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Revista de Matemática Teoría y Aplicaciones

Print version ISSN 1409-2433

Rev. Mat vol.29 n.1 San José Jan./Jun. 2022

http://dx.doi.org/10.15517/rmta.v29i1.48408 

Artículo

Influencia del efecto allee débil en las presas en un modelo de depredación del tipo Leslie-Gower con respuesta funcional sigmoidea

Influence of the weak allee effect on prey in a Leslie-Gower type predation model with sigmoid functional response

Sebastián Valenzuela-Figueroa1 

Eduardo González-Olivares2 

Alejandro Rojas-Palma3 

1Universidad Austral de Chile, Centro de Docencia Superior en Ciencias Básicas, Puerto Montt, Chile; sebastian.valenzuela@uach.cl

2Pontificia Universidad Católica de Valparaíso, Instituto de Matemáticas, Valparaíso, Chile; ejgonzal@ucv.cl

3Universidad Católica del Maule, Departamento de Matemática, Física y Estadística, Facultad de Ciencias Básicas, Talca, Chile; amrojas@ucm.cl

Resumen

Utilizando un sistema topológicamente equivalente al original, dependiente sólo de cuatro parámetros, en este trabajo analizamos un modelo de depredación del tipo Leslie-Gower, considerando que el consumo de los depredadores es modelado por una respuesta funcional sigmoidea. Además, asumimos que las presas están afectadas por un efecto Allee y que los depredadores son generalistas. Mostramos que el sistema de ecuaciones diferenciales ordinarias que describe el modelo puede tener hasta cuatro puntos de equilibrio positivos. Dadas las dificultades para obtener explícitamente las coordenadas de estos puntos analizamos parcialmente el sistema considerando que la población de presas está afectada por un efecto Allee débil. Entre los resultados más importantes obtenidos, se demuestra la existencia de una curva separatriz, dividiendo el comportamiento de las soluciones o trayectorias del sistema en el plano de fase. Dos soluciones muy cercanas pero en un lado diferente de esa separatriz, tendrían ω − limites diferentes y distantes. Esto implica que teniendo un mismo tamaño poblacional de las presas, para distintos tamaños poblacionales de depredadores, pero muy cercanos, ambas poblaciones podrían coexistir o las presas podrían ir a la extinción.

Palabras clave: modelo depredador-presa; bifurcación; ciclo límite; curva separatriz; estabilidad; respuesta funcional.

Abstract

Using a topologically equivalent system to the original, dependent only on four parameters, in this work we analyze a Leslie-Gower type predation model, considering that predator consumption is modeled by a sigmoid functional response. Furthermore, we assume that the prey are affected by an Allee effect and that the predators are generalists. We show that the system of ordinary differential equations (ODE) that the model describes can have up to four positive equilibrium points. Given the difficulties in obtaining explicitly the coordinates of these points, we partially analyze the system considering that the prey population is affected by a weak Allee effect. Among the most important results obtained, the existence of a separator curve is demonstrated, dividing the behavior of the solutions or trajectories of the system in the phase plane. Two very close solutions, but on a different side of that separatrix, would have different and distant ω − limit sets. This implies that having the same population size of the prey, for different population sizes of predators, but very close, both populations could coexist or the prey could go to extinction. We estimate that the analytical results obtained have an adequate ecological interpretation, under the underlying assumptions in the modeling of the predation interaction with ODEs.

Keywords: predator-prey model; bifurcation; limit cycle; separatrix curve; stability; functional response.

Mathematics Subject Classification: 92D25; 34C23; 58F14; 58F21.

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Agradecimientos

Los autores agradecen a los árbitros anónimos por expresar comentarios y sugerencias que permitieron mejorar este manuscrito.

Referencias

P, Aguirre; E, González-Olivares; E, Sáez. Three limit cycles in a LeslieGower predator-prey model with additive Allee effect, SIAM Journal on Applied Mathematics 69(2009) 1244-1262. Doi: 10.1137/070705210 [ Links ]

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, Singapore (2011) pp.146-162. Doi: 10.1142/9789814343435_0010 [ Links ]

N, Bacaër. A Short History of Mathematical Population Dynamics, Springer, New York NY, 2011. Doi: 10.1007/978-0-85729-115-8 [ Links ]

A ,D, Bazykin . Nonlinear Dynamics of interacting populations, World Scientific, Singapore, 1998. Doi: 10.1142/2284 [ Links ]

R, Becerra-Klix; E, González-Olivares. A Leslie-Gower type predation model considering double Allee effect on prey and a sigmoid functional response, in J, Vigo-Aguiar (Ed.), Proceedings of the 17th International Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE, Cádiz, España, 2017 pp.252-263. In: http://www.rd.unir.net/sisi/research/ resultados/1526642421Proceedings_CMMSE_2017_vol_ 1_6.desprotegido_%20BEEBOTS.PDFLinks ]

