tailieunhanh - Dynamics of a predator-prey system with a mate-finding Allee effect on prey

The mathematical results show that the system is highly sensitive to the parameters and initial status. It exhibits a stable limit cycle, or different types of heteroclinic curves, or a homoclinic loop when parameters take suitable values. | Turk J Math (2017) 41: 585 – 604 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Dynamics of a predator-prey system with a mate-finding Allee effect on prey Ruiwen WU, Xiuxiang LIU∗ School of Mathematical Sciences, South China Normal University, Guangzhou, . China Received: • Accepted/Published Online: • Final Version: Abstract: We consider a predator–prey system with nonmonotonic functional response and a hyperbolic type of matefinding Allee effect on prey. A detailed mathematical analysis of the system, including the stability and a series of bifurcations (a saddle-node, a Hopf, and a Bogdanov–Takens bifurcation), has been given. The mathematical results show that the system is highly sensitive to the parameters and initial status. It exhibits a stable limit cycle, or different types of heteroclinic curves, or a homoclinic loop when parameters take suitable values. Key words: Predator–prey system, mate-finding Allee effect, nonmonotonic functional response, Bogdanov–Takens bifurcation, heteroclinic curve, homoclinic loop 1. Introduction The dynamical behaviors of predator–prey systems subject to Allee effects have gained a lot of attention in the current study of mathematical biology [6, 7, 11, 13, 30]. Allee effects describe a positive relationship between (a component of) individual fitness and population density [1, 2, 32], also known as inverse density dependence, positive density dependence, or depensation and depensatory dynamics in fisheries literature. The Allee effect implies a scenario in which, at low densities, a population subject to an Allee effect may go to extinction; thus, the study of predator–prey systems subject to Allee effects has a profound significance on conservation [14, 18, 19], the management of endangered species [10], biological invasions [6, 8], pest control [17], etc. A range of mechanisms may lead to Allee effects, .