Timo Eirola, Alexandr V. Osipov + and Gunnar Soderbacka o Chaotic Regimes in a Dynamical System of the Type Many Predators One Prey, Helsinki Universi

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1 Chaotic Regimes in a Dynamical System of the Type Many Predators One Prey Timo Eirola, Alexandr V. Osipov, and Gunnar Soderbacka Helsinki University of Technology Institute of Mathematics Research Reports A 386, June 996

2 Timo Eirola, Alexandr V. Osipov + and Gunnar Soderbacka o Chaotic Regimes in a Dynamical System of the Type Many Predators One Prey, Helsinki University of Technology, Institute of Mathematics, Research Reports A368 (996). Abstract Amultidimensional system of dierential equations of the type many predators-one prey is examined. The system has been introduced by S.B. Hsu, S.P. Hubell and P. Waltman and the possibility for coexistence of the predators has been established by dierent authors. Most of the results here are for the three and four dimensional cases when two or three predators are present. We examine the stability of limit cycles and nd parameter regions for dierent types of coexistence, for example, periodic and chaotic. Bifurcations in the dynamics are also examined. In the three dimensional case a simple one-dimensional map is suggested as model for the dynamics of the Poincare map on the attractor. This model is supported not only by experiments but mainly as a result of analytical estimates of the trajectories. The model does not work for all parameter values. Examples are given of cases when the model does not work. There the Poincare maps seem not to be approximated by simple maps and the attractor is more complicated. In the four dimensional case most of the results are experimental. AMS subject classications 58F, 34C, 92D25 ISBN ISSN TKK OFFSET, 996 Helsinki University oftechnology, Institute of Mathematics Otakaari, FIN-25 Espoo, Finland Timo.Eirolahut. + Department of Ordinary Dierential Equations Faculty of Mathematics and Mechanics St Petersburg State University Ulianovskaya,StPetergof 9894 St Petersburg, Russia o Department of Applied Mathematics Lulea University oftechnology S-9787 Lulea, Sweden 2

3 Introduction. Let us consider a (n + )-dimensional system of dierential equations of the form (.) x i = f i(s)x i i = n s = h(s) ; (s)x ; ; n (s)x n in the region x i s The system describes the interaction of n predators x i exploiting the same prey s. (.2) We assume that f i (s) = s ; i s + a i i (s) = s s + a i h(s) =(; s)s where a i i > are parameters. The form of the three dimensional system can be derived from the more standerd form of the system ds dt = S( ; S=K) ; M B X S (A + S) ; M 2 B 2 X 2 S (A 2 + S) ; ; M n B n dx i dt = M ix i S (A i + S) ; D ix i i = n examined in [2, 6, 9, 25]. After the transformations s = S=K i = A i D i K(M i ; D i ) X n S (A n + S) t = T m i = M i ; D i i = A i =K x i = M ix i KB i in [9], the system takes the corresponding form in (.)-(.2) if m i =. For the biological backround of this system see [8]. The two dimensional system obtained by restriction of system (.)-(.2) to a coordinate plane, where only one predator is present is examined in [7, 3, 4,9,2,25,3]. We are specially interested in solutions where all the predators coexist and call such solutions inner solutions. More precisely a solution is called inner if the closure of the corresponding trajectory does not intersect the n-dimensional subspaces where one predator is absent. We review and give some results from which one can determine the existence or absence of inner solution (extinction of one predator). We also discuss the behaviour of these inner solutions and try to nd parameter regions where there is only one simple periodic inner solution (it is then stable), 3

4 where there is a period doubling bifurcation, and where the inner solutions are chaotic (there are no inner equilibrium points). In [5]itisproved that the probability in the parameter space of nding coexistence is rapidly decreasing when the dimension of the system is increasing. Inner solutions of the three dimensional system have also been examined in [2,, 6, 8, 9, 2, 25]. The system with periodic variations in the coecients has also been examined in [6]. Analogous systems, where the prey is a substrate or nutrient and has linear growth instead of logistic are examined in [3, 5, 7] and with periodic variations, day-night in[26] or periodic death (washout) in [3]. A good suvey of results on related systems can be found in [27]. In section 2 we nd bounded regions such that all solutions will enter them and remain there. In section 3 we will discuss the main properties of the two dimensional case (system of one predator-one prey) and in section 4 the properties of the three dimensional system. In section 5 we introduce a model for the Poincare map in the three dimensional case and examine the conditions under which itworks. In section 6 we study situations when the model map does not work. In section 7 we also consider the three dimensional case and discuss the stability properties of the limit cycles in the two dimensional coordinate planes and the eects of their stability onthe existence of inner solutions. In section 8 we discuss the properties of the four dimensional system. Section 9 is devoted to the numerical methods of our experiments. Sections and contain the proofs of the main theorems. 2 Dissipativity of the system A system is dissipative (see V. A. Pliss [24]) if there is a bounded set such that any solution eventually enters it and remains there. (This is not to be confused with the other common denition of dissipativity requiring the divergence being negative). The system (.)-(.2) is dissipative Theorem 2. Let V = x =q + x 2 =q x n =q n + s, where q i = a i ; i + 2 i = n All solutions of the system (.)-(.2) starting in x i s i = n enter the region f(x x n s)j V x i s g and remain there. The proof of the theorem uses the ideas of [8] and is based on the fact that V =(s ; ) x (s + ) q (s + a ) + + x n(s + n ) q n (s + a n )! +(; V )s < for V > and s<. The following theorem gives another result for the region, where the solutions remain. 4