L, Berec; E, Angulo; F, Courchamp. Multiple Allee effects and population management, Trends in Ecology & Evolution, 22(2007) 185-191. Doi: 10.1016/j.tree.2006.12.002 [ Links ]

A,A, Berryman; A,P, Gutierrez; R, Arditi. Credible parsimonious and useful predator-prey models- A reply to Abrams, Gleesson, and Sarnelle, Ecology 76(1995) 1980-1985. Doi: 10.2307/1940728 [ Links ]

D,S, Boukal; L, Berec. Single-species models and the Allee effect: Extinction boundaries, sex ratios and mate encounters, Journal of Theoretical Biology 218(2002), 375-394. Doi: 10.1006/jtbi.2002.3084 [ Links ]

C,W ;Clark Mathematical Bioeconomic: The Optimal Management of Renewable Resources, 2nd Ed., John Wiley and Sons, New York NY, 1990. Doi: 10.1137/1020117 [ Links ]

C, Chicone. Ordinary Differential Equations with Applications no.2, Texts in Applied Mathematics 34, Springer, 2006. Doi: 10.1007/0-387-35794-7 [ Links ]

K,-S,-Cheng. Uniqueness of a limit cycle for a predator-prey system, SIAM Journal of Mathematical Analysis 12(1981), no. 4, 541-548. Doi: 10.1137/0512047 [ Links ]

F, Courchamp; L, Berec; J, Gascoigne. Allee effects in ecology and conservation, Oxford University Press, Oxford, 2008. Doi: 10.1093/acprof:oso/9780198570301.001.0001 [ Links ]

Y, Dai; Y, Zhao; B, Sang. Four limit cycles in a predator-prey system of Leslie type with generalized Holling type III functional response, Nonlinear Analysis: Real World Applications 50(2019) 218-239. Doi: 10.1016/J.NONRWA.2019.04.003 [ Links ]

F, Dumortier; J, Llibre; J,C, Artés. Qualitative theory of planar differential systems, Springer, Berlin, 2006. Doi: 10.1007/978-3-540-32902-2 [ Links ]

H I, Freedman. Deterministic Mathematical Model in Population Ecology, Marcel Dekker, New York NY, 1980. Doi: 10.2307/3556198 [ Links ]

V A, Gaiko. Global Bifurcation Theory and Hilbert’s Sixteenth Problem, Mathematics and its Applications 559, Springer, Boston, MA, 2003. Doi: 10.1007/978-1-4419-9168-3 [ Links ]

V A, Gaiko; V, Vuik. Global dynamics in the Leslie-Gower model with the Allee effect, International Journal of Bifurcation and Chaos, 28(2018), no. 12. Doi: 10.1142/S0218127418501511 [ Links ]

B-S, Goh. Management and Analysis of Biological Populations, Elsevier Scientific Publ. Co., 1980. [ Links ]

E, González-Olivares; B, González-Yañez; J, Mena-Lorca; R, Ramos-Jiliberto. Modelling the Allee effect: Are the different mathematical forms proposed equivalents?, in: R, Mondaini (Ed.) Proceedings of the International Symposium on Mathematical and Computational Biology BIOMAT 2006, E-papers Serviços Editoriais Ltda., Rio de Janeiro, Brazil (2007), pp. 53-71. [ Links ]

E, González-Olivares; A, Rojas-Palma. Multiple limit cycles in a Gause type predator-prey model with Holling type III functional response and Allee effect on prey, Bulletin of Mathematical Biology 35 (2011) 366-381. Doi: 10.1007/s11538-010-9577-5 [ Links ]

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, Applied Mathematical Modelling 35(2011) 366-381. Doi: 10.1016/j.apm.2010.07.001 [ Links ]

E, González-Olivares; P, Tintinago-Ruiz; A, Rojas-Palma. A Leslie-Gower type predator-prey model with sigmoid funcional response, International Journal of Computer Mathematics 93(2015), no. 9, 1895-1909. Doi: 10.1080/00207160.2014.889818 [ Links ]

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, Mathematical Biosciences and Engineering 16(2019), no. 6, 7995-8024. Doi: 10.3934/mbe.2019403 [ Links ]

E, González-Olivares; E A, Rojas-Palma. Global stability in a modified Leslie-Gower type predation model assuming mutual interference among generalist predators, Mathematical Biosciences and Engineering, 17(2020), no. 6, 7708-7731. Doi: 10.3934/mbe.2020392 [ Links ]

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; R, Dilao (Eds.), BIOMAT 2006 - International Symposium on Mathematical and Computational Biology (Manaus, Brazil), World Scientific, Singapour: 2007 pp.359-384. Doi: 10.1142/9789812708779_0023 [ Links ]

C S,Holling. The components of predation as revealed by a study of smallmammal predation of the European pine sawfly, Canadian Entomologist 91(1959) 293-320. Doi: 10.4039/Ent91293-5 [ Links ]

Y A, Kuznetsov. Elements of Applied Bifurcation Theory (3rd ed.), Springer, Cham, 2004. Doi: 10.1007/978-1-4757-3978-7 [ Links ]

