Global qualitative analysis of a discrete host‐parasitoid model with refuge and strong Allee effects

• Published in: Mathematical methods in the applied sciences Vol. 41; no. 5; pp. 2039 - 2062
• Main Authors: Wu, Daiyong, Zhao, Hongyong
• Format: Journal Article
• Language: English
• Published: Freiburg Wiley Subscription Services, Inc 30.03.2018
• Subjects: Allee effect; Bifurcations; Computer simulation; Economic models; Equilibrium; global asymptotic stability; host‐parasitoid model; Manifolds; Mathematical models; Qualitative analysis; refuge; transcritical bifurcation; uniform persistence
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.4731

In this paper, we discuss the qualitative behavior of a discrete host‐parasitoid model with the host subject to refuge and strong Allee effects. More precisely, we study the local and global asymptotic stability, stable manifolds and unstable manifolds of boundary equilibrium points, existence and unique positive equilibrium point, local and global behavior of the positive equilibrium point, and the uniform persistence for the model with the host subject to the refuge or both refuge and strong Allee effects. It is also proved that the model undergoes a transcritical bifurcation in a small neighborhood of the boundary equilibrium point. Some numerical simulations are given to support our theoretical results. We can obtain that the addition of the refuge may make the parasitoids go extinct while the hosts survive or may stabilize the host‐parasitoid interaction; the addition of both refuge and strong Allee effects has either a negative or positive impact on the coexistence of both populations.