Aly, Shaban

ELETTREBY, Mohamed

Miskolc Mathematical Notes

Shaban Aly

Mohamed Elettreby

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Optimal control of a two teams prey-predator interaction model

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Miskolc Mathematical Notes HU e-ISSN 1787-2413Vol. 16 (2015), No. 2, pp. 793–803 DOI: 10.18514/MMN.2015.1079OPTIMAL CONTROL OF A TWO TEAMS PREY-PREDATORINTERACTION MODELM. F. ELETTREBY AND SHABAN ALYThis paper is dedicated to the memory of Professor Miklo´s Farkas .Received 11 December, 2013Abstract. In this paper, a model of two teams of predators interacting with two teams of preysis proposed. The predator teams help each other and so do the prey teams. Therefore, somestability analysis are carried out on the model to detect the stability and instability of the system’sequilibrium solutions. Also, the persistence of the solutions is investigated. Finally, the optimalcontrol is applied to the unstable equilibrium solutions. Some numerical simulations are carriedout to verify the analytic results.2010 Mathematics Subject Classification: 34C23; 35K57; 92D25; 92D30Keywords: predator, prey, equilibrium solutions, asymptotic stability, persistence, optimal con-trol1. INTRODUCTIONThe work of Malthus [10] and Verhulst [13] is the base to the field of the growthand decline of a population [3,16]. In the nature, species do not exist alone. So, therewill be interactions with the other species. When species interact, the populationdynamics of each species is affected. There are three main types of interaction [7,11]:(1) If the growth rate of one population is increased while the growth rate of theother population decreased, then the two populations are in a predator–preysituation.(2) If the growth rate of each population is decreased affected by the other spe-cies, then the two populations are in a competition case.(3) If the growth rate of each population is enhanced affected by the other spe-cies, then it is a mutualism situation.The work that was proposed independently by Lotka [9] and Volterra [14] opened thegate for studying the predator–prey interaction. Lotka-Volterra model is the simplestmodel of these interactions. Since their pioneering work, many other notable contri-butions were made [12, 15].c 2015 Miskolc University Press794 M. F. ELETTREBY AND SHABAN ALYMany creatures form teams. This has, at least, two main advantages: the first isthe improvement in foraging, since looking for food in a team is more efficient thandoing it alone. The second advantage is that living in a team reduces predation riskdue to early spotting of predators and that existing in a team gives a higher probabilitythat the predator will attack another member of the team. In this paper, a model isgiven where two teams of predators interact with two teams of preys. The teams ofeach group (predators or preys) help each other.A biological realization of our study is represented by herds of zebras and gazellesliving side by side and attacked by two types of predators (like lions and tigers).Multi-team game theory has been studied previously in [1, 2].The model is presented in Section 2. The equilibrium points, their local stabilityand persistence are