TY - JOUR

T1 - Consequences of double Allee effect on the number of limit cycles in a predatorprey model

AU - González-Olivares, Eduardo

AU - González-Yaez, Betsabé

AU - Mena Lorca, Jaime

AU - Rojas-Palma, Alejandro

AU - Flores, José D.

N1 - Funding Information:
The authors wish to thank the members of the Grupo de Ecología Matemática of the Instituto de Matemáticas at the Pontificia Universidad Católica de Valparaíso, for their valuable comments and suggestions. This work is partially financed by Project DIEA-PUCV 047.336/2011 . JDF 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 - 2011/11

Y1 - 2011/11

N2 - The main goal of this work is to show a comparative analysis of simple continuous time predatorprey models considering the Allee effect affecting the prey population, also known as depensation in fisheries sciences. This phenomenon may be expressed by different mathematical forms, yielding a distinct number of limit cycles surrounding a positive equilibrium point, when two of these different formalizations are considered in the same system. It is known that the Volterra predation model, using the most usual form to express the Allee effect, has a unique limit cycle. In this work, considering a more complex mathematical expression, the existence of two limit cycles is proved, by means of the Lyapunov quantities. We argue that the second equation explains the existence of two Allee effects affecting the same population, which could justify the difference observed between the models. These results imply that the election of mathematical formulation can have consequences on the fit of the observed data, thus leading to mistakes for ecologists. We conclude that the oscillatory behaviors and overall dynamics depend strongly on the algebraic expression of the Allee effect, making difficult the proposition of general results. Nevertheless, the techniques reviewed in this paper emerge as key tools to analyze the existence of limit cycles in the presence of multiple Allee effects.

AB - The main goal of this work is to show a comparative analysis of simple continuous time predatorprey models considering the Allee effect affecting the prey population, also known as depensation in fisheries sciences. This phenomenon may be expressed by different mathematical forms, yielding a distinct number of limit cycles surrounding a positive equilibrium point, when two of these different formalizations are considered in the same system. It is known that the Volterra predation model, using the most usual form to express the Allee effect, has a unique limit cycle. In this work, considering a more complex mathematical expression, the existence of two limit cycles is proved, by means of the Lyapunov quantities. We argue that the second equation explains the existence of two Allee effects affecting the same population, which could justify the difference observed between the models. These results imply that the election of mathematical formulation can have consequences on the fit of the observed data, thus leading to mistakes for ecologists. We conclude that the oscillatory behaviors and overall dynamics depend strongly on the algebraic expression of the Allee effect, making difficult the proposition of general results. Nevertheless, the techniques reviewed in this paper emerge as key tools to analyze the existence of limit cycles in the presence of multiple Allee effects.

KW - Allee effect

KW - Bifurcation

KW - Limit cycles

KW - Predatorprey model

KW - Separatrix curve

KW - Stability

UR - http://www.scopus.com/inward/record.url?scp=80054709237&partnerID=8YFLogxK

U2 - 10.1016/j.camwa.2011.08.061

DO - 10.1016/j.camwa.2011.08.061

M3 - Article

AN - SCOPUS:80054709237

SN - 0898-1221

VL - 62

SP - 3449

EP - 3463

JO - Computers and Mathematics with Applications

JF - Computers and Mathematics with Applications

IS - 9

ER -