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 [ Links ]

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: 10.2307/2333294 [ Links ]

M, Liermann; R, Hilborn. Depensation: evidence, models and implications, Fish and Fisheries 2(2001), no. 1, 33-58. Doi: 10.1046/j.1467- 2979.2001.00029.x [ Links ]

N, Martínez-Jeraldo; E, Rozas-Torres; E, González-Olivares. Un modelo de depredación del tipo Leslie-Gower considerando depredadores generalistas y efecto Allee en las presas, Selecciones Matemáticas 8(2021), no. 1, 147-160. Doi: 10.17268/sel.mat.2021.01.14 [ Links ]

J, Mena-Lorca; E, González-Olivares; B, González-Yañez. The LeslieGower predator-prey model with Allee effect on prey: A simple model with a rich and interesting dynamics, in: Mondaini R. (Ed.), Proceedings of the 2006 International Symposium on Mathematical and Computational Biology BIOMAT 2006, E-papers Serviços Editoriais Ltda., Rio de Janeiro, Brazil (2007) pp. 105-132. Doi: 10.1142/6483 [ Links ]

P, Monzón. Almost global attraction in planar systems, Systems & Control Letters 54(2005) 753-758. Doi: 10.1016/j.sysconle.2004.11.014 [ Links ]

L, Perko. Differential equations and dynamical systems(3rd ed.), Springer, New York NY, 2001. Doi: 10.1007/978-1-4613-0003-8 [ Links ]

L, Puchuri-Medina; E, González-Olivares; A, Rojas-Palma. Multistability in a Leslie-Gower type predation model with rational nonmonotonic functional response and generalist predators, Computational and Mathematical Methods 2(2020). Doi: 10.1002/cmm4.1070 [ Links ]

A, Rantzer. A dual to Lyapunov’s stability theorem, Systems and Control Letters 42(2001), no. 3, 161-168. Doi: 10.1016/S0167-6911(00)00087-6 [ Links ]

F J, Reyes-Bahamón. Sobre la dinámica de algunos modelos depredadorpresa tipo Leslie con respuesta funcional no monótona y efecto Allee en las presas, Tesis de Maestría en Ciencias Matemática Aplicada, Universidad Nacional de Colombia, 2017. In: https://www.repositorio.unal. edu.co/handle/unal/59746 Links ]

A, Rojas Palma; P C, Tintinago-Ruiz; E, González-Olivares. Bifurcaciones en un modelo de depredación tipo Leslie-Gower modificado con una respuesta funcional Holling tipo III racional, en preparación, 2021. [ Links ]

E, Sáez; E, González-Olivares. Dynamics on a Predator-prey Model, SIAM J. Applied Mathematics 59(1999) 1867-1878. Doi: https://doi. org/10.1137/S003613999731845710.1137/S0036139997318457 [ Links ]

P A, Stephens; W J, Sutherland; R P, Freckleton. What is the Allee effect?, Oikos 87(1999) 185-190. Doi: 10.2307/3547011 [ Links ]

J, Sugie; K, Miyamoto; K, Morino. Absence of limits cycle of a predatorprey system with a sigmoid functional response, Applied Mathematical Letter 9(1996), no. 4, 85-90. Doi: 10.1016/0893-9659(96)00056-0 [ Links ]

R J, Taylor. Predation, Chapman and Hall, London, 1984. In: https: //www.worldcat.org/title/predation/oclc/12453276Links ]

P, Tintinago-Ruiz; L, Restrepo-Alape; E, González-Olivares. Consequences of weak Allee effect in a Leslie-Gower type predator-prey model with a generalized Holling type III functional response, in: G, Olivar Tost; O, Vasilieva. (Eds.) Analysis, Modelling, Optimization, and Numerical Technique, Springer Proceedings in Mathematics & Statistics, 121(2015), no. 6, 89-103. Doi: 10.1007/978-3-319-12583-1_6 [ Links ]

P C, Tintinago-Ruiz; E, González-Olivares; A, Rojas-Palma. Dinámicas de un modelo de depredación del tipo Leslie-Gower considerando respuesta funcional sigmoidea y alimento alternativo para los depredadores, en preparación, 2021. [ Links ]

P, Turchin. Complex Population Dynamics: A Theoretical/Empirical Synthesis, Princeton University Press, Princeton NJ, 2003. Doi: 10.1515/9781400847280 [ Links ]

S, Valenzuela-Figueroa. Modelos de depredación del tipo Leslie-Gower con respuesta funcional sigmoidea y efecto Allee en las presas, Tesis Magister en Matemática, Pontificia Universidad Católica de Valparaíso, 2013. En: http://www.repositorio.conicyt.cl/handle/ 10533/194403Links ]

Recibido: 10 de Septiembre de 2021; Revisado: 06 de Diciembre de 2021; Aprobado: 10 de Diciembre de 2021

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