Analysis of a harvested discrete-time biological models

Document Type : Research Paper


Department of Mathematics, College of Science, University of Baghdad, Baghdad, Iraq


This work aims to analyze a three-dimensional discrete-time biological system, a prey-predator model with a constant harvesting amount. The stage structure lies in the predator species.  This analysis is done by finding all possible equilibria and investigating their stability. In order to get an optimal harvesting strategy, we suppose that harvesting is to be a non-constant rate. Finally, numerical simulations are given to confirm the outcome of mathematical analysis.


[1] A. S. Ackleh and P. Zhang, Competitive exclusion in a discrete stage-structured two species model, Math. Model.
Nat. Phenom. 4 (2009) 156-–175.
[2] D.C. Alberto and H. Onesimo The maximum principle for discrete-time control systems and applications to
dynamic games, J. Math. Anal. Appl. 475 (2019) 253–277.
[3] W.G. Aiello and H.I. Freedman A time-delay model of single-species growth with stage structure, Math. Biosci.
101 (1990) 139-153.
[4] C. Clark Mathematical Bioeconomics: The Optimal Management of Renewable Resources, 2nd edition, Wiley,
[5] J. Cui, L. Chen and W. Wang The effect of dispersal on population growth with stage-structure, Comput. Math.
Appl. 39 (2000) 91–102.
[6] J.M. Cushing An Introduction to Structured Population Dynamics, CBMSNSF Regional Conf. Ser. Appl. Math.
71, SIAM, Philadelphia, 1998.
[7] S.N. Elaydi Discrete Chaos with Applications in Science and Engineering, Chapman and Hall/CRC, 1999.
[8] S.A. Gourley and Y. Kuang A stage structured predator-prey model and its dependence on through stage delay
and death rate, J. Math. Biol. 49 (2004) 188–200.
[9] Ghosoon M. and Sadiq Al-Nassir Dynamics and an Optimal Policy for A Discrete Time System with Ricker
Growth, Iraqi J. Sci. 60(1) (2019) 135–142.
[10] E. Jung, S. Lenhart, V. Protopopescu and C.F. Babbs, Optimal control theory applied to a difference equation
model for cardiopulmonary resuscitation, Math. Mod. Meth. Appl. Sci. 15 (2005) 1519–1531.
[11] S. Lenhart and J. Workman, Optimal Control Applied to Biological Models, Chapman Hall/CRC,Boca Raton,
[12] V. Leis, D. Vrabie and V.L. Symos, Optimal Control, Wiley-Interscience, New York, 1986.
[13] R.S. Mustafa and Orlando Merino, Discrete Dynamical Systems and Difference Equations with Mathematica,
Chapman, and Hall/CRC, 2002.
[14] D. Kamel Dynamics in a discrete-time three dimensional cancer system, Int. J. Appl. Math. 49(4) (2019) IJAM49-4-31.
[15] O. K. Shalsh and S. Al-Nassir, Dynamics and optimal Harvesting strategy for biological models with Beverton
Holt growth, Iraqi J. Sci. 2020 (Special Issue) 223–232.
[16] L. S. Pontryagin V. S. Boltyanskii R. V. Gamkrelidze and E. Mishchenko The Mathematical Theory of Optimal
Processes, Wiley-Inter science, New York, 1962.
[17] S.P. Sethi and G. L. Thompson, Optimal Control Theory: Applications to Management Science and Economics,
2nd edition, Springer 2009.
[18] S. Tang, and L. Chen, The effect of seasonal harvesting on stage-structured population models, J. Math. Biol. 48
(2004) 357-374.
[19] S. Tang, R.A. Cheke and Y. Xiao, Optimal impulsive harvesting on non-autonomous Beverton-Holt difference
equations, Nonlinear Anal. 65 (2006) 2311–2341.
[20] O. Tahvonen Economics of harvesting age-structured fish populations, J. Envir. Econ. Manag. 58(3) (2009) 281–
[21] W.D. Wang, and L.S. Chen, A predator-prey system with stage-structure for predator, Comput. Math. Appl. 33
(1997) 83–91.
[22] D. Wandi, S. Lenhart and H. Behncke, Discrete time optimal harvesting of fish populations with age structure,
Lett. Biomath. Int. J. 2 (2014) 193–207.
[23] X.A. Zhang, L.S. Chen and A.U. Neumann, The stage-structured predator-prey model and optimal harvesting
policy, Math. Biosci. 168 (2000) 201–210.
Volume 12, Issue 2
November 2021
Pages 2235-2246
  • Receive Date: 01 April 2021
  • Revise Date: 16 June 2021
  • Accept Date: 12 August 2021