5 Theorem 2.2 Any solution of (.)-(.2), where a i i <, enters the region x + x x n < 6 and thereafter remains there. The proof of this theorem is given in section. This theorem implies that any attractor must be in the region x + x x n < 6 s<. Note that Theorem 2.. already says that an attractor must be in x + x 2 + x n < 2 in this case. 3 Behaviour of the two dimensional system Let us consider the system (.)-(.2) in a two dimensional coordinate plane where only one predator is present, for example, let x 2 = = x n =. After rescaling the time the two dimensional system can be written in the form x =(s ; )x s = (( ; s)(s + a) ; x)s where x = x = a= a. The system always has two equilibria the origin and the point ( ). The origin is always a saddle with stable mainfold on the positive x-axis. The unstable manifold contains the part of the s-axis between the origin and the point ( ). For > the equilibrium ( ) is a sink attracting the whole P = f(x s)jx > s>g. For <the point ( ) becomes a saddle. Then there is also a third equilibrium (( ; )( + a) )inp. It attracts the whole P for 2 + a>. For 2 + a< there is a unique periodic solution in P which attracts all P except for the source (( ; )( + a) ) inside itself. This is proved in [4, 3, 4, 2, 3]. Most of the results, however, can also be obtained as corollaries from [3]-[33]. At 2 + a = there is Hopf bifurcation. For more details of the two-dimensional behaviour see [7, 3, 4,9,2,25,3]. In [6, 3] the relaxation behaviour of the trajectories is also studied when a small parameter is introduced. 4 Behaviour of the three dimensional system Consider the four dimensional subsystem of (.)-(.2) written in the form x = s ; s + a x (4.) y = s ; 2 y s + a 2 s = ( ; s ; x=(s + a ) ; y=(s + a 2 ) ; z=(s + a 3 )) s 5

6 The system always has two equilibria the origin and the point ( ). The origin is always a saddle the stable manifold of which contains the xy-plane. The unstable manifold is the line from the origin to the point ( ). If 2 >, the point ( ) attracts the whole set f(x y s)jx y s > g. If < there is an equilibrium (( ; )( + a ) ) in the xs-plane, which isasinkif2 + a > and 2 >, a saddle with one dimensional unstable manifold if 2 +a > and 2 <, a saddle with two dimensional unstable manifold if 2 + a < and 2 >, and a source if 2 + a < and 2 <. The limit cycle in the xs-plane is always stable in that plane but can be stable or unstable saddle for the three dimensional system. When 2 < wehave an analogous situation in the ys-plane. Suppose now that a >a 2 (case a 2 <a is analogous). In [2] isproved that if < 2 then all solutions in f(x y s)jx y s > g approach the plane y = and if a 2 (a 2 +) > a a (a ; a 2 )+a 2 then all solutions in f(x y s)jx y s > g approach the plane x =. Thus there is nothing interesting inside the set f(x y s)jx y s > g under these parameter conditions. Hence, we will assume the opposite (4.2) 2 < < a 2 (a 2 +) a a (a ; a 2 )+a 2 We observe that (for a >a 2 ) a 2 (a 2 + ))(a a (a ; a 2 )+a 2 ) ; < 2 a =a 2 Experiments show that it is worth to conjecture that for xed 2 a 2 and a,if 2 < ( ; a 2 )=2, then there exist a and b such that 2 a < b ( 2 a )=a 2 and such that if < a, then all solutions in x y s > tend to y = and if > b, then all solutions tend to x = and for a << b there exists an inner solution which isperiodicorchaotic. These parameter regions are illustrated in gures and 2 for some xed a and a 2. There we have approximatively plotted for xed a and a 2 the regions in the 2 - parameter plane, where the predator x becomes extinct (region 4), where the predator y becomes extinct (region ), where the attractor is a simple periodic trajectory (region 2) and where the attractor is more complicated (region 3). On the boundary between the regions 2 and 3 there is a period doubling bifurcation. We have considered that one of the predators becomes extinct if the corresponding coordinate becomes less than ; before we have iterated the map times and even less for the following iterate of the Poincare map. Wehave started our trajectory from the point (25 25 ) and taken the rst intersection with the dention range of the Poincare map as the initial value. That means that the regions and 4 may in reality be slightly larger than those in the gures. 6

