Flip bifurcation and Neimark-Sacker bifurcation in a discrete predator-prey model with harvesting. (English) Zbl 1433.37082
Summary: In this paper, a difference-algebraic predator-prey model is proposed, and its complex dynamical behaviors are analyzed. The model is a discrete singular system, which is obtained by using Euler scheme to discretize a differential-algebraic predator-prey model with harvesting that we establish. Firstly, the local stability of the interior equilibrium point of proposed model is investigated on the basis of discrete dynamical system theory. Further, by applying the new normal form of difference-algebraic equations, center manifold theory and bifurcation theory, the Flip bifurcation and Neimark-Sacker bifurcation around the interior equilibrium point are studied, where the step size is treated as the variable bifurcation parameter. Lastly, with the help of Matlab software, some numerical simulations are performed not only to validate our theoretical results, but also to show the abundant dynamical behaviors, such as period-doubling bifurcations, period 2, 4, 8, and 16 orbits, invariant closed curve, and chaotic sets. In particular, the corresponding maximum Lyapunov exponents are numerically calculated to corroborate the bifurcation and chaotic behaviors.
MSC:
| 37N25 | Dynamical systems in biology |
| 92D25 | Population dynamics (general) |
| 39A28 | Bifurcation theory for difference equations |
Software:
MatlabReferences:
| [1] | Chen, L. S., Mathematical Models and Methods in Ecology, 2nd edn. (Science Press, Beijing, 2017) (in Chinese). |
| [2] | Murray, J. D., Mathematical Biology: I. An Introduction, 3rd edn. (Springer, New York, 2002). · Zbl 1006.92001 |
| [3] | Turchin, P., Complex Population Dynamics: A Theoretical/Empirical Synthesis (Princeton University Press, Princeton, 2003). · Zbl 1062.92077 |
| [4] | Das, K. P., Kundu, K. and Chattopadhyay, J., A predator-prey mathematical model with both the populations affected by diseases, Ecol. Complex.8 (2011) 68-80. |
| [5] | Tang, S., Liang, J., Xiao, Y. and Cheke, R. A., Sliding bifurcations of Filippov two stage pest control models with economic thresholds, SIAM J. Appl. Math.72 (2012) 1061-1080. · Zbl 1256.34038 |
| [6] | Zhang, G. D., Zeng, Z. G. and Hu, J. H., New results on global exponential dissipativity analysis of memristive inertial neural networks with distributed time-varying delays, Neural Netw.97 (2018) 183-191. · Zbl 1442.34121 |
| [7] | Zhang, G. D., Hu, J. H. and Zeng, Z. G., New criteria on global stabilization of delayed memristive neural networks with inertial item, IEEE Trans. Cyberne (2019). https://doi.org/10.1109/TCYB.2018.2889653 |
| [8] | Berryman, A. A., The origins and evolution of predator-prey theory, Ecology73 (1992) 1530-1535. |
| [9] | van Baalen, M., Krivan, V., van Rijn, P. C. J. and Sabelis, M. W., Alternative food, switching predators, and the persistence of predator-prey systems, Ameri. Nat.157 (2001) 512-524. |
| [10] | van Rijn, P. C. J., van Houten, Y. M. and Sabelis, M. W., How plants benefit from providing food to predators even when it is also edible to herbivores, Ecology83 (2002) 2664-2679. |
| [11] | Wang, X. and Song, X. Y., A predator-prey system with two impulses on the diseased prey and a Beddington-DeAngelis response, Math. Methods Appl. Sci.33 (2010) 303-312. · Zbl 1188.34061 |
