Access the full text.
Sign up today, get DeepDyve free for 14 days.
R. Ghaziani, J. Alidousti, A. Eshkaftaki (2016)
Stability and dynamics of a fractional order Leslie-Gower prey-predator modelApplied Mathematical Modelling, 40
P. Turchin (2003)
Complex Population Dynamics
P. Neher, C. Clark (1978)
Mathematical Bioeconomics. The Optimal Management of Renewable Resources.The Economic Journal, 88
E. Doedel (2007)
Lecture Notes on Numerical Analysis of Nonlinear Equations
Anuraj Singh, S. Gakkhar (2014)
Stabilization of Modified Leslie–Gower Prey–Predator ModelDifferential Equations and Dynamical Systems, 22
M. Haque, N. Ali, S. Chakravarty (2013)
Study of a tri-trophic prey-dependent food chain model of interacting populations.Mathematical biosciences, 246 1
Pablo Aguirre, E. González‐Olivares, E. Sáez (2009)
Three Limit Cycles in a Leslie--Gower Predator-Prey Model with Additive Allee EffectSIAM J. Appl. Math., 69
J. Guckenheimer, P. Holmes (1983)
Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, 42
M. Rosenzweig (1971)
Paradox of Enrichment: Destabilization of Exploitation Ecosystems in Ecological TimeScience, 171
K. McCann, P. Yodzis (1995)
Bifurcation Structure of a Three-Species Food-Chain ModelTheoretical Population Biology, 48
D. Mukherjee (2014)
The Effect of Prey Refuges on a Three Species Food Chain ModelDifferential Equations and Dynamical Systems, 22
A. Hastings, T. Powell (1991)
Chaos in a Three-Species Food ChainEcology, 72
Swarnali Sharma, G. Samanta (2014)
Dynamical Behaviour of a Two Prey and One Predator SystemDifferential Equations and Dynamical Systems, 22
E. González‐Olivares, E. Sáez, Eduardo Stange, I. Szántó (2005)
Topological Description of a Non-Differentiable Bioeconomics ModelRocky Mountain Journal of Mathematics, 35
E. González‐Olivares, Betsabé González-Yañez, J. Mena-Lorca, J. Flores (2013)
Uniqueness of limit cycles and multiple attractors in a Gause-type predator-prey model with nonmonotonic functional response and Allee effect on prey.Mathematical biosciences and engineering : MBE, 10 2
J. Françoise, R. Roussarie (1990)
Bifurcations of planar vector fields, 1455
Banshidhar Sahoo, S. Poria (2014)
Oscillatory Coexistence of Species in a Food Chain Model With General Holling InteractionsDifferential Equations and Dynamical Systems, 22
K. Has'ik (2010)
On a predator–prey system of Gause typeJournal of Mathematical Biology, 60
Y. Kuznetsov, O. Feo, S. Rinaldi (2001)
Belyakov Homoclinic Bifurcations in a Tritrophic Food Chain ModelSIAM J. Appl. Math., 62
Pablo Aguirre, B. Krauskopf, H. Osinga (2013)
Global Invariant Manifolds Near Homoclinic Orbits to a Real Saddle: (Non)Orientability and Flip BifurcationSIAM J. Appl. Dyn. Syst., 12
P. Leslie (1948)
SOME FURTHER NOTES ON THE USE OF MATRICES IN POPULATION MATHEMATICSBiometrika, 35
E. González‐Olivares, J. Mena-Lorca, A. Rojas‐Palma, J. Flores (2011)
Dynamical complexities in the Leslie–Gower predator–prey model as consequences of the Allee effect on preyApplied Mathematical Modelling, 35
Marina Schroder (2016)
Nonlinear Dynamics Of Interacting Populations
E. Sáez, I. Szántó (2007)
A polycycle and limit cycles in a non-differentiable predator-prey modelProceedings Mathematical Sciences, 117
Pablo Aguirre (2014)
A general class of predation models with multiplicative Allee effectNonlinear Dynamics, 78
R. Hannesson (1993)
Bioeconomic Analysis of Fisheries
(2010)
AUTO-07p Version 0.7: Continuation and Bifurcation Software for Ordinary Differential Equations
J. Kuznecov (1998)
Elements of applied bifurcation theory
Kie Saputra, L. Veen, Gilles Quispel (2010)
The saddle-node-transcritical bifurcation in a population model with constant rate harvestingarXiv: Dynamical Systems
C. Chiu, S. Hsu (1998)
Extinction of top-predator in a three-level food-chain modelJournal of Mathematical Biology, 37
(1986)
Adaptive Management of Renewable Fisheries
(2006)
Equilibriumizing all food chain through reproductive efficiency
B. Deng, G. Hines (2002)
Food chain chaos due to Shilnikov's orbit.Chaos, 12 3
E. Sáez, Eduardo Stange, I. Szántó, E. González‐Olivares, M. Falconi (2015)
Chaotic dynamics and coexistence in a three species interaction modelInternational Journal of Biomathematics, 08
Pablo Aguirre (2015)
Bifurcations of Two-Dimensional Global Invariant Manifolds near a Noncentral Saddle-Node Homoclinic OrbitSIAM J. Appl. Dyn. Syst., 14
E. González‐Olivares, A. Rojas‐Palma (2013)
Allee Effect in Gause Type Predator-Prey Models: Existence of Multiple Attractors, Limit cycles and Separatrix Curves. A Brief ReviewMathematical Modelling of Natural Phenomena, 8
We study a model of three interacting species in a food chain composed by a prey, an specific predator and a generalist predator. The capture of the prey by the specific predator is modelled as a modified Holling-type II non-differentiable functional response. The other predatory interactions are both modelled as Holling-type I. Moreover, our model follows a Leslie-Gower approach, in which the function that models the growth of each predator is of logistic type, and the corresponding carrying capacities depend on the sizes of their associated available preys. The resulting model has the form of a set of nonlinear ordinary differential equations which includes a non-differentiable term. By means of topological equivalences and suitable changes of parameters, we find that there exists an Allee threshold for the survival of the prey population in the food chain, given, effectively, as a critical level for the generalist predator. The dynamics of the model is studied with analytical and computational tools for bifurcation theory. We present two-parameter bifurcation diagrams that contain both local phenomena (Hopf, saddle-node transcritical, cusp, Bogdanov-Takens bifurcations) and global events (homoclinic and heteroclinic connections). In particular, we find that two types of heteroclinic cycles can be formed, both of them containing connections to the origin. One of these cycles is planar involving the absence of the specific predator. In turn, the other heteroclinic cycle is formed by connections in the full three-dimensional phase space.
Acta Applicandae Mathematicae – Springer Journals
Published: Feb 24, 2020
Read and print from thousands of top scholarly journals.
Already have an account? Log in
Bookmark this article. You can see your Bookmarks on your DeepDyve Library.
To save an article, log in first, or sign up for a DeepDyve account if you don’t already have one.
Copy and paste the desired citation format or use the link below to download a file formatted for EndNote
Access the full text.
Sign up today, get DeepDyve free for 14 days.
All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.