studied in Section 3. In Section 4, an optimal control to the modelis investigated.2. THE MODELConsider two teams of preys with densities x1; x2 interacting with two teams ofpredators with densities y1; y2; respectively. We assume that, in the absence of thepredators, each team of preys grows logistically, ai xi .1 xi /. Each team of the preyswill contribute to the growth of the other team, ai xj ; i ¤ j . Also, the two teamsof preys help each other against the predator. The effect of the predation is to reducethe prey growth rate by a term proportional to the prey and predator populations,xi .y1Cy2/. Finally, in the absence of any prey for sustenance, the predators deathrate results in inverse decay, ci yi . Using these assumptions, the following model isproposed [6];dx1dtD a1 Œx1.1 x1/Cx2 x1.y1Cy2/;dx2dtD a2 Œx2.1 x2/Cx1 x2.y1Cy2/; (2.1)dy1dtD c1y1C .x1Cx2/.y1Cy2/;dy2dtD c2y2C .x1Cx2/.y1Cy2/;where the positive coefficients a1 and a2 represent the intrinsic growth rates of thepreys densities x1 and x2; respectively. Both the positive parameters c1 and c2 arethe death rates of the predator y1 and y2, respectively. It is clear that the two preys(predators) help each other e.g. in foraging and (in the case of preys) in early warningagainst predation.OPTIMAL CONTROL OF A TWO TEAMS PREY-PREDATOR INTERACTION MODEL 7953. THE ANALYSIS OF THE MODELThe steady state (equilibrium) solutions are important [4]. The system (2.1) hasthe following equilibria. The trivial one E1 D .0;0;0;0/, the boundary one E2 D.2;2;0;0/ and the coexistence (interior) E3 D .x1;x2;y1;y2/: The components ofthe interior solutions are given by the relations;y2 D c1c2y1; y1 D a1cx1c1 c x21; x2 D .c x1/ ; (3.1)andx31   c .1CA2/ x21   c .1  cA2/ x1C c2A1 D 0; (3.2)whereA1 D a1a1Ca2 ; A2 Da2a1Ca2 and c Dc1 c2c1C c2 :Proposition 1. The system (2.1) has at least one and at most two interior equilib-rium solutions [6].Proposition 2. Local stability analysis [4] shows that the equilibrium state E1 D.0;0;0;0/ is unstable, while E2 D .2;2;0;0/ is stable under the conditions c1Cc2 >8; and 4.c1C c2/ < c1 c2:Proof. The Jacobian matrix of the system (2.1) is given by J D2664a1 .1 2x1/  .y1Cy2/ a1  x1  x1a2 a2 .1 2x2/  .y1Cy2/  x2  x2.y1Cy2/ .y1Cy2/ .x1Cx2/  c1 .x1Cx2/.y1Cy2/ .y1Cy2/ .x1Cx2/ .x1Cx2/  c23775 :At the equilibrium point E1 D .0;0;0;0/; the above Jacobian matrix becomes;J D2664a1 a1 0 0a2 a2 0 00 0  c1 00 0 0  c23775 ;whose eigenvalues are D 0; c1; c1;a1Ca2; which has one positive eigenvalue,a1Ca2 > 0. So, it is unstable equilibrium point.Similarly, at the equilibrium pointE2D .2;2;0;0/, The Jacobian matrix becomes;J D2664 3a1 a1  2  2a2  3a2  2  20 0 4  c1 40 0 4 4  c23775 ;whose eigenvalues satisfy the characteristic equations;2C3.a1Ca2/C8a1a2 D 0;796 M. F. ELETTREBY AND SHABAN ALYand2  .8  c1  c2/C c1 c2 4.c1C c2/D 0:The first equation has two negative eigenvalues since 3.a1Ca2/ > 0, 8a1a2 > 0 andthe two other eigenvalues come from the second equation which are negative underthe conditionsc1C c2 > 8; 4.c1C c2/ < c1 c2:Definition 1. A system is persistent if there exists a compact region V subset ofthe interior of the state space such that all solutions with positive initial conditionsare attracted to V [8].Proposition 3. The system (2.1) is persistent if at least one of the above conditionsis not satisfied [6].The characteristic equation of the eigenvalue of the internal equilibrium point E3is given by, 4CD13CD22CD3CD4 D 0; whereD1 D . 