Chaos and adaptive control in two prey, one predator system with nonlinear feedback

Chaos and adaptive control in two prey, one predator system with nonlinear feedback Awad El-Gohary, a, and A.S. Al-Ruzaiza a a Department of Statistics and O.R., College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia Accepted 13 March 2006. Communicated by Professor Gevardo Ioane. Available online 5 June 2006. Abstract We show that the continuous time three species prey predator populations can be asymptotically stabilized using a nonlinear feedback control inputs. The necessary feedback control law for asymptotic stability of this system is obtained. The system appears to exhibit a chaotic behavior for a range of parametric values. The range of the system parameters for which the subsystems converge to limit cycles is determined. The results of some other models in the literature can be obtained as special cases of the present model. Numerical examples and analysis of the results are presented. Article Outline 1. Introduction 2. The model system 3. Nondimensional model and stability analysis 4. Adaptive control problem 5. Analysis and numerical simulation 6. Conclusion Acknowledgements References 1. Introduction

The subjects of chaos and adaptive control are growing rapidly in many different fields such as biological systems, structural engineering, ecological models, aerospace science, and others. It is very hard to keep track of all the new research papers produced in these fields all over the world. One could devote several volumes to an adequate converge to these papers [9] and [21]. In the recent years, interest in adaptive control systems has increased rapidly along with interest and progress in control topics. The adaptive control has a variety of specific meanings, but it often implies that the system is capable of accommodating unpredictable environmental changes, whether these changes arise within the system or external to it. Adaptation is a fundamental characteristic of living organisms such as prey predator systems and many other biological models since these systems attempt to maintain physiological equilibrium in the midst changing environmental conditions [21]. Prey predator phenomena have many important applications in many different field, such as biology, economic, ecology and others sciences. The study of prey predator phenomena is now a dominant problem in many ecological sciences (see e.g. Refs. [1], [2], [3], [6], [7] and [20]). The mathematical ecology has emerged and developed rapidly. A variety of mathematical methods can be used in ecological science. One of the main problems of ecosystems is to study the stability and instability of these systems [26], [27] and [28]. A model was introduced by Volterra for community in which organisms of one population provided food for those of the other is very important. Similar phenomena can be observed in communities with one population parasitizing on the organisms of another species. The communities of such type are usually termed prey predator or host parasite models [12], [15], [22] and [25]. A stochastic analysis of the Lotka Volterra model for the prey predator when the birth rate of the prey and death rate of the predator are perturbed by independent white noises and others related models are presented in [4], [5], [6], [8], [16], [23] and [28]. The evolution of this model is also investigated for a large length of the time interval. One of the first successes of mathematical ecology was the demonstration of population periodic oscillations in a stationary medium. The biological literature abounds in works where systems are either observed in nature or simulated on models populations in laboratory conditions [14], [17], [24] and [25].

The global asymptotic stability of a prey predator population with Holling type and others types are investigated in [19] and [27] and also the conditions of unique positive equilibrium and the conditions of local asymptotic stability of some equilibrium states are derived. The stochastic process model of human aging and mortality to estimate the effect of the competing risk is introduced in [30]. Also a competing risk model with a weaker assumption of conditional independence of the time to death, given an assumed stochastic covariant process, is developed and applied to cause specific mortality data from the Famingham heart study. The problem of optimal control and synchronization of Lotka Volterra population is studied in [8]. Optimal control of the equilibrium states of prey predator model using the Lyapunov function technique is studied by El-Gohary and Al-Ruzuiza [7]. The optimal control functions are obtained from the conditions that ensure the asymptotic stability of these equilibrium states. Also El-Gohary and Bukhari have studied the problem of optimal control of spatial stochastic prey predator population during finite and infinite time intervals. They were concerned with the time intervals of the control process [6]. Many important studies about the chaotic behavior of nonlinear systems can be found in [10], [13] and [29]. This paper is organized as follows. We start in Section 2 defining the three species population that consists two preys and one predator. The nonlinear system of differential equations that govern this system is introduced. In Section 3 we obtained the nondimensional system of differential equations that describe the time evolution of the three species prey predator population and reduces the number of the system parameters. In this section the stability analysis of the prescribed system is investigated. Further, the convergence of the subsystems of this system to limit cycles is studied. The ranges of the system parameters for which the system converges to limit cycles are determined. The chaotic behavior of this system is investigated. The problem of adaptive control of three species prey predator population is studied in Section 4. Also in this section, the Lyapunov asymptotic stability of the controlled system is proved and the necessary control inputs for this asymptotic stability is obtained as nonlinear feedback of the population densities. Extensive numerical examples and simulation are introduced in Section 5. 2. The model system

