Bifurcations and global dynamics in a predator-prey model with a strong Allee effect on the prey, and a ratio-dependent functional response

被引:46
作者
Aguirre, Pablo [1 ,2 ]
Flores, Jose D. [3 ]
Gonzalez-Olivares, Eduardo [4 ]
机构
[1] Univ Tecn Federico Santa Maria, Dept Med, Grp Anal & Math Modeling Valparaiso AM2V, Valparaiso, Chile
[2] Univ Tecn Federico Santa Maria, Dept Med, Grp Anal & Math Modeling Valparaiso AM2V, Valparaiso, Chile
[3] Univ S Dakota, Dept Math Sci, Vermillion, SD 57069 USA
[4] Pontificia Univ Catolica Valparaiso, Inst Matemat, Grp Ecol Matemat, Valparaiso, Chile
基金
美国国家科学基金会;
关键词
LIMIT-CYCLES; EXTINCTION; CONSEQUENCES; DRIVEN; HETEROGENEITY; SYSTEM;
D O I
10.1016/j.nonrwa.2013.10.002
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
We extend a previous study of a predator-prey model with strong Allee effect on the prey in which the functional response is a function of the ratio of prey to predator. We prove that the solutions are always bounded and non-negative, and that the species can always tend to long-term extinction. By means of bifurcation analysis and advanced numerical techniques for the computation of invariant manifolds of equilibria, we explain the consequences of the (dis)appearance of limit cycles, homoclinic orbits, and heteroclinic connections in the global arrangement of the phase plane near a Bogdanov-Takens bifurcation. In particular, we find that the Allee threshold in the two-dimensional system is given as the boundary of the basin of attraction of an attracting positive equilibrium, and determine conditions for the mutual extinction or survival of the populations. (C) 2013 Elsevier Ltd. All rights reserved.
引用
收藏
页码:235 / 249
页数:15
相关论文
共 58 条
[1]   Stochastic predator-prey model with Allee effect on prey [J].
Aguirre, Pablo ;
Gonzalez-Olivares, Eduardo ;
Torres, Soledad .
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS, 2013, 14 (01) :768-779
[2]   INVESTIGATING THE CONSEQUENCES OF GLOBAL BIFURCATIONS FOR TWO-DIMENSIONAL INVARIANT MANIFOLDS OF VECTOR FIELDS [J].
Aguirre, Pablo ;
Doedel, Eusebius J. ;
Krauskopf, Bernd ;
Osinga, Hinke M. .
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS, 2011, 29 (04) :1309-1344
[3]   THREE LIMIT CYCLES IN A LESLIE-GOWER PREDATOR-PREY MODEL WITH ADDITIVE ALLEE EFFECT [J].
Aguirre, Pablo ;
Gonzalez-Olivares, Eduardo ;
Saez, Eduardo .
SIAM JOURNAL ON APPLIED MATHEMATICS, 2009, 69 (05) :1244-1262
[4]   Two limit cycles in a Leslie-Gower predator-prey model with additive Allee effect [J].
Aguirrea, Pablo ;
Gonzalez-Olivares, Eduardo ;
Saez, Eduardo .
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS, 2009, 10 (03) :1401-1416
[5]  
[Anonymous], 2004, ELEMENTS APPL BIFURC
[6]  
[Anonymous], 2010, AUTO-07P: Continuation and Bifurcation Software for Ordinary Differential Equations
[7]  
[Anonymous], 2003, The Struggle for Existence
[8]   THE BIOLOGICAL-CONTROL PARADOX [J].
ARDITI, R ;
BERRYMAN, AA .
TRENDS IN ECOLOGY & EVOLUTION, 1991, 6 (01) :32-32
[9]   COUPLING IN PREDATOR PREY DYNAMICS - RATIO-DEPENDENCE [J].
ARDITI, R ;
GINZBURG, LR .
JOURNAL OF THEORETICAL BIOLOGY, 1989, 139 (03) :311-326
[10]   EMPIRICAL-EVIDENCE OF THE ROLE OF HETEROGENEITY IN RATIO-DEPENDENT CONSUMPTION [J].
ARDITI, R ;
SAIAH, H .
ECOLOGY, 1992, 73 (05) :1544-1551