International Journal of Advanced and Applied Sciences

Int. j. adv. appl. sci.

EISSN: 2313-3724

Print ISSN: 2313-626X

Volume 4, Issue 7  (July 2017), Pages:  74-79

Title:  Global stability of two-species mutualism model with proportional harvesting

Author(s):  Rusliza Ahmad *


Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA, Perak Branch, Tapah Campus, 35400 Tapah, Perak, Malaysia

Full Text - PDF          XML


This paper deals with the study on a mathematical model consisting of mutualistic interactions among two-species with proportional harvesting. Harvesting function is introduced to describe the rate of removal of the species. The local stability analysis shows that the unique positive equilibrium point is asymptotically stable when certain conditions are satisfied. Global stability is discussed by constructing Lyapunov function. It has been shown that the unique positive equilibrium point is globally asymptotically stable. Finally, numerical simulations supporting theoretical results are also included. 

© 2017 The Authors. Published by IASE.

This is an open access article under the CC BY-NC-ND license (

Keywords: Global stability, Lyapunov function, Mutualism model, Proportional harvesting

Article History: Received 2 March 2017, Received in revised form 28 May 2017, Accepted 7 June 2017

Digital Object Identifier:


Ahmad R (2017). Global stability of two-species mutualism model with proportional harvesting. International Journal of Advanced and Applied Sciences, 4(7): 74-79


Benz R (2000). Ecology and evolution: Islands of change. NSTA Press, Virginia, USA.
Boyce WE and DiPrima RC (1992). Elementary differential equations and boundary value problems. John Wiley and Sons, Hoboken, USA.
Cheng KS, Hsu SB, and Lin SS (1981). Some results on global stability of a predator-prey system. Journal of Mathematical Biology, 12(1): 115-126.
Do KD and Pan J (2009). Control of ships and underwater vehicles. Springer Science and Business Media, Berlin, Germany.
Fay TH and Greeff JC (2006). Lion, wildebeest and zebra: a predator-prey model. Ecological Modeling, 196(1): 237-244.
Georgescu P, Zhang H, and Maxin D (2016). The global stability of coexisting equilibria for three models of mutualism. Mathematical Biosciences and Engineering, 13(1): 101-118.
Goh BS (1979). Stability in models of mutualism. The American Naturist, 113(2): 261-275.
Haberman R (1998). Mathematical models: Mechanical vibrations, population dynamics and traffic flow. Society for Industrial and Applied Mathematics, Pennsylvania, USA.
Hsu FC and Ho CP (2006). Global stability for the lotka-volterra mutualistic system with time delay. Tunghai Science, 8: 81-107.
Janzen DH (1985). The natural history of mutualisms. In: Boucher DL (Ed.), The biology of mutualism: Ecology and evolution: 40-99. Oxford University Press, New York, USA.
Kot M (2001). Elements of mathematical ecology. The Press Syndicate of the University of Cambridge, Cambridge University Press, Cambridge, UK.
León CVD (2012). Lyapunov function for two-species cooperative systems. Applied Mathematics and Computation, 219(5): 2493-2497.
León CVD (2015). Lyapunov functional for global stability of lotka-volterra cooperative systems with discrete delays. Abstraction and Application, 12: 42-50.
Morin PJ (2011). Community ecology. John Wiley and Sons, Hoboken, USA.
Ouncharoen R, Pinjai S, Dumrongpokaphan T, and Lenbury Y (2012). Global stability analysis of predator-prey model with harvesting and delay. Thai Journal of Mathematics, 8(3): 589-605.
Reddy BR, Narayan KL, and Pattabhiramacharyulu NC (2011). On global stability of two mutually interacting species with limited resources for both the species. International Journal of Contemporary Mathematical Sciences, 6(9): 401-407.
Rockwood LL (2015). Introduction to population ecology. John Wiley and Sons, Hoboken, USA.
Saito Y (2002). The necessary and sufficient condition for global stability of a lotka-volterra cooperative or competition system with delays. Journal of Mathematical Analysis and Applications, 268(1): 109-124.
Starr C, Taggart R, Evers C, and Starr L (2015). Biology: The unity and diversity of life. Nelson Education, Toronto, Canada.