In this section, we will describe the three species prey predator system which consists of two competing preys and one predator. Such system can be describe by the following set of nonlinear differential equations: (2.1) where α, r i, k i, e i and c i, i = 1, 2 are the model parameters assuming only positive values, and the functions Φ i (x, y), i = 1, 2 represent a functional response of predator to preys. The variables x, y represent the densities of the two prey species and z represents the density of the predator species. The predator z consumes the preys x, y according to the response functions [18]: (2.2) where a i, i = 1, 2 are the search rates of a predator for the preys x, y respectively, while b i = h i a i, i = 1, 2 where h i, i = 1, 2 are the expected handing times spent with the preys x, y, respectively. The parameters e 1 and e 2 represent, the conversion rates of the preys x, y to predator z. Obviously, when b 1 and b 2 are very small the functional of response Φ i, (i = 1, 2) become linear response see Volttera functional response [8]. In the other hand as one of both b 1 and b 2 tends to zero the system approaches to hyperbolic Holling type II [26]. The prescribed model characterized by nonlinear response since amount of food consumed by predator per unit time depends upon the available food sources from the two preys x and y. 3. Nondimensional model and stability analysis To reduce the number of the system parameters we will transform the system (2.1) to the nondimensional form by using the following transformation of the variables: (3.1) Using the transformation (3.1) the system (2.1) takes the nondimensional form:

(3.2) where the relations between the nondimensional and dimensional parameters given by: (3.3) The system (3.2) is more simplicity than (2.1) for the mathematical study since the number of system parameters has been reduced from 13 to 10 only. For a biological food web model to be logically credible, it must satisfy the following conditions: 1. The equations should be invariant under identification of identical species. 2. The system of equations for a food web must be separate into independent subsystems if the community splits into disconnected sub-webs. The interaction of two competing prey and one predator given by (3.2) satisfy the above conditions. So the system (3.2) can be separated into two independent subsystems. The first system is obtained by assuming the absence of the second prey x 2 (3.4) and the second is obtained when the first prey is absent: (3.5) It is easy to show that all solutions of (3.4) and (3.5) are bounded in the future and remain in the regions {(x 1, x 3 ) : x 1 > 0 and x 3 > 0} for (3.4) and {(x 2, x 3 ) : x 2 > 0 and x 3 > 0} for (3.5).

In what follows we examine the behavior of the trajectories of the subsystems (3.4) and (3.5) near the equilibrium points. The first subsystem (3.4) has thee equilibrium points which are given by (3.6) The characteristic equation of the first equilibrium point has the eigenvalues λ 1 = 1 and λ 2 = µ 1 < 0 which are real and λ 2 lies in the left-half plane and λ 1 lies in the right-half plane. Thus the first equilibrium point of the subsystem (3.4) is unstable saddle point. The necessary condition for linear stability of E 12 is (3.7) and the necessary condition for linear stability of E 13 is (3.8) Therefore E 13 is the only stable equilibrium point of the subsystem (3.4) if the system parameters satisfy the condition: (3.9) while this subsystem has a limit cycle (see Fig. 1) for the system parameters range (see also Fig. 2) (3.10) Similarly the subsystem (3.5) has the following three equilibrium points which given by (3.11)

