Existence and Uniqueness of Solutions of Mathematical Models of Predator Prey Interactions

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1 Punjab University Journal of Mathematics (ISSN ) Vol. 49()(017) pp Existence an Uniqueness of Solutions of Mathematical Moels of Preator Prey Interactions Muhamma Shakil 1, Hafiz Abul Wahab 1, Saee Ur Rahman, Saira Bhatti, Muhamma Shahza 1, Muhamma Naeem 3 1 Department of Mathematics, Hazara University, Manshera, Pakistan Department of Mathematics, COMSATS Institute of Information Technology, Abbottaba, Pakistan 3 Department of Information Technology, AUST, Abbottaba Pakistan ( corresponing author: wahabmaths@yahoo.com/wahab@hu.eu.pk) Receive: 15 December, 016 / Accepte: 17 February, 017 / Publishe online: 6 May, 017 Abstract. In the stuy of ecological sciences preator-prey moels are very beneficial an they are frequently use because ynamics of animal populations can easily be observe an researchers can also preict that how they will evelop over with the passage of time. The objective of this paper is to verify the existence an uniqueness of solutions of mathematical moels of preator prey interactions etermine in Shakil et al. [19, 0] through a quasi-chemical approach in orer to stuy the preator prey populations an iscover the tenencies visible. Each case starte with a set of initial an bounary conitions that forme ifferent consequences for the functions of the populations of preator (foxes) an prey (rabbits). AMS (MOS) Subject Classification Coes: 35S9; 40S70; 5U09 Key Wors: Preator prey interactions; existence an uniqueness of solution; moeling of fox an rabbit interaction. 1. INTRODUCTION 1.1. Existence, Uniqueness, an Geometry of Solutions. The Lotka Voltera moels are a couple of first orer nonlinear ifferential equations which were originally introuce by Alfre J. Lotka [14] an Vito Volterra [1], these equations are also famous as the preator prey moels. If F is the number of preator (foxes) population an R is the number of prey (rabbits) population, then this moel will have a conventional mathematical representation as F = AF R BF, t R = CR DF R. (1. 1) t 75

2 76 Muhamma Shakil, Hafiz Abul Wahab, Saee Ur Rahman, Saira Bhatti, Muhamma Shahza an Muhamma Naeem The erivatives F R t an t represent the growth of two populations with respect to time,a, B, C an D are parameters escribing the two species interaction in the following way: A-represents how many foxes can live off of the rabbits per year. B-represents how many ie off naturally per year. C- represents how quickly rabbits reprouce. D- is how many rabbits get eaten by foxes per year. Differential equations have been a funamental tool for moeling the natural worl. At the opening moment of the universe if we are familiar accurately with the laws of nature an with the situation of the universe, the situation at a subsequent of the same universe can easily be preicte. These equations have provie key insights into catastrophic shifts in ecosystems, ynamics of isease outbreaks, mechanisms maintaining bioiversity, an stabilizing forces in foo webs. For these equations, one might be intereste in unerstaning how the species ensity changes in time an how these temporal changes epen on its initial ensity. Often, ecological systems involve many moving parts with multiple types of interacting iniviuals in which case escribing their ynamics involves systems of orinary ifferential equations. A classic example of this type is the Lotka-Volterra preator-prey equations that escribe the ynamics between a prey an its preator. After writing own a ifferential equation moel of an ecological system, a moeler wants to know how the variables of interest change in time. For instance, how are ensities of the prey an preator changing relative to one another? Moreover, how oes this ynamic epen on the initial ensities of both populations? To answer these types of questions, one nees to fin solutions to the ifferential equation. If a solution to the initial/bounary value problem exists, one has to woner whether it is unique. After all, if it is not unique, one may never know if all solutions have been uncovere an which, if any, are biologically relevant. Biologically plausible things can still happen. Even if solutions to the initial/bounary value problems exist an are unique, they might not be efine for all time.. HISTORICAL BACKGROUND OF WORK ON EXISTENCE AND UNIQUENESS OF PREDATOR PREY MODELS Here in this article we provie criteria for the existence an uniqueness of solutions for the preator prey mathematical moels of two species uner certain initial an bounary conitions. The prime objective of the article is to exten an sharpen prevailing consequences on the existence an uniqueness of solutions to preator prey two species moels. We provie conitions for existence an uniqueness an we give improve bouns for two species moels hence these improve bouns will sharpen the known results. The Lotka-Volterra moel is criticize ue to the hypothesis of the infinite growth of the prey population in the absence of a preator an as being impractical generally for its structural instability. However, it is a constructive tool having the essential properties of the actual preator-prey systems, an assists as a robust basis from which more sophisticate moels can be evelope, Murray [16].

