How do predator interference, prey herding and their possible retaliation affect prey-predator coexistence?

Francesca Acotto; Ezio Venturino; Francesca Acotto; Ezio Venturino

doi:10.3934/math.2024831

AIMS Mathematics

2024, Volume 9, Issue 7: 17122-17145. doi: 10.3934/math.2024831

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Research article

How do predator interference, prey herding and their possible retaliation affect prey-predator coexistence?

Francesca Acotto ^{1
,
,},
Ezio Venturino ^1,2

1.
Department of Mathematics "Giuseppe Peano'', University of Turin, via Carlo Alberto 10, Turin, 10123, Italy
2.
Laboratoire Chrono-environnement, Université de Franche-Comté, 16 route de Gray, Besançon, 25030, France

Received: 11 March 2024 Revised: 03 May 2024 Accepted: 10 May 2024 Published: 16 May 2024
MSC : 92D25, 92D40

In this paper, focusing on individualistic generalist predators and prey living in herds which coexist in a common area, we propose a generalization of a previous model, namely, a two-population system that accounts for the prey response to predator attacks. In particular, we suggest a new prey-predator interaction term with a denominator of the Beddington-DeAngelis form and a function in the numerator that behaves as $ N $ for small values of $ N $, and as $ N^{\alpha} $ for large values of $ N $, where $ N $ denotes the number of prey. We can take the savanna biome as a reference example, concentrating on large herbivores inhabiting it and some predators that feed on them. Only two conditionally stable equilibrium points have emerged from the model analysis: the predator-only equilibrium and the coexistence one. Transcritical bifurcations from the former to the latter type of equilibrium, as well as saddle-node bifurcations of the coexistence equilibrium have been identified numerically by using MATLAB. In addition, the model was found to exhibit bistability. Bistability is studied by using the MATLAB toolbox bSTAB, paying particular attention to the basin stability values. Comparison of coexistence equilibria with other prey-predator models in the literature essentially shows that, in this case, prey thrive in greater numbers and predators in smaller numbers. The population changes due to parameter variations were found to be significantly less pronounced.
- herd behavior,
- boundary interactions,
- individualistic attacks,
- generalist predator,
- predator satiation,
- prey retaliation,
- prey-induced fear,
- predator interference,
- bistability,
- bSTAB
Citation: Francesca Acotto, Ezio Venturino. How do predator interference, prey herding and their possible retaliation affect prey-predator coexistence?[J]. AIMS Mathematics, 2024, 9(7): 17122-17145. doi: 10.3934/math.2024831

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Abstract

In this paper, focusing on individualistic generalist predators and prey living in herds which coexist in a common area, we propose a generalization of a previous model, namely, a two-population system that accounts for the prey response to predator attacks. In particular, we suggest a new prey-predator interaction term with a denominator of the Beddington-DeAngelis form and a function in the numerator that behaves as $ N $ for small values of $ N $, and as $ N^{\alpha} $ for large values of $ N $, where $ N $ denotes the number of prey. We can take the savanna biome as a reference example, concentrating on large herbivores inhabiting it and some predators that feed on them. Only two conditionally stable equilibrium points have emerged from the model analysis: the predator-only equilibrium and the coexistence one. Transcritical bifurcations from the former to the latter type of equilibrium, as well as saddle-node bifurcations of the coexistence equilibrium have been identified numerically by using MATLAB. In addition, the model was found to exhibit bistability. Bistability is studied by using the MATLAB toolbox bSTAB, paying particular attention to the basin stability values. Comparison of coexistence equilibria with other prey-predator models in the literature essentially shows that, in this case, prey thrive in greater numbers and predators in smaller numbers. The population changes due to parameter variations were found to be significantly less pronounced.

