Dynamics in a predator-prey model with memory effect in predator and fear effect in prey

Ruizhi Yang; Dan Jin; Ruizhi Yang; Dan Jin

doi:10.3934/era.2022069

Electronic Research Archive

2022, Volume 30, Issue 4: 1322-1339. doi: 10.3934/era.2022069

Previous Article Next Article

Research article

Dynamics in a predator-prey model with memory effect in predator and fear effect in prey

Ruizhi Yang ,
Dan Jin ^,

Department of Mathematics, Northeast Forestry University, Harbin 150040, Heilongjiang, China

Academic Editor: Lixia Duan

Received: 14 January 2022 Revised: 02 March 2022 Accepted: 13 March 2022 Published: 18 March 2022

The spatial memory effect in predator and fear effect in prey are incorporated in a diffusive predator-prey model. We are interested in studying the dynamics generated by the memory effect and fear effect, and mainly study the local stability of coexisting equilibrium, the existence of Hopf bifurcation and the property of Hopf bifurcation. Through the numerical simulations, we show that increasing memory-based diffusion coefficient is not conducive to the stability of the coexisting equilibrium, and the fear effect has both stabilizing and destabilizing effect on the coexisting equilibrium under different parameters.
- predator-prey,
- delay,
- memory effect,
- fear effect,
- Hopf bifurcation
Citation: Ruizhi Yang, Dan Jin. Dynamics in a predator-prey model with memory effect in predator and fear effect in prey[J]. Electronic Research Archive, 2022, 30(4): 1322-1339. doi: 10.3934/era.2022069

Related Papers:

Abstract

The spatial memory effect in predator and fear effect in prey are incorporated in a diffusive predator-prey model. We are interested in studying the dynamics generated by the memory effect and fear effect, and mainly study the local stability of coexisting equilibrium, the existence of Hopf bifurcation and the property of Hopf bifurcation. Through the numerical simulations, we show that increasing memory-based diffusion coefficient is not conducive to the stability of the coexisting equilibrium, and the fear effect has both stabilizing and destabilizing effect on the coexisting equilibrium under different parameters.