3 Existence an Uniqueness of Solutions of Mathematical Moels of Preator Prey Interactions 77 A great eal of work is one on the preator prey competition moels, particularly by taking the case stuy of two species. A few of the illustrative articles are [l-15, 17, ]. About the work prior to 1986, a more etaile bibliography is ebate in [1] by Ali. The articles, by Blate an Brown [], Dancer [8], Li an Logan [13] provie a qualitative treatment uner irichlet bounary conitions for two species moel. A very etaile an comprehensive analysis is presente in Dancer [8] but it still nees to answer some open questions. For three species moels a few relate complications are iscusse in Lakos [1]. Many species case of the existence of a coexistence states uner Neumann bounary conitions are treate in the articles, Ali [1], Brown [3,4], Mckeznna [15], Korman an Leung [10]. But theoretically the case of Neumann bounary conitions is easier comparatively because Neumann bounary conitions are always satisfie by the solutions of the equivalent system of orinary ifferential equations an in the reaction-iffusion system they are use for the purpose of assessment. For two competing species case a reasonably simple an quantitative conitions are provie in the articles [5-7, 11, 15, 17, ] for the coefficients which involve stability, existence an uniqueness of positive steay states. For two species moel a few results yieling uniqueness are given in, Cantrell an Cosner [6], Cixner [7], Mckeznna [15], Korman an Leung [11]. Further, we refer the reaer for the existence an uniqueness of solutions for problems to [7-8] There is a qualitative ifference between the case of two species an the case of many species moel, because with two species moel it is possible to make the system quasimonotone but with three or more species it will be not possible. Thus, quasimonotonicity methos cannot be applie to larger systems. The articles, Blate an Brown [], Cixner [7], Dancer [8], Cantrell an Cosnero [5] ebate the cases in which the Lotka-Volterra moel retains numerous coexistence states. The stability analysis of preator-prey population moel with time elay an constant rate of harvesting can be seen in [4]. Further, an Alternative approach to the persistence in a 3 species preator-prey moal is given in [5]. The survival curves for the moel of growth an ecay of tumor cab be seen in [6]. For coexistence states the question of uniqueness is moerately elicate. In this article conitions for stability an uniqueness of solutions of the coexistence state for the case of two species are given an some quantitative outcomes are improve. An estimate is obtaine here for a quantity befalling in the assumptions of some of the uniqueness results. Our assessment improves some of the existing results in the stuies of ecological systems an partially negative answers are given to the conjectures of many researchers. These result yiels an algebraically computable criterion for the positive coexistence of competing species of animals in many biological moels. 3. MAIN IDEAS 3.1. A Simple Cell Jump Moel, The Origin Of Our Work. Let us consier that our space is ivie into two cells, a system represente as a chain of cells each with homogeneous composition an elementary transfer process (for us there are only two cells) between them. Let us numerate these cells by the Roman numbers I an II an mark all the components an quantities relate to them by the upper inex I or II, corresponingly (Fig. 1). On each cell we have some concentrations for these processes. Let c I is the vector of concentration in the first cell I an c II is the vector of concentration in the secon cell II.