References

[1]	L. B. Hutley, S. A. Setterfield, Savanna, Encyclopedia of Ecology, Academic Press, (2008), 3143–3154. https://doi.org/10.1016/B978-008045405-4.00358-X
[2]	S. L. Lima, Back to the basics of anti-predatory vigilance: the group-size effect, Anim. Behav., 49 (1995), 11–20. https://doi.org/10.1016/0003-3472(95)80149-9 doi: 10.1016/0003-3472(95)80149-9
[3]	G. Roberts, Why individual vigilance declines as group size increase, Anim. Behav., 51 (1996), 1077–1086. https://doi.org/10.1006/anbe.1996.0109 doi: 10.1006/anbe.1996.0109
[4]	T. M. Caro, Antipredator defenses in birds and mammals, University of Chicago Press, 2005.
[5]	V. Ajraldi, M. Pittavino, E. Venturino, Modeling herd behavior in population systems, Nonlinear Anal. Real World Appl., 12 (2011), 2319–2338. https://doi.org/10.1016/j.nonrwa.2011.02.002 doi: 10.1016/j.nonrwa.2011.02.002
[6]	I. M. Bulai, E. Venturino, Shape effects on herd behavior in ecological interacting population models, Math. Comput., 141 (2017), 40–55. https://doi.org/10.1016/j.matcom.2017.04.009 doi: 10.1016/j.matcom.2017.04.009
[7]	S. P. Bera, A. Maiti, G. Samanta, Modelling herd behavior of prey: analysis of a prey-predator model, WJMS, 11 (2015), 3–14.
[8]	C. Berardo, I. M. Bulai, E. Venturino, Interactions obtained from basic mechanistic principles: prey herds and predators, Mathematics, 9 (2021), 2555. https://doi.org/10.3390/math9202555 doi: 10.3390/math9202555
[9]	P. A. Braza, Predator–prey dynamics with square root functional responses, Nonlinear Anal. Real World Appl., 13 (2012), 1837–1843. https://doi.org/10.1016/j.nonrwa.2011.12.014 doi: 10.1016/j.nonrwa.2011.12.014
[10]	S. Djilali, C. Cattani, L. N. Guin, Delayed predator-prey model with prey social behavior, Eur. Phys. J. Plus, 136 (2021), 940. https://doi.org/10.1140/epjp/s13360-021-01940-9 doi: 10.1140/epjp/s13360-021-01940-9
[11]	J. Tan, W. Wang, J. Feng, Transient dynamics analysis of a predator-prey system with square root functional responses and random perturbation, Mathematics, 10 (2022), 4087. https://doi.org/10.3390/math10214087 doi: 10.3390/math10214087
[12]	S. Belvisi, E. Venturino, An ecoepidemic model with diseased predators and prey group defense, Simul. Model. Pract. Theory, 34 (2013), 144–155. https://doi.org/10.1016/j.simpat.2013.02.004 doi: 10.1016/j.simpat.2013.02.004
[13]	G. Gimmelli, B. W. Kooi, E. Venturino, Ecoepidemic models with prey group defense and feeding saturation, Ecol. Complex., 22 (2015), 50–58. https://doi.org/10.1016/j.ecocom.2015.02.004 doi: 10.1016/j.ecocom.2015.02.004
[14]	S. Saha, G. P. Samanta, Analysis of a predator-prey model with herd behavior and disease in prey incorporating prey refuge, Int. J. Biomath., 12 (2019), 1950007. https://doi.org/10.1142/S1793524519500074 doi: 10.1142/S1793524519500074
[15]	F. Acotto, E. Venturino, Modeling the herd prey response to individualistic predators attacks, Math. Meth. Appl. Sci., 46 (2023), 13436–13456. https://doi.org/10.1002/mma.9262 doi: 10.1002/mma.9262
[16]	S. C. Hayley, J. T. Craig, I. H. K. Graham, Prey morphology and predator sociality drive predator-prey preferences, J. Mammal., 97 (2016), 919–927. https://doi.org/10.1093/jmammal/gyw017 doi: 10.1093/jmammal/gyw017
[17]	M. Chen, Y. Takeuchi, J. F. Zhang, Dynamic complexity of a modified Leslie-Gower predator-prey system with fear effect, Commun. Nonlinear Sci. Numer. Simul., 119 (2023), 107109. https://doi.org/10.1016/j.cnsns.2023.107109 doi: 10.1016/j.cnsns.2023.107109
[18]	M. Das, G. P. Samanta, A delayed fractional order food chain model with fear effect and prey refuge, Math. Comput., 178 (2020), 218–245. https://doi.org/10.1016/j.matcom.2020.06.015 doi: 10.1016/j.matcom.2020.06.015
[19]	S. Garai, N. C. Pati, N. Pal, G. C. Layek, Organized periodic structures and coexistence of triple attractors in a predator-prey model with fear and refuge, Chaos Solit., 165 (2022), 112833. https://doi.org/10.1016/j.chaos.2022.112833 doi: 10.1016/j.chaos.2022.112833