Similar results can be obtained for the equilibrium points (3.11) of the subsystem (3.5). The necessary condition for linear stability of E 22 is (3.12) and the necessary condition for linear stability of E 23 is (3.13) Also E 23 is the only stable equilibrium point of the subsystem (3.5) if the system parameters satisfy following condition (3.14) and this subsystem has a limit cycle if the system parameters have the range (3.15) Limit cycles and attractors occur in many different biological and electronic systems as prey predator system, Lorenz system and others. Limit cycles have a distinct geometric configuration in the phase plane portrait, namely, that of an isolated closed path in the phase plane. A limit cycle represents a steady-state oscillation, from which all trajectories nearby will converge or diverge. A limit cycle in a nonlinear system describes the amplitude and period of a selfsustained oscillation [11]. Full-size image (49K) Fig. 1. For the subsystem (3.4) that converges to outside limit cycle in (a) while its converges to inside limit cycle in (b) where the parameters in these cases given by α 1 = 0.5, α 2 = 2.6, β 1 = 1.5,

β 2 = 5.4, γ 1 = 1.5, γ 2 = 0.9, ν 1 = 2.1, ν 2 = 2.3, µ 1 = 0.2, µ 2 = 3.9 and α 1 = 1.2, α 2 = 3.6, β 1 = 1.5, β 2 = 5.4, γ 1 = 1.5, γ 2 = 0.9, ν 1 = 2.1, ν 1 = 2.3, µ 1 = 0.2, µ 2 = 4.9, respectively, and initial densities x 1 (0) = 0.3, x 2 (0) = 0.5, x 3 (0) = 0.5. View Within Article Full-size image (39K) Fig. 2. For the subsystem (3.5) that converges to outside limit cycle in (a) while its converges to inside limit cycle in (b) where the parameters in these cases given by α 1 = 6.3, α 2 = 2.6, β 1 = 1.5, β 2 = 1.4, γ 1 = 1.5, γ 2 = 2.9, ν 1 = 2.1, ν 2 = 2.3, µ 1 = 0.2, µ 2 = 3.9 and α 1 = 6.3, α 2 = 2.6, β 1 = 1.5, β 2 = 1.4, γ 1 = 1.5, γ 2 = 2.9, ν 1 = 2.1, ν 1 = 2.3, µ 1 = 0.5, µ 2 = 3.9, respectively, and initial densities x 1 (0) = 0.3, x 2 (0) = 1.5, x 3 (0) = 0.5. View Within Article Next, we discuss the problem of adaptive control of two preys, one predator system using nonlinear feedback approach. 4. Adaptive control problem This section devoted to study the adaptive control of the three species population that consists of two preys and one predator. In order to study the adaptive control of the three species prey predator system using nonlinear feedback control approach, we start by assuming that the system (3.2) can be written in the following suitable form (4.1) where u 1, u 2 and u 3 are control inputs that will be suitably choice to make the trajectory of the whole system (3.2) that specified by the equilibrium states E 1 (0, 0, 0), E 2 (1, 0, 0), E 3 (0, 1, 0) and to be asymptotically stabile about these equilibrium states of the uncontrolled system. Note the form (4.1) is more suitable for study the adaptive control of the system (3.2).

If u i = 0, (i = 1, 2, 3) then the system (3.2) and (4.1) have absolutely an unstable special solution x i =0,(i=1,2,3) where x i = 0, (i = 1, 2, 3) is the trivial equilibrium point of the uncontrolled system (3.2). The eigenvalues of the characteristic equation for the trivial equilibrium point (4.2) has the following real values (4.2) (4.3) Hence this trivial equilibrium point is absolutely unstable. In this study we will asymptotically stabilized this point using the control inputs u i, (i = 1, 2, 3). In what follows we study the adaptive control of the system (4.1). Adaptation is a fundamental characteristic of living organisms such as prey predator systems and host parasite model and many other ecological models since they attempt to maintain physiological equilibrium in the midst of changing environmental conditions. An approach to the design of adaptive systems is then to consider the adaptive aspects of human or animal behavior and to develop systems which behave somewhat analogously. The following theorem derives the adaptive nonlinear feedback control inputs u 1, u 2 and u 3 that asymptotically stabilized the three species prey predator system (3.2) about its equilibrium states. Theorem 4.1 Using the nonlinear feedback controllers (4.4) the system (4.1) will be asymptotically stable in the Lyapunov sense about its equilibrium state (4.2). Proof