7 Figure Parameter regions of coexistence for a =2 a 2 = Figure 2 Parameter regions of coexistence for a =3 a 2 =2 7

8 ln(y=x) Figure 3 Bifurcation diagram for a = = a 2 =3 2 =4 Figure 3 illustrates the bifurcation diagram for the three dimensional system with parameters a = = anda 2 =3 and 2 =4. The vertical axis corresponds to the ln(y=x)-coordinates of some iterates of a point under the Poincare map in the surface s = s < and the horizontal axis represents the parameter. Under suitable conditions for small a i and i all inner solutions will intersect a two-dimensional set D dened by u ; <x+ y<u + and s =constant in the region s < andthus their behaviour is described by thepoincare map there. This Poincare map at least exists and describes the behaviour of the inner solutions well in the case when all trajectories starting in D behave the following way The s-coordinate on the trajectory rst decreases to a minimum where x + y is also small, then the s-coordinate increases to a value a bit less than while x+y still remains small, and then x+y strongly increases and the s-coordinate decreases and nally the trajectory hits D. The Poincare map is illustrated in gure 4, where P 2 is the image of P which is the image of P and Q 2 is the image of Q which is the image of Q. This behaviour of the solutions imply that the inner solutions are contained in a 'thick' cylinder containing the range of denition of the Poincare map. We use this fact and some other properties to construct a model map for the behaviour of the solutions on the attractor in section 5. Not only experiments but also theoretical estimates (see also [8]) indicate the existence of parameter values (or functions i and f i ) for which the assumptions given in section 5 are fullled. Anyhow there are a lot of examples when a Poincare 8

9 s y Q P P P Q 2 Q 2 D x Figure 4 Illustration of Poincare map. map of the type given in section 5 does not work. An example of these is given in section 6. 5 A model map in the three dimensional case Not only from numerical experiments but also from theoretical estimates (see [8] and sections and ) one can be convinced that the Poincare map dened on D in section 4 is usually strongly contracting in the x + y- direction. Thus the behaviour of the Poincare map is well described by aone dimensional map of the y=x coordinate. Here we construct such a map. If we denote by v the ln(y=x)-coordinate and by u the x + y-coordinate, then the model map f takes the form where and f(v) = + v ; k + k 2 e v +e v u k =( ; )=a k 2 =( ; )=(a 2 ) = 2a a 2 = ( ; 2)( + a ) ( ; )( + a 2 ) From (4.2) it follows that for the interesting parameter regions we have <<<a =a 2 9

10 Analysis of the map f shows that it can have an attractor, which can be axedpoint, a periodic orbit or a chaotic set. The attractor corresponds to a periodic solution or to chaotic inner solutions for the original system. We can also use the \reconstruction theorem" of F. Takens [28]-[29] to get results about the real map itself. In [8] the system (4.) and the bifurcations leading to chaos through period doubling are examined in more detail in the case a = =2 anda 2 =3 and 2 =6, where 2. To see why andhow the model map is working we prove some results about the behaviour of the trajectories of the system and the orbits of the model map. Let s < " minfa a 2 2 g "< Consider a trajectory starting at a point (x y s ), where s < Denote the next intersection with the plane s = s by(x y s ) s = s and the next intersection with the plane s =; " by (x 2 y 2 s 2 ). Continuing from the point (x 2 y 2 s 2 ) denote the next intersection with the plane s = s 2 =; " by (x 3 y 3 s 3 ) s 3 = s 2 and the next intersection with the plane s = s by(x 5 y 5 s 5 ) s 5 = s. Denote u i = x i + y i v i =ln(y i =x i ) i = 5 A trajectory starting at a point (x y s ) x + y > 7 is called a g{ trajectory, ifitintersects the plane s =;" before it leaves the region s > and thereafter intersects the the plane s = s for x + y>7 beforeitleaves the region s <. We have strong numerical and analytical evidence that most of the trajectories are g{trajectories in a suitable range of the parameters a i i so that the model map constructed below is appropriate. The trajectories for small s have the strongest inuence on the model map and the result of that inuence can be seen from the proof of the following theorem. Theorem 5. Suppose <<a =a 2 a < 2 and "<. Then any g{trajectory satises where y 2 x y x 2 y 5 x 3 y 3 x 5 x3 x x 2 x k + k 2 e v v 5 = K + v ; qu +e v ; <e K < y 2x y x 2 y 5 x 3 y 3 x 5 q =+" + " 2 + ; x3 x x 2 x (+"=4); Further, " 2! when "!, and "! when " a! The exact dependence of " and " 2 on " and a can be seen in the proof of the theorem given in section. Both numerical experiments and analytical