2 a2/ x1C .c1C c2 a1 a2C2 a2 c 2c/C .2a1 c/ x 11D2 D .2 a1 a21 2 a1 a2/ x21C .4 a2 cC2 a1 a2 cCa1 a2Ca21 2 a1 c 2 a2 c1 2 a2 c2/ x1  .4 a1a2cC4 a2c2Ca1c1Ca1c2Ca2c1Ca2c2 2 a2cc1 2 a2cc2 2 a2c/C .a1 c/ .2 c1C2 c2C2 a2 c a1 a2 2 c/ x 11 C .a21c2/ x 21D3 D .2 a21/ x31Ca1 .4 a2 cC2 c1 2 a2c1 2 a2c2 a1c1 a1c2/ x21 a1 .c .c1C c2/C .a1Ca2C2 a2 c/.2 c  c1  c2// x1C .2 a1 c/.2 a2 .2 c  c1  c2/  c1/C .a1 c/..2 c  c1  c2/.a1Ca2 2 a2 c/C c .c1C c2/ 2 a1 c/ x 11 C .a21 c c1 c2/ x 21D4 D .2 a1/.a1 c2 a2 c1Ca2 c2/ x31   .a1 c/.a1 .c1C c2/C2 a2 .c2  c1// x21C .2 a1 c/.a1Ca2/.c1  c2/ x1C .2 a1 a2 c2/.c2  c1/  .2 a21 c2 c1/ x 11C .a21 c3/.c1C c2/ x 21The Routh-Hurwitz criteria [4] shows that the interior point E3.x1;x2;y1;y2/ islocally asymptotically stable if the determinants of all Hurwitz matrices are positivewhere the Hurwitz matrices are given by;H1 D .D1/ H2 DD1 1D3 D2H3 D0@ D1 1 0D3 D2 D10 D4 D31A H4 D0BB@D1 1 0 0D3 D2 D1 10 D4 D3 D20 0 0 D41CCAOPTIMAL CONTROL OF A TWO TEAMS PREY-PREDATOR INTERACTION MODEL 797Using different values of the constants a1;a2; c1 and c2 to find the roots of equa-tion (3.1), one gets that one of them is negative and one of the other two positiveroots is greater than c i.e x2 will be negative. So, we have only one interior equilib-rium point.Let the initial values of the preys and the predators are, respectively, x1.0/D 0:1;x2.0/D 0:4; y1.0/D 0:3; y2.0/D 0:7; and the values of the parameters of the modelare a1 D 1:2, a2 D 1:7, c1 D 0:3 and c2 D 0:5. Substituting by these values in equa-tions (3.2, 3.1), we get the interior equilibrium point E3 D .0:079; 0:1085; 1:7208;1:0325/, which is asymptotically stable since there are two negative real eigenvaluesD  3:019684381;  0:4022762114 and another two conjugate complex eigenval-ues with negative real parts D 0:08409198332˙1:009538530I . We simulate themodel (2.1) for the mentioned initial values of the densities, we found that the systemconverges to the asymptotically stable point E3 as shown in Fig. 1 and Fig. 2.        0 100 200 300t0.00.20.40.60.8x10 100 200 300t0.00.20.40.60.8x20 100 200 300t0.40.81.21.62.02.4y10 100 200 300t0.40.81.21.62.02.4y2FIGURE 1. The densities of the preys and predators with time at theparameter values a1 D 1:2, a2 D 1:7, c1 D 0:3, c2 D 0:5 and theinitial densities x1.0/D 0:1; x2.0/D 0:4; y1.0/D 0:3; y2.0/D 0:7:4. OPTIMAL CONTROL PROBLEMIn some cases the presence of the predators is harmful. So, we should keep thesituation in which the predator does not exist. Such as the case when the predatorsare viruses or tumors. Also, there are some animals that present in few numbers, like798 M. F. ELETTREBY AND SHABAN ALY0.0 0.1 0.2 0.3 0.4 0.5x0.00.20.40.60.8x12  0.0 0.1 0.2 0.3 0.4 0.5x0.40.81.21.62.02.4y11  0.0 0.1 0.2 0.3 0.4 0.5x0.40.81.21.62.02.4y12  0.4 0.8 1.2 1.6 2.0 2.4y0.40.81.21.62.02.4y12  FIGURE 2. The system converges to the stable pointE3 for the para-meter values a1 D 1:2, a2 D 1:7, c1 D 0:3, c2 D 0:5 and the initialdensities x1.0/D 0:1; x2.0/D 0:4; y1.0/D 