compiles/6a43080106cbd1fa83482983593961c6/output.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 7, No. 1, 2014, 37-44 ISSN 1307-5543 – www.ejpam.com Modeling the Prey Predator Problem by a Graph Differential Equation J. Vasundhara Devi∗, R. V. G. Ravi Kumar Department of Mathematics, GVP-Prof.V.Lakshmikantham Institute for Advanced Studies, GVP Col- lege of Engineering, Visakhapatnam, AP, India. Abstract. In this paper, we introduce new concepts like a pseudo simple graph, product of two graphs and obtain a sufficient condition which will guarantee that the solution of the IVP of a graph differential equation has the same nature as its graph of initial conditions. Further, we model the well known pre- predator problem by graph differential equations and show that the nonlinearity is naturally preserved. 2010 Mathematics Subject Classifications: 05C20 05C22 05C50 34AB2 92D25 Key Words and Phrases: Simple Graph, Pseudo simple graph, product of graphs, graph differential equation and prey predator model. 1. Introduction Any natural or a man made system involves interconnections between its constituents, thus forming a network, which can be expressed by a graph [2, 3]. Graphs arise naturally when trying to model organizational structures in social sciences. It has been noted that a graph which is static in nature is not suitable for social phenomena whose changes with time are natural. This led to the introduction of a dynamic graph and a Graph Differential Equation (GDE) in [3]. The introduced concepts were successfully utilized to study stability of complex dynamic systems through its associated adjacency matrix [3]. In [2] we have utilized the concepts defined in [3] including a graph linear space and its associated matrix linear space. Using the notion of a dynamic graph and the graph differential equations we observed that the study of GDEs falls into the realm of differential equations in abstract spaces. This study, through highly mathematical, would be of little use for practical problems. On the other hand, if we consider the associated matrix differential equation(MDE) then the approach appeared more reasonable and practical for the study of GDEs. Hence in [2], we considered a weighted directed simple graph as the basic element and developed ∗Corresponding author. Email address: jvdevi@gmail.com (J. Vasundhara Devi) http://www.ejpam.com 37 c© 2014 EJPAM All rights reserved. J. Devi, R. Kumar / Eur. J. Pure Appl. Math, 7 (2014), 37-44 38 the theory. We have obtained existence and uniqueness of solutions of a GDE through its associated MDE using the monotone iterative technique. In [2] through we have developed significant results, the basic concept involved was weighted directed simple graph. Since a simple graph has no loops, this fact when trans- lated into differential equations frame work states that there is no way to accommodate the rate of change of an edge eii and its relation with other edges including the edge eii . This is a drawback that had to be handled to model physical phenomena using graph differential equations, which called for a new concept that we plan to introduce in this paper. Further, since there exists an isomorphism between graphs and their adjacency matrices, we successfully exploited it and defined the product of two graphs. A good example , will go a long way in support of the theory, we have considered the prey predator problem and developed the corresponding matrix differential equation and showed how the nonlinearity is preserved in this set up. The rest of the paper is as follows. In section two, we introduce the concepts of pseudo simple graph and product of two graphs and have obtained a result, that can be of practical importance in this set up. In section three we obtained the matrix differential equation for prey predator problem and extended it to three species and further generalized it. In section four we conclude our work. 2. Main Results In this section, we begin with the concept of a pseudo simple and later introduce the product of two graphs. Definition 1 (Pseudo simple graph). A simple graph having loops is called as a pseudo simple graph. Parallel to the definitions and theory developed in [2] we proceed to state the results in this set up. We avoid the details for fear of repetition. Let v1, v2, . . . , vN be N verticess, N fixed. Let DN be the set of all weighted directed pseudo simple graphs D = (V, E). Then (DN ,+, .) is a linear space with the definitions given in [3] and [2]. Let the set of all corresponding adjacency matrices be EN . Then (EN ,+, .) is a matrix linear space where ’+’ denotes matrix addition and ’.’ indicates scalar multiplication. With this basic structures defined, the comparison theorems, existence and uniqueness results of solutions of MDE and the corresponding GDE follow as in [2]. Taking cue from matrix multiplication we define the product of two graphs as follows. Product of graphs: Let G1 and G2 be two graphs with edges (ei j)N×N and (di j)N×N respec- tively. Then the product of the two graphs G1 and G2 is the graph G in which the weight gi j of the edge from v j to vi is the dot product of the vectors one having the weights of the edges inwards to vi and the other having weights of the edges outwards from v j . We now proceed to develop a result on the nature of solutions of a graph differential equation. J. Devi, R. Kumar / Eur. J. Pure Appl. Math, 7 (2014), 37-44 39 Let D′ = g(t, D) (1) be a graph differential equation. Now if possible suppose g(t, D) can be written as a product of two graphs C D where C is a graph having constant weights. Then the GDE (1) can be written in the form D′ =C D Dt0 =D0 (2) where C is a graph called a coefficient graph and D 0 is the initial graph. Let E′ =AE Et0 =E0 (3) be the corresponding IVP of the MDE where E 0 is the adjacency matrix corresponding to the initial graph D 0 . Then we have the following result relating to the solutions of MDE and hence to that of GDEs. Theorem 1. Let E(t) be a solution of the IVP (3). Suppose there exists a non singular matrix P such that P−1AP = H is a diagonal matrix. Then the solution E(t) has the same nature as that of E 0 . In other words, the solution of the IVP of the GDE (2) has the same nature as that of the initial graph D 0 . Proof. Suppose there exist a matrix P such that P−1AP = H. Then we know from the theory of linear algebra that A and H have the same eigen values. Further we know that the solution of the IVP of MDE (3) is same as the solution of the MDE E′ = HE, Et0 = E0. The solution of the MDE (3) is given by E(t) = eHt E 0 , where eHt is a diagonal matrix. Thus it is clear that E(t) enjoys the same character as of E 0 . Using the fact that there exists an isomorphism between matrices and graphs. We can easily conclude that the IVP of GDE (2) has a solution D(t) having the same nature as that of D0. Remark 1. Corresponding to the matrices P−1 and P we can find two graphs GP−1 and Gp such that GP−1 GAGP= GH is a graph having only loops. Please see Figures 1a through 1d. J. Devi, R. Kumar / Eur. J. Pure Appl. Math, 7 (2014), 37-44 40 (a) Gp (b) Gp −1 (c) GA (d) Gp −1GAG Figure 1: Graphs J. Devi, R. Kumar / Eur. J. Pure Appl. Math, 7 (2014), 37-44 41 3. Modeling of the Prey-Predator Problem In this section, we formulate a matrix differential equation for the famous prey predator model and later extend it to three species and N -species. Let x denote the prey population and y denote the predator population, Then the rate of change of prey and that of predator gives rise to a system of nonlinear differential equations given by d x d t = ax + bx y, a > 0, b < 0, (4) d y d t = c y x + d y, c > 0, d < 0. (5) It is well known that the above differential equations are linearized and solved as a linear system of differential equations. We now express the above system as a Graph Differential Equation and consider the cor- responding Matrix Differential Equation. We show that the nonlinearity is preserved in this set up. Let the vertex v1 denote the prey and v2 denote the predator. Set e11 = x as population of the prey and e22 = y as the population of the predator. It can be seen that e12 is the edge going outward from v2 and is incident on v1. This means that e12 denotes the interaction between predator and prey. Actually, e12 gives the status of predators finding the prey. Similarly e21 denotes the edge outward from v1 and incident on v2. In terms of our model, this edge indicates the status of prey that fall prey to predators. Now the graph of the prey predator model is of the form Figure 2: The Graph of the prey-predator problem and its adjacency matrix is given by � e11 e12 e21 e22 � . The equations (4) and (5) reduce to the form e′11 = ae11+ be21, (6) J. Devi, R. Kumar / Eur. J. Pure Appl. Math, 7 (2014), 37-44 42 e′22 = ae12+ be22. (7) Our aim is to obtain a Matrix Differential Equations of the form � e11 e12 e21 e22 �′ = A � e11 e12 e21 e22 � where A2×2 is the coefficient matrix. It can be easily seen that � e11 e22 �′ = � a b c d �� e11 e22 � and hence we propose to choose A= � a b c d � and obtain matrix