11 K u v v 2 v v Figure 5 estimates show that the changes in K are small compared with the other eects in the expression for v 5. Also u can usually be approximated by a constant on the attractor. Because for constant parameters we cannot let "! ifwewant tohave enough g{trajectories, the model cannot be made arbitrarily good, but anyhow it reects main properties. In practice the model map works even with parameter values for which Theorem 5. does not give good estimates. In Figure 5 we see how v 5 depends on v in the case " =3 for the parameter values a = = a 2 = 75 2 = on the attractor. Also the dependence of u 5 = x 5 + y 5 and K on v is plotted. Both on the attractor. Because K and u 5 are usually not very strongly dependent on v the model map f (where these are replaced by constants and u) describes the behaviour reasonably well. Let us now state some immediate results for the model map, which canbe used to nd inner solutions of periodic or chaotic type. (Observe thatfrom the parameter conditions it follows that k 2 >k.) Theorem 5.2 If (k 2 ; k )u>4 then f has two extrema a maximum at v ; and a minimum at v +, where v ; <v +. If >uk 2 then v!and if <uk then v!;when the map is iterated. If uk <<uk 2 the map hasaxedpoint and an attractor in a bounded set. The xed point has a period doubling for k 2 ; k = u(=u ; k )(k 2 ; =u)=2.

12 s y x Figure 6 Part of the attractor for a = =25 a 2 =6 2 =63 The statement follows by direct calculations. A chaotic attractor is thus expected after the period doubling in the region k 2 ; k >u(=u ; k )(k 2 ; =u)=2. 6 When the model map does not work Figure 6 illustrates an attractor in the case when a Poincare map of the type considered in section 6 does not work. The problem arises because the intersection of the attractor with the plane s = constant canintersect the curve s = or have points very close to it. Thus the intersection of the attractor cannot be well approximated by the graph of a function of the y=x coordinate which is seen in gures 7-. Another diculty is connected with the fact that the Poincare map becomes usually discontinous at points mapped to the curve s =. In gures 7- we describe the behaviour of the Poincare map on the intersection of the attractor with the plane s =5 in the case a =25 = 25 a 2 = 2 =2. The points a;f in gure 7 are mapped to the corresponding points in gure 8 and the pieces of the curve between two nearby points are mapped to the attractor without making folds. Analogously the points a ; e in gure 9 are mapped to the corresponding points in the gure. If we decrease a 2 and 2 then the type of the crosssection of the attractor is changed as shown in gures -3. The behaviour of the one dimensional stable manifold of the saddle equilibrium in x = y s > clearly strongly 2

13 x + y.4.2 d e f.8 c.6 a b ln(y=x) Figure 7 Intersection of the attractor for a = =25 a 2 = 2 = 2 with the plane s =5 s > with given points a ; f eects the type of the attractor. For the parameter values in gures 7- the stable manifold comes from the equilibrium in the xs-plane, but for the parameter values in gure it comes from the innity. For some parameters between these it is contained in the unstable manifold of the limit cycle in the xs-plane. For parameters near to these special values the behaviour of the attractor becomes more complicated as is seen in gures Location and stability of cycles in the two dimensional coordinate planes In this section we discuss the stability of the cycles in the planes x = and y = for the three dimensional system of type two predators-one prey. The following theorem is proved in section. From the lemmas used in the proof we can easily nd estimates for the location of the limit cycle in the plane x =. Theorem 7. Suppose i < 2a i < i a i <"<5, i = 2 and > 4=3 ; " ln " ; 2" +2" ln " Then the limit cycle in the plane x =is unstable. 3

14 x + y.4.2 c a.8.6 d e f.4.2 b ln(y=x) Figure 8 Intersection of the attractor for a = =25 a 2 = 2 = 2 with the plane s =5 s > with images of the points a ; f in the previous gure. x + y.4.2 b e a.8.6 c.4.2 d ln(y=x) Figure 9 Intersection of the attractor for a = =25 a 2 = 2 = 2 with the plane s =5 s > withgiven points a ; e 4

15 x + y.4 c d.2 a.8.6 e b ln(y=x) Figure Intersection of the attractor for a = =25 a 2 = 2 = 2 with the plane s =5 s > with images of the points a ; e from the previous gure. x + y ln(y=x) Figure Intersection of the attractor for a = =25 a 2 =6 2 = 2 with s =2 s > 5

16 x + y ln(y=x) Figure 2 Intersection of the attractor for a = = 25 a 2 = 75 2 =5 with s =2 s > x + y ln(y=x) Figure 3 Intersection of the attractor for a = =25 a 2 =7 2 = 4 with s =5 s > 6

17 x + y s'= ln(y=x) Figure 4 Intersection of the attractor for a = =25 a 2 =7 2 = 4 with s =5 ;2 < ln(y=x) < 2 x+ y<2 This means that if the parameters are small and the dierence between a and a 2 is not small the limit cycle in the x = plane will be unstable for most between and a 2 =a (the region where there may be inner solutions). On the other hand if the limit cycle in the plane y = is not intersecting the line s = 2 or only have a small part below it then the cycle is clearly unstable (y is positive or mostly positive). When both cycles are unstable we usually expect inner solutions. In connection with these results we have the following questions Question 7. Does it always mean that all other predators go extinct if one cycle in a coordinate plane is locally stable? The question can also be posed for the case when there is only a stable equilibrium in one of the coordinate planes and no cycle (i.e. 2 i + a i > ). Question 7.2 Does it always mean that all other predators go extinct if one equilibrium in a coordinate plane is locally stable? A partial answer is given in [22]. Compare also with Figures -2. 7

