Global stability in a modified Leslie-Gower type predation model assuming mutual interference among generalist predators

Eduardo Gonzalez-Olivares; Alejandro Rojas-Palma; Eduardo Gonzalez-Olivares; Alejandro Rojas-Palma

doi:10.3934/mbe.2020392

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 6: 7708-7731. doi: 10.3934/mbe.2020392

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Research article

Global stability in a modified Leslie-Gower type predation model assuming mutual interference among generalist predators

Eduardo Gonzalez-Olivares ^{1
,
,},
Alejandro Rojas-Palma ²

1.
Pontificia Universidad Católica de Valparaíso, Chile
2.
Departamento de Matemáticas, Física y Estadística, Facultad de Ciencias Básicas, Universidad Católica del Maule, Talca, Chile

Received: 25 June 2020 Accepted: 25 October 2020 Published: 05 November 2020

In the ecological literature, mutual interference (self-interference) or competition among predators (CAP) to effect the harvesting of their prey has been modeled through different mathematical formulations. In this work, the dynamical properties of a Leslie-Gower type predation model is analyzed, incorporating one of these forms, which is described by the function $g\left(y\right) = y^{\beta }$, with $0 < \beta < 1$. This function $g$ is not differentiable for $y = 0$, and neither the Jacobian matrix of the system is not defined in the equilibrium points over the horizontal axis ($x-axis$). To determine the nature of these points, we had to use a non-standard methodology. Previously, we have shown the fundamental properties of the Leslie-Gower type model with generalist predators, to carry out an adequate comparative analysis with the model where the competition among predators (CAP) is incorporated. The main obtained outcomes in both systems are: (ⅰ) The unique positive equilibrium point, when exists, is globally asymptotically stable (g.a.s), which is proven using a suitable Lyapunov function. (ⅱ) There not exist periodic orbits, which was proved constructing an adequate Dulac function.
- predator-prey model,
- functional response,
- bifurcation,
- limit cycle,
- separatrix curve,
- stability
Citation: Eduardo Gonzalez-Olivares, Alejandro Rojas-Palma. Global stability in a modified Leslie-Gower type predation model assuming mutual interference among generalist predators[J]. Mathematical Biosciences and Engineering, 2020, 17(6): 7708-7731. doi: 10.3934/mbe.2020392

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Abstract

In the ecological literature, mutual interference (self-interference) or competition among predators (CAP) to effect the harvesting of their prey has been modeled through different mathematical formulations. In this work, the dynamical properties of a Leslie-Gower type predation model is analyzed, incorporating one of these forms, which is described by the function $g\left(y\right) = y^{\beta }$, with $0 < \beta < 1$. This function $g$ is not differentiable for $y = 0$, and neither the Jacobian matrix of the system is not defined in the equilibrium points over the horizontal axis ($x-axis$). To determine the nature of these points, we had to use a non-standard methodology. Previously, we have shown the fundamental properties of the Leslie-Gower type model with generalist predators, to carry out an adequate comparative analysis with the model where the competition among predators (CAP) is incorporated. The main obtained outcomes in both systems are: (ⅰ) The unique positive equilibrium point, when exists, is globally asymptotically stable (g.a.s), which is proven using a suitable Lyapunov function. (ⅱ) There not exist periodic orbits, which was proved constructing an adequate Dulac function.