The proof of this theorem can be reached by using Lyapunov stability theorem which gives sufficient conditions for asymptotic stability. Substituting (4.4) into (4.1) one can get the following nonlinear system of differential equations: (4.5) Let us consider the Lyapunov function for the system (4.5) in the form (4.6) Obviously, the functions (4.6) is a positive definite form with respect to the variables x 1, x 2 and x 3 and its time derivative along the trajectory of the system (4.4) is given by (4.7) Since the densities x 1 (t), x 2 (t) and x 3 (t) are usually positive and the parameters α 1, γ 2, ν 2 and µ 1 are also take only positive values then (4.8) Using the variable gradient method we find that (4.9) Using the inequalities (4.8) we can verify that the function in (4.9) and so (4.7) are negative definite forms which proves the asymptotic stability of the system (4.4) in the Lyapunov sense. Therefore the coupled system (4.1) is asymptotically stable with the nonlinear feedback controllers (4.4), which completes the proof. Now we can easily conclude that the three species prey predator populations can be asymptotically stabilized using nonlinear feedback controllers about its trivial equilibrium point.

5. Analysis and numerical simulation The main objective of the numerical simulation is to explore the possibility of the chaotic behavior and the effect of the adaptive control to this behavior. Extensive numerical examples for uncontrolled and controlled three species two preys and one predator system were carried out for various parameters values and different initial densities. Firstly, we start by studying the density behavior of the two preys and the predator with the time of the uncontrolled system. Also the phase plot of the three dimensional trajectory of the uncontrolled system will investigate for some values of the system parameters (see Fig. 3, Fig. 4 and Fig. 5). Full-size image (43K) Fig. 3. Displays the densities of the first prey x 1 (t), the second prey x 2 (t) and the predator x 3 (t) of the uncontrolled system for the system parameter values α 1 = 6.3, α 2 = 2.9, β 1 = 3.5, β 2 = 0.1, γ 1 = 10, γ 2 = 2.9, ν 1 = 3.01, ν 2 = 2.3, µ 1 = 1.5, µ 2 = 10.9 and to initial densities x 1 (0) = 0.3, x 2 (0) = 1.5, x 3 (0) = 0.5. (a) First prey density of uncontrolled system. (b) Second prey density of uncontrolled system. (c) Predator density of uncontrolled system. View Within Article Full-size image (70K) Fig. 4. Displays the densities of the first prey x 1 (t), the second prey x 2 (t) and the predator x 3 (t) of uncontrolled system for the system parameters α 1 = 1.3, α 2 = 2.9, β 1 = 3.5, β 2 = 2.1, γ 1 = 1, γ 2 = 2.9, ν 1 = 3.1, ν 2 = 2.3, µ 1 = 1.5, µ 2 = 15 and initial densities x 1 (0) = 0.8, x 2 (0) = 0.5, x 3 (0) = 1.5. (a) First prey density of uncontrolled system. (b) Second prey density of uncontrolled system. (c) Predator density of uncontrolled system.

View Within Article Full-size image (44K) Fig. 5. Display the three dimensional phase plot of uncontrolled three species prey predator system for the system parameter values and initial densities α 1 = 3, α 2 = 5, β 1 = 1.5, β 2 = 2, γ 1 = 0.3, γ 2 = 0.3, ν 1 = 1.5, ν 2 = 3.5, µ 1 = 0.85, µ 2 = 6, x 1 (0) = 0.1, x 2 (0) = 1.5, x 3 (0) = 0.5 and α 1 = 3, α 2 = 5, β 1 = 1.5, β 2 = 2, γ 1 = 0.6, γ 2 = 4, ν 1 = 1.5, ν 2 = 3.5, µ 1 = 0.85, µ 2 = 6, x 1 (0) = 0.8, x 2 (0) = 0.5, x 3 (0) = 1, respectively. View Within Article Secondly, we display the density of the controlled three species prey predator system and control inputs time behavior for different values of the system parameters (see Figs. 6 and 7 and Figs. 8 and 9). Full-size image (61K) Figs. 6 and 7. Display the densities of the first prey x 1 (t), the second prey x 2 (t) and the predator x 3 (t) and the control inputs u 1 (t), u 2 (t) and u 3 (t) for the system parameter values α 1 = 5.1, α 2 = 5.9, β 1 = 2.30, β 2 = 4.5, γ 1 = 5, γ 2 = 2.9, ν 1 = 5, ν 2 = 10.3, µ 1 = 1.5, µ 2 = 0.9, and initial densities x 1 (0) = 0.3, x 2 (0) = 0.5, x 3 (0) = 1. View Within Article Full-size image (62K)

