of fear and inverted biomass pyramids, Oikos, 129 (2020), 787-798.  doi: 10.1111/oik.06948.

[13]

R. M. May, Biological populations with nonoverlapping generations: Stable points, stable cycles, and chaos, Science, 186 (1974), 645-647.  doi: 10.1126/science.186.4164.645.

[14]

J. D. Murray, Mathematical Biology: I. An Introduction, 3$^rd$ edition, Interdisciplinary Applied Mathematics, 17. Springer-Verlag, New York, 2002.

[15]

S. PalN. PalS. Samanta and J. Chattopadhyay, Effect of hunting cooperation and fear in a predator–prey model, Ecological Complexity, 39 (2019), 100770.  doi: 10.1016/j.ecocom.2019.100770.

[16]

P. Panday, N. Pal, S. Samanta and J. Chattopadhyay, Stability and bifurcation analysis of a three-species food chain model with fear, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 28 (2018), 1850009, 20 pp. doi: 10.1142/S0218127418500098.

[17]

N. C. Pati, S. Garai, M. Hossain, G. C. Layek and N. Pal, Fear induced multistability in a predator-prey model, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 31 (2021), 2150150, 21 pp. doi: 10.1142/S0218127421501509.

[18]

N. C. Pati, G. C. Layek and N. Pal, Bifurcations and organized structures in a predator-prey model with hunting cooperation, Chaos Solitons Fractals, 140 (2020), 110184, 11 pp. doi: 10.1016/j.chaos.2020.110184.

[19]

W. E. Ricker, Stock and recruitment, Journal of the Fisheries Board of Canada, 11 (1954), 559-623.  doi: 10.1139/f54-039.

[20]

J. P. SuraciM. ClinchyL. M. DillD. Roberts and L. Y. Zanette, Fear of large carnivores causes a trophic cascade, Nature Communications, 7 (2016), 10698.  doi: 10.1038/ncomms10698.

[21]

R. Underwood, Vigilance behaviour in grazing African antelopes, Behaviour, 79 (1982), 81-107.  doi: 10.1163/156853982X00193.

[22]

X. WangL. 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.

[23]

L. Y. ZanetteA. F. WhiteM. C. Allen and M. Clinchy, Perceived predation risk reduces the number of offspring songbirds produce per year, Science, 334 (2011), 1398-1401.  doi: 10.1126/science.1210908.

[24]

X. Zhao, Dynamical Systems in Population Biology, 2$^nd$ edition, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer, Cham, 2017. doi: 10.1007/978-3-319-56433-3.

show all references

References:
[1]

A. S. AcklehM. I. HossainA. Veprauskas and A. Zhang, Persistence and stability analysis of discrete-time predator–prey models: A study of population and evolutionary dynamics, J. Difference Equ. Appl., 25 (2019), 1568-1603.  doi: 10.1080/10236198.2019.1669579.

[2]

Z. AlSharawiS. PalN. Pal and J. Chattopadhyay, A discrete-time model with non-monotonic functional response and strong Allee effect in prey, J. Difference Equ. Appl., 26 (2020), 404-431.  doi: 10.1080/10236198.2020.1739276.

[3]

J. S. BrownJ. W. Laundré and M. Gurung, The ecology of fear: Optimal foraging, game theory, and trophic interactions, Journal of Mammalogy, 80 (1999), 385-399. 

[4]

S. CreelP. Schuette and D. Christianson, Effects of predation risk on group size, vigilance, and foraging behavior in an African ungulate community, Behavioral Ecology, 25 (2014), 773-784.  doi: 10.1093/beheco/aru050.

[5]

J. K. Hale, Dissipation and attractors, In International Conference on Differential Equations, (Berlin, 1999), World Sci. Publ., River Edge, NJ, 1, 2 (2000), 622–637.

[6]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, volume 25 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1988. doi: 10.1090/surv/025.

[7]

J. Hofbauer and J. W-H So, Uniform persistence and repellors for maps, Proc. Amer. Math. Soc., 107 (1989), 1137-1142.  doi: 10.1090/S0002-9939-1989-0984816-4.

[8]

J. HuangS. LiuS. Ruan and D. Xiao, Bifurcations in a discrete predator–prey model with nonmonotonic functional response, J. Math. Anal. Appl., 464 (2018), 201-230.  doi: 10.1016/j.jmaa.2018.03.074.

[9]

T. KimbrellR. D. Holt and P. Lundberg, The influence of vigilance on intraguild predation, J. Theoret. Biol., 249 (2007), 218-234.  doi: 10.1016/j.jtbi.2007.07.031.

[10]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 3$^rd$ edition, Applied Mathematical Sciences, 112. Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-3978-7.

[11]

S. L. Lima and L. M. Dill, Behavioral decisions made under the risk of predation: A review and prospectus, Canadian Journal of Zoology, 68 (1990), 619-640.  doi: 10.1139/z90-092.

