2D-Volterra-Lotka Modeling For 2 Species

Transcription

1 Majalat Al-Ulum Al-Insaniya wat - Tatbiqiya 2D-Volterra-Lotka Modeling For 2 Species Alhashmi Darah 1 University of Almergeb Department of Mathematics Faculty of Science Zliten Libya. Abstract The purpose of this paper is to model multi-species interactions using Volterra Lotka equations in two dimensions which is a family of first order autonomous ordinary differential equations. The model for a pair of interacting populations of the predator and prey is studied and the stability points have been viewed. Lyapunov function is used to view the stable points. The population dynamics of the resulting systems are analyzed in terms of stability around equilibrium points and within invariant surfaces. Introduction The Volterra lotka (VL) problem, originally introduced in 192 by A. Lotka and later applied by V. Volterra to model the prey predator interactions. It is determine the predator-prey system with continuous of non-linear predator-prey interaction, the predicts oscillatory behaviour of populations with constant amplitudes dependent on the initial conditions [1], [3]. The discussion starts by presenting the basic exponential growth 2D VL. of differential equations and analyzing it in terms of stability of stationary points [5]. In this paper we examine some linear and non-linear two dimensional systems that have been used as mathematical model of the growth of the species showing a common environment [4]. In the first unit, which treats only a single species variation mathematical a summations on the growth rate are discussed. In the second unit, the simplest types of equations that model predator-prey ecology are investigated [2], [3]. The object is to find out the long-run qualitative behavior of these dynamical systems. 1 Alhashmi Darah, PO Box 675 Zliten Libya, << a_darah@hotmail.com >>. 63

2 2D-Volterra-Lotka modeling for 2 species n Definition: LetM, the differentiable function φ : M M is called dynamical system if: 1- φ(, x) x. 2- φ( t, φ( s, x)) φ( t s, x) for every t, s, x M. Definition: An equilibrium solution of the system: dx x ( t) f ( x( t)), (1) dt n x : a vector valued function, is a point x such that f ( x). where Definition: x(t) is said to be stable if for any given, there exists n ( ), such that, for any other solution y( t) of (1) satisfying x t ) y( ), then x t ) y( ) for all t t, t. ( t ( t Definition: x(t) is said to be asymptotically stable if it is stable and if there exists a constant b such that, if x( t ) y( ) b, then lim x ( t ) y( t ). t 1. One Specie t The birth-rate of human population is usually given in terms of the number of births per thousand in one year [1]. Similarly, one can define death rate and the growth rate = (birth rate death rate)/ total population. The growth rate is thus the net change in population per unit of time divided by the total population at the beginning of the time period. The growth rate is: y y( t) lim. yt y( t) t 64

3 Majalat Al-Ulum Al-Insaniya wat - Tatbiqiya If the growth rate is a constant =, then: Thus, y y d (ln. dt t y( t) e y, y y(, is the predator population at the time t where ) exponential growth rate. If a ( ), a,, we have an then the growth rate is positive if, and negative if, while in case, the growth rate equal zero. The solution of the differential equation: dy dt a( ) y( t) (1) is: y ( t ) exp[( a ( ) t ]. If we assume that is the limit of the population, then we assume that the growth rate is proportional to ( y ): c(, c, Then differential equation (1) takes the form: dy c( y, c, (2) dt The equilibrium of (2) accumulates at y = and at y =, the equilibrium at is asymptotically stable since the derivative of cy( - at is -c which is negative. A more realistic mold of single species is y ym (, where M denotes an arbitrary function. (3) 65

4 2D-Volterra-Lotka modeling for 2 species 2- Volterra-Lotka System We consider the Predator-Prey Interaction of two species; the predator y and its prey x. The prey population is the total food supply for the predator at any given time, then: y a( x ) y, a,, which can be written in the form: y ( Cx D) y, C, D. We investigate the growth rate of the prey. At each small time, number of prey is eaten [6]. The prey species is assumed to have a constant per a capita food supply available sufficient to increase its population in the absence of predators. Therefore the prey is subject to differential equation of the form: x ( A B A, B. Now we arrive to the predator-prey equation of Volterra and Lotka: x ( A B ( 4) y ( Cx D) y, where x(t), y(t) are the prey and predator populations at time t respectively, A, D are the natural growth and decay coefficients, and A, B, C and D are positive numbers. The line terms A -Dy model the natural growth and decay of the prey and predator respectively, the quadratic terms Bxy and Cxy model the effects of interaction on the rates of change of the two species. This system has stationary points at P1 ( (, ) and at P ( ( D C, A B). 2 To determine the stability of the stationary points we linearize the system by taking the partial derivative of the system. The stability of points in linear system of equation 4 can be determined by finding the Eigenvalues of the matrix for the linear system at the points and applying the following theorem: 66

