The impact of fear factor and self-defence on the dynamics of predator-prey model with digestion delay

Jiang Li; Xiaohui Liu; Chunjin Wei; Jiang Li; Xiaohui Liu; Chunjin Wei

doi:10.3934/mbe.2021277

Mathematical Biosciences and Engineering

2021, Volume 18, Issue 5: 5478-5504. doi: 10.3934/mbe.2021277

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Research article

The impact of fear factor and self-defence on the dynamics of predator-prey model with digestion delay

School of Science, Jimei University, Xiamen Fujian 361021, China

Academic Editor:Xingfu Zou

Received: 14 April 2021 Accepted: 16 June 2021 Published: 21 June 2021

In this paper, we propose both deterministic and stochastic predator-prey models with digestion delay, incorporating fear factor and self-defence. For the deterministic model, the existence and stability of the equilibrium are investigated and the occurrence of Hopf bifurcation is studied. For the stochastic model, we investigate the existence of a unique global positive solution of the model and analyze the asymptotic behavior of the global solution around the equilibriums of the deterministic model. Finally, numerical simulations are carried out to verify our analytical results, which indicate that the intensity of white noise, fear factor and self-defence have a significant relationship with the dynamics of the predator-prey model and expand the theoretical analyses.
- predator-prey,
- digestion delay,
- fear factor,
- self-defence,
- Hopf bifurcation,
- asymptotic behaviors
Citation: Jiang Li, Xiaohui Liu, Chunjin Wei. The impact of fear factor and self-defence on the dynamics of predator-prey model with digestion delay[J]. Mathematical Biosciences and Engineering, 2021, 18(5): 5478-5504. doi: 10.3934/mbe.2021277

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Abstract

In this paper, we propose both deterministic and stochastic predator-prey models with digestion delay, incorporating fear factor and self-defence. For the deterministic model, the existence and stability of the equilibrium are investigated and the occurrence of Hopf bifurcation is studied. For the stochastic model, we investigate the existence of a unique global positive solution of the model and analyze the asymptotic behavior of the global solution around the equilibriums of the deterministic model. Finally, numerical simulations are carried out to verify our analytical results, which indicate that the intensity of white noise, fear factor and self-defence have a significant relationship with the dynamics of the predator-prey model and expand the theoretical analyses.