18 8 The behaviour of the four dimensional system Consider the four dimensional subsystem of (.)-(.2) written in the form x = s ; s + a x (8.3) y = s ; 2 s + a 2 y z = s ; 3 s + a 3 z s = ( ; s ; x=(s + a ) ; y=(s + a 2 ) ; z=(s + a 3 ))s Parameter regions for (8.3), where one predator goes extinct can be found in the same way as in[2] by only interchanging indices. Thus for example if we suppose a >a 2 >a 3 there cannot be any inner solutions outside the region determined by the inequalities 2 < < 2 a =a 2 3 < < 3 a =a 3 3 < 2 < 3 a 2 =a 3 But even in this parameter region inner solutions are hard to nd. An analog of the model map in section 5 is not seen to give any stable coexistence. This analog is the two dimensional map dened by u! 2 +u;g 2 = + e u = 2 + e v = 3 +e u + e v v! 3 +v;g 3 = + e u = 2 + e v = 3 +e u + e v where g i =(; i ) =a and i = i a =( a i ) i=2 3 Clearly this map has no xed points (where u v 6= ), except for some special relations between the parameters. Mostly also either u! orv!. Anyhow experimentally we can show that stable inner solutions exist for a narrow parameter region (a part of such a region is plotted in the parameter space in [2]). In Figure 5 we see a bifurcation diagram for the x-coordinate varying a parameter r, where there always seems to be inner solutions (also the y- and z-coordinates are nonzero). One value of the parameter r corresponds to a point on a straight line in the six dimensional parameter space given by the points a =25 =3 a 2 =5 2 = 8 a 3 = 2 3 = 3 (here r = ) and a = 25 = 3 a 2 = 6 2 =2 a 3 =25 3 =6 (here r =). Figure 6 shows an period eight solution from dierent perspectives in the case r = 64=999. Figure 7 shows the intersection of the inner attractor with the surface s = s < for the parameter r =99=499. In the bifurcation diagram we can observe except for period doubling also Hopf bifurcations of the Poincare map. 8

19 x Figure 5 Bifurcation diagram in the four dimensional case. r s.8 s x..2.3 z y.5..5 s x.4 z.2.5 x Figure 6 Period eight solution for r = 64=999. 9

20 ln(z=x) ln(y=x) Figure 7 Intersection of the attractor with s = for r = 99=499. Unstable inner solutions seem to be more frequent although also they are rare. We have no examples where we have seen two dierent attractors. So the following question is open Question 8. Can there be more than one attractor for the system (8.3)? The question is open even in the three dimensional case even if the results above do not give somuch hope for the existence of many attractors in this case. In the examples of inner solutions illustrated in the gures 5-7 two of the predators always coexist if the third is absent. So also the following question is open Question 8.2 Can the system (8.3) have stable inner solutions even in the case where one of the two predators go extinct if some third predator is not present? 9 About numerical methods. Some of our results have been veried only numerically. For the numerical integration (using the Runge{Kutta{Fehlberg 4/5 {method) we rst write the equations for the logarithms of x y z and sto speed up computations 2

21 we approximate the trajectories near equilibria by the linearized ow and in the region with small s using the estimates of Lemmas. and.2. For good cases well known program libraries (see [] for references) can also be used eectively for examining the limit cycles and the period doubling etc. In bad cases (i.e. with small parameters) these have to be modied. Proof of Theorem 5.. In this section we assume a i i < 2 i= 2 <<a =a 2 and denote by ;(s) the ratio f 2 (s)=f (s). Observe that ;() =. Theorem 5.. follows from the following technical theorem. Theorem. Consider a trajectory starting at a point (x y s ), where s <"a i " i i= 2 and "< and u = x + y > 7. Let (x y s ) be the next intersection with the plane s = s and (x 2 y 2 ;") and (x 3 y 3 ;") be the next intersections with the plane s =; ". Let us denote Then O = y x x y O 3 = y 3x 2 x 3 y 2 where and K e ;q (;)(x =a +y =(a 2 )) <O O 3 <K 2 e ;q 2(;)(x =a +y =(a 2 )) K = x3 x K2 x3 x = x 2 x ; x 2 x 2; 2 = ( + "=4) q =(+2") + 3" ; q 2 =(; e ;5=a )( ; ") ; 2"( + ) ;!! To prove the theorem we use three lemmas. The rst one gives estimates for x, the second one gives estimates for the ratio y =x and the third one estimates the ratio y 3 =x 3. Lemma. In the situation of Theorem. the following estimates are true x <x e ;(;e;582=a )(;")(x =a +y =(a 2 )) (.) and (.2) x >x e ;(+59"=44)(x =a +y =(a 2 )) 2