5 Majalat Al-Ulum Al-Insaniya wat - Tatbiqiya n Theorem (Principle of Linearzed Stabilit: Let F C 1 ( U, ) for n U with F ( P ). then the non-linear system a F(a) the following is true: 1- Re( ( F( P ))) P is asymptotically stable. 2- P is stable Re( ( F( P ))). According to the linear part of the vector field (4): A By Bx F ( x,. Cy Cx D The Jacobian at the point (, ): A F (, ), D The point (,) has Eigenvalues A and C, and we have A > and C > so the Eigenvalue is A, then the point (,) is saddle point, figure.1, so the system is unstable at the point (,). This is not surprising which stases the prey population increases in the absence of predators. Fig. 1 The Jacobian at the point ( D C, A B ) is: 67

6 2D-Volterra-Lotka modeling for 2 species BA B BD C A B CD C A F( D C, A B), C D BD C. AC B The trace of this point P is zero, the matrix has the Eigenvalues i AD 2, the real part of these values is zero, so the point could be either stable or instable. Another method of analysis is needed to find out more. Theorem 2: (Liapunov's theorem): Let ~ x W be an equilibrium for the system x f (x). Let V : U be a continuous function defined on a neighborhood U W of differentiable on U ~ x, such that: 1- V (x ~ ) and V (x) if x ~ x ; 2- V in U ~ x. Then ~ x is stable. Furthermore, if also 3- V inu ~ x, Then ~ x is asymptotically stable. Suppose that the Liapunov function is in the following form: v( F( G(. Then,. d v( v[ x( t), y( t)], dt df dg x y, dx dy df dg ( A B ( Cx D). dx dy Let v, then: df x y dg = constant. dx Cx D By A dy 68

7 Majalat Al-Ulum Al-Insaniya wat - Tatbiqiya If we put this constant = 1, we get: By integrating: df dx dg dy C D x, B A y. F( x) Gx D log G( By Alog y. Then: v ( Cx Dlog x By Alog y, x, y, By take the derivative with respect to x ( to y ) we get: d v y C D x, dx d vd C, A B, dx d v y B A y, dy d vd C, A B. dy Thus P is absolute minimum of v, then v is Liapunov function and P is 2 2 stable equilibrium point. Since v, therefore, every trajectory of the Volterra-Lotka is closed orbit except the equilibrium points, and then there is no limit cycle. The phase portrait is shown in Figure 2. 69

8 2D-Volterra-Lotka modeling for 2 species y Therefore, for any given initial population (x(), y()) with x() and y() other than P 2, the population of predator and prey will oscillate cyclically, no matter what the number of prey and predator are, neither species will die out, nor will grow indefinitely. On the other hand except for the state P 2, which is improbable, the population will not remain constant. Example: Assume that the initial number of foxes and rabbits are = 1 and, = 5 respectively, and that the coefficients, A 2, B. 1, D. 8, C. 2 are used to form the system of D. E.'s dx dt Fig. 2. Ax Bxy 2 x. 1xy, and x dy dt Cxy Dy. 2xy. 8y Solve the system of D. E.'s for x(t)and y(t) over the interval a t b. 7

9 Majalat Al-Ulum Al-Insaniya wat - Tatbiqiya Solution: We used the Rung-Kutta numerical method with initial data: x 5, y 1, a., b 12., n 6. The graph of the solution, over a larger time interval, the graph traces over itself, because the solution is periodic. Fig 3. Fig. 3. Fig. 4. The number of rabbits. 71

10 2D-Volterra-Lotka modeling for 2 species Fig. 5. The number of foxes. The periodic behavior is clearly present in the latter two graphs. Notice the phase lag for the number of prey. 3- Predator-prey equation of species with limited (Overcrowding) This model is more realistic from the biological point of view. Two species may compete for a resource in short supply. One can model for competitive interaction with overcrowding is: x ( A By x) y ( Cx D y, ( 4) where λ, μ, A, B, C and D are positive numbers, and x >, y >. We divide the positive quadratic Q (x >, y > ) into sectors by two lines L and M: L : A By x M : Cx D y. We get two cases according to the position of L and M. Case1: (they are not intersection) We can see that in the figure (6), 72

11 Majalat Al-Ulum Al-Insaniya wat - Tatbiqiya A B L M x x y y x y A D C We obtain two of equilibrium points P,, P A,. of (4) is A By 2x F( Cy F( P ), 1 Fig By, Cx D 2y The linear part A, D then the point P is a saddle as in the previous section. 1 The linearization at the point P : 2 A AB F ( P 2 ), AC ( D) then the point P is stable. Note, from the figure 4, one can be seen that, 2 the point P ( A, ) is globally asymptotically stable, because every 2 trajectory coming from the position quadrant ends to this point. 73