| [12] | Swift, R. J., A stochastic predator-prey model, Irish Math. Soc. Bull.48 (2002) 57-63. · Zbl 1267.60084 |
| [13] | Mukhopadhyay, B. and Bhattacharyya, R., Effects of deterministic and random refuge in a prey-predator model with parasite infection, Math. Biosci.239 (2012) 124-130. · Zbl 1312.92036 |
| [14] | Xiao, D. and Jennings, L., Bifurcations of a ratio-dependent predator-prey system with constant rate harvesting, SIAM J. Appl. Math.65 (2005) 737-753. · Zbl 1094.34024 |
| [15] | Sharma, S. and Samanta, G. P., Optimal harvesting of a two species competition model with imprecise biological parameters, Nonlinear Dyn.77 (2014) 1101-1119. · Zbl 1331.92134 |
| [16] | Zhang, G. D., Shen, Y. and Chen, B. S., Hopf bifurcation of a predator-prey system with predator harvesting and two delays, Nonlinear Dyn.73 (2013) 2119-2131. · Zbl 1281.92076 |
| [17] | Zhang, G. D., Shen, Y. and Chen, B. S., Positive periodic solutions in a non-selective harvesting predator-prey model with multiple delays, J. Math. Anal. Appl.395 (2012) 298-306. · Zbl 1256.34073 |
| [18] | Zhang, G. D., Zhu, L. L. and Chen, B. S., Hopf bifurcation in a delayed differential-algebraic biological economic system, Nonlinear Anal. Real World Appl.12 (2011) 1708-1719. · Zbl 1221.34227 |
| [19] | Zhang, G. D., Zhu, L. L. and Chen, B. S., Hopf bifurcation and stability for a differential-algebraic biological economic system, Appl. Math. Comput.217 (2010) 330-338. · Zbl 1197.92051 |
| [20] | Al-Omari, J. F. M., The effect of state dependent delay and harvesting on a stage-structured predator-prey model, Appl. Math. Comput.271 (2015) 142-153. · Zbl 1410.92093 |
| [21] | Panja, P., Stability and dynamics of a fractional-order three-species predator-prey model, Theory Biosci.138 (2019) 251-259. |
| [22] | Panja, P. and Mondal, S. K., Stability analysis of coexistence of three species prey-predator model, Nonlinear Dyn.81 (2015) 373-382. · Zbl 1347.37139 |
| [23] | Panja, P., Poria, S. and Mondal, S. K., Analysis of a harvested tritrophic food chain model in the presence of additional food for top predator, Int. J. Biomath.11 (2018) 1850059. https://doi.org/10.1142/S1793524518500596 · Zbl 1390.92118 |
| [24] | Beddington, J. R., Free, C. A. and Lawton, J. H., Dynamic complexity in predator-prey models framed in difference equations, Nature255 (1975) 58-60. |
| [25] | Fazly, M. and Hesaaraki, M., Periodic solutions for a discrete time predator-prey system with monotone functional responses, C. R. Acad. Sci. Paris, Ser. I345 (2007) 199-202. · Zbl 1122.39005 |
| [26] | Hu, Z. Y., Teng, Z. D. and Zhang, L., Stability and bifurcation analysis of a discrete predator-prey model with nonmonotonic functional response, Nonlinear Anal. Real World Appl.12 (2011) 2356-2377. · Zbl 1215.92063 |
| [27] | Chen, Q. L. and Teng, Z. D., Codimension-two bifurcation analysis of a discrete predator-prey model with nonmonotonic functional response, J. Difference Equ. Appl.23 (2017) 2093-2115. · Zbl 1382.92216 |
| [28] | Wen, B. Y., Wang, J. P. and Teng, Z. D., A discrete-time analog for coupled within-host and between-host dynamics in environmentally driven infectious disease, Adv. Difference Equ.69 (2018) 1-25. · Zbl 1445.92284 |
| [29] | Agiza, H. N., ELabbasy, E. M., EL-Metwally, H. and Elsadany, A. A., Chaotic dynamics of a discrete prey-predator model with Holling type II, Nonlinear Anal. Real World Appl.10 (2009) 116-129. · Zbl 1154.37335 |