4 78 Muhamma Shakil, Hafiz Abul Wahab, Saee Ur Rahman, Saira Bhatti, Muhamma Shahza an Muhamma Naeem FIGURE 1. Cell Jump Moel Where c I, c II represent the concentrations of the components of A I, A II respectively an N I, N II represent the vectors of composition of the respective cells. 3.. Simple Diffusion. The simplest mechanism of iffusion between any two cells is the process of jumping of particles from one cell to another neighbouring cell. This type of mutually inverte an mutually inverse process can be written as, Gorban et al. [9], A I A II, A II A I. (3. ) Suppose that wr I an wr II represent the reaction rates in the respective cells an in the continuous limit we get the Fick law Gorban et al. [9] as the first Taylor approximation. To get the continuous limit, we take c I = c (x),c II = c (x + l) an use the Taylor expansion: c (x + l) = c (x) + l x c + o ( l ). Then by using the mass action law, total flux J F will be calculate as J F = kl [ c II c I] = D c. (3. 3) l In this approximation, kl = D, where l is the cell size. 4. A QUASI CHEMICAL BASED APPROACH FOR PREDATOR-PREY INTERACTIONS Let us suppose that our space is ivie on two territories, on each territory we have some concentrations (number of foxes an rabbits) for these processes. Let F I represents (fox/ foxes in the first territory) an R I represents (rabbit/ rabbits in first territory) is the vector concentration in territory I an F II represents (fox/ foxes in secon territory) an R II represents (rabbit/ rabbits in secon territory) is the vector of concentration in the secon territory II. The territories for the foxes are consiere to be the simple cells. The interactions between preator (fox/foxes) an its prey (rabbit/rabbits) are represente by the chemical reactions which obey the mass action law. Here we specially refer this moeling by Gorban et al. [9] who evelope the iea an moele iffusion equations for ifferent mechanisms an prove the issipation inequalities. The cell-jump moels may be consiere as the proper iffusion moels by themselves, the territorial animal like fox is given a simple cell as its territory. The sense in which the iscrete equations for cells converge to the partial ifferential equations of iffusion is that the cell moels give the semi-iscrete approximation of the partial ifferential equations for iffusion which result in a system of orinary ifferential equations in cells. There are jumps of concentrations on the bounary of cells. The system of semi iscrete moels for

5 Existence an Uniqueness of Solutions of Mathematical Moels of Preator Prey Interactions 79 cells with no-flux bounary conitions has all the nice properties of the chemical kinetic equations for close systems. Uner the proper relations between coefficients, like complex balance or etaile balance, this system emonstrate globally stable ynamics Mechanism Of Circulation. Between two foxes/rabbits the mechanism of circulation is the proceure when a fox/ rabbit move from that territory which is efine for it, to another territory which is efine for another fox/rabbit an vise versa. The foxes are territorial animals; they efine proper areas for them to stay. This mechanism is illustrate as F I F II, F II F I. (4. 4) Let us suppose that c I /c II be the concentrations of foxes in the first /secon cell an I / II be the concentrations of rabbits in the first/secon cells respectively. Then by using the law of mass action, mathematical moel for this mechanism is etermine by Shakil et al. [19] as c = D c. (4. 5) With initial an bounary conitions as c (x, 0) = c 0 (x), c (0, t) = 0, c (l, t) = 0. (4. 6) In this approximation kl = D, where k is a constant, l is the cell size an is Laplace operator. Our above moel is a special case of Rahman [18] an existence an uniqueness of solution for this moel is alreay prove by Rahman [18]. 4.. Mechanism Of Sharing Place. Let us suppose that a fox Fi I is present in territory one an another fox Fj II is present in territory two, if fox Fi I leaves territory one an moves to territory two an vise versa. This mechanism will be escribe through the equations of the form Fi I + F j II Fj I + F i II, Fi II + Fj I F i I + F j II (4. 7). Mathematical moels for this mechanism are etermine by Shakil et al. [19] as c i = kl [c j c i c i c j ], (4. 8) c j = kl [c i c j c j c i ], (4. 9) with initial an bounary conitions as c i (x, 0) = c 0 (x), c i (0, t) = 0, c i (l, t) = 0, c j (x, 0) = c 1 (x), c j (0, t) = 0, c j (l, t) = 0. For the existence an uniqueness of solution of the above moels, we move as; Aing equations (4. 8 ) an (4. 9 ), we get (4. 10) c i + c j = 0 (c i + c j ) = 0. (4. 11)

6 80 Muhamma Shakil, Hafiz Abul Wahab, Saee Ur Rahman, Saira Bhatti, Muhamma Shahza an Muhamma Naeem Let us suppose that (c i + c j ) = u, then we get u an initial an bounary conitions will reuce to = 0, (4. 1) u (x, 0) = u 0 (x), u (0, t) = 0, u (l, t) = 0. (4. 13) Proposition 1. If u is the solution of system (4. 1 ) an (4. 13 ) then the concentrations of u satisfies sup u L = u 0 L. 0 t T Theorem 1. Assume u o H (Ω) then the system (4. 1 ) an (4. 13 ) has a unique classical solution u(r, t)on (0, T ) where H (Ω) represents Hilbert Space an Ω = [0, l]. In orer to prove the theorem, first we nee to prove the proposition. Proof. Taking inner prouct of equation (4. 1 ) with (u) an integrating, we have u u Ω x = 0 u x = 0 t Ω t u L = 0 sup 0 t T u L = u 0 L. (4. 14) This proves the existence of solution of the above moel. For uniqueness of solution, let us suppose that u 1 an u be two solutions of the equation (4. 1 ). Therefore, u 1 = u = 0, an this implies (u 1 u ) = 0. Let us suppose that w = (u 1 u ) an w (x, 0) = 0, then w = 0. Taking inner prouct with w an integrating, we have w w Ω x = 0 t w L = 0 w L = 0 u 1 u L = 0 u 1 u = 0 u 1 = u. (4. 15)

