On Lotka-Volterra Evolution Law

Transcription

1 Advanced Studies in Biology, Vol. 3, 0, no. 4, 6 67 On Lota-Volterra Evolution Law Farruh Muhaedov Faculty of Science, International Islaic University Malaysia P.O. Box, 4, 570, Kuantan, Pahang, Malaysia farruh_@iiu.edu.y Mansoor Saburov Faculty of Science, International Islaic University Malaysia P.O. Box, 4, 570, Kuantan, Pahang, Malaysia saburov@gail.co Abstract In this paper we study a Lota-Volterra (LV) evolution law. We give a atheatical odel of the LV evolution law. This odel includes into itself the classical LV predator-prey odel as a particular case. One of the basic results is that under suitable conditions the classical LV predator-prey odel can exhibit any asyptotical behavior such as equilibriu states, periodic cycles, and attractors. In this paper we provide soe LV type odels in which one can observe any asyptotical behavior such as equilibriu states, periodic cycles, and attractors. Keywords: LV evolution law, LV type odel, equilibriu state. Introduction Predator-prey odels have been studied for a long tie. Many biologists believe that if the unique positive equilibriu point of a predator-prey syste is local asyptotically stable, then it is global asyptotically stable. However, this is not always true. It is found that a unique positive local asyptotically stable equilibriu point has at least one liit cycle surrounding the equilibriu point under suitable condition. Therefore, any atheaticians try to use soe well-nown ethods to find conditions for global stability for the equilibriu point of predator-prey systes.

2 6 F. Muhaedov and M. Saburov The Lota-Volterra (in short LV) odel is the siplest odel of predator-prey interactions. It is based on linear per capita growth rates and written as follows x& = x( b py), () y& = y( d + rx) where y is the nuber of soe predators (for exaple, wolves); x is the nuber of its preys (for exaple, rabbits); y& and x& represent the growth of the two populations against the tie; t represents the tie; b, p, d, r 0 are paraeters representing the interaction of the two species. The equation () has periodic solutions which do not have a siple expression in ters of the usual trigonoetric functions. At any given tie in the phase plane, the syste is in a liit cycle and lies soewhere on the inside of these elliptical solutions Fig.. Periodic activity generated by the LV odel A discrete analogy of the LV odel can be considered as follows xn+ xn = xn( b pyn) xn+ = xn( + b pyn) or yn+ yn = yn( d + rxn) yn+ = yn( d + rxn). () Under suitable conditions, the discrete LV odel () has an application in genetic population systes. One of the siplest cases of the discrete LV odel () is the following evolution operator V : S S of the population on the one S = ( x, y) R : x, y 0, x+ y = diensional siplex { } x = x( ay) V :, (3) y = y( + ax)

3 On Lota-Volterra evolution law 63 where V( x, y) = ( x, y ) is an iage of the point ( x, y ) under the operator V. In the series of wors [-3], a quadratic LV odel V : S S defined by x ( Vx) = x + aixi, =,, (4) i= deeply studied on the ( ) diensional siplex S = x = ( x, L, x) R : xi =, xi 0, i =, (5) i= A,, = ai a i, = i is a Vx = x = x,, L x is an iage of the point as the generalization of the odel (3), where ( ) [ ] sew-syetric atrix and ( ) x It is worth entioning that a quadratic LV odel (4) has a fascinating application in genetic population systes [5]. Let us briefly ention it, in order to describe ain properties of the quadratic LV odel (4). S. Consider a population consisting of species. Let x ( x x ) =,, L be the probability distribution of species in the initial generations and P ij, the probability that individuals in the i -th and j -th species interbreed to produce an individual fro -th specie. We assue that in each generation an individual can inherit only species of its parents. In other words, it eans that Pij, = 0, { i, j}. Then the probability distribution x = ( x,, L x ) of the species in the next generation can be found by the total probability given by (4), where ai = Pi,. This eans that the association x x defines a apping V so-called an evolution operator of the population syste. Theore.[-3] The evolution of the population syste is given by (4). Then the following stateents hold true. (i) The population syste has at least equilibriu states. The nuber of equilibriu states is finite. In typical conditions of the nature, equilibriu states are not local asyptotically stable. (ii) If the population syste is at the equilibriu state then it has only an odd nuber of species. In other words only an odd nuber of species lie one, three, and so forth can survive during the evolution. However, all species would not disappear. (iii) The population syste does not have any periodic cycle. (iv) If the population syste is not at the equilibriu state then in the future, soe species of the population syste will disappear.

