American Institute of Mathematical Sciences

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September  2008, 10(4): 857-871. doi: 10.3934/dcdsb.2008.10.857

Global dynamics of a Predator-Prey model with Hassell-Varley Type functional response

Sze-Bi Hsu 1, , Tzy-Wei Hwang 2, and  Yang Kuang 3,

1. 

Department of Mathematics, National Tsing Hua University, Hsinchu 300
2. 

Department of Mathematics, National Chung Cheng University, Min-Hsiung, Chia-Yi 621, Taiwan
3. 

Department of Mathematics, Arizona State University, Tempe, AZ 85287-1804

Received  August 2007 Revised  February 2008 Published  August 2008

Predator-prey models with Hassell-Varley type functional response are appropriate for interactions where predators form groups and have applications in biological control. Here we present a systematic global qualitative analysis to a general predator-prey model with Hassell-Varley type functional response. We show that the predator free equilibrium is a global attractor only when the predator death rate is greater than its growth ability. The positive equilibrium exists if the above relation reverses. In cases of practical interest, we show that the local stability of the positive steady state implies its global stability with respect to positive solutions. For terrestrial predators that form a fixed number of tight groups, we show that the existence of an unstable positive equilibrium in the predator-prey model implies the existence of an unique nontrivial positive limit cycle.
Keywords: limit cycles, extinction., Functional response, predator-prey model, global stability.
Mathematics Subject Classification: 34D05, 34D20, 92D2.
Citation: Sze-Bi Hsu, Tzy-Wei Hwang, Yang Kuang. Global dynamics of a Predator-Prey model with Hassell-Varley Type functional response. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 857-871. doi: 10.3934/dcdsb.2008.10.857
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