Abstract
The main peculiarity of the Leslie–Gower type models is the predator growth equation is the logistic type, in which the environmental carrying capacity is proportional to the prey population size. This assumption implies the predators are specialists. Considering that the predator is generalist, the environmental carrying capacity is modified adding a positive constant. In this work, the two simple classes of Leslie–Gower type predator-prey models are analyzed, considering a non-usual functional response, called Rosenzweig or power functional responses, being its main feature that is non-differentiable over the vertical axis. Just as Volterra predator-prey model, when the Rosenzweig functional response is incorporated, the systems describing the models have distinctive properties from the original one; moreover, differences between them are established. One of the main properties proved is the existence of a wide set of parameter values for which a separatrix curve, dividing the phase plane in two complementary sectors. Trajectories with initial conditions upper this curve have the origin or a point over the vertical axis as their \(\omega \)-limit. Meanwhile those trajectories with initial conditions under this curve can have a positive equilibrium point, or a limit cycle or a heteroclinic curve as their \(\omega \)-limit. The marked differences between the two cases studied shows as a little change in the mathematical expressions to describe the models can produce rich dynamics. In other words, little perturbations over the functions representing predator interactions have significant consequences on the behavior of the solutions, without change the general structure in the classical systems.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Ajraldi, V., Pittavino, M., Venturino, E.: Modeling herd behavior in population systems. Nonlinear Anal.: R. World Appl. 12, 2319–2338 (2011)
Arancibia-Ibarra, C., González-Olivares, E.: A modified Leslie-Gower predator-prey model with hyperbolic functional response and Allee effect on prey. In: Mondaini, R. (ed.) BIOMAT 2010 International Symposium on Mathematical and Computational Biology, pp. 146–162. World Scientific Co. Pte. Ltd., Singapore
Ardito, A., Ricciardi, P.: Lyapunov functions for a generalized Gause-type model. J. Math. Biol. 33, 816–828 (1995)
Aziz-Alaoui, M.A., Daher Okiye, M.: Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes. Appl. Math. Lett. 16, 1069–1075 (2003)
Bazykin, A.D.: Nonlinear Dynamics of Interacting Populations. World Scientific Publishing Co. Pte. Ltd., Singapore (1998)
Bera, S.P., Maiti, A., Samanta, G.P.: Modelling herd behavior of prey: analysis of a prey-predator model. World J. Model. Simul. 11, 3–14 (2015)
Berryman, A.A., Gutierrez, A.P., Arditi, R.: Credible, parsimonious and useful predator-prey models - a reply to Abrams, Gleeson, and Sarnelle. Ecology 76, 1980–1985 (1995)
Bravo, J.L., Fernández, M., Gámez, M., Granados, B., Tineo, A.: Existence of a polycycle in non-Lipschitz Gause-type predator-prey models. J. Math. Anal. Appl. 373, 512–520 (2011)
Chicone, C.: Ordinary Differential Equations with Applications. Texts in Applied Mathematics, 2nd edn. Springer, Berlin (2006)
Clark, C.W.: Mathematical Bioeconomics: The Optimal Management of Renewable Resources, 2nd edn. Wiley, New York (1990)
Dumortier, F., Llibre, J., Artés, J.C.: Qualitative Theory of Planar Differential Systems. Springer, Berlin (2006)
Freedman, H.I.: Deterministic Mathematical Model in Population Ecology. Marcel Dekker, New York (1980)
Gause, G.F.: The Struggle for Existence. Dover, New York (1934)
González-Olivares, E., Sáez, E., Stange, E., Szantó, I.: Topological description of a non-differentiable bio-economics model. Rocky Mt. J. Math. 35(4), 1133–1155 (2005)
González-Olivares, E., Mena-Lorca, J., Rojas-Palma, A., Flores, J.D.: Dynamical complexities in the Leslie-Gower predator-prey model as consequences of the Allee effect on prey. Appl. Math. Model. 35, 366–381 (2011)
