TY - JOUR

T1 - Multiple stability and uniqueness of the limit cycle in a Gause-type predatorprey model considering the Allee effect on prey

AU - González-Olivares, Eduardo

AU - Meneses-Alcay, Héctor

AU - González-Yaez, Betsabé

AU - Mena-Lorca, Jaime

AU - Rojas-Palma, Alejandro

AU - Ramos-Jiliberto, Rodrigo

PY - 2011/12

Y1 - 2011/12

N2 - In this work, a bidimensional differential equation system obtained by modifying the well-known predatorprey RosenzweigMacArthur model is analyzed by considering prey growth influenced by the Allee effect. One of the main consequences of this modification is a separatrix curve that appears in the phase plane, dividing the behavior of the trajectories. The results show that the equilibrium in the origin is an attractor for any set of parameters. The unique positive equilibrium, when it exists, can be either an attractor or a repeller surrounded by a limit cycle, whose uniqueness is established by calculating the Lyapunov quantities. Therefore, both populations could either reach deterministic extinction or long-term deterministic coexistence. The existence of a heteroclinic curve is also proved. When this curve is broken by changing parameter values, then the origin turns out to be an attractor for all orbits in the phase plane. This implies that there are plausible conditions where both populations can go to extinction. We conclude that strong and weak Allee effects on prey population exert similar influences on the predatorprey model, thereby increasing the risk of ecological extinction.

AB - In this work, a bidimensional differential equation system obtained by modifying the well-known predatorprey RosenzweigMacArthur model is analyzed by considering prey growth influenced by the Allee effect. One of the main consequences of this modification is a separatrix curve that appears in the phase plane, dividing the behavior of the trajectories. The results show that the equilibrium in the origin is an attractor for any set of parameters. The unique positive equilibrium, when it exists, can be either an attractor or a repeller surrounded by a limit cycle, whose uniqueness is established by calculating the Lyapunov quantities. Therefore, both populations could either reach deterministic extinction or long-term deterministic coexistence. The existence of a heteroclinic curve is also proved. When this curve is broken by changing parameter values, then the origin turns out to be an attractor for all orbits in the phase plane. This implies that there are plausible conditions where both populations can go to extinction. We conclude that strong and weak Allee effects on prey population exert similar influences on the predatorprey model, thereby increasing the risk of ecological extinction.

KW - Allee effect

KW - Bifurcations

KW - Ecological extinction

KW - Heteroclinic curve

KW - Limit cycle

KW - Predatorprey model

UR - http://www.scopus.com/inward/record.url?scp=80052031437&partnerID=8YFLogxK

U2 - 10.1016/j.nonrwa.2011.04.003

DO - 10.1016/j.nonrwa.2011.04.003

M3 - Article

AN - SCOPUS:80052031437

VL - 12

SP - 2931

EP - 2942

JO - Nonlinear Analysis: Real World Applications

JF - Nonlinear Analysis: Real World Applications

SN - 1468-1218

IS - 6

ER -