References

[1]	A. A. Berryman, A. P. Gutierrez, R. Arditi, Credible, parsimonious and useful predator-prey models - A reply to Abrams, Gleeson and Sarnelle, Ecology, 76 (1995), 1980-1985. doi: 10.2307/1940728
[2]	R. M. May, Stability and complexity in model ecosystems (2nd edition), Princeton University Press, 2001.
[3]	P. Turchin, Complex population dynamics. A theoretical/empirical synthesis, Monographs in Population Biology 35 Princeton University Press, 2003.
[4]	N. Bacaër, A short history of Mathematical Population Dynamics, Springer-Verlag, 2011.
[5]	A. D. Bazykin, Nonlinear Dynamics of interacting populations, World Scientific Publishing Co. Pte. Ltd., 1998.
[6]	V. Volterra, Variazioni e fluttuazioni del numero d'individui in specie animali conviventi, Memorie della R. Accademia dei Lincei, S.VI, IT, II (1926), 31-113.
[7]	J. D. Murray, Mathematical Biology, Springer - Verlag New-York, 1989.
[8]	C. Chicone, Ordinary differential equations with applications (2nd edition), Texts in Applied Mathematics 34, Springer, 2006.
[9]	F. Dumortier, J. Llibre, J. C. Artés, Qualitative theory of planar differential systems, Springer, 2006.
[10]	F. M. Scudo, J. R. Ziegler, The golden age of Theoretical Ecology 1923-1940. Lecture Notes in Biomathematics 22. Springer-Verlag, Berlin 1978.
[11]	G. F. Gause, The Struggle for existence, Dover, 1934.
[12]	C. S. Coleman, Hilbert's 16th. Problem: How many cycles? In: M. Braun, C. S. Coleman and D. Drew (Eds.), Differential Equations Models, Springer-Verlag, (1983), 279-297.
[13]	P. H. Leslie, Some further notes on the use of matrices in population mathematics, Biometrika, 35 (1948), 213-245. doi: 10.1093/biomet/35.3-4.213
[14]	H. I. Freedman, Deterministic Mathematical Model in Population Ecology, Marcel Dekker, 1980.
[15]	P. Aguirre, E. González-Olivares, E. Sáez, Two limit cycles in a Leslie-Gower predator-prey model with additive Allee effect, Nonlinear Anal. Real World Appl. N, 10 (2009), 1401-1416. doi: 10.1016/j.nonrwa.2008.01.022
[16]	E. González-Olivares, J. Gallegos-Zuñiga, Stability in a modified Leslie-Gower type predation model considering competence among predators (Estabilidad en un modelo de depredación del tipo Leslie-Gower modificado considerando competencia entre los depredadores), Selecciones Matemáticas, 7 (2020), 10-24 (in spanish).
[17]	E. Sáez, E. González-Olivares, Dynamics on a predator-prey model, SIAM J. Appl. Math., 59 (1999), 1867-1878. doi: 10.1137/S0036139997318457
[18]	C. Arancibia-Ibarra, E. González-Olivares, A modified Leslie-Gower predator-prey model with hyperbolic functional response and Allee effect on prey, In R. Mondaini (Ed.) BIOMAT 2010 International Symposium on Mathematical and Computational Biology, World Scientific Co.Pte. Ltd., Singapore, (2011), 146-162.
[19]	P. C. Tintinago-Ruiz, L. M. Gallego-Berrío, E. González-Olivares, A class of Leslie-Gower type predator model with a non-monotonic rational functional response and alternative food for the predators (Una clase de modelo de depredación del tipo Leslie-Gower con respuesta funcional racional no monotónica y alimento alternativo para los depredadores), Selecciones Matemáticas, 6 (2019), 204-216 (in spanish).
[20]	E. González-Olivares, C. Arancibia-Ibarra, A. Rojas-Palma, B. González-Yañez, Bifurcations and multistability on the May-Holling-Tanner predation model considering alternative food for the predators, Math. Biosci. Eng., 16 (2019), 4274-4298. doi: 10.3934/mbe.2019213
[21]	E. González-Olivares, C. Arancibia-Ibarra, A. Rojas-Palma, B. González-Yañez, Dynamics of a Leslie-Gower predation model considering a generalist predator and the hyperbolic functional response, Math. Biosci. Eng., 16 (2019), 7995-8024. doi: 10.3934/mbe.2019403
[22]	C. S. Holling, The components of predation as revealed by a study of small-mammal predation of the European pine sawfly, Can. Entomol., 91 (1959), 293-320. doi: 10.4039/Ent91293-5
[23]	R. J. Taylor, Predation, Chapman and Hall, 1984.
[24]	D. K. Arrowsmith, C. M. Place, Dynamical System. Differential equations, maps and chaotic behaviour, Chapman and Hall, 1992.
[25]	L. M. Gallego-Berrío, E. González-Olivares, The Holling-Tanner predation model with a special weak Allee effect on prey, In J. Vigo-Aguiar (Ed.) Proceedings of the 14th International Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE 2014, 2 (2014), 585-596.