7 Existence an Uniqueness of Solutions of Mathematical Moels of Preator Prey Interactions 81 This proves the uniqueness of solution of the above moel. Again, u L = c i + c j L = u 0 L c i L c j L u 0 L c i L u 0 L c j L u 0 L. (4. 16) Now by taking foxes an rabbit s case together, this mechanism will be escribe through the equations of the form F I i F II i + R II j F I i + R I j F I i, (4. 17) + R I j F II i + R II j F II i. (4. 18) The above state mechanism (4. 17 ) illustrates that a fox is present in territory one an if a rabbit from territory two moves to territory one, fox present in that territory will prey that rabbit an a similar case for territory two is illustrate in mechanism (4. 18 ). Mathematical moels for this mechanism are etermine by Shakil et al. [19] as c = kl [ c c + l c], (4. 19) = kl [c c + l c ], (4. 0) with initial an bounary conitions as c (x, 0) = c 0 (x), c (0, t) = 0, c (l, t) = 0, (x, 0) = 0 (x), (0, t) = 0, (l, t) = 0. (4. 1) To fin the existence an uniqueness of solution of the above moels, we a the equations (4. 19 ) an (4. 0 ), to get c + = kl c (c + ) = kl c. (4. ) Let us suppose that c + = v, then we get v = kl c. (4. 3) The initial an bounary conitions will reuce to v (x, 0) = v 0 (x), v (0, t) = 0, v (l, t) = 0. (4. 4) Proposition. If v is the solution of system (4. 3 ) an (4. 4 ) then the concentrations of v satisfies v L = v 0 L. sup 0 t T Theorem. Assume v 0 H (Ω) then the system (4. 3 ) an (4. 4 ) has a unique classical solution v(r, t) on (0, T ) where H (Ω) represents Hilbert Space an Ω = [0, l].

8 8 Muhamma Shakil, Hafiz Abul Wahab, Saee Ur Rahman, Saira Bhatti, Muhamma Shahza an Muhamma Naeem Proof. Taking inner prouct of equation (4. 3 ) with ( v) an using integration by parts, we get 1 ( v) x = kl v c x t = kl ( v) ( c )x. (4. 5) Now, 4 c = ( c + ) ( c ) 4 c ( v) + ( c + ) 4 c ( v) c ( v). (4. 6) Then equation (4. 5 ) implies that 1 t v L kl ( v) ( v) x. (4. 7) Let us suppose that ( v ( v) ) = v ( v) + v ( v) ( v ( v) ) ( v) 3 = v ( v). (4. 8) Then equation (4. 7 ) implies that ( t v L kl v ( v) ) x ( v) 3 x t v L 0 ( v) 3 x t v L + ( v) 3 x 0 t v L 0 v L v 0 L. (4. 9) Using Poincare inequality, we have Since v = c +, then, v 0 L. Thus existence of solution of the above moel fol- Also this implies L lows. v L v 0 L. c L L c + L c L L < v 0 L c L v 0 L. (4. 30)

9 Existence an Uniqueness of Solutions of Mathematical Moels of Preator Prey Interactions 83 For uniqueness of solution, let us suppose that v 1 an v be two solutions of equation (4. 3 ) therefore, v 1 kl ( v 1 ), v kl ( v ), which gives (v 1 v ) kl [ ( v 1 ) ( v ) ]. (4. 31) For the case, ( v 1 ) ( v ), we have, (v 1 v ) 0. Let R = v 1 v, then R 0. Now taking the inner prouct with R an integrating, we have This proves the uniqueness of solution. R R Ω x 0 t R L 0 R L 0 v 1 v L 0 v 1 v 0 v 1 = v. (4. 3) 4.3. Mechanism Of Attraction. The mechanisms of attraction in the case stuy of foxes are unusual but in winter seasons they pair up. If two foxes are present in a same territory an a fox from another territory inters in that territory, then it is given by the following equations: F I + F II 3F I, F I + F II 3F II. Mathematical moel for this mechanism is etermine by Shakil et al. [19] as With initial an bounary conitions as (4. 33) c = kl 1 3 c3. (4. 34) c (x, 0) = c 0 (x), c (0, t) = 0, c (l, t) = 0. (4. 35) Proposition 3. If c is the solution of system (4. 34 ) an (4. 35 ) then the concentrations of csatisfies sup c L = c 0 L. 0 t T Theorem 3. Assume c 0 H (Ω) then the system (4. 34 ) an (4. 35 ) has a unique classical solution c(r, t) on (0, T ) where H (Ω) represents Hilbert Space an Ω = [0, l].