| [30] | Liu, X. and Xiao, D., Complex dynamic behaviors of a discrete-time predator-prey system, Chaos Solitons Fractals32 (2007) 80-94. · Zbl 1130.92056 |
| [31] | Chen, B. S. and Chen, J. J., Complex dynamic behaviors of a discrete predator-prey model with stage-structure and harvesting, Int. J. Biomath.10 (2017) 1-25. · Zbl 1366.92101 |
| [32] | Wanga, P. and Zhang, J., Monotone iterative technique for initial-value problems of nonlinear singular discrete systems, J. Comput. Appl. Math.221 (2008) 158-164. · Zbl 1159.65101 |
| [33] | Zhuk, S. M., Minimax state estimation for linear discrete-time differential-algebraic equations, Automatica46 (2010) 1785-1789. · Zbl 1218.93098 |
| [34] | Kuznetsov, Y., Elements of Applied Bifurcation Theory, 3rd edn. (Springer, 2004). · Zbl 1082.37002 |
| [35] | Gukenheimer, J. and Holmes, P., Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (Springer, 1983). · Zbl 0515.34001 |
| [36] | Wiggins, S., Introduction to Applied Nonlinear Dynamical Systems and Chaos (Springer, 2003). · Zbl 1027.37002 |
| [37] | Carr, J., Application of Center Manifold Theory (Springe, 1982). |
| [38] | Leslie, P. H., Some further notes on the use of matrices in population mathematics, Biometrika35 (1948) 213-245. · Zbl 0034.23303 |
| [39] | Zhou, S. R., Liu, Y. F. and Wang, G., The stability of predator-prey systems subject to the Allee effects, Theor. Popul. Biol.67 (2005) 23-31. · Zbl 1072.92060 |
| [40] | Mankiw, N. G., Principles of Economics (Peking University Press, Beijing, 2015). |
| [41] | Fan, M. and Wang, K., Harvesting problem with price changed with demand and supply, J. Biomath.16 (2001) 411-415(in Chinese). · Zbl 1058.91564 |
| [42] | Boothby, W. M., An Introduction to Differential Manifolds and Riemannian Geometry, 2nd edn. (Academic Press, 1986). · Zbl 0596.53001 |
| [43] | Chen, B. S., Liao, X. X. and Liu, Y. Q., Normal forms and bifurcations for the differential-algebraic systems, Acta Math. Appl. Sin.23 (2000) 429-443(in Chinese). · Zbl 0960.34004 |
| [44] | Petzold, L., Differential/algebraic equations are not ODE’s, SIAM J. Sci. Stat. Comput.3 (1982) 367-384. · Zbl 0482.65041 |
| [45] | Venkatasubramanian, V., Schättler, H. and Zaborszky, J., Local bifurcation and feasibility regions in differential-algebraic systems, IEEE Trans. Automat. Contr.40 (1995) 1992-2013. · Zbl 0843.34045 |
| [46] | Ilchmann, A. and Reis, T. (Eds.), Surveys in Differential-Algebraic Equations I (Springer, 2013). · Zbl 1261.65003 |
| [47] | Ilchmann, A. and Reis, T. (Eds.), Surveys in Differential-Algebraic Equations II-III (Springer, 2015). · Zbl 1333.65004 |
| [48] | Ilchmann, A. and Reis, T. (Eds.), Surveys in Differential-Algebraic Equations IV (Springer, 2017). · Zbl 1369.65004 |
| [49] | May, R. M., Simple mathematical models with very complicated dynamics, Nature261 (1976) 459-467. · Zbl 1369.37088 |
| [50] | May, R. M., Biological populations with nonoverlapping generations: Stable points, stable cycles, and chaos, Science186 (1974) 645-647. |
| [51] | Liu, W. and Jiang, Y. L., Modeling and dynamics of an ecological-economic model, Int. J. Biomath.12 (2019) 1950030. https://doi.org/10.1142/S179352451950030X · Zbl 1416.92139 |
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