[26]	S. Geritz, M. Gyllenberg, A mechanistic derivation of the DeAngelis-Beddington functional response, J. Theor. Biol., 314 (2012), 106-108. doi: 10.1016/j.jtbi.2012.08.030
[27]	Y. Vera-Dámian, C. Vidal, E. González-Olivares, Dynamics and bifurcations of a modified Leslie-Gower type model considering a Beddington-DeAngelis functional response, Math. Methods Appl. Sci., 42 (2019), 3179-3210. doi: 10.1002/mma.5577
[28]	E. González-Olivares, J. Cabrera-Villegas, F. Córdova-Lepe, A. Rojas-Palma, Competition among predators and Allee effect on prey: their influence on a Gause-type predation model, Math. Probl. Eng., 2019 (2019), 3967408.
[29]	C. W. Clark, Mathematical Bioeconomic: The optimal management of renewable resources, (2nd edition). John Wiley and Sons, 1990.
[30]	C. W. Clark, The worldwide crisis in fisheries: Economic models and human behavior, Cambridge Univerity Press, 2006.
[31]	H. I. Freedman, Stability analysis of a predator-prey system with mutual interference and density-dependent death rates, Bull. Math. Biol., 41 (1979), 67-78. doi: 10.1016/S0092-8240(79)80054-3
[32]	J. Díaz-Avalos, E. González-Olivares, A class of predator-prey models with a non-differentiable functional response, in J. Vigo-Aguiar (editor), Proceedings of the 17th International Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE 2017, 3 (2017), 765-776.
[33]	E. González-Olivares, E. Sáez, E. Stange, I. Szantó, Topological description of a non-differentiable bio-economics model, Rocky Mountain J. Math., 35 (2005), 1133-1155. doi: 10.1216/rmjm/1181069680
[34]	P. H. Leslie, J. C. Gower, The properties of a stochastic model for the predator-prey type of interaction between two species, Biometrika, 47 (1960), 219-234. doi: 10.1093/biomet/47.3-4.219
[35]	A. Erbach, F. Lutscher, G. Seo, Bistability and limit cycles in generalist predator-prey dynamics, Ecol. Complexity, 14 (2013), 48-55. doi: 10.1016/j.ecocom.2013.02.005
[36]	B-S. Goh, Management and Analysis of Biological Populations, Elsevier Scientific Publishing Company, 1980.
[37]	J. M. Epstein, Nonlinear Dynamics, Mathematical Biology, and Social Science, Addison-Wesley, 1997.
[38]	M. W. Hirsch, S. Smale, R. L. Devaney, Differential equations, dynamical systems, and an introduction to chaos (2nd edition) Elsevier, 2004.
[39]	L. Perko, Differential Equations and Dynamical Systems, Springer-Verlag, 1991.
[40]	A. A. Korobeinikov, Lyapunov function for Leslie-Gower predator-prey models, Appl. Math. Lett., 14, (2001), 697-699. doi: 10.1016/S0893-9659(01)80029-X
[41]	G. D. Birkhoff, G. C. Rota. Ordinary Differential Equations (4th edition), John Wiley and Sons, New York, 1989.
[42]	B. González-Yañez, E. González-Olivares, J. Mena-Lorca, Multistability on a Leslie-Gower Type predator-prey model with nonmonotonic functional response, In R. Mondaini and R. Dilao (eds.), BIOMAT 2006 - International Symposium on Mathematical and Computational Biology, World Scientific Co. Pte. Ltd., (2007), 359-384.
[43]	P. Aguirre, E. González-Olivares, E. Sáez, Three limit cycles in a Leslie-Gower predator-prey model with additive Allee effect, SIAM J. Appl. Math., 69 (2009), 1244-1262. doi: 10.1137/070705210
[44]	D. S. Boukal, L. Berec, Single-species models and the Allee effect: Extinction boundaries, sex ratios and mate encounters, J. Theor. Biol., 218 (2002), 375-394. doi: 10.1006/jtbi.2002.3084
[45]	F. Courchamp, L. Berec, J. Gascoigne, Allee effects in Ecology and Conservation, Oxford University Press, 2008.
[46]	J. Gallegos-Zuñiga, Modelo depredador-presa del tipo Leslie-Gower considerando interferencia entre los depredadores, Trabajo para optar al grado de Licenciado en Matemática, Instituto de Matemáticas, Pontificia Universidad Católica de Valparaíso, Chile, 2014 (in spanish).
[47]	E. González-Olivares, J. Mena-Lorca, A. Rojas-Palma, J. D. Flores, Dynamical complexities in the Leslie-Gower predator-prey model as consequences of the Allee effect on prey, Appl. Math. Model., 35 (2011), 366-381. doi: 10.1016/j.apm.2010.07.001
[48]	J. T. Tanner, The stability and the intrinsic growth rates of prey and predator populations, Ecology, 56 (1975), 855-867 doi: 10.2307/1936296

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