References

[1]	R. Yang, C. Zhang, Dynamics in a diffusive predator-prey system with a constant prey refuge and delay, Nonlinear Anal. Real World Appl., 31 (2016), 1–22. https://doi.org/10.1016/j.nonrwa.2016.01.005 doi: 10.1016/j.nonrwa.2016.01.005
[2]	R. Yang, Q. Song, Y. An, Spatiotemporal dynamics in a predator-prey model with functional response increasing in both predator and prey densities, Mathematics, 10 (2022), 17. https://doi.org/10.11948/20190295 doi: 10.11948/20190295
[3]	Y. Liu, D. Duan, B. Niu, Spatiotemporal dynamics in a diffusive predator-prey model with group defense and nonlocal competition, Appl. Math. Lett., 103 (2020), 106175. https://doi.org/10.1016/j.aml.2019.106175 doi: 10.1016/j.aml.2019.106175
[4]	R. Yang, L. Wang, D. Jin, Hopf bifurcation analysis of a diffusive nutrient-phytoplankton model with time delay, Axioms, 11 (2020), 56. https://doi.org/10.3390/axioms11020056 doi: 10.3390/axioms11020056
[5]	D. Geng, W. Jiang, Y. Lou, H. Wang, Spatiotemporal patterns in a diffusive predator-prey system with nonlocal intraspecific prey competition, Stud. Appl. Math., 148 (2022), 396–432. https://doi.org/10.1111/sapm.12444 doi: 10.1111/sapm.12444
[6]	R. Yang, X. Zhao, Y. An, Dynamical analysis of a delayed diffusive predator-prey model with additional food provided and anti-predator behavior, Mathematics, 10 (2022), 469. https://doi.org/10.3390/math10030469 doi: 10.3390/math10030469
[7]	S. Lima, L. Dill, Behavioral decisions made under the risk of predation: a review and prospectus, Can. J. Zool., 68 (1990), 619–640. https://doi.org/10.1139/z90-092 doi: 10.1139/z90-092
[8]	K. B. Altendorf, J. W. Laundr$\acute{e}$, C. A. L. Gonz$\acute{a}$lez, J. S. Brown, Assessing effects of predation risk on foraging behavior of mule deer, J. Mammal., 82 (2001), 430–439. https://doi.org/10.1644/1545-1542(2001)082<0430:AEOPRO>2.0.CO;2 doi: 10.1644/1545-1542(2001)082<0430:AEOPRO>2.0.CO;2
[9]	S. Creel, D. Christianson, S. Liley, J. A. Winnie, Predation risk affects reproductive physiology and demography of elk, Science, 315 (2007), 960. https://doi.org/10.1126/science.1135918 doi: 10.1126/science.1135918
[10]	P. Panday, S. Samanta, N. Pal, J. Chattopadhyay, Delay induced multiple stability switch and chaos in a predator-prey model with fear effect, Math. Comput. Simul., 172 (2019), 134–158. https://doi.org/10.1016/j.matcom.2019.12.015 doi: 10.1016/j.matcom.2019.12.015
[11]	W. F. Fagan, M. A. Lewis, M. Auger-M$\acute{e}$th$\acute{e}$, T. Avgar, S. Benhamou, G. Breed, et al., Spatial memory and animal movement, Ecol. Lett., 16 (2014), 1316–1329. https://doi.org/10.1111/ele.12165 doi: 10.1111/ele.12165
[12]	B. Abrahms, E. L. Hazen, E. O. Aikens, M. S. Savoca, J. A. Goldbogen, S. J. Bograd, et al., Memory and resource tracking drive blue whale migrations, Proc. Natl. Acad. Sci., 116 (2019), 5582–5587. https://doi.org/10.1073/pnas.1819031116 doi: 10.1073/pnas.1819031116
[13]	W. F. Fagan, Migrating whales depend on memory to exploit reliable resources, Proc. Natl. Acad. Sci., 116 (2019), 5217–5219. https://doi.org/10.1073/pnas.1901803116 doi: 10.1073/pnas.1901803116
[14]	J. Shi, C. Wang, H. Wang, X. Yan, Diffusive spatial movement with memory, J. Dyn. Differ. Equations, 32 (2020), 979–1002. https://doi.org/10.1007/s10884-019-09757-y doi: 10.1007/s10884-019-09757-y
[15]	P. R. Moorcroft, M. A. Lewis, R. L. Crabtree, Home range analysis using amechanistic home range model, Ecology, 80 (1999), 1656–1665. https://doi.org/10.1890/0012-9658(1999)080[1656:HRAUAM]2.0.CO;2 doi: 10.1890/0012-9658(1999)080[1656:HRAUAM]2.0.CO;2
[16]	M. A. Lewis, J. D. Murray, Modelling territoriality and wolf-deer interactions, Nature, 366 (1993), 738–740. https://doi.org/10.1038/366738a0 doi: 10.1038/366738a0
[17]	J. R. Potts, M. A. Lewis, Spatial memory and taxis-driven pattern formation in model ecosystems, Bull. Math. Biol., 81 (2019), 2725–2747. https://doi.org/10.1007/s11538-019-00626-9 doi: 10.1007/s11538-019-00626-9
[18]	J. R. Potts, M. A. Lewis, How memory of direct animal interactions can lead to territorial pattern formation, J. R. Soc. Interface, 13 (2016), 20160059. https://doi.org/10.1098/rsif.2016.0059 doi: 10.1098/rsif.2016.0059
[19]	Q. An, C. Wang, H. Wang, Analysis of a spatial memory model with nonlocal maturation delay and hostile boundary condition, Discrete Contin. Dyn. Syst., 40 (2020), 5845–5868. https://doi.org/10.3934/dcds.2020249 doi: 10.3934/dcds.2020249
[20]	J. Shi, C. Wang, H. Wang, Diffusive spatial movement with memory and maturation delays, Nonlinearity, 32 (2019), 3188–3208. https://doi.org/10.1088/1361-6544/ab1f2f doi: 10.1088/1361-6544/ab1f2f
[21]	Q. Shi, J. Shi, H. Wang, Spatial movement with distributed memory, J. Math. Biol., 82 (2021), 33. https://doi.org/10.1007/s00285-021-01588-0 doi: 10.1007/s00285-021-01588-0
[22]	Y. Song, S. Wu, H. Wang, Spatiotemporal dynamics in the single population model with memory-based diffusion and nonlocal effect, J. Differ. Equations, 267 (2019), 6316–6351. https://doi.org/10.1016/j.jde.2019.06.025 doi: 10.1016/j.jde.2019.06.025
[23]	Y. Song, Y. Peng, T. Zhang, The spatially inhomogeneous Hopf bifurcation induced by memory delay in a memory-based diffusion system, J. Differ. Equations, 300 (2021), 597–624. https://doi.org/10.1016/j.jde.2021.08.010 doi: 10.1016/j.jde.2021.08.010

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)