0:3; y2.0/D 0:7:Panda, we have to keep them from extinction or predation. In cases like these, it isrequired to make these equilibrium solutions stable.We saw that the first equilibrium solution E1 is unstable and the second equilib-rium solutionE2 is unstable if one of the conditions c1Cc2 >8 or 4.c1Cc2/ < c1 c2is not satisfied. Then, we will use the optimal control technique to stabilize the secondsolution E2 in this section.Here, we will use the same feedback control approach which is presented in [5]for a different model. Consider the above system in the following form;dx1dtD a1 Œx1.1 x1/Cx2 x1.y1Cy2/Cu1;dx2dtD a2 Œx2.1 x2/Cx1 x2.y1Cy2/Cu2; (4.1)dy1dtD c1y1C .x1Cx2/.y1Cy2/Cu3;dy2dtD c2y2C .x1Cx2/.y1Cy2/Cu4;OPTIMAL CONTROL OF A TWO TEAMS PREY-PREDATOR INTERACTION MODEL 799where ui ; .i D 1; 2; 3; 4/, are the control inputs. Some performance measure isused to determine these inputs from the conditions of asymptotic stability of the sys-tem (2.1) about its unstable equilibrium states.Let´1 D x1 x1; ´2 D x2 x2; w1 D y1 y1; w2 D y2 y2; (4.2)where ´1; ´2; w1; w2 represent perturbed states about its unstable equilibrium statesand .x1; x2; y1; y2/ denote to the coordinates of the unstable equilibrium states ofsystem (2.1). Substituting by (4.2) into the system (4.1) taking into account the iden-tities that satisfied by the equilibrium states, we get the following system,P´1 D .2a1x1Cy1Cy2/´1Ca1 .´1C´2 ´21/  .x1C´1/.w1Cw2/Cu1;P´2 D .2a2x2Cy1Cy2/´2Ca2 .´1C´2 ´22/  .x2C´2/.w1Cw2/Cu2;Pw1 D c1w1C .´1C´2/.w1Cw2Cy1Cy2/C .x1Cx2/.w1Cw2/Cu3;Pw2 D c2w2C .´1C´2/.w1Cw2Cy1Cy2/C .x1Cx2/.w1Cw2/Cu4:(4.3)The system (4.3) admits the trivial solution,´1 D ´2 D w1 D w2 D 0; u1 D u2 D u3 D u4 D 0: (4.4)This solution (from equation (4.2)) represents the equilibrium states of the system (2.1).Theorem 1. Using the optimal nonlinear control inputs;u1 D a1 .´1C´2 ´21/C .x1C´1/.w1Cw2/;u2 D a2 .´1C´2 ´22/C .x2C´2/.w1Cw2/; (4.5)u3 D .´1C´2/.w1Cw2Cy1Cy2/  .x1Cx2/.w1Cw2/;u4 D .´1C´2/.w1Cw2Cy1Cy2/  .x1Cx2/.w1Cw2/;the unstable equilibrium states of the system (2.1) will be optimally asymptoticallystable with respect to the performance measure;I DZ 10˝ dtDZ 10Œu1C a1 .´1C´2 ´21/  .x1C´1/.w1Cw2/2C Œu2C a2 .´1C´2 ´22/  .x2C´2/.w1Cw2/2 (4.6)C Œu3C .´1C´2/.w1Cw2Cy1Cy2/C .x1Cx2/.w1Cw2/2C Œu4C .´1C´2/.w1Cw2Cy1Cy2/C .x1Cx2/.w1Cw2/2C .2a1x1Cy1Cy2/´21C .2a2x2Cy1Cy2/´22C c1w21C c2w22dt:800 M. F. ELETTREBY AND SHABAN ALYProof. We will use Bellman dynamic programming technique to obtain the op-timal control inputs and Liapunov asymptotic stability theorem to prove our theorem.Suppose that we have the nonlinear system (4.3) with the four perturbed state vari-ables ´1; ´2; w1; w2 and the four control inputs ui ; .i D 1; 2; 3; 4/ with P´ i ; Pwi ; aredefinite functions of ´i ; wi ; i D 1; 2; and uj ; j D 1; 2; 3; 4. If we consider the min-imization of an integral of the form (4.6) we can write the condition for minimizationof the integral of the form;min !u@V@´1P´1C @V@´2P´2C @V@w1Pw1C @V@w2Pw2C Œu1C a1 .´1C´2 ´21/  .x1C´1/.w1Cw2/2C Œu2C a2 .´1C´2 ´22/  .x2C´2/.w1Cw2/2 (4.7)C Œu3C .´1C´2/.w1Cw2Cy1Cy2/C .x1Cx2/.w1Cw2/2C Œu4C .