# American Institute of Mathematical Sciences

• Previous Article
Input-to-state stable synchronization for delayed Lurie systems via sampled-data control
• DCDS-B Home
• This Issue
• Next Article
Stochastic differential equation with piecewise continuous arguments: Markov property, invariant measure and numerical approximation
doi: 10.3934/dcdsb.2022017
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## 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. Ackleh, M. I. Hossain, A. 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. AlSharawi, S. Pal, N. 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. Brown, J. 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. Creel, P. 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. Huang, S. Liu, S. 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. Kimbrell, R. 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. Malone, A. 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. Pal, N. Pal, S. 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. Suraci, M. Clinchy, L. M. Dill, D. 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. 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. [23] L. Y. Zanette, A. F. White, M. 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. Ackleh, M. I. Hossain, A. 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. AlSharawi, S. Pal, N. 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. Brown, J. 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. Creel, P. 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. Huang, S. Liu, S. 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. Kimbrell, R. 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. Malone, A. 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. Pal, N. Pal, S. 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. Suraci, M. Clinchy, L. M. Dill, D. 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. 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. [23] L. Y. Zanette, A. F. White, M. 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 (a) shows the region $\Omega_1$ when $\bar x_2\geq M_1$. Figure (b) shows the region $\Omega_\infty$ when $\bar x_2<x^*$
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 $c$
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
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)
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
The figure shows variation of the densities of prey and predator populations in $v-r$ bi-parameter space
(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
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)
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
 [1] Shanshan Chen, Jianshe Yu. Stability and bifurcation on predator-prey systems with nonlocal prey competition. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 43-62. doi: 10.3934/dcds.2018002 [2] Ming Liu, Dongpo Hu, Fanwei Meng. Stability and bifurcation analysis in a delay-induced predator-prey model with Michaelis-Menten type predator harvesting. Discrete and Continuous Dynamical Systems - S, 2021, 14 (9) : 3197-3222. doi: 10.3934/dcdss.2020259 [3] Lizhi Fei, Xingwu Chen. Bifurcation and control of a predator-prey system with unfixed functional responses. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021292 [4] Qizhen Xiao, Binxiang Dai. Heteroclinic bifurcation for a general predator-prey model with Allee effect and state feedback impulsive control strategy. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1065-1081. doi: 10.3934/mbe.2015.12.1065 [5] Yunshyong Chow, Sophia Jang. Neimark-Sacker bifurcations in a host-parasitoid system with a host refuge. Discrete and Continuous Dynamical Systems - B, 2016, 21 (6) : 1713-1728. doi: 10.3934/dcdsb.2016019 [6] Xiaoyuan Chang, Junjie Wei. Stability and Hopf bifurcation in a diffusive predator-prey system incorporating a prey refuge. Mathematical Biosciences & Engineering, 2013, 10 (4) : 979-996. doi: 10.3934/mbe.2013.10.979 [7] Jicai Huang, Sanhong Liu, Shigui Ruan, Xinan Zhang. Bogdanov-Takens bifurcation of codimension 3 in a predator-prey model with constant-yield predator harvesting. Communications on Pure and Applied Analysis, 2016, 15 (3) : 1041-1055. doi: 10.3934/cpaa.2016.15.1041 [8] Jicai Huang, Yijun Gong, Shigui Ruan. Bifurcation analysis in a predator-prey model with constant-yield predator harvesting. Discrete and Continuous Dynamical Systems - B, 2013, 18 (8) : 2101-2121. doi: 10.3934/dcdsb.2013.18.2101 [9] Qing Zhu, Huaqin Peng, Xiaoxiao Zheng, Huafeng Xiao. Bifurcation analysis of a stage-structured predator-prey model with prey refuge. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 2195-2209. doi: 10.3934/dcdss.2019141 [10] Tongtong Chen, Jixun Chu. Hopf bifurcation for a predator-prey model with age structure and ratio-dependent response function incorporating a prey refuge. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022082 [11] Xiao He, Sining Zheng. Bifurcation analysis and dynamic behavior to a predator-prey model with Beddington-DeAngelis functional response and protection zone. Discrete and Continuous Dynamical Systems - B, 2020, 25 (12) : 4641-4657. doi: 10.3934/dcdsb.2020117 [12] Eric Avila-Vales, Gerardo García-Almeida, Erika Rivero-Esquivel. Bifurcation and spatiotemporal patterns in a Bazykin predator-prey model with self and cross diffusion and Beddington-DeAngelis response. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 717-740. doi: 10.3934/dcdsb.2017035 [13] Bing Zeng, Shengfu Deng, Pei Yu. Bogdanov-Takens bifurcation in predator-prey systems. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 3253-3269. doi: 10.3934/dcdss.2020130 [14] Xiaoling Zou, Dejun Fan, Ke Wang. Stationary distribution and stochastic Hopf bifurcation for a predator-prey system with noises. Discrete and Continuous Dynamical Systems - B, 2013, 18 (5) : 1507-1519. doi: 10.3934/dcdsb.2013.18.1507 [15] Shu Li, Zhenzhen Li, Binxiang Dai. Stability and Hopf bifurcation in a prey-predator model with memory-based diffusion. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022025 [16] Yiwen Tao, Jingli Ren. The stability and bifurcation of homogeneous diffusive predator–prey systems with spatio–temporal delays. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 229-243. doi: 10.3934/dcdsb.2021038 [17] Hongyong Zhao, Daiyong Wu. Point to point traveling wave and periodic traveling wave induced by Hopf bifurcation for a diffusive predator-prey system. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 3271-3284. doi: 10.3934/dcdss.2020129 [18] Zuolin Shen, Junjie Wei. Hopf bifurcation analysis in a diffusive predator-prey system with delay and surplus killing effect. Mathematical Biosciences & Engineering, 2018, 15 (3) : 693-715. doi: 10.3934/mbe.2018031 [19] Haiying Jing, Zhaoyu Yang. The impact of state feedback control on a predator-prey model with functional response. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 607-614. doi: 10.3934/dcdsb.2004.4.607 [20] Sílvia Cuadrado. Stability of equilibria of a predator-prey model of phenotype evolution. Mathematical Biosciences & Engineering, 2009, 6 (4) : 701-718. doi: 10.3934/mbe.2009.6.701

2021 Impact Factor: 1.497

## Tools

Article outline

Figures and Tables