10 84 Muhamma Shakil, Hafiz Abul Wahab, Saee Ur Rahman, Saira Bhatti, Muhamma Shahza an Muhamma Naeem Proof. For proving the existence an uniqueness of solutions for these moels taking inner prouct of equation (4. 34 ) by c an integrating by parts, we have 1 t c L = kl c c 3 x 3 t c L + 3 kl c c 3 x = 0 t c L 0 sup 0 t T c L c 0 L. (4. 36) This proves the existence of solution for the moele equation of mechanism of attraction an uniqueness of solution of the above moel follows from previous moels Mechanism Of Repulsion. In winter seasons if two foxes are alreay present in a same territory an a fox from another territory inters that territory then the foxes present in that territory will have a fight with that fox, as a result of which this fox will leave that territory an returns to his initial position then this mechanism will be escribe by the stoichiometric equations as 3F I F I + F II, 3F II F I + F II (4. 37). Mathematical moel for this mechanism is etermine by Shakil et al. [19] as c = kl c3. (4. 38) The initial an bounary conitions are c (x, 0) = c 0 (x), c (0, t) = 0, c (l, t) = 0. (4. 39) This moel is a special case of mechanism of attraction an existence an uniqueness of solution for this moel follows from the moele equation of mechanism of attraction Autocatalysis Mechanism. Self reprouction process is calle autocatalysis process. In our case stuy, in Shakil et al. [0] we stuie that how foxes/ rabbits interact together in an autocatalysis way to enhance their populations. When in winter months two foxes couple together in territory one/two, we suppose that a male fox F (m) an a female fox F (f) interact together, as a result of that interaction n number of foxes are born, this mechanism will be escribe as F I n 1 F I, F II n F II, (4. 40) where n 1, n >. Mathematical moel for this mechanism is etermine by Shakil et al. [0] as c = klc c, (4. 41) with initial an bounary conitions as c (x, 0) = c 0 (x), c (0, t) = 0, c (l, t) = 0. (4. 4)

11 Existence an Uniqueness of Solutions of Mathematical Moels of Preator Prey Interactions 85 Proposition 4. If c is the solution of system (4. 41 ) an (4. 4 ) then the concentrations of c satisfies sup c L = c 0 L. 0 t T Theorem 4. Assume c 0 H (Ω) then the system (4. 41 ) an (4. 4 ) has a unique classical solution c(r, t) on (0, T ) where H (Ω) represents Hilbert Space an Ω = [0, l]. Proof. Taking inner prouct of equation (4. 41 ) with c an integrating, we have c c x = kl c c Ω t c L = kl c( c )x t c L + kl c( c )x = 0 t c L 0 (4. 43) c L c 0 L 0 c L c 0 L. (4. 44) Thus, existence of solution for the above mechanism follows an uniqueness of solution for the above moel also follows from previous moels. Remarks: Some of our mathematical moels etermine by Shakil et al. [19, 0] can be treate as special cases of the above iscusse moels, so existence an uniqueness of solutions for those mathematical moels follows from the above moels. 5. CONCLUSIONS The consequences an analysis of this article uner the suppositions mae emonstrate the existence an uniqueness of solutions of simple preator-prey moels an some fascinating biological consequences have been erive an justifie. Each case starte with a set of initial an bounary conitions that forme ifferent significances for the functions of the populations of preator prey interactions. Subsequently, in more etaile moels it will be useful to observe the robustness of these results. This approach merely elivers a short time results which lea to the unfavorable conclusion. A similar approach can easily be extene to systems with several prey species an single preator, or several preator species an single prey, an an equivalent conclusion will hol. These results lea to a better unerstaning an give a new insight for the movements of ifferent mechanisms of preator prey interactions. The consequences of the paper exhibit that, uner the raise suppositions, a species may be abolishe (efficiently in finite interval of time). The consequences of the stuy lea to the conclusion that in the process of evolution, preation, competition is not the only reason for estruction of certain species. Our conclusions lea to the results that, not only in significance of ineffective competition, but even when competition is inattentive, the original species are certainly riven to estruction. For example, in the aforementione foxes an rabbits moel, the rabbits may be exclue entirely by preation alone in a finite time.

