Dynamics and spatio-temporal patterns in a prey–predator system with aposematic prey

Sourav Kumar Sasmal; Jeet Banerjee; Yasuhiro Takeuchi; Sourav Kumar Sasmal; Jeet Banerjee; Yasuhiro Takeuchi

doi:10.3934/mbe.2019191

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 5: 3864-3884. doi: 10.3934/mbe.2019191

Previous Article Next Article

Research article Special Issues

Dynamics and spatio-temporal patterns in a prey–predator system with aposematic prey

1.
Department of Physics and Mathematics, Aoyama Gakuin University, Kanagawa 252-5258, Japan
2.
Department of Electronics and Electrical Communication Engineering, Indian Institute of Technology Kharagpur, Kharagpur 721302, India

Received: 20 February 2019 Accepted: 26 April 2019 Published: 01 May 2019

We analyze the impact of aposematic time and searching efficiency of prey on the temporal and spatio-temporal dynamics of a diffusive prey-predator system. Here, our assumption is that the prey population primarily invests its total time in two activities——(ⅰ) defense against predation and (ⅱ) searching for food, followed by growth-induced reproduction, whereas, predators do not involve in self-defense. Moreover, we consider that the reproduction rate of prey and the rate of predation have a negative linear correlation with the amount of time invested for aposematism. Based on the presumptions, we find that unlike searching efficiency of prey, the aposematic time can diminish the proportion in which prey and predator coexist when it crosses a certain threshold, and at the extreme aposematism, the entire population drives into the extinction. The proposed dynamics undergoes Hopf-bifurcation with respect to the searching efficiency of prey. We examine the individual effect of aposematic time and searching efficiency on the formation of regular Turing patterns——the low to medium to high values of defense-time and food searching efficiency generate 'spots' to 'stripes' to 'holes' pattern, respectively; however, the combined impact of both presents only non-Turing 'spot' pattern with the 'predominance of predators, ' which happens through the Turing-Hopf bifurcation.

Keywords:

Citation: Sourav Kumar Sasmal, Jeet Banerjee, Yasuhiro Takeuchi. Dynamics and spatio-temporal patterns in a prey–predator system with aposematic prey[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 3864-3884. doi: 10.3934/mbe.2019191

Related Papers:

[1]	Yuhong Huo, Gourav Mandal, Lakshmi Narayan Guin, Santabrata Chakravarty, Renji Han . Allee effect-driven complexity in a spatiotemporal predator-prey system with fear factor. Mathematical Biosciences and Engineering, 2023, 20(10): 18820-18860. doi: 10.3934/mbe.2023834
[2]	Yue Xing, Weihua Jiang, Xun Cao . Multi-stable and spatiotemporal staggered patterns in a predator-prey model with predator-taxis and delay. Mathematical Biosciences and Engineering, 2023, 20(10): 18413-18444. doi: 10.3934/mbe.2023818
[3]	Yongli Cai, Malay Banerjee, Yun Kang, Weiming Wang . Spatiotemporal complexity in a predator--prey model with weak Allee effects. Mathematical Biosciences and Engineering, 2014, 11(6): 1247-1274. doi: 10.3934/mbe.2014.11.1247
[4]	Fang Liu, Yanfei Du . Spatiotemporal dynamics of a diffusive predator-prey model with delay and Allee effect in predator. Mathematical Biosciences and Engineering, 2023, 20(11): 19372-19400. doi: 10.3934/mbe.2023857
[5]	Ranjit Kumar Upadhyay, Swati Mishra . Population dynamic consequences of fearful prey in a spatiotemporal predator-prey system. Mathematical Biosciences and Engineering, 2019, 16(1): 338-372. doi: 10.3934/mbe.2019017
[6]	Sangeeta Kumari, Sidharth Menon, Abhirami K . Dynamical system of quokka population depicting Fennecaphobia by Vulpes vulpes. Mathematical Biosciences and Engineering, 2025, 22(6): 1342-1363. doi: 10.3934/mbe.2025050
[7]	Meiling Zhu, Huijun Xu . Dynamics of a delayed reaction-diffusion predator-prey model with the effect of the toxins. Mathematical Biosciences and Engineering, 2023, 20(4): 6894-6911. doi: 10.3934/mbe.2023297
[8]	Tingting Ma, Xinzhu Meng . Global analysis and Hopf-bifurcation in a cross-diffusion prey-predator system with fear effect and predator cannibalism. Mathematical Biosciences and Engineering, 2022, 19(6): 6040-6071. doi: 10.3934/mbe.2022282
[9]	Guangxun Sun, Binxiang Dai . Stability and bifurcation of a delayed diffusive predator-prey system with food-limited and nonlinear harvesting. Mathematical Biosciences and Engineering, 2020, 17(4): 3520-3552. doi: 10.3934/mbe.2020199
[10]	Swadesh Pal, Malay Banerjee, Vitaly Volpert . Spatio-temporal Bazykin’s model with space-time nonlocality. Mathematical Biosciences and Engineering, 2020, 17(5): 4801-4824. doi: 10.3934/mbe.2020262