22 Proof. From the system we get the equations (.3) and (.4) (s ; )ds (s + a )( ; s)s =! dx = x ; (s + a )( ; s) ; y (s + a 2 )( ; s) x dx x ; dx ( ; s)(s + a ) ; (s ; )dy ( ; s)(s + a )(s ; 2 ) (s ; 2 )ds (s + a 2 )( ; s)s = y ; (s + a 2 )( ; s) ; (s + a )( ; s) dy y ;! x dy = y dy ( ; s)(s + a 2 ) ; (s ; 2 )dx ( ; s)(s + a 2 )(s ; ) Integrating (.3) and (.4) along the trajectory in s<"from (x y s ) to (x y s ) the left hand side of the equations will become zero (because the s-values are the same). Thus we get the integrals (.5) and (.6) ln( x x )= ln( y y )= Z x x Z y y dx ( ; s)(s + a ) + Z y y dy ( ; s)(s + a 2 ) + Z x x (s ; )dy ( ; s)(s + a )(s ; 2 ) (s ; 2 )dx ( ; s)(s + a 2 )(s ; ) Using the fact that s is small we can use these integrals to get estimates for x and y. Because s<"a i " i <<a =a 2 and "< we get the estimates (.7) (.8) (.9) ; " a < ; " a < ( + ")a i < ( + ")a 2 < ( ; s)(s + a i ) < a i < a 2 ; s (a + s)( 2 ; s)( ; s) < a 2 ( ; ")( ; 2") < + 59"=44 a 2 (We use that ( ;s)=( 2 ;s) is an increasing function of s for <s< 2 and ( i ; s)=(a i + s) is a decreasing function of s for <s< i.) Furthermore, (.) (.) ; 2" < ; " a +" a < 2 ; s (a 2 + s)( ; s)( ; s) < a ( ; ")( ; 2") < + 59"=44 a 2 22

23 Using estimates (.7)-(.) in integrals (.5)-(.6) we get (.2) x <x e (;")(u ;u )=a (.3) y <y e (;2")(u ;u )=a where u i = x i + y i. The estimates (.2) and (.3) together give (.4) u <u e (;2")(u ;u )=a Using u > 7, a < 2 and "< in (.4) we get u < 53. Substituting u < 53 into (.2) and (.3) and using "< we get (.5) x <x e ;582=a y <y e ;57=a But using (.5) and (.7)-(.9) when integrating (.3) gives inequalities (.) and (.2). Lemma is proved. y x x x x x ; Lemma.2 Consider the trajectory in Lemma.. Then the ratio y =x satises 2 ; y < < y (.6) x x where =(; 2") and 2 =(+2"=9). Proof. For s<" (.7) (.8) =(; 2") < ; " +" < 2 ; s a < ;(s) a 2 + s < 2 a + s < +" a 2 ; s ; " <( + 2"=9) = 2 (( i ; s)=(a i + s) is decreasing in s for <s< i.) By the system equations we have (.9) dy dx =;(s)y x Integrating gives (.6) and the lemma is proved. y 2 x3 x 2 x 2 x3 x 22; Lemma.3 In the situation of Theorem. the following estimates are true ; y 3 < < y 2 (.2) x 3 x 2 where = ;() and 2 = ( + "=4). Proof. Estimating ;(s) for s> ; " we get (.2) ;(s) > ;() = 23

24 ((s + a )=(s + a 2 )and(s ; 2 )=(s ; ) are both decreasing for s> ; ") and ;(s) < ; 2 ; " + a (.22) = ( + "=4) = 2 +a 2 ; " ; (We have usedthat(s ; i )=(s + a i ) increases with s for s>and that ( ; " + a )=( ; " ; )=((+a )=( ; ))( + (A ; B)"=(B(B ; "))) where A =+a B =; "< anda < 2.) Integrating (.9) again and using the estimates (.22) and (.2) we get (.2). The lemma is proved. Proof of Theorem.. Combining (.6) from Lemma.2 and (.2) from Lemma.3 we get K x <O x (.23) O 3 <K 2 x 2; x ;2 where K = ; x3 x x 2 x Using (.7) and (.22) we get Thus we get ; 2 =( ; ) and K 2 = ; ; 2 > ( ; ) ; Analogously using (.8) and (.2) we get 2 ; <( ; ) x3 x x 2 x 2; 2" + "=4 ; 2"( + ) ; + 2"=9 ; Combining these estimates with the results of Lemma. we get where and K e ;q (;)(x =a +y =(a 2 )) <O O 3 <K 2 e ;q 2(;)(x =a +y =(a 2 )) q = ( + 59"=44) + q 2 =(; e ;582=a )( ; ") ;!! 2" 9( ; )!! 2"( + ) ; The theorem follows now directly from these estimates. Let us now see how Theorem 5. follows from Theorem.. We observe that v 5 = v +ln(o O 3 )+lna, where A = x y 2 x 3 y 5 (y x 2 y 3 x 5 ) ;.From the estimate of O O 3 from below in Theorem. we get v 5 >v +ln(k A) ; q ( ; ) 24 x a + y a 2!!