12 86 Muhamma Shakil, Hafiz Abul Wahab, Saee Ur Rahman, Saira Bhatti, Muhamma Shahza an Muhamma Naeem The consequences of the paper propose an justify our new approach towars the stuy of ecological problems of preator prey types of interactions. Usually biological stuy requires the introuction of preator prey populations increasing or ecreasing to an aequate level. This preator-prey system after a few time will reach to its coexisting equilibrium, so only a short term results are achieve through this approach. Our results state that this approach will be further effective if, instantaneously with introucing the preators, a prey species is introuce in orer to provie a longstaning foo source. For the introuce prey it is not essential to contest with the pest for resources. The resources can be a theme of fortification, so if they o not compete it will be much better for them, but the prey species introuce certainly have a better survival capability. Uner the circumstances, in finite time the actual pest species will be entirely annihilate. REFERENCES [1] W. A. Ali, Steay states in the iffusive Volterra-Lotka moel for several competing species, Dissertation, University of Miami, [] I. K. Argyros, Uniqueness-Existence theorems for the solutions of polynomial equations in Banach space, Punjab University Journal of Mathematics 14, (1986) [3] J. Blat an K. J. Brown, Bifurcation of steay state solutions in preator-prey an competition systems, Proc. Roy. Sot. Einburgh Sect. A. 97, (1984) [4] P. N. Brown, Decay to uniform states in ecological interactions, SIAM J. Appl. Math. 38, (1980) -37. [5] P. N. Brown, Decay to uniform states in competitive systems, SIAM J. Math. Anal. 14, (1983) [6] R. S. Cantrell an C. Cosnero, On the steay-state problem for the Volterra-Lotka competition moel with iffusion, Houston J. Math. 13, (1987) [7] R. S. Cantrell an C. Cosnero,On the uniqueness an stability of positive solutions in the Lotka-Volterra competition moel with iffusion, Houston J. Math. 15, (1989) [8] C. Cixner an A. C. Lazer, Stable coexistence states in the Volterra-Lotka competition moel with iffusion, SIAM J. Appl. Math. (1984) [9] E. N. Dancer, On the existence an uniqueness of positive solutions for competing species moels with iffusion, Trans. Amer. Math. Sot. 36, (1991) [10] I. H. Elmabruk, Alternative approach to the persistence in a 3 species preator-prey moal,punjab University Journal of Mathematics 39, (007) [11] A. N. Gorban, H. P. Sargsyan an H. A. Wahab, Quasichemical moels of multicomponent nonlinear iffusion, Mathematical Moeling of Natural Phenomena 6, (011) [1] T. Jankowski, Existence an uniqueness of solutions for problems with a parameter, Punjab University Journal of Mathematics 6, (1993) 1-6. [13] K. Khan, Some mathematical moels an survival curves for growth an ecay of tumor, Punjab University Journal of Mathematics 35, (00) [14] P. Korman an A. Leung, A general monotone scheme for elliptic systems with applications to ecological moels, Proc. Roy. Sot. Einburgh Sect. A. 10, (1986) [15] P. Korman an A. Leung,On existence an uniqueness of positive steay states in the Volterra-Lotka ecological moels with iffusion, Appl. Anal. 6, (1987) [16] N. Lakos, Existence of steay state solutions for one preator-two prey system, preprint. Lotka-Volterra moels with iffusion [17] L. Li an R. Logan, Positive solutions to general elliptic competition moels, J. Differential an Integral Equations 4, (1991) [18] A. J. Lotka, Unamme oscillations erive from the law of mass action, J. Am. Chem. Soc. 4, No. 8 (190) [19] P. J. Mckeznna an W. Walter, On the Dirichlet problem for elliptic systems, Appl. Anal. 1, (1986) [0] J. D. Murray, Mathematical biology I: An Introuction, Germany, Springer-Verlag, 003. [1] C. V. Pao, Coexistence an stability of a competition-iffusion system in population ynamics, I. Math. Anal. Appl. 83, (1981)

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