References

[1]	A. J. Lotka, Analytical note on certain rhythmic relations in organic systems, Proc. Natl. Acad. Sci., 6 (1920), 410–415. doi: 10.1073/pnas.6.7.410
[2]	V. Volterra, Variazioni e fluttuazioni del numero dindividui in specie animali conviventi, Mem. Acad. Lincei Roma, 2 (1926), 31–113.
[3]	C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, Mem. Entomol. Soc. Can., 45 (1965), 5–60.
[4]	M. Kot, Elements of Mathematical Ecology, Cambridge University Press, Cambridge, 2002.
[5]	Q. Khan, E. Balakrishnan, G. C. Wake, Analysis of a predator-prey system with predator switching, Bull. Math. Biol., 66 (2004), 109–123. doi: 10.1016/j.bulm.2003.08.005
[6]	S. Liu, E. Beretta, A stage-structured predator-prey model of Beddington-Deangelis type, SIAM J. Appl. Math., 66 (2006), 1101–1129. doi: 10.1137/050630003
[7]	K. Chakraborty, S. Jana, T. Kar, Global dynamics and bifurcation in a stage structured prey-predator fishery model with harvesting, Appl. Math. Comput., 218 (2012), 9271–9290.
[8]	W. Cresswell, Predation in bird populations, J. Ornithol., 152 (2011), 251–263. doi: 10.1007/s10336-010-0638-1
[9]	S. Creel, D. Christianson, Relationships between direct predation and risk effects, Trends Ecol. Evol., 23 (2008), 194–201. doi: 10.1016/j.tree.2007.12.004
[10]	S. L. Lima, Predators and the breeding bird: behavioral and reproductive flexibility under the risk of predation, Biol. Rev., 84 (2009), 485-513. doi: 10.1111/j.1469-185X.2009.00085.x
[11]	N. Pettorelli, T. Coulson, S. M. Durant, J. M. Gaillard, Predation, individual variability and vertebrate population dynamics, Oecologia, 167 (2011), 305–314. doi: 10.1007/s00442-011-2069-y
[12]	S. Creel, D. Christianson, S. Liley, J. A. Winnie, Predation risk affects reproductive physiology and demography of elk, Science, 315 (2007), 960–960. doi: 10.1126/science.1135918
[13]	S. D. Peacor, B. L. Peckarsky, G. C. Trussell, J. R. Vonesh, Costs of predator-induced phenotypic plasticity: a graphical model for predicting the contribution of nonconsumptive and consumptive effects of predators on prey, Oecologia, 171 (2013), 1–10. doi: 10.1007/s00442-012-2394-9
[14]	M. J. Sheriff, R. Boonstra, The sensitive hare: sublethal effects of predator stress on reproduction in snowshoe hares, J. Anim. Ecol., 78 (2009), 124C1258.
[15]	L. Y. Zanette, A. F. White, M. C. Allen, M. Clinchy, Perceived predation risk reduces the number of offspring songbirds produce per year, Science, 334 (2011), 1398–1401. doi: 10.1126/science.1210908
[16]	X. Wang, L. Zanette, X. Zou, Modelling the fear effect in predator-prey interactions, J. Math. Biol., 73 (2016), 1179–1204. doi: 10.1007/s00285-016-0989-1
[17]	X. Wang, X. Zou, Pattern formation of a predator-prey model with the cost of anti-predator behavbiors, Math. Biosci. Eng., 15 (2017), 775–805.
[18]	S. K. Sasmal, Population dynamics with multiple allee effects induced by fear factors-a mathematical study on prey-predator interactions, Appl. Math. Model., 64 (2018), 1–14. doi: 10.1016/j.apm.2018.07.021
[19]	A. Sha, S. Samanta, M. Martcheva, J. Chattopadhyay, Backward bifurcation, oscillations and chaos in an eco-epidemiological model with fear effect, J. Biol. Dyn., 13 (2019), 301–327. doi: 10.1080/17513758.2019.1593525
[20]	M. Hossain, N. Pal, S. Samanta, J. Chattopadhyay, Impact of fear on an eco-epidemiological model, Chaos Solitons Fractals, 134 (2020), 109718. doi: 10.1016/j.chaos.2020.109718
[21]	K. Kundu, S. Pal, S. Samanta, A. Sen, N. Pal, Impact of fear effect in a discrete-time predator-prey system, Bull. Calcutta Math. Soc., 110 (2018), 245–264.
[22]	S. Mondal, A. Maiti, G. Samanta, Effects of fear and additional food in a delayed predator-prey model, Biophys. Rev. Lett., 13 (2018), 157–177. doi: 10.1142/S1793048018500091
[23]	D. Duan, B. Niu, J. Wei, Hopf-hopf bifurcation and chaotic attractors in a delayed diffusive predator-prey model with fear effect, Chaos Solitons Fractals, 123 (2019), 206–216. doi: 10.1016/j.chaos.2019.04.012
[24]	S. Chen, Z. Liu, J. Shi, Nonexistence of nonconstant positive steady states of a diffusive predator-prey model with fear effect, J. Nonlinear Model. Anal., 1 (2019), 47–56.
[25]	Y. Wang, X. Zou, On a predator-prey system with digestion delay and anti-predation strategy, J. Nonlinear Sci., 30 (2020), 1579–1605. doi: 10.1007/s00332-020-09618-9
[26]	H. Qiu, M. Liu, K. Wang, Y. Wang, Dynamics of a stochastic predator-prey system with Beddington-DeAgelis functional response, Appl. Math. Comput., 219 (2012), 2303–2312.
[27]	Q. Liu, L. Zu, D. Jiang, Dynamics of stochastic predator-prey models with Holling II functional response, Commun. Nonlinear Sci. Numer. Simul., 37 (2016), 62–76. doi: 10.1016/j.cnsns.2016.01.005
[28]	X. Meng, L. Fei, S. Gao, Global analysis and numerical simulations of a novel stochastic eco-epidemiological model with time delay, Appl. Math. Comput., 339 (2018), 701–726.
[29]	D. Higham, Analgorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525–546. doi: 10.1137/S0036144500378302
[30]	S. E. Francis, Descartes rule of signs, Math Fun Facts, Available from: https://www.math.hmc.edu/funfacts.
[31]	K. L. Cooke, Z. Grossman, Discrete delay, distributed delay and stability switches, J. Math. Anal. Appl., 86 (1982), 592–627. doi: 10.1016/0022-247X(82)90243-8
[32]	X. Mao, Stochastic Differential Equations and Applications, 2$^nd$ edition, Horwood, New York, 1997.
[33]	Z. Li, Y. Mu, H. Xiang, Mean persistence and extinction for a novel stochastic turbidostat model, Nonlinear Dynam., 97 (2019), 185–202. doi: 10.1007/s11071-019-04965-z
[34]	Y. Xiao, L. Chen, Global stability of a predator-prey system with stage structure for the predator, Acta Math. Sin., 20 (2004), 63–70. doi: 10.1007/s10114-002-0234-2
[35]	G. Lan, C. Wei, S. Zhang, Long time behaviors of single-species population models with psychological effect and impulsive toxicant in polluted environments, Phys. A, 521 (2019), 828–842. doi: 10.1016/j.physa.2019.01.096

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