A class of Leslie-Gower type predator model with a non-monotonic rational functional response and alternative food for the predators
Keywords:Predator-prey model, functional response, bifurcation, limit cycle, separatrix curve, stability
The interactions between two species are basic in the study of complex food chains, particularly the relation among the predators and their prey.
The analysis of simple models, described by continuous-time systems, in which some ecological phenomena are incorporated giving lights about this interesting interrelationship.
In this work, a Leslie-Gower type predator-prey model is analyzed, considering two aspects: the prey defends from the predation, forming group defense and the predators have an alternative food. So, a rational Holling type IV functional response and a modification of the predators carrying capacity are assumed, to describe each phenomenon.
We determine conditions on the parameter space for the existence of the equilibria and their nature.
Using the Lyapunov quantities method, we also establish conditions on the parameter values for which there exist a unique positive equilibrium point, which is stable and surrounded by two limit cycle, the innermost unstable and the outermost sable.
We conclude that the parameter indicating the existence of alternative food for predator has a great importance on the dynamic of model, because appear new equilibrum points and separatrix curves in the phase plane.Some simulations are given to reinforce our findings the ecological interpretations of resultas are given.
Aguilera-Moya A. and González-Olivares E. A Gause type model with a generalized class of non-monotonic functional response, In R. Mondaini (ed.), Proceedings of the Third Brazilian Symposium on Mathematical and Computational Biology, E-Papers Serviços Editoriais, Ltda., Rio de Janeiro, 2004; 2:206-217.
Aguilera-Moya A., González-Yañez B. and González-Olivares E. Existence of multiple limit cycles on a predator-prey with generalized non-monotonic functional response, In R. Mondaini (ed.), Proceedings of the Fourth Brazilian Symposium on Mathematical and Computational Biology, E-Papers Serviços Editoriais, Ltda., Rio de Janeiro, 2005; 2:196-210.
Aguirre, P., González-Olivares, E., Sáez, E. Two limit cycles in a Leslie-Gower predator-prey model with additive Allee effect, Nonlinear Analysis: Real World Applications, 2009; 10:1401-1416.
Aguirre,P., González-Olivares, E., Sáez, E. Three limit cycles in a Leslie-Gower predator-prey model with additive Allee effect, SIAM Journal on Applied Mathematics, 2009; 69(5):1244-1269.
Arancibia-Ibarra, C. and González-Olivares, E. 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.
Arrowsmith, D. K. and Place,C. M. Dynamical System.Differential equations, maps and chaotic behaviour, Chapman and Hall, 1992.
Aziz-Alaoui, M. A. and Daher Okiye, M. Boundedness and global stability for a predator-prey model with modied Leslie-Gower and Holling-type II schemes, Applied Mathematics Letters, 2003; 16: 1069-1075.
Bazykin, A.D. Nonlinear Dynamics of interacting populations, World Scientific Publishing Co. Pte. Ltd., 1998.
A. A. Berryman, A.A., Gutierrez, A.P. and Arditi, R. Credible, parsimonious and useful predator-prey models - A reply to Abrams, Gleeson, and Sarnelle, Ecology, 1995; 76:1980-1985.
Cheng, K. S. Uniqueness of a limit cycle for a predator-prey system, SIAM Journal of Mathematical Analysis. 1981; 12:541-548.
Chicone, C. Ordinary differential equations with applications, Texts in Applied Mathematics, 34, Springer, 1999.
Dumortier, F., Llibre, J. and Artés, J. C. Qualitative theory of planar differential systems, Springer, 2006.
H. I. Freedman, Deterministic mathematical models in populations ecology, Marcel Dekker, Inc. New York 1980.
Freedman, H. I. and Wolkowicz, G. S. K. Predator-prey systems with group defence: The paradox of enrichment revisted. Bulletin of Mathematical Biology. 1986; 48:493-508.
Gaiko, V. A. Global Bifurcation theory and Hilbert´s sixteenth problem, Mathematics and its Applications 559, Kluwer Academic Publishers, 2003.
Gause, G. F. The Struggle for existence, Dover, 1934.
González-Olivares, E. A predator-prey model with nonmonotonic consumption function, In R. Mondaini (Ed.) Proceedings of the Second Brazilian Symposium on Mathematical and Computational Biology, E-papers Serviços Editoriais Ltda. Río de Janeiro, (2003) 23-39.