4 64 F. Muhaedov and M. Saburov (v) Prehistory of the population syste can be uniquely recovered. There exists a unique equilibriu state, fro which the history of any state begins. In other words ancestors of any two distinct generations are the sae. (vi) The end or final state of the evolution is unpredictable. More precisely, the possibility of end or final states of the evolution is infinite. In other words the future of the evolution is unpredictable. It sees the picture of the continuous LV odel is different fro the discrete LV one. For exaple, in continuous case, a syste could have a periodic cycle. In contrast, in discrete case there is not any periodic state. The ai of this paper is that to provide soe nonlinear LV type odels whose properties different fro the properties of the quadratic LV odel (4). Soe LV type Models In recent years, a penetration of atheatical odel in biological probles is staidly rising. Both continuous and quadratic LV odel describe the following evolution law: LV evolution law - the absent species of the initial state will not appear during the future evolution. In [4] it is obtained an explicit for of all odels which are caring an LV evolution law. Let diensional siplex defined by (5) and I S be an ( ) = {,, L, } be a finite set. The set { α x S : x 0, α} Γ = = is called a face of the siplex. The set ri ( ) { α x S : x 0, α} Γ = = is called a relatively interior of Γ α. A apping V : S S is said to be a nonlinear LV type odel if it is defined as follows x = Vx = x + f ( x), =,, x S ( ) ( ) where a apping F : S x F( x) ( f ( x),, f ( x) ) following conditions (i) A apping F : S R is continuous; (ii) For every x S one has f ( x), =, ; (iii) For every (iv) For every x S one has x f( x) = 0; = α I one has ( ) L R satisfies the f x > for every ( ) x ri Γ α and α.

5 On Lota-Volterra evolution law 65 Theore. [4] A odel V : S S describes an LV evolution law if and only if it is a nonlinear LV type odel. It is worth entioning that nonlinear LV type odels were deeply studied 5 whenever a apping F : S R satisfies the following condition ( F x F( y) x y) ( ), 0. (6) Such ind of nonlinear LV type odels are called F onotone LV type odels. It is easy to chec that a quadratic LV odel (4) is a F onotone LV type odel with F ( x) = A x. An analogy of Theore. was proven (see [4]) for F onotone LV type odels. Proble. What is the eaning of inequality (6) in the population syste? Which ind of laws does inequality (6) describe? In this paper we study a nonlinear LV type odel which is not F onotone. Let us consider the following nonlinear LV type odel V : S S where, ε [, ] \ { 0}. Vε ( x) = x x xi,,, + = i= (7) ( ) ε Theore.3 Suppose the LV evolution law of the population syste is given by (7). Then the following stateents hold true. (i) The population syste has nuber of equilibriu states. The population syste does not have any equilibriu state except the equally distributed states. All equilibriu states are local as well as global asyptotically stable. (ii) The future of the evolution is predictable. (iii) If ε > 0 then the species of the initial state doinating in ters of the nuber will survive forever during the evolution and the rest species will disappear. At the end of the evolution the surviving species will be equally distributed. (iv) If ε < 0 then all involving species of the initial state will survive forever during the evolution and their portion will be equal at the end of the evolution. Let us consider the following nonlinear LV type odel W : S S ( Wx) = x + x + xi xi 3 x ix j i= i= i, j= i< j ε. (8)

6 66 F. Muhaedov and M. Saburov Theore.4 If the LV evolution law of the population syste is given by (8) then the prehistory of the evolution cannot be recovered. In other words, there are different ancestors who have the sae generation. Suppose one diensional siplex S has the following decoposition S = CUCUC3UC4U C5 where C = x, x = x S : a x a, i =,5, {( ) + } i i i 9 9 ( a, L, a6) = 0,,,,, Let us consider the following nonlinear LV type odel V : S S ( Vx) = x( + f( x, x) ), (9) ( Vx) = x( + f( x, x) ) x 9 x C x, 0x, 4x 0x 3 f( x, x) = x, f( x, x) = 4x. 3 x 4 0x 9 x C4 x 0x 5 x 5 Theore.5 If the LV evolution law of the population syste is given by (9) then the population syste has periodic cycles. Acnowledgents The Authors acnowledge Research Endowent Grant B (EDW B ) of IIUM and the MOSTI grant SF0079. References [] R.N. Ganihodzhaev, Quadratic stochastic operators, Lyapunov functions and tournaents, Rus. Acad. Sci. Sbor. Math., 76 (993),

7 On Lota-Volterra evolution law 67 [] R.N. Ganihodzhaev, Map of fixed points and Lyapunov functions for one class of discrete dynaical systes, Matheatical Notes, 56 (994), 5-3 [3] R.N.Ganihodzhaev and D. Eshaatova, Quadratic autoorphiss of siplex and asyptotical behavior of their trajectories, Vladiavaz Math. Jour. 8 (006), -3. [4] R.N.Ganihodzhaev and M. Saburov, A generalized odel of nonlinear operators of Volterra type and Lyapunov functions, J. Sib. Fed. Math & Phys. (008), [5] Yu.Lyubich, Matheatical structures in population genetics, Springer-Verlag, Berlin, 99. Received: May, 0