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Control Schemes to Reduce Risk of Extinction in the Lotka-Volterra Predator-Prey Model

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DOI: 10.4236/jamp.2014.27071    3,457 Downloads   4,881 Views   Citations
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ABSTRACT

The Lotka-Volterra predator-prey model is widely used in many disciplines such as ecology and economics. The model consists of a pair of first-order nonlinear differential equations. In this paper, we first analyze the dynamics, equilibria and steady state oscillation contours of the differential equations and study in particular a well-known problem of a high risk that the prey and/or predator may end up with extinction. We then introduce exogenous control to reduce the risk of extinction. We propose two control schemes. The first scheme, referred as convergence guaranteed scheme, achieves very fine granular control of the prey and predator populations, in terms of the final state and convergence dynamics, at the cost of sophisticated implementation. The second scheme, referred as on-off scheme, is very easy to implement and drive the populations to steady state oscillation that is far from the risk of extinction. Finally we investigate the robustness of these two schemes against parameter mismatch and observe that the on-off scheme is much more robust. Hence, we conclude that while the convergence guaranteed scheme achieves theoretically optimal performance, the on-off scheme is more attractive for practical applications.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Li, J. (2014) Control Schemes to Reduce Risk of Extinction in the Lotka-Volterra Predator-Prey Model. Journal of Applied Mathematics and Physics, 2, 644-652. doi: 10.4236/jamp.2014.27071.

References

[1] Lotka, A.J. (1910) Contribution to the Theory of Periodic Reaction. The Journal of Physical Chemistry, 14, 271-274.
http://dx.doi.org/10.1021/j150111a004
[2] Volterra, V. (1931) Variations and Fluctuations of the Number of Individuals in Animal Species Living Together. In: Chapman, R.N., Ed., Animal Ecology, McGraw-Hill, New York, 409-448.
http://dx.doi.org/10.1086/284409
[3] Holt, R.D. and Pickering, J. (1985) Infectious Disease and Species Coexistence: A Model of Lotka-Volterra Form. The American Naturalist, 126, 196-211.
[4] Takeuchi, Y. (1996) Global Dynamical Properties of Lotka-Volterra Systems. World Scientific, Singapore.
[5] Mollison, D. (1991) Dependence of Epidemic and Population Velocities on Basic Parameters. Mathematical Biosciences, 107, 255-287. http://dx.doi.org/10.1016/0025-5564(91)90009-8
http://www.ma.hw.ac.uk/~denis/epi/velocities.pdf
[6] Wikipedia (2014) Lotka-Volterra Equation.
[7] Leung, A.W. (1995) Optimal Harvesting-Coefficient Control of Steady-State Prey-Predator Diffusive Volterra-Lotka Systems. Applied Mathematics & Optimization, 32, 219-241.
http://dx.doi.org/10.1007/BF01182789
[8] El-Gohary, A. and Yassen, M.T. (2001) Optimal Control and Synchronization of Lotka-Volterra Model. Chaos, Solitons & Fractals, 12, 2087-2093. http://dx.doi.org/10.1016/S0960-0779(00)00023-0
[9] Chen, F. (2005) Positive Periodic Solutions of Neutral Lotka-Volterra System with Feedback Control. Applied Mathematics and Computation, 162, 1279-1302. http://dx.doi.org/10.1016/j.amc.2004.03.009

  
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