Bifurcations of a singular prey-predator economic model with time delay and stage structure. (English) Zbl 1198.37138

Summary: We study a singular prey-predator economic model with time delay and stage structure. Compared with other researches on dynamics of prey-predator population, this model is described by differential-algebraic equations due to economic factor. For zero economic profit, this model exhibits three bifurcational phenomena: transcritical bifurcation, Hopf bifurcation and singular induced bifurcation. For positive economic profit, the model undergoes a saddle-node bifurcation at critical value of positive economic profit, and the increase of delay destabilizes the positive equilibrium point of the system and bifurcates into small amplitude periodic solution. Finally, by using Matlab software, numerical simulations illustrate the effectiveness of the results.
Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.


37N25 Dynamical systems in biology
92D25 Population dynamics (general)
34K18 Bifurcation theory of functional-differential equations
91B55 Economic dynamics


Full Text: DOI


[1] Ayasun, S.; Nwankpa, C. O.; Kwatny, H. G., Computation of singular and singularity induced bifurcation points of differential-algebraic power system model, IEEE Trans Circ Syst - I: Fundam Theor Appl, 51, 8, 1525-1538 (2004) · Zbl 1374.34027
[2] Riaza, R., Singularity-induced bifurcations in lumped circuits, IEEE Trans Circ Syst - I: Fundam Theor Appl, 52, 7, 1442-1450 (2005) · Zbl 1374.94923
[3] Luenberger, D. J., Nonsingular descriptor system, J Econ Dynam Contr, 1, 219-242 (1979)
[4] Krishnan, H.; McClamroch, N. H., Tracking in nonlinear differential-algebra control systems with applications to constrained robot systems, Automatica, 30, 1885-1897 (1994) · Zbl 0815.93057
[5] Bloch, A. M.; Reyhanoglu, M.; McClamroch, N. H., Control and stabilization of nonholonomic dynamic systems, IEEE Trans Automat Contr, 37, 1746-1757 (1992) · Zbl 0778.93084
[6] Ling, L.; Wang, W. M., Dynamics of a Ivlev-type predator-prey system with constant rate harvesting, Chaos, Solitons & Fractals, 41, 4, 2139-2153 (2009) · Zbl 1198.34061
[7] Liu, Z. H.; Yuan, R., Stability and bifurcation in a harvested one-predator-two-prey model with delays, Chaos, Solitons & Fractals, 27, 5, 1395-1407 (2006) · Zbl 1097.34051
[8] Zhang, L.; Wang, W. H.; Xue, Y. K., Spatiotemporal complexity of a predator-prey system with constant harvest rate, Chaos, Solitons & Fractals, 41, 1, 38-46 (2009) · Zbl 1198.35127
[9] Das, T.; Mukherjee, R. N.; Chaudhuri, K. S., Harvesting of a prey-predator fishery in the presence of toxicity, Appl Math Model (2008) · Zbl 1185.91120
[10] Gakkhar, S.; Singh, B., The dynamics of a food web consisting of two preys and a harvesting predator, Chaos, Solitons & Fractals, 34, 4, 1346-1356 (2007) · Zbl 1142.34335
[11] Kar, T. K.; Pahari, U. K., Non-selective harvesting in prey-predator models with delay, Commun Nonlinear Sci Numer Simulat, 11, 4, 499-509 (2006) · Zbl 1112.34057
[12] Gordon, H. S., Economic theory of a common property resource: the fishery, J Polit Econ, 63, 116-124 (1954)
[13] Zhang, X.; Zhang, Q. L.; Zhang, Y., Bifurcations of a class of singular biological economic models, Chaos, Solitons & Fractals, 40, 3, 1309-1318 (2009) · Zbl 1197.37129
[14] Xiao, Y. N.; Cheng, D. Z.; Tang, S. Y., Dynamic complexities in predator-prey ecosystem models with age-structure for predator, Chaos, Solitons & Fractals, 14, 9, 1403-1411 (2002) · Zbl 1032.92033
[15] Gao, S. J.; Chen, L. S.; Sun, L. H., Optimal pulse fishing policy in stage-structured models with birth pulses, Chaos, Solitons & Fractals, 25, 5, 1209-1219 (2005) · Zbl 1065.92056
[16] Gao, S. J.; Chen, L. S., The effect of seasonal harvesting on a single-species discrete population model with stage structure and birth pulses, Chaos, Solitons & Fractals, 24, 4, 1013-1023 (2005) · Zbl 1061.92059
[17] Bandyopadhyay, M.; Banerjee, S., A stage-structured prey-predator model with discrete time delay, Appl Math Comput, 182, 2, 1385-1398 (2006) · Zbl 1102.92044
[18] Guckenheimer, J.; Holmes, P., Nonlinear oscillations, dynamical systems, and bifurcations of vector fields (1983), Springer: Springer New York · Zbl 0515.34001
[19] Venkatasubramanian, V.; Schättler, H.; Zaborszky, J., Local bifurcation and feasibility regions in differential-algebraic systems, IEEE Trans Automat Contr, 40, 12, 1992-2013 (1995) · Zbl 0843.34045
[20] Li, F.; Woo, P. Y., Fault detection for linear analog IC - the method of short-circuit admittance parameters, IEEE Trans Circ Syst - I: Fundam Theor Appl, 49, 1, 105-108 (2002)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.