differential equation of the form � e11 e12 e21 e22 �′ = � a b c d �� e11 e12 e21 e22 � (8) The system (8) yields the equations (6), (7) and the following two differential equations given by e′12 = ae12+ be22, (9) e′21 = ce11+ de21. (10) The equation (9) describe the rate of change of predator finding prey and it is positively pro- portional to the predator finding prey and negatively proportional to the predator population. The equation (10) gives the rate of change prey coming in way of predator and this is pos- itively proportional to prey available and negatively proportional to prey falling to predator. Hence it can be seen that all the four equations given by (6), (7), (9) and (10) are consis- tent with the standard prey predator problem. The beauty in this set up is that the nonlinearity is preserved and effectively used. The sys- tem obtained reduces to a Matrix linear differential equation and the solution is immediately given by � e11(t) e12(t) e21(t) e22(t) � = eA(t−t0) E0 where E0 is the given matrix of initial conditions at t = t0, see [1]. Observe that eA(t−t0) is a matrix. If A is diagonalizable then eA(t−t0) can be replaced by the diagonal matrix eH(t−t0), where H = diag[λ1,λ2] where λ1 and λ2 are the eigen values of A and the matrix has the form eH(t−t0) = � eλ1(t−t0) 0 0 eλ2(t−t0) � . J. Devi, R. Kumar / Eur. J. Pure Appl. Math, 7 (2014), 37-44 43 Thus it has been effectively shown that a physical phenomena can be described through a graph and using the standard models we can preserve the nonlinearity and obtain more in- formation using its associated matrix differential equation. Next we consider a three species model given by d x d t = ax + bx y + cxz, d y d t = d y x + e y + f yz, dz d t = gzx + hz y + kz. Working parallel to the prey predator problem, we consider three vertices v1, v2 and v3 repre- senting x , y and z respectively. Proceeding as in the prey predator problem, we arrive at the linear matrix differential equation of the form    e11 e12 e13 e21 e22 e23 e31 e32 e33    ′ =    a b c d e f g h k       e11 e12 e13 e21 e22 e23 e31 e32 e33.    It can be observed that we will get six additional equations and hence more information is known. The solution for this system can be immediately given by E(t) = eA(t−t0)C , where C is the matrix of initial conditions. Clearly, this approach can be extended suitably to a N -species model. Remark 2. It can be observed that if the rate of change of an edge ei j is proportional only to the edges that are incident outward from v j then we obtain a matrix differential equation. On the other hand, if the rate of change of edge ei j is proportional to all the N × N edges or to some of them (without any structure) then we can treat the N × N edges as an N2 vector and consider a vector differential equation of the X ′ = AX where A is an N × N matrix. 4. Conclusion In this paper we have introduced the notions of a pseudo simple graph and the product of two graphs we have given sufficient conditions under which a solution of a Graph Differential Equation has the same nature as its graph of initial conditions. Further, we have obtained a matrix differential equation for a prey predator probleml and explicitly gave its solutions preserving the nonlinearity. From the model, it is clear that the nonlinearity in the prey predator problem is preserved by using a graph differential equation. As long as the rate of change of the species X i is proportional to linear interactions between itself and X j species it will be possible to obtain a linear graph differential equation. In other words, in the above discussed prey-predator problem if the rate of change of prey population X w.r.t time is proportional to the interaction of the square of prey population (X 2) and square REFERENCES 44 of product population (Y 2) and similarly with predator population then the problem reduces to the form d x d t = ax + bx2 y2, d y d t = c y2 x2+ d y. This cannot be immediately modeled by a linear graph differential equation and needs further investigations. ACKNOWLEDGEMENTS This work was done under the project no. 2/48(8)/2011/-R&D II/1600 sanctioned by National Board of Higher Mathematics, Department of Atomic Energy, Government of India. The authors acknowledge their support. References [1] S.G. Deo, V. Lakshmikantham, and V. Raghavendra. Textbook of Ordinary Differential Equations, 2nd Edition. Tata McGraw Hill Education Private Limited, New Delhi. 1998. [2] J.V. Devi, R.V.G.R. Kumar, and N.G. Babu. 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