Z-type control of populations for Lotka-Volterra model with exponential convergence. (English) Zbl 1369.92108
Summary: The population control of the Lotka-Volterra model is one of the most important and widely investigated issues in mathematical ecology. In this study, assuming that birth rate is controllable and using the Z-type dynamic method, we develop Z-type control laws to drive the prey population and/or predator population to a desired state to keep species away from extinction and to improve ecosystem stability. A direct controller group is initially designed to control the prey and predator populations simultaneously. Two indirect controllers are then proposed for prey population control and predator population control by exerting exogenous measure on another species. All three control laws possess exponential convergence performances. Finally, the corresponding numerical simulations are performed. Results substantiate the theoretical analysis and effectiveness of such Z-type control laws for the population control of the Lotka-Volterra model.
Keywords:
Lotka-Volterra model; Z-type controller; population control; exponential convergence; numerical simulationReferences:
| [1] | Shchekinova, E. Y.; Loeder, M. G.J.; Boersma, M.; Wiltshire, K. H., Facilitation of intraguild prey by its intraguild predator in a three-species Lotka-Volterra model, Theor. Popul. Biol., 92, 55-61 (2014) · Zbl 1284.92117 |
| [2] | Lacitignola, D.; Tebaldi, C., Effects of ecological differentiation on Lotka-Volterra systems for species with behavioral adaptation and variable growth rates, Math. Biosci., 194, 1, 95-123 (2005) · Zbl 1063.92054 |
| [3] | Murray, J. D., Mathematical Biology (1989), Springer · Zbl 0682.92001 |
| [4] | Waniewski, J.; Jedruch, W., Individual based modeling and parameter estimation for a Lotka-Volterra system, Math. Biosci., 157, 1-2, 23-36 (1999) |
| [5] | Mickens, R. E., A nonstandard finite-difference scheme for the Lotka-Volterra system, Appl. Numer. Math., 45, 2-3, 309-314 (2003) · Zbl 1025.65047 |
| [6] | Haque, M.; Sarwardi, S.; Preston, S.; Venturino, E., Effect of delay in a Lotka-Volterra type predator-prey model with a transmissible disease in the predator species, Math. Biosci., 234, 1, 47-57 (2011) · Zbl 1239.92080 |
| [7] | White, A.; Bowers, R. G., Adaptive dynamics of Lotka-Volterra systems with trade-offs: the role of interspecific parameter dependence in branching, Math. Biosci., 193, 1, 101-117 (2005) · Zbl 1062.92078 |
| [8] | Gouze, J. L., Global behavior of n-dimensional Lotka-Volterra systems, Math. Biosci., 113, 2, 231-243 (2005) · Zbl 0782.92021 |
| [9] | Ma, Z.; Chen, F.; Wu, C.; Chen, W., Dynamic behaviors of a Lotka-Volterra predator-prey model incorporating a prey refuge and predator mutual interference, Appl. Math. Comput., 219, 15, 7945-7953 (2013) · Zbl 1293.34062 |
| [10] | El-Gohary, A.; Yassen, M. T., Optimal control and synchronization of Lotka-Volterra model, Chaos Solitons Fract., 12, 11, 2087-2093 (2001) · Zbl 1004.92037 |
| [11] | Pchelkina, I.; Fradkov, A. L., Control of oscillatory behavior of multispecies populations, Ecol. Modell., 227, 1-6 (2012) |
| [12] | Dong, L.; Takeuchi, Y., Impulsive control of multiple Lotka-Volterra systems, Nonlin. Anal., 14, 2, 1144-1154 (2013) · Zbl 1266.34105 |
| [13] | Hsu, S. B.; Hwang, T. W.; Yang, K., A ratio-dependent food chain model and its applications to biological control, Math. Biosci., 181, 1, 55-83 (2003) · Zbl 1036.92033 |
| [14] | Chen, F., Positive periodic solutions of neutral Lotka-Volterra system with feedback control, Appl. Math. Comput., 162, 3, 1279-1302 (2005) · Zbl 1125.93031 |
| [15] | Fatmawati; Tasman, H., An optimal control strategy to reduce the spread of malaria resistance, Math. Biosci., 262, 73-79 (2015) · Zbl 1315.92077 |
| [16] | Loeuille, N.; Loreau, M., Nutrient enrichment and food chains: can evolution buffer top-down control?, Theor. Popul. Biol., 65, 285-298 (2004) · Zbl 1109.92056 |
| [17] | Thingstad, T. F.; Sakshuag, E., Control of phytoplankton growth in nutrient recycling ecosystems, Theory terminology, Mar. Ecol. Prog. Ser., 63 (1990), 261-172 |
| [18] | Zhang, Y.; Yi, C., Zhang Neural Networks and Neural-dynamic Method (2011), Nova Science |
| [19] | Liao, B.; Zhang, Y., Different complex ZFs leading to different complex ZNN models for time-varying complex generalized inverse matrices, IEEE Trans. Neural Netw. Learn. Syst., 25, 9, 1621-1631 (2014) |
| [20] | Guo, D.; Zhang, Y., Neural dynamics and Newton-Raphson iteration for nonlinear optimization, ASME J. Comput. Nonlinear Dyn., 9, 2, 021016-1-021016-10 (2014) |
| [21] | Zhang, Y.; Jiang, D.; Wang, J., A recurrent neural network for solving sylvester equation with time-varying coefficients, IEEE Trans. Neural Netw., 13, 5, 1053-1063 (2002) |
| [22] | Zhang, Y.; Li, Z., Zhang neural network for online solution of time-varying convex quadratic program subject to time-varying linear-equality constraints, Phys. Lett., 373, 18, 1639-1643 (2009) · Zbl 1229.92008 |
| [23] | Shankar, S., Nonlinear Systems: Analysis, Stability, and Control (1999), Springer · Zbl 0924.93001 |
| [24] | Zhang, Z.; Zhang, Y., Design and experimentation of acceleration-level drift-free scheme aided by two recurrent neural networks, IET Control Theory Appl., 7, 1, 25-42 (2013) |
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