[12]

M. A. MaloneA. H. Halloway and J. S. Brown, The ecology of fear and inverted biomass pyramids, Oikos, 129 (2020), 787-798.  doi: 10.1111/oik.06948.

[13]

R. M. May, Biological populations with nonoverlapping generations: Stable points, stable cycles, and chaos, Science, 186 (1974), 645-647.  doi: 10.1126/science.186.4164.645.

[14]

J. D. Murray, Mathematical Biology: I. An Introduction, 3$^rd$ edition, Interdisciplinary Applied Mathematics, 17. Springer-Verlag, New York, 2002.

[15]

S. PalN. PalS. Samanta and J. Chattopadhyay, Effect of hunting cooperation and fear in a predator–prey model, Ecological Complexity, 39 (2019), 100770.  doi: 10.1016/j.ecocom.2019.100770.

[16]

P. Panday, N. Pal, S. Samanta and J. Chattopadhyay, Stability and bifurcation analysis of a three-species food chain model with fear, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 28 (2018), 1850009, 20 pp. doi: 10.1142/S0218127418500098.

[17]

N. C. Pati, S. Garai, M. Hossain, G. C. Layek and N. Pal, Fear induced multistability in a predator-prey model, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 31 (2021), 2150150, 21 pp. doi: 10.1142/S0218127421501509.

[18]

N. C. Pati, G. C. Layek and N. Pal, Bifurcations and organized structures in a predator-prey model with hunting cooperation, Chaos Solitons Fractals, 140 (2020), 110184, 11 pp. doi: 10.1016/j.chaos.2020.110184.

[19]

W. E. Ricker, Stock and recruitment, Journal of the Fisheries Board of Canada, 11 (1954), 559-623.  doi: 10.1139/f54-039.

[20]

J. P. SuraciM. ClinchyL. M. DillD. Roberts and L. Y. Zanette, Fear of large carnivores causes a trophic cascade, Nature Communications, 7 (2016), 10698.  doi: 10.1038/ncomms10698.

[21]

R. Underwood, Vigilance behaviour in grazing African antelopes, Behaviour, 79 (1982), 81-107.  doi: 10.1163/156853982X00193.

[22]

X. WangL. 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.

[23]

L. Y. ZanetteA. F. WhiteM. C. Allen and M. Clinchy, Perceived predation risk reduces the number of offspring songbirds produce per year, Science, 334 (2011), 1398-1401.  doi: 10.1126/science.1210908.

[24]

X. Zhao, Dynamical Systems in Population Biology, 2$^nd$ edition, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer, Cham, 2017. doi: 10.1007/978-3-319-56433-3.

Figure 1.  Figure (a) shows the region $ \Omega_1 $ when $ \bar x_2\geq M_1 $. Figure (b) shows the region $ \Omega_\infty $ when $ \bar x_2<x^* $
Figure 2.  This figure shows the geometric illustration for establishing a positively invariant region. The top blue line segment represents $ L(\gamma_{c_1}(t)), $ while the two other blue line segments represent $ L(\gamma_{c}(t)) $ for two random choices of $ c $
Figure 3.  This figure illustrates the stability scenarios of the coexistence equilibrium $ (\bar x_2,\bar y_2) $ based on Lemma 3.1 and the values of $ X: = \frac{a}{1+v}\bar x_2, $ $ Y: = \frac{mp}{k+v}\bar y_2. $ The shaded dashed-region reflects the local stability region of the coexistence equilibrium
Figure 4.  The figure shows a bifurcation diagram (left) and maximum Lyapunov exponent (right) of the system (1.3) with respect to the vigilance parameter $ v $, where other parameters are same as equation (5.1)
Figure 5.  The figure shows the region of survival of the populations in $ v-r $ bi-parameter space. In region $ I $, populations show chaotic and quasiperiodic oscillations. In region $ II $, both prey and predator show stable coexistence. In region $ III $, the prey population shows stable behavior, but predator populations extinct from the system. In region $ IV $, both prey and predator extinct from the system
Figure 6.  The figure shows variation of the densities of prey and predator populations in $ v-r $ bi-parameter space
Figure 7.  (A) Phase portrait of the system showing period-$ 12 $ and period-$ 13 $ attractors with initial conditions $ (0.5, 0.2) $, and $ (0.1, 0.2) $, (B) phase portrait of the system showing period-$ 13 $ and period-$ 14 $ attractors with initial conditions $ (0.1, 0.2) $, and $ (0.7, 0.2) $, (C) basins of attraction for the coexisting period-$ 12 $ and period-$ 13 $ attractors, (D) basins of attraction for the coexisting period-$ 13 $ and period-$ 14 $ attractors
Figure 8.  This figure shows bifurcation diagram (left) and maximum Lyapunov exponent (right) of the system (1.3) with respect to the vigilance parameter $ v $, where other parameters are same as equation (5.2)
Figure 9.  The figure shows the region of survival of the populations in $ v-r $ bi-parameter space. The meanings of the regions are the same as in figure 4
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