Some new observations for a stage-structured predator-prey model

• Main Authors: Li Wei, Li Xianyi
• Format: Article
• Language: English
• Published: Academic Journals Center of Shanghai Normal University 2017-06-01
• Series: Journal of Shanghai Normal University (Natural Sciences)
• Subjects: stage-structured predator-prey model; center manifold theorem; dynamics at infinity; Poincar&#233 compactification; infinite heteroclinic orbit
• Online Access: http://qktg.shnu.edu.cn/zrb/shsfqkszrb/ch/reader/view_abstract.aspx?file_no=20170311&flag=1

A 3D stage-structured predator-prey model, whose necessary and sufficient conditions for the permanence of two species and the extinction of one species or two species were previously obtained, is revisited in this paper. By using the center manifold theorem, we show that the nonnegative equilibrium point of this system is also locally asymptotically stable in the critical case <i>a</i> = <i>b</i> + <i>ce</i>. Our main new discovery is that the parameter <i>d</i> in this model irrelevant to finite dynamical behaviors of this system plays a role in the dynamical behaviors at infinity of this system. More specially, by using the Poincar&#233; compactication in R³ we make a global analysis for this model, including the complete description of its dynamic behavior on the sphere at infinity. Combining analytical and numerical techniques we show that for the parameters satisfying <i>a</i> ≤ <i>b</i> and 0 < <i>d</i> < 1, the system presents two infinite heteroclinic orbits.