Figs. 8 and 9. Display the densities of the first prey x 1 (t), the second prey x 2 (t) and the predator x 3 (t) and the control inputs u 1 (t), u 2 (t) and u 3 (t) for the system parameter values α 1 = 3, α 2 = 4.2, β 1 = 5.2, β 2 = 5, γ 1 = 5, γ 2 = 9, ν 1 = 20, ν 2 = 12, µ 1 = 1.5, µ 2 = 4, and initial densities x 1 (0) = 1, x 2 (0) = 1.5, x 3 (0) = 1.3. View Within Article Finally, we conclude that, controlled three species prey predator model is asymptotically stable for arbitrary values of the system parameters and initial prey predator densities. Further all the densities of the first and second prey and the predator are converge exponential to the equilibrium states through positive values which is biologically meaningful. 6. Conclusion This paper generalizes and unifies many previous results in this area see for example [6] and [7]. The chaotic behavior of continuous time three species prey predator system is investigated. The ranges of the system parameters for which subsystems converge to limit cycles are determined. The problem of adaptive control of three species two prey and one predator is studied. The asymptotic stability of the controlled system is proved using the Lyapunov function. The necessary control inputs for this asymptotic stability is obtained as nonlinear feedback. Finally, extensive numerical examples and simulation are introduced. Acknowledgement This research was supported by the College of Science Research Center at King Saud University. References [1] A. Bazykin, A. Khibnik and E.A. Aponina, Model of evolutionary appearance of dissipative structure in ecosystems, J Math Biol 18 (1983), pp. 13 23. MathSciNet View Record in Scopus Cited By in Scopus (1) [2] J. Chattopadhyay, R. Sarkar and G. Ghosal, Removal of infected prey prevent limit cycle oscillations in an infected prey predator system A mathematical study, Ecol Model 156 (2002), pp. 113 121. Article PDF (200 K) View Record in Scopus Cited By in Scopus (7)

[3] J. Chattopadhyay and O. Arino, A predator prey model with disease in the prey, Non-linear Anal (36) (1999), pp. 747 766. Abstract PDF (133 K) MathSciNet View Record in Scopus Cited By in Scopus (66) [4] M. Christophr and X. Jorge, A simple vaccination model with multiple endemic states, Math Biosci (164) (2000), pp. 183 201. [5] A. El-Gohary, Optimal control of the genital herpes epidemic, Chaos, Solitons & Fractals 12 (2001), pp. 1817 1822. Article PDF (84 K) View Record in Scopus Cited By in Scopus (15) [6] A. El-Gohary and F. Bukhari, Optimal control of stochastic prey predator models, Appl Math Comput 146 (2) (2004), pp. 403 416. [7] A. El-Gohary and A. Al-Ruzaiza, Optimal control of equilibrium state of prey predator model, Non-linear Dyn Psychol Life Sci 6 (1) (2002), pp. 27 36. Full Text via CrossRef [8] A. El-Gohary and M. Yassen, Optimal control and synchronization of Lotka Volterra model, Chaos, Solitons & Fractals 12 (2001), pp. 2087 2093. Article PDF (96 K) View Record in Scopus Cited By in Scopus (8) [9] El-Naschie MS, Al-Athel S. Stress stability, and chaos in structural engineering: An energy approach. University of Libraries, King Saud University, 1989. [10] S. Gakkhar and R. Naji, Chaos in seasonally perturbed ratio-dependent, Chaos, Solitons & Fractals 15 (2003), pp. 107 118. Article PDF (882 K) View Record in Scopus Cited By in Scopus (33) [11] S. Gakkhar and R. Naji, Existence chaos in two-prey, one predator system, Chaos, Solitons & Fractals 17 (2003), pp. 639 649. Article PDF (189 K) View Record in Scopus Cited By in Scopus (15) [12] S. Gakkhar and R. Naji, Order and Chaos in a food web consisting of a predator and two independent preys, Commun Nonlinear Sci Numer Simul 10 (2) (2005), pp. 105 120. Article PDF (475 K) View Record in Scopus Cited By in Scopus (10) [13] J. Gamarra, R. Sole and D. Alonso, Control, synchrony and the persistence of chaotic populations, Chaos, Solitons & Fractals 12 (2001), pp. 235 249. Article PDF (624 K) MathSciNet View Record in Scopus Cited By in Scopus (11)