Hesaaraki, M., Moghadas, S.M.: Existence of limit cycles for predator-prey systems with a class of functional responses. Ecol. Model. 142, 1–9 (2001)
Korobeinikov, A.: A Lyapunov function for Leslie-Gower predator-prey models. Appl. Math. Lett. 14, 697–699 (2001)
Leslie, P.H.: Some further notes on the use of matrices in population mathematics. Biometrica 35, 213–245 (1948)
Leslie, P.H., Gower, J.C.: The properties of a stochastic model for the predator-prey type of interaction between two species. Biometricka 47, 219–234 (1960)
May, R.M.: Stability and Complexity in Model Ecosystems, 2nd edn. Princeton University Press, Princeton (2001)
Melchionda, D., Pastacaldi, E., Perri, C., Banerjee, M., Venturino, E.: Social behavior-induced multistability in minimal competitive ecosystems. J. Theor. Biol. 439, 24–38 (2018)
Monzón, P.: Almost global attraction in planar systems. Syst. Control Lett. 54, 753–758 (2005)
Myerscough, M.R., Darwen, M.J., Hogarth, W.L.: Stability, persistence and structural stability in a classical predator-prey model. Ecol. Model. 89, 31–42 (1996)
Perko, L.: Differential Equations and Dynamical Systems, 3rd edn. Springer, Berlin (2001)
Ramos-Jiliberto, R., González-Olivares, E.: Regulaci de la tasa intrínseca de crecimiento poblacional de los depredadores: modificación auna clase de modelos de depredación. Rev. Chil. Hist. Nat. 69, 271–280 (1996). (in spanish)
Rantzer, A.: A dual to Lyapunov’s stability theorem. Syst. Control Lett. 42(3), 161–168 (2001)
Rivera-Estay, V.: Un modelo de Leslie-Gower con respuesta funcional no diferenciable (A Leslie-Gower type model with non-differentiable functional respose). Instituto de Matemáticas at the Pontificia Universidad Católica de Valparaí so, Licenciate final work (2013). in spanish
Rosenzweig, M.L.: Paradox of enrichment: destabilization of exploitation ecosystem in ecological time. Science 171, 385–387 (1971)
Sáez, E., González-Olivares, E.: Dynamics on a predator-prey model. SIAM J. Appl. Math. 59, 1867–1878 (1999)
Sáez, E., Szántó, I.: A polycycle and limit cycles in a non-differentiable predator-prey model. Proc. Indian Acad. Sci. Math. Sci. 117, 219–231 (2007)
Turchin, P.: Complex Population Dynamics: A Theoretical/Empirical Synthesis. Mongraphs in Population Biology. Princeton University Press, Princeton (2003)
Venturino, E., Petrovskii, S.: Spatiotemporal behavior of a prey-predator system with a group defense for prey. Ecol. Complex. 14, 37–47 (2013)
Vilches-Ponce, K., Dinámicas de un modelo de depredaci ón del tipo Gause con respuesta funcional no diferenciable (Dynamics of a Gause type predator-prey model with non-differentiable functional response) Master thesis, Instituto de Matemáticas at the Pontificia Universidad Católica de Valparaíso (2009), in spanish
Vilches, K., González-Olivares, E., Rojas-Palma, A.: Prey herd behavior modeled by a generic non-differential functional response. Math. Model. Nat. Phenom. 13(3), 26 (2018)
Acknowledgements
This work has been sponsorship by Mathematical Modeling and Pattern Recognition (GMMRP), Chile (www.biomatematica.cl). The second author was partially financed by the DIEA-PUCV 124.730/2012 project. The fourth author was supported by Conicyt PAI-Academia 79150021 project.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Rivera-Estay, V., González-Olivares, E., Rojas-Palma, A., Vilches-Ponce, K. (2020). Dynamics of a Class of Leslie–Gower Predation Models with a Non-Differentiable Functional Response. In: Dutta, H., Peters, J. (eds) Applied Mathematical Analysis: Theory, Methods, and Applications. Studies in Systems, Decision and Control, vol 177. Springer, Cham. https://doi.org/10.1007/978-3-319-99918-0_14
Download citation
DOI: https://doi.org/10.1007/978-3-319-99918-0_14
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-99917-3
Online ISBN: 978-3-319-99918-0
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)