González-Olivares, E., Mena-Lorca, J., Rojas-Palma, A. and Flores, J. D. Dynamical complexities in the Leslie-Gower predator-prey model considering a simple form to the Allee effect on prey, Applied Mathematical Modelling, 2011; 35:366-381.
González-Olivares, E. and Rojas-Palma, A. Allee effect in Gause type predator-prey models: Existence of multiple attractors, limit cycles and separatrix curves. A brief review, Mathematical Modelling of Natural Phenomena. 2013; 8(6):143-164.
González-Olivares, E., Tintinago-Ruiz, P. and Rojas-Palma, A. A Leslie-Gower type predator-prey model with sigmoid funcional response, International Journal of Computer Mathematics, 2015; 93(9):1895-1909.
González-Olivares, E., Gallego-Berrío, L. M., González-Yañez, B. and Rojas-Palma,A. Consequences of weak Allee effect on prey in the May-Holling-Tanner predator-prey model, Mathematical Methods in the Applied Sciences, 2016; 39:4700-4712.
González-Olivares, E., Arancibia-Ibarra, C., Rojas-Palma, A. and González-Yañez, B. Bifurcations and multistability on the May-Holling-Tanner predation model considering alternative food for the predators, Mathematical Biosciences and Engineering, 2019; 16(5):4274-4298.
Guckenheimer J. and Holmes, P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, 1983.
Hanski, I., Hentonnen, H., Korpimaki, E., Oksanen, L. and Turchin, P. Small-rodent dynamics and predation, Ecology, 2001; 82:1505-1520.
Hasík, K. On a predator-prey system of Gause type, Journal of Mathematical Biology, 1010; 60:59-74.
Holling, C. S. The components of predation as revealed by a study of small-mammal predation of the European pine sawfly, The Canadian Entomologist, 1959; XCI:293-320.
Leslie, P. H. Some further notes on the use of matrices in population mathematics, Biometrika, 1948; 35:213-245.
Leslie, P. H., Gower, J. C. The properties of a stochastic model for the predator-prey type of interaction between two species, Biometrika, 1960; 47:219-234
May, R. M. Stability and complexity in model ecosystems, Princeton University Press, 1974.
Martínez-Jeraldo, N. and Aguirre, P. Allee effect acting on the prey species in a Leslie-Gower predation model, Nonlinear Analysis: Real World Applications, 2019; 45:895-917.
Monzón, P. Almost global attraction in planar systems, System and Control Letter, 2005; 54:753-758.
Perko, L. Differential equations and dynamical systems, Springer-Verlag, 1991.
Rantzer, A. A dual to Lyapunov’s stability theorem, System and Control Letter, 2001; 42:161-168.
Rojas-Palma A. and González-Olivares, E. Gause type predator-prey models with a generalized rational non-monotonic functional response, In J. Vigo-Aguiar (Ed.) Proceedings of the 14th International Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE 2014, 4:1092-1103. ISBN: 978-84-616-9216-3.
Ruan, S. and Xiao, D. Global analysis in a predator-prey system with nonmonotonic functional response, SIAM Journal on Applied. Mathematics, 2001; 61:1445-1472.
Sáez, E. González-Olivares, E. Dynamics on a Predator-prey Model, SIAM Journal of Applied Mathematics, 1999; 59(5):1867-1878.
Taylor, R. J. Predation, Chapman and Hall, 1984.
Turchin, P. Mongraphs in population biology: Vol. 35. Complex population dynamics. A theoretical/empirical synthesis, 2003.
Venturino, E. and Petrovskii, S. Spatiotemporal behavior of a prey-predator system with a group defense for prey, Ecological Complexity, 2013; 14:37-47.
Vilches, K., González-Olivares, E. and Rojas-Palma, A. Prey herd behavior modeled by a generic non-differentiable functional response, Mathematical Modelling of Natural Phenomena, 2018; 13:26.
Wolkowicz, G. S. W. Bifurcation analysis of a predator-prey system involving group defense, SIAM Journal on Applied Mathematics, 1988; 48:592-606.
Zhu, H., Campbell, S. A. andWolkowicz, G. S. K. Bifurcation analysis of a predator-prey system with nonmonotonic functional response, SIAM Journal on Applied Mathematics, 2002; 63:636-682.
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