## Bifurcations and periodic solutions for an algae-fish semicontinuous system.(English)Zbl 1470.34115

Summary: We propose an algae-fish semicontinuous system for the Zeya Reservoir to study the control of algae, including biological and chemical controls. The bifurcation and periodic solutions of the system were studied using a Poincaré map and a geometric method. The existence of order-1 periodic solution of the system is discussed. Based on previous analysis, we investigated the change in the location of the order-1 periodic solution with variable parameters and we described the transcritical bifurcation of the system. Finally, we provided a series of numerical results to illustrate the feasibility of the theoretical results. These results may help to facilitate a better understanding of algal control in the Zeya Reservoir.

### MSC:

 34C25 Periodic solutions to ordinary differential equations 34A36 Discontinuous ordinary differential equations 34C60 Qualitative investigation and simulation of ordinary differential equation models 92D40 Ecology 34A37 Ordinary differential equations with impulses
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 [1] Lakshmikantham, V.; Baĭnov, D. D.; Simeonov, P. S., Theory of Impulsive Differential Equations. Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics, 6, xii+273, (1989), Teaneck, NJ, USA: World Scientific, Teaneck, NJ, USA · Zbl 0719.34002 [2] Baĭnov, D. D.; Simeonov, P. S., Impulsive Differential Equations: Asymptotic Properties of the Solutions. Impulsive Differential Equations: Asymptotic Properties of the Solutions, Series on Advances in Mathematics for Applied Sciences, 28, xii+230, (1995), River Edge, NJ, USA: World Scientific, River Edge, NJ, USA · Zbl 0828.34002 [3] Bainov, D.; Covachev, V., Impulsive Differential Equations with a Small Parameter. Impulsive Differential Equations with a Small Parameter, Series on Advances in Mathematics for Applied Sciences, 24, x+268, (1994), River Edge, NJ, USA: World Scientific, River Edge, NJ, USA · Zbl 0828.34001 [4] Dai, C.; Zhao, M.; Chen, L., Complex dynamic behavior of three-species ecological model with impulse perturbations and seasonal disturbances, Mathematics and Computers in Simulation, 84, 83-97, (2012) · Zbl 1257.92040 [5] Liu, X.; Chen, L., Complex dynamics of Holling type II Lotka-Volterra predator-prey system with impulsive perturbations on the predator, Chaos, Solitons & Fractals, 16, 2, 311-320, (2003) · Zbl 1085.34529 [6] Baek, H.; Do, Y., Seasonal effects on a Beddington-DeAngelis type predator-prey system with impulsive perturbations, Abstract and Applied Analysis, 2009, (2009) · Zbl 1187.34058 [7] Shao, Y.; Li, P.; Tang, G., Dynamic analysis of an impulsive predator-prey model with disease in prey and Ivlev-type functional response, Abstract and Applied Analysis, 2012, (2012) · Zbl 1246.92028 [8] Yu, H.; Zhong, S.; Agarwal, R. P., Mathematics and dynamic analysis of an apparent competition community model with impulsive effect, Mathematical and Computer Modelling, 52, 1-2, 25-36, (2010) · Zbl 1201.34018 [9] Chen, L. S., Pest control and geometric theory of semi-dynamical systems, Journal of Beihua University (Natural Science), 12, 1-9, (2011) [10] Dai, C.; Zhao, M.; Chen, L., Homoclinic bifurcation in semi-continuous dynamic systems, International Journal of Biomathematics, 5, 6, (2012) · Zbl 1305.34065 [11] Simeonov, P. S.; Baĭnov, D. D., Orbital stability of periodic solutions of autonomous systems with impulse effect, International Journal of Systems Science, 19, 12, 2561-2585, (1988) · Zbl 0669.34044 [12] Baek, H., The dynamics of a predator-prey system with state-dependent feedback control, Abstract and Applied Analysis, 2012, (2012) · Zbl 1243.34061 [13] Tang, S.; Cheke, R. A., State-dependent impulsive models of integrated pest management (IPM) strategies and their dynamic consequences, Journal of Mathematical Biology, 50, 3, 257-292, (2005) · Zbl 1080.92067 [14] Dai, C.; Zhao, M.; Chen, L., Dynamic complexity of an Ivlev-type prey-predator system with impulsive state feedback control, Journal of Applied Mathematics, 2012, (2012) · Zbl 1256.34032 [15] Nie, L.; Peng, J.; Teng, Z.; Hu, L., Existence and stability of periodic solution of a Lotka-Volterra predator-prey model with state dependent impulsive effects, Journal of Computational and Applied Mathematics, 224, 2, 544-555, (2009) · Zbl 1162.34007 [16] Bianca, C.; Rondoni, L., The nonequilibrium Ehrenfest gas: a chaotic model with flat obstacles?, Chaos, 19, 1, (2009) · Zbl 1311.76118 [17] Rasband, S. N., Chaotic Dynamics of Nonlinear Systems, x+230, (1990), New York, NY, USA: John Wiley & Sons, New York, NY, USA · Zbl 0863.58053 [18] Bianca, C.; Pennisi, M., The triplex vaccine effects in mammary carcinoma: a nonlinear model in tune with SimTriplex, Nonlinear Analysis: Real World Applications, 13, 4, 1913-1940, (2012) · Zbl 1401.92139
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