12 2D-Volterra-Lotka modeling for 2 species y x y x y x y A / λ Fig. 7. x Case 2: (L intersects M) In this case we add the point z ( which is the intersection of L and M to the equilibrium points P 1 and P 2, figure 8. Then the Jacobian at the point z is: A B y 2x B x F (. C y C x D 2y We have from L and M that: then A By x, Cx D y, 74

13 Majalat Al-Ulum Al-Insaniya wat - Tatbiqiya x B x F ( x,, C y y Therefore, z ( is asymptotically stable (stable node). It is not easy to know whether there is any limited cycle, but if we assume that there is a rectangle whose corners are (,), (P,), (P,Q) and (,Q) with: P A, Q A, and ( P, Q) M. Every trajectory at boundary points of either enters is positively invariant or is a part of the boundary. Therefore, is positively invariant. Figure 8. (,Q) (P,Q) Г L M z D/C Fig. 8. A/λ (P,) 4- Competing Species We consider two species x and y, which compete for some thing (common food supply for example).the equations of the growth of the two species are formed as: x M ( y N( y, ( 5) 75

14 2D-Volterra-Lotka modeling for 2 species where the growth rates M and N are C 1 functions of non-negative variables x and y with the following three assumptions: (1) If either species is increasing, the growth rate of the other goes down. Hence, M y N x, and. (2) If either population is very large, neither species can multiply. Hence there exists K, such that: M (, and N(, if K x or K y (3) In the absence either species, the other has positive growth rate up to a certain population, and negative growth rate beyond it. 1 By 1 and 3 each vertical line {x}r meats the set M () exactly once if x a and not at all if x > a. By (1) and the implicit function 1 theorem, μ is the growth rate of non-negative C map f:[,a] R, 1 such that f () a. Then M > below the curve μ and M < above it. Fig. 6-a. 1 In the same way, the set v N () is smooth curve of the form {(: 1 x=g(}, where g: [,b] R is non-negative C map with 1 g () b. The function N is positive to the left of v and negative to the right. Fig. 9-a, b. 76

15 Majalat Al-Ulum Al-Insaniya wat - Tatbiqiya μ ν M < N > M > N > Fig. 9-a. Fig. 9-b. 4.1-Case: 1 ( and do not intersect and below ) With this assumption we have three equilibrium points: (, ), ( a, ) and (, b ) The equilibrium point (, b ) is asymptotically stable, figure 1. 77

16 2D-Volterra-Lotka modeling for 2 species (,b) x y x y Fig Case: 2 ( and are intersections) x y (a,) is a finite set, then the Suppose that and are intersection and curves and, and the coordinate axes bound a finite number of connected open sets in the positive quadrant, these sets where x, and y, figure 11. (1) Points in the set (vertices). (2)Points on or on but not on both and not on the coordinate axes. (Ordinary points). (3)Points on the axes. A vertex is an equilibrium, (let us denote by p, q), and the other equilibrium points are (,), (a,), (,b). At an ordinary point B, the vector ( y ) is either vertical ( if ), or horizontal ( if v ), it points either into or out of B since μ has no 78

17 Majalat Al-Ulum Al-Insaniya wat - Tatbiqiya vertical tangents and ν has no horizontal tangents. We call ω an inward or out ward point of B, accordingly. (,b) x y q x y x y p x y Figure 11. x y (a,) Conclusion The two-dimensional Volterra-Lotka system exhibits stable periodic behavior for all non-zero initial conditions. These trajectories run along closed paths around the stationary point ( D C, A B), which is non asymptotically stable. The other stationary point is at (; ), for which both populations are extinct. This point is instable. If only the predator population y is extinct, then the prey population x grows without bound. If only the prey population x is extinct, then the predator population y approaches extinction. Constants A;B;C;D are positive by dentition, so no alteration of the constants changes this behavior. Periodic stability is present for all possible combinations of variables. 79

18 2D-Volterra-Lotka modeling for 2 species References [1] Farkas H. and Noszyiczius Z., Generalized Lotka-Volterra of Two Dimensional Exploratory Cores and their Liapunov Bifurcations., J. Chem. Soc., Faraday Trans. 2, 1985, 81, [2] Farkas H. and Noszticzius Z., Two Dimensional Exploratory. 2. Global Analysis of Lotka-Volterra-Brusselator (LVB) Model. Act Physica., Hungary 66, 23-22, [3] Guckenheimer J. and Holmes P., Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer-Verlag, New York, [4] Hirsch M. and Smale S., differential Equations. Dynamical System, and Linear Algebra, Academic Press, INC., London, [5] Krasnov M. L., Kiselyov A. I and Makarenko G. I., A Book of Problems in ordinary Differential Equations., Mir Publishers, Mosco, [6] Wiggins S., Introduction to Applied Nonlinear Dynamical System and Chaos., Springer-Verlag. New York, Inc.,