TY - JOUR

T1 - Dynamical complexities in the Leslie-Gower predator-prey model as consequences of the Allee effect on prey

AU - González-Olivares, Eduardo

AU - Mena-Lorca, Jaime

AU - Rojas-Palma, Alejandro

AU - Flores, José D.

N1 - Funding Information:
The authors thank to the members of the Mathematical Ecology Group of the Instituto de Matemáticas at the Pontificia Universidad Católica de Valparaı´so, for their valuable comments and suggestions. EGO was partially supported by the project DII PUCV N°124.720/2009 and JF has been partially supported by grants from the National Science Foundation (NSF – Grant DMPS-0838704 ), the National Security Agency (NSA – Grant H98230-09-1-0104 ), the Alfred P. Sloan Foundation and the Office of the Provost of Arizona State University.

PY - 2011/1

Y1 - 2011/1

N2 - This work deals with the analysis of a predator-prey model derived from the Leslie-Gower type model, where the most common mathematical form to express the Allee effect in the prey growth function is considered.It is well-known that the Leslie-Gower model has a unique globally asymptotically stable equilibrium point. However, it is shown here the Allee effect significantly modifies the original system dynamics, as the studied model involves many non-topological equivalent behaviors.None, one or two equilibrium points can exist at the interior of the first quadrant of the modified Leslie-Gower model with strong Allee effect on prey. However, a collapse may be seen when two positive equilibrium points exist.Moreover, we proved the existence of parameter subsets for which the system can have: a cusp point (Bogdanov-Takens bifurcation), homoclinic curves (homoclinic bifurcation), Hopf bifurcation and the existence of two limit cycles, the innermost stable and the outermost unstable, in inverse stability as they usually appear in the Gause-type predator-prey models.In contrast, the system modelling an special of weak Allee effect, may include none or just one positive equilibrium point and no homoclinic curve; the latter implies a significant difference between the mathematical properties of these forms of the phenomenon, although both systems show some rich and interesting dynamics.

AB - This work deals with the analysis of a predator-prey model derived from the Leslie-Gower type model, where the most common mathematical form to express the Allee effect in the prey growth function is considered.It is well-known that the Leslie-Gower model has a unique globally asymptotically stable equilibrium point. However, it is shown here the Allee effect significantly modifies the original system dynamics, as the studied model involves many non-topological equivalent behaviors.None, one or two equilibrium points can exist at the interior of the first quadrant of the modified Leslie-Gower model with strong Allee effect on prey. However, a collapse may be seen when two positive equilibrium points exist.Moreover, we proved the existence of parameter subsets for which the system can have: a cusp point (Bogdanov-Takens bifurcation), homoclinic curves (homoclinic bifurcation), Hopf bifurcation and the existence of two limit cycles, the innermost stable and the outermost unstable, in inverse stability as they usually appear in the Gause-type predator-prey models.In contrast, the system modelling an special of weak Allee effect, may include none or just one positive equilibrium point and no homoclinic curve; the latter implies a significant difference between the mathematical properties of these forms of the phenomenon, although both systems show some rich and interesting dynamics.

KW - Allee effect

KW - Bifurcations

KW - Functional response

KW - Leslie-Gower predator-prey models

KW - Limit cycle

KW - Separatrix curves

UR - http://www.scopus.com/inward/record.url?scp=77955713687&partnerID=8YFLogxK

U2 - 10.1016/j.apm.2010.07.001

DO - 10.1016/j.apm.2010.07.001

M3 - Article

AN - SCOPUS:77955713687

VL - 35

SP - 366

EP - 381

JO - Applied Mathematical Modelling

JF - Applied Mathematical Modelling

SN - 0307-904X

IS - 1

ER -