´1C´2/.w1Cw2Cy1Cy2/C .x1Cx2/.w1Cw2/2C .2a1x1Cy1Cy2/´21C .2a2x2Cy1Cy2/´22C c1w21C c2w22D 0:Now, the function V is a function of ´i ; wi ; i D 1; 2; and it represents the minimumvalue of the integral performance measure (4.6) achievable by starting from time tand state .´1; ´2; w1; w2/. Let us assume that V is the solution of the (4.7) takes theform,V D ´21C´22Cw21Cw22 ; (4.8)which is a positive definite function of the variables .´1; ´2; w1; w2/ and so will beused to prove the asymptotic stability of the solution (4.4).Substituting by (4.3) and (4.8) in (4.7) and differentiating the resulting expressionwith respect to ui and setting it to zero, we obtain the optimal control inputs ui asgiven by (4.5).Differentiating the function V along the trajectory of the system (4.3) taking intoaccount the control inputs (4.5) we get;PV D 2´1 P´1C2´2 P´2C2w1 Pw1CC2w1 Pw1D 2.2a1x1Cy1Cy2/´21C .2a2x2Cy1Cy2/´22C c1w21C c2w22D 2˝ (4.9)where ˝ is the value of the objective function ˝ at the optimal control inputs.Obviously, the function PV is a negative definite function of .´1; ´2; w1; w2/ onlyfor positive equilibrium states Nxi ; Nyi > 0;.i D 1; 2/ and so the positive equilibriumstates (4.4) are asymptotically stable in the Liapunov sense if the conditions of sta-bility are satisfied. OPTIMAL CONTROL OF A TWO TEAMS PREY-PREDATOR INTERACTION MODEL 801Some numerical analysis are achieved to show the stabilization behavior of thesystem (2.1) at the second unstable equilibrium solution E2: This numerical analysissupports the above analytical analysis. The model parameters are fixed as follows:a1 D 0:2, a2 D 0:7, c1 D 0:3, c2 D 0:5. Note that the conditions of stability of E2are not satisfied. The initial densities of preys are ´1 D 0:1 and ´2 D 0:4 while theinitial densities of predators are w1 D 0:3 and w2 D 0:7. A lot of simulations fordifferent values of the parameters and the initial values of the densities are done. Fig.3 shows that the preys and predator densities perturbations decay to zero. In Fig. 4,the components of the control inputs tend to zero. So, we conclude that the unstableequilibrium solution E2 of system (2.1) can be stabilized by using nonlinear controlparameters (4.1)FIGURE 3. The densities perturbations of the preys and predatorswith time converges to zero at the parameter values a1 D 0:2, a2 D0:7, c1D 0:3, c2D 0:5 and the initial densities ´1.0/D 0:1; ´2.0/D0:4; w1.0/D 0:3; w2.0/D 0:7:802 M. F. ELETTREBY AND SHABAN ALYFIGURE 4. The components of the control inputs converges to thestable point state .0;0;0;0/ at the parameter values a1 D 0:2, a2 D0:7, c1D 0:3, c2D 0:5 and the initial densities ´1.0/D 0:1; ´2.0/D0:4; w1.0/D 0:3; w2.0/D 0:7:5. CONCLUSIONMany animals live in groups. Different groups share one habitat hence thesegroups may cooperate, compete with each other or form predator-prey system. Inthis work we present a model for the interaction between predator-prey teams. Equi-librium solutions are derived, their local stability and persistence. An optimal controlis used to stabilized the unstable equilibrium solution. The boundary unstable equi-librium solution E2 is stabilized using nonlinear control inputs.ACKNOWLEDGMENTSThe authors would like to thank the editor as well as the anonymous referees verymuch for their invaluable and comprehensive comments which helped in improv-ing the paper. We are very grateful to all members of Applied Mathematics GroupOPTIMAL CONTROL OF A TWO TEAMS PREY-PREDATOR INTERACTION MODEL 803(Department of Mathematics, Faculty of Science, King Khalid University) for usefuldiscussions on the topics investigated in this paper.REFERENCES[1] E. Ahamed and A. S. Hegazi, “On dynamical multi-team and signaling games,” Applied Mathem-atics and Computation, vol. 172, pp. 524–530, 2006, doi: 10.1016/j.amc.2005.02.030.