25 and and x! + y = u + p p = e v a a 2 +p a a 2 + q =+" + " 2 ; " =2" " 2 = (3" +6"2 ) + < 3" +6" 2 An estimate for v 5 from above is obtained analogously (the main dierence is that here " and " 2 depend on a ). Theorem 5. follows from these estimates. Proofs of Theorems 2.2 and 7.. Proof of Theorem 2.2. Let a be the greatest of a i i= n. Set A = x =(s + a )+x 2 =(s + a 2 )+ + x n =(s + a n ) and u = x + + x n Then du=dt < sa and ds=dt =(; s ; A)s. Because A u=(s + a) a< andds=du < ( ; s ; A)=A (in s < <u ) which is greatest for smallest A, theu-value of the trajectory starting in s < u > certainly will grow slower than for the trajectory in the system (.) ds=dt =(; s)(s +) ; u du=dt = u Consider the function V = s ; cu ; bus, where c = ;=6 andb =2. The sign of dv=dt with respect to (.) on V = is determined by a fourth degree polynomial and examination shows that it is negative. Thus a trajectory starting in V < cannot leave the region. We nowshow that any trajectory not on the s-axis must enter the region V <. Any such trajectory starts in some region dened by s < ;As for some A>which can be considered arbitrarily small. If the solution starts in the region R dened by s < ;As and s<=2 it cannot remain there forever because u < ;u=(2a + ). But when it leaves R it enters the region s > ;As. A solution starting in the region R 2 dened by s < ;As and s =2 cannot remain there forever because s < ;A=2 <. But solutions leaving R 2 enter either the region s > ;As or the region R.Thus any trajectory enters the region s. But clearly the region s is inside the region dened by u<( ; s)(s +) which is inside the region V<. This means that any trajectory will come into V < and thereafter stay in that region. But then the x + + x n -value cannot be greater than.6. Proof of Theorem 7.. To prove the theorem we need some lemmas. The rst ve lemmas consider the position of dierent parts of the trajectory for the limit cycle in the ys-plane. The others give estimates for the trajectories in a neighbourhood of the limit cycle. We suppose i < 2a i <, i = 2. In Lemmas.-.4 we will consider trajectories in the y s-plane x = only. In these cases we will denote 2 and a 2 simply by and a. 25

26 Lemma. Atrajectory starting in the region s 95 intersects s =4 next time for an y-value, where y>846. Proof. The y-value of the trajectory will certainly grow faster than for a system where = anda =. Butwe shall show that the estimate holds even for that system. Let us again consider the function V = s ; cy ; bys and let c = ;8 b =4. The sign of the derivative ofv with respect to the system is again determined by a fourth degree polynomial and is positive on V = k =95 for 4 s 95. On V = k we have s =(k +cy)=(;by). s is decreasing in y 2 [ ] and s>4 for y =846. Because a trajectory starting in s 95 cannot intersect s =(k + cy)=( ; by) for 4 s 95 it intersects s =4 only when y>846. Lemma.2 Atrajectory starting in s =4 y>846 intersects s =2 next time for an y-value, where y>. Proof. The proof is analogous to the proof of Lemma., where we choose c = ;6696 b=7 k=7296. Lemma.3 Any trajectory starting in s =2 y> intersects s = for y>83. Proof. Because the y-value is increasing for <s<2, clearly the trajectory intersects s = for y>. For s<and s < ds dy > 35 ; y s s ; y because H(s) <H() < 35 where H(s) =(; s)(s + a). Thus the trajectory cannot intersect the solution of ds=s =(35 ; y)dy=y through s = y = 35 ln(y) ; y + ln(s) ; s = ln() ; ; To nd where the solution intersects s = we have tosolve L (y) =35 ln(y) ; y = (ln() ; 9=) ; =L 2 () Because L (y) < fory>35 and L 2 () > the greatest y-solution is obtained for the greatest =. This solution is clearly greater than.83. Lemma.4 Atrajectory starting at a point (y ) on s =, where y > 83, intersects s = next time at a point (y ), where (.2) y < 83e ; 69 a 26

27 Proof. From ds s = H(s) ; y dy y follows Z y H(s) ; y dy = y y In s< we have H(s) <s+ a<2a from which follows H(s)=y ; < 2a=y ; and y <y e y ;83 (.3) 2a If for y<83 F (y a) = 83 e 83 2a ye ; y 2a > then y>y. Because F (y a) >F(y 5) and F ( 5) > we get y <. Thus (.2) follows from (.3). Lemma.5 There is a neighbourhood of the point ( y 2 ), where y<8e ;69 a 2 such that if a trajectory starts at a point (x y 2 ) in that neighbourhood thes-coordinate of the trajectory will increase until it reaches the point (x 2 y 2 95), where 5 (.4) 39y <y 2 <y a 34 2 and (.5) 39x <x 2 <x 5 a 34 Proof. In this proof we use the notations a and for a 2 and 2 resp. Suppose (.6) y<h(95) and y<a on the trajectory between the two points. Then B(s) =(H(s) ; y)=h(s) > 9. Because (.7) (.8) dy yds = s ; (H(s) ; y ; x(s + a)=(s + a ))s s ; = (B(s)H(s) ; x(s + a)=(s + a ))s where x can be supposed arbitrarily small, and because (.9) s ; H(s)s = ; as + + a a( + a) s + a + ; +a ; s 27

