Research Article Open Access Khan et al., Fish Aquac J 2015, 6:4 DOI: 10.4172/2150-3508.1000142 Short Communication Open Access Volume 6 • Issue 4 • 1000142 Fish Aquac J ISSN: 2150-3508 FAJ, an open access journal Keywords: Chaos; Chaos control; reshold mechanism; Hastings and Powell model; Maximum sustainable yield Introduction An extremely common phenomenon in nonlinear dynamical systems arising from a variety of disciplines is chaos. Chaotic dynamics is interesting to analyse but for technological processes like optimization of the production in a production farm chaos is highly unwanted or even harmful. erefore, strategies are required to devise control algorithms capable of achieving the desired type of behaviour from a chaotic system. ere has been many techniques for designing effective control of chaotic systems but very few of these methods are applicable to control a biological systems like food chain models [1- 4]. Many authors proposed harvesting model for ecological systems [5-8]. But none of these are very useful from applied point of view because harvesting at every time is not realistic and meaningful for many biological systems. ose methods also required knowledge of functional response and knowledge of system parameters. Now, we shall discuss a new harvesting mechanism. Consider a general N-dimensional food chain model ( ) dX FX dt = Where X=(x 1 , x 2 ,…….., x N ) are the state variables. Let the variable x i is chosen for harvesting. e threshold harvesting strategy is as follows. We shall check the population size represented by the state variable x i at regular interval of time. At the time of checking if the population of variable x i exceeds a critical population x i * , then harvesting will be done and collect (x i - x i * ) number of fishes otherwise do not collect any fish. is assumption is natural because in fishery because harvesting of fish takes place at regular interval of time. Hastings and Powell [1] introduced a continuous time model of a food chain incorporating nonlinear functional responses and shown that model exhibits chaotic dynamics in long term behaviour when biologically reasonable parameter values are chosen. Wilson et al. [4] had obtained chaotic dynamics in a multi-species fishery. Managing such a chaotic fishery system, demands different approach for controlling chaos. Chattopadhyay et al. [5] interpret the population of first, second and third species of the Hastings and Powell [1] model as the population of the toxin producing phytoplankton (TPP), zooplankton and fish respectively. en according to Hastings and Powell [1], the fish population will vary chaotically for some biologically significant parameter region. In this work, we have shown that threshold harvesting strategy can be applied for controlling chaotic fish population and to obtain regular fish population dynamics e.g., steady state, limit cycle, period-2, period-4 etc. Here the thresholding variable is chosen as the fish population variable of the system. In section-2, the threshold harvesting mechanism is discussed. In section-3, application of threshold harvesting for controlling chaotic dynamics of fish population is demonstrated for chaotic Hastings and Powell [1] model. In section-4, numerical simulation results are discussed. Finally a conclusion is drawn in section-5. reshold Harvesting Mechanism Consider a general N dimensional dynamical system, described by the following evolution equations 1 1 1 2 3 ( , , , ......., ; ), N dx f x x x x t dt = 2 2 1 2 3 ( , , , ......., ; ), N dx f x x x x t dt = 1 2 3 ( , , , ......., ; ), N N N dx f x x x x t dt = Where X=(x 1 , x 2 , x 3 ,……, x N )are the state variables. Let the variable x i , i ϵ 1, 2,….., N is chosen as the monitored variable which we want to control. e mechanism for threshold action in this system is as follows. Control will be triggered aſter a finite time interval. Whenever the value of the monitored variable exceeds a critical threshold x* and the variable x i will then be reset to x*, i.e. if * i x x ≤ then no harvesting if > * i x x then * i x x → . e dynamics continues undisturbed until x i exceeding the threshold value. When it exceeds then control resets its value to x* again. As the system parameters are leſt invariant by this method therefore it acts *Corresponding author: Mohammad Ali Khan, Department of Mathematics, Ramananda College, Bishnupur, Bankura, West Bengal, India, Tel: 03244-252059; E-mail: mdmaths@gmail.com Received June 22, 2015; Accepted August 19, 2015; Published August 26, 2015 Citation: Khan MA, Ghosh J, Sahoo B (2015) Controlling Chaos in a Food Chain Model through Threshold Harvesting. Fish Aquac J 6: 142. doi:10.4172/2150- 3508.1000142 Copyright: © 2015 Sogbesan OA, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Abstract In this paper, we propose a new harvesting strategy namely the harvesting for controlling chaotic population in a food chain model. In particular, we have taken the three species Hastings and Powell food chain model for demonstration. We have shown threshold harvesting strategy can be effectively employed to obtain a steady or cyclic behaviour from chaotic fish population by varying either the frequency of harvesting or the amount of harvesting of fish population. Numerical simulation results are presented to show the effectiveness of the scheme. We obtain steady state; limit cycle, period-2 and period-4 behaviour from chaotic Hastings and Powell model. This threshold harvesting strategy will be very useful for species conservation and fishery management. Controlling Chaos in a Food Chain Model through Threshold Harvesting Mohammad Ali Khan 1* , Joydev Ghosh 2 and Banshidhar Sahoo 2 1 Department of Mathematics, Ramananda College, Bishnupur, Bankura, West Bengal, India 2 Department of Applied Mathematics, University of Calcutta, Kolkata, West Bengal, India F i s h e r i e s a n d A q u a c ul t u r e J o u r n a l ISSN: 2150-3508 Fisheries and Aquaculture Journal