[2] E. Ahmed, A. S. Hegazi, M. F. Elettreby, and S. S. Askar, “On multi-team games,” Physica A, vol.369, pp. 809–816, 2006, doi: 10.1016/j.physa.2006.02.011.[3] J. Cai and D. Mei, “Influence of strong noise on the decline and propagation of population in thedelayed malthus-verhulst model,” Physica A, vol. 389, pp. 4715–4724, 2010.[4] L. Edelstein-Keshet, Mathematical Models in Biology. SIAM, 2005.[5] A. El-Gohary, “Chaos and optimal control of self-remission and tumor system steady states,”Chaos, Solitons & Fractals, vol. 37, pp. 1305–1316, 2008, doi: 10.1016/j.chaos.2006.10.060.[6] M. F. Elettreby, S. H. Saker, and E. Ahmed, “On multi-team predator-prey models,” ictp, pp. 5–23,2006.[7] M. Farkas, Mathematical Models in Biology. San Diego, CA: Academic Press, 2001.[8] J. Hofbauer and K. Sigmund, Evolutionary games and population dynamics. Cambridge: Cam-bridge University Press, 1998.[9] A. J. Lotka, Elements of Physical Biology. Williams and Wilkins: Baltimore, 1925.[10] T. R. Malthus, An Essay on The Principle of Population. London: J Johnson, 1798.[11] J. D. Murray, Mathematical Biology I: An Introduction. Springer 3rd edition, 2007.[12] G. Seo and D. L. DeAngelis, “A predator-prey model with a holling type i: Functional responseincluding a predator mutual interference,” Journal of Nonlinear Science, vol. 21, pp. 811–833,2011, doi: 10.1007/s00332-011-9101-6.[13] P. F. Verhulst, “Notice sur la loi que la population suit dans son accroissement,” CorrespondanceMathematique et Physique, vol. 10, pp. 113–121, 1838.[14] V. Volterra, Variations and Fluctuations of a Number of Individuals in Animal Species LivingTogether. Newyork: McgrawHill, 1931.[15] Q. Zhang, C. Liu, and X. Zhang, “Complex dynamical behavior in bio-economic prey–predatormodels with competition for prey,” Lecture Notes in Control and Information Sciences, vol. 421,pp. 185–201, 2012, doi: 10.1007/978-1-4471-2303-3˙13.[16] M. Zhou and Z. Liu, “Hopf bifurcations in a ricardo-malthus model,” Applied Mathematics andComputation, vol. 217, pp. 2425–2432, 2010, doi: 10.1016/j.amc.2010.07.043.Authors’ addressesM. F. ElettrebyCurrent address: King Khalid University, Faculty of Science, Department of Mathematics, AppliedMathematics Group, Abha 9004, Saudi ArabiaMansoura University, Faculty of Science, Department of Mathematics, Mansoura 35516, EgyptE-mail address: mohfathy@mans.edu.egShaban AlyCurrent address: King Khalid University, Faculty of Science, Department of Mathematics, AppliedMathematics Group, Abha 9004, Saudi ArabiaAl-Azhar University, Faculty of Science, Mathematics Department, Assiut 71511, EgyptE-mail address: shhaly70@yahoo.com

OPTIMAL CONTROL OF A TWO TEAMS PREY-PREDATOR INTERACTION MODEL

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