Abstract

References

[1]	R. M. May, Biological populations with nonoverlapping generations: stable points, stable cycles, and chaos, Science, 186 (1974), 645–647.
[2]	J. Banerjee, S. K. Sasmal and R. K. Layek, Supercritical and subcritical Hopf-bifurcations in a two-delayed prey–predator system with density-dependent mortality of predator and strong Allee effect in prey, BioSystems, 180 (2019), 19–37.
[3]	S. K. Sasmal, Population dynamics with multiple Allee effects induced by fear factors–A mathe-matical study on prey–predator interactions, Appl. Math. Model., 64 (2018), 1–14.
[4]	J. Banerjee, T. Ranjan and R. K. Layek, Dynamics of cancer progression and suppression: A novel evolutionary game theory based approach, 37th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), (2015), 5367–5371.
[5]	J. Banerjee, T. Ranjan and R. K. Layek, Stability Analysis of Population Dynamics Model in
[6]	Microbial Biofilms with Non-participating Strains, 7th ACM International Conference on Bioin-formatics, Computational Biology, and Health Informatics, (201, 220–230.
[7]	6. M. Wang, A. L. Schaefer, A. A. Dandekar, et al., Quorum sensing and policing of Pseudomonas aeruginosa social cheaters, Proceed. Nat. Aca. Sci., 112 (2015), 2182191.
[8]	7. J. Banerjee, R. K. Layek, S. K. Sasmal, et al., Delayed evolutionary model for public goods competition with policing in phenotypically-variant bacterial biofilms, Europhys. Let., (2019) (in press).
[9]	8. M. Stevens and G. D. Ruxton, Linking the evolution and form of warning coloration in nature, Proceed. Royal Soc. B Biol. Sci., 27(2011), 417–426.
[10]	9. J. Skelhorn, C. G. Halpin and C. Rowe, Learning about aposematic prey, Behav. Ecol., 27 (2016), 955–964.
[11]	10. J. Skelhorn and C. Rowe, Predators' toxin burdens influence their strategic decisions to eat toxic prey, Curr. Biol., 17 (2007), 1479–1483.
[12]	11. C. G. Halpin, J. Skelhorn and C. Rowe, Predators' decisions to eat defended prey depend on the size of undefended prey, Animal Behav., 85 (2013), 1315–1321.
[13]	12. K. E. Smith, C. G. Halpin and C. Rowe, Body size matters for aposematic prey during predator aversion learning, Behav. Process., 109 (2014), 173–179.
[14]	13. J. E. Huheey, Mathematical models of mimicry, Am. Nat., 131 (1988), S22–S41.
[15]	14. J. C. Santos, L. A. Coloma and D. C. Cannatella, Multiple, recurring origins of aposematism and diet specialization in poison frogs, Proceed. Nat. Aca. Sci., 100 (2003), 12792–12797.
[16]	15. R. A. Saporito, R. Zuercher, M. Roberts, et al., Experimental evidence for aposematism in the dendrobatid poison frog Oophaga pumilio, Copeia, 2007 (2007), 1006–1011.
[17]	16. J. C. Santos and D. C. Cannatella, Phenotypic integration emerges from aposematism and scale in poison frogs, Proceed. Nat. Aca. Sci., 108 (2011), 6-6180.
[18]	17. E. D. Brodie III, Differential avoidance of coral snake banded patterns by free-ranging avian predators in Costa Rica, Evolution, 47 (1993), 227–235.
[19]	18. Y. Takeuchi, W. Wang, S. Nakaoka, et al., Dynamical adaptation of parental care, Bull. Math. Biol., 71 (2009), 931–951.
[20]	19. S. Chakraborty, P. K. Tiwari, S. K. Sasmal, et al., Interactive effects of prey refuge and additional food for predator in a diffusive predator-prey system, Appl. Math. Model., 47 (7), 128–140.
[21]	20. D. Alonso, F. Bartumeus and J. Catalan, Mutual interference between predators can give rise to Turing spatial patterns, Ecology, 83 (2002), 28–34.
[22]	21. J. R. Meyer, S. P. Ellner, N. G. Hairston, et al., Prey evolution on the time scale of predator–prey dynamics revealed by allele-specific quantitative PCR, Proceed. Nat. Aca. Sci., 103 (2006), 10690–10695.
[23]	22. L. Lindström, R. V. Alatalo, J. Mappes, et al., Can aposematic signals evolve by gradual change?, Nature, 397 (1999), 249–251.
[24]	23. J. Gohli and G. Högstedt, Explaining the evolution of warning coloration: Secreted secondary defence chemicals may facilitate the evolution of visual aposematic signals, PLoS One, 4 (2009), e5779.
[25]	24. J. B. Barnett, C. Michalis, N. E. Scott-Samuel, et al., Distance-dependent defensive coloration in the poison frog Dendrobates tinctorius, Dendrobatidae, Proceed. Nat. Aca. Sci., 115 (2018), 6416–6421.

This article has been cited by:

1.	Sourav Kumar Sasmal, Yasuhiro Takeuchi, Evolutionary dynamics of single species model with Allee effects and aposematism, 2021, 58, 14681218, 103233, 10.1016/j.nonrwa.2020.103233
2.	Sourav Kumar Sasmal, Balram Dubey, Diffusive patterns in a predator–prey system with fear and hunting cooperation, 2022, 137, 2190-5444, 10.1140/epjp/s13360-022-02497-x
3.	Sourav Kumar Sasmal, Yasuhiro Takeuchi, Editorial: Mathematical Modeling to Solve the Problems in Life Sciences, 2020, 17, 1551-0018, 2967, 10.3934/mbe.2020167
4.	Balram Dubey, Sourav Kumar Sasmal, Anand Sudarshan, Consequences of fear effect and prey refuge on the Turing patterns in a delayed predator–prey system, 2022, 32, 1054-1500, 123132, 10.1063/5.0126782
5.	Balram Dubey, Study of a cannibalistic prey–predator model with Allee effect in prey under the presence of diffusion, 2024, 182, 09600779, 114797, 10.1016/j.chaos.2024.114797
6.	Ankit Kumar, Sourav Kumar Sasmal, Complex dynamics of a stage structured prey-predator model with parental care in prey, 2024, 112, 0924-090X, 15623, 10.1007/s11071-024-09821-3

Reader Comments

Your name:*

Email:*
© 2019 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)