[14] T. Gard, Top predator persistence in differential equations models of food chains: The effect of omnivory and external forcing of lower trophic levels, J Math Biol 14 (1982), pp. 285 299. MathSciNet Full Text via CrossRef View Record in Scopus Cited By in Scopus (8) [15] S. Jang and J. Baglama, A nutrient-prey predator model with intratrophic predation, Appl Math Comput 129 (2002), pp. 517 536. Article PDF (271 K) View Record in Scopus Cited By in Scopus (7) [16] R.Z. Khasminskh and F.C. Klebaner, Long term behavior of solutions of the Lotka Volterra system under small random perturbations, Ann Appl Probab 11 (3) (2001), pp. 952 963. [17] V. Krivan and J. Eisner, Optimal foraging and predator prey dynamics III, Theoret Population Biol 63 (2003), pp. 269 279. Article PDF (580 K) View Record in Scopus Cited By in Scopus (25) [18] Y. Kuang, Basic properties of mathematical population models, Biomathematics 17 (2002), pp. 129 142. [19] A. Lipowski and D. Lipowska, Nonequilibrium phase transition in a lattice prey predator system, Physica A 276 (2000), pp. 456 464. Article PDF (150 K) MathSciNet View Record in Scopus Cited By in Scopus (30) [20] D. Mukherjee, Uniform persistence in a generalized prey predator system with parasitic infection, BioSystem 47 (1998), pp. 149 155. Article PDF (90 K) View Record in Scopus Cited By in Scopus (12) [21] K. Ogata, Modern control engineering, Prentice-Hall, Inc., Englewood Cliffs, NJ (1970). [22] J. Pitchforeed and J. Brindley, Intratrophic predation in simple prey predator models, Bull Math Biol 60 (1998), pp. 937 953. [23] S.N. Rasband, Chaotic dynamics nonlinear systems, John Wiley & Sons, Inc. (1990). [24] J.E. Satulovky and T. Tome, Stochastic lattice gas model for a predator prey system, Phys. Rev. E 49 (1994), pp. 5073 5079. [25] R. Schinazi, Predator prey and hast parasite spatial stochastic models, Ann Appl Probab 7 (1) (1997), pp. 1 9. MathSciNet View Record in Scopus Cited By in Scopus (11) [26] J. Sugie and M. Katayama, Global asymptotic stability of a predator prey system of Holling type, Non-linear Anal 38 (1999), pp. 105 121. Abstract Record in Scopus Cited By in Scopus (14) PDF (124 K) MathSciNet View

[27] Y. Svirezlev and D.O. Logofet, Stability of biological communities, Mir Publishers, Moscow (1983). [28] A. El-Gohary, Optimal control of stochastic lattice of prey-predator models, Appl Math Comput 160 (1) (2005), pp. 15 28. Article PDF (221 K) View Record in Scopus Cited By in Scopus (2) [29] J.M. Thompson and H.B. Stewart, Nonlinear dynamics and chaos, John Wiley & Sons Ltd. (1987). [30] A. Yashin, K. Manton and E. Stalleard, Dependent competing risks: A stochastic process model, Biology (1986). Corresponding author. Permanent address: Department of Mathematics, College of Science, Mansoura University, Mansoura 35516, Egypt. Fax: +20 5034 6781. Chaos, Solitons & Fractals Volume 34, Issue 2, October 2007, Pages 443-453