[20]	S. Kim, K. Antwi-Fordjour, Prey group defense to predator aggregated induced fear, Eur. Phys. J. Plus, 137 (2022), 704. https://doi.org/10.1140/epjp/s13360-022-02926-x doi: 10.1140/epjp/s13360-022-02926-x
[21]	S. K. Sasmal, Y. Takeuchi, Dynamics of a predator-prey system with fear and group defense, J. Math. Anal. Appl., 481 (2020), 123471. https://doi.org/10.1016/j.jmaa.2019.123471 doi: 10.1016/j.jmaa.2019.123471
[22]	J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Anim. Ecol., 44 (1975), 331–340. https://doi.org/10.2307/3866 doi: 10.2307/3866
[23]	D. L. DeAngelis, R. A. Goldstein, R. V. O'Neill, A model for tropic interaction, Ecology, 56 (1975), 881–892. https://doi.org/10.2307/1936298 doi: 10.2307/1936298
[24]	D. Borgogni, L. Losero, E. Venturino, A more realistic formulation of herd behavior for interacting populations, R.P. Mondaini (eds) Trends in Biomathematics: Modeling Cells, Flows, Epidemics, and the Environment, BIOMAT 2019 (2020), Springer, Cham., Chapter 2, 9–21. https://doi.org/10.1007/978-3-030-46306-9_2
[25]	E. Venturino, Y. Caridi, V. Dos Anjos, G. D'Ancona, On some methodological issues in mathematical modeling of interacting populations, J. Biol. Syst., 31 (2023), 169–184. https://doi.org/10.1142/S0218339023500080 doi: 10.1142/S0218339023500080
[26]	M. Stender, N. Hoffmann, bSTAB: an open-source software for computing the basin stability of multi-stable dynamical systems, Nonlinear Dyn., 107 (2022), 1451–1468. https://doi.org/10.1007/s11071-021-06786-5 doi: 10.1007/s11071-021-06786-5
[27]	P. J. Menck, J. Heitzig, N. Marwan, J. Kurths, How basin stability complements the linear-stability paradigm, Nat. Phys., 9 (2013), 89–92. https://doi.org/10.1038/nphys2516 doi: 10.1038/nphys2516
[28]	P. J. Menck, J. Heitzig, J. Kurths, H. J. Schellnhuber, How dead ends undermine power grid stability, Nat. Commun., 5 (2014), 3969. https://doi.org/10.1038/ncomms4969 doi: 10.1038/ncomms4969
[29]	K. A. Johnson, R. S. Goody, The original Michaelis constant: translation of the 1913 Michaelis-Menten paper, Biochem., 50 (2011), 8264–8269. https://doi.org/10.1021/bi201284u doi: 10.1021/bi201284u
[30]	C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, Mem. Ent. Soc. Can., 97 (1965), 5–60. https://doi.org/10.4039/entm9745fv doi: 10.4039/entm9745fv
[31]	B. Noble, Applied Linear Algebra, Englewood Cliffs: Prentice-Hall, 1969.
[32]	R. Woods, Analytic Geometry, New York: Mac Millan, 1939.
[33]	L. Perko, Differential Equations and Dynamical Systems, New York: Springer, 2001. https://doi.org/10.1007/978-1-4613-0003-8
[34]	A. Erbach, F. Lutscher, G. Seo, Bistability and limit cycles in generalist predator-prey dynamics, Ecol. Complex., 14 (2013), 48–55. https://doi.org/10.1016/j.ecocom.2013.02.005 doi: 10.1016/j.ecocom.2013.02.005
[35]	S. Garai, S. Karmakar, S. Jafari, N. Pal, Coexistence of triple, quadruple attractors and Wada basin boundaries in a predator-prey model with additional food for predators, Commun. Nonlinear Sci. Numer. Simul., 121 (2023), 107208. https://doi.org/10.1016/j.cnsns.2023.107208 doi: 10.1016/j.cnsns.2023.107208
[36]	R. López-Ruiz, D. Fournier-Prunaret, Indirect Allee effect, bistability and chaotic oscillations in a predator-prey discrete model of logistic type, Chaos Soliton. Fract., 24 (2005), 85–101. https://doi.org/10.1016/j.chaos.2004.07.018 doi: 10.1016/j.chaos.2004.07.018
[37]	Rajni, B. Ghosh, Multistability, chaos and mean population density in a discrete-time predator-prey system, Chaos Soliton. Fract., 162 (2022), 112497. https://doi.org/10.1016/j.chaos.2022.112497 doi: 10.1016/j.chaos.2022.112497
[38]	D. Melchionda, E. Pastacaldi, C. Perri, M. Banerjee, E. Venturino, Social behavior-induced multistability in minimal competitive ecosystems, J. Theor. Biol., 439 (2018), 24–38. https://doi.org/10.1016/j.jtbi.2017.11.016 doi: 10.1016/j.jtbi.2017.11.016

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