28 we get (.) dy yds < ; as +! + a a( + a) s + a + ; +a ; s 9 Integrating along the trajectory (from (y ) to(y s) )we get (.) y < y s!k s + a + a (k+)=9 ; ; s! ; 9(+a) where k = =a. For s = s 2 the right hand side of (.) is less than (.2) 95!k (k+)=9 k a ( ; 95) =9 =2=9 C k a (k+)=9 where C ==95. Using k = =a we get (.3) k a (k+)=9 = where q(k) =(k +)=9 ; k. Thus we get (ka)k a = (k+)=9 kk a q(k) (.4) y 2 k k a q(2);q(k) < 2 =9Ck y a q(2) = 2=9 e g(k a) a q(2) where g(k a) =k ln(c)+k ln(k)+(q(2);q(k)) ln(a) <g(k 5) <g(2 5). But now because q(2) < 34 we get (.5) y 2 y < 2=9 e g(2 5) a 34 < 5 a 34 Thus the second part of (.4) is proved provided (.6) holds. Let us now convince ourself that (.6) is satised. Since we can suppose we get from (.5) y < 83e ;69 a y 2 < 83 5 ;69 a a34e which satises (.6) for a<5. Clearly (.6) is satised at the endpoints of the trajectory and because the y-value rst decreases and then increases (.6) is satised on the whole trajectory. Thus we have proved the estimate from above fory 2 =y. The analogous estimate for x 2 =x can be proved in the same way. Let us now prove the estimate for y 2 =y from below. As above we get (.6) dy yds > ; 9as + + a a( + a) s + a + ; +a ; s 28!

29 After integration we get (.7) y 2 > y 95! 9a 95 + a k+ + a +a ; ; 95! ; +a k 9 Because <2a < and thus k<2 we get (.8) y 2 y > k+ 5 (2 99) 9 5 k=9 a (k+)=5 The right hand side of (.8) can be rewritten as (.9) BA k (ka)k a k+ = Be g(k a) where g(k a) =k ln(a)+kln(k)+(k ; k ; ) ln(a). For our values of k and a the minimum of g is g(2 5) and thus we get (.2) y 2 y >Be g(2 5) > 39 Again the proof for the lower estimate for x 2 =x is analogous and thus the proof of the lemma is nished. Lemma.6 There is a neighbourhood ofthepoint ( y 2 ) where y> 8 such that if a trajectory starts at a point (x y 2 ) in that neighbourhood it will intersect s = 2 next time at a point (x y 2 ), where (.2) x x > y y!g and G = 4 3. Proof. The proof follows from dy=dx =;(s)(y=x), where ;(s) = 2 ; s s + a 2 s + a ; s and for s< 2 ;(s) > 2 ; 2 2a 2 + a 2 a = 3 4 Lemma.7 Suppose a trajectory starts at a point (x 2 y 2 95), where s > and let (x 3 y 3 95) be the next intersection with s =95. Then (.22) x 3 x 2 > y 3 y 2!A and A =; 2"=95, where ">max(a a 2 2 ) 29

30 Proof. The proof is analogous to the proof of Lemma (.6). We have only to convince ourself that for s>95 ;(s) < 95 + a 95 ; and thus where ">a. ;(s) ; > ; 2" 95 + " > ; 2"=95 Lemma.8 Suppose a trajectory goes from the point (x 4 y 4 ) to (x 5 y 5 2 ) and lies wholly in the region y>8. Then x 5 x 4 > 2! =(3a ) Proof. Because s< y>8 anda 2 < we get s < ( ; y=(s + a 2 ))s <( ; 8=2)s = ;3s and x > ; x=a. The lemma follows by integration. Lemma.9 Consider a trajectory starting at a point (x y 2 ) and denote by (x y 2 ) the following intersection with s = 2, by (x 2 y 2 95) and (x 3 y 3 95) the following intersections with s =95 in that order, by (x 4 y 4 ) the next intersection with s = and nally by (x 5 y 5 2 ) the next intersection with s = 2 in s <. There exists a neighbourhood of the point ( y 2 ), where y > 83, such that any trajectory starting in that neighbourhood with coordinates (x y 2 ) in the notations above has the following properties (.23) x 3 x 5 95 > x 2 x 6 5 a34 2 e 69(A;G)=a 2 (.24) (.25) and (.26) x 2 > 39 x x 4 > x 3 x 5 x 4 > 2! =(3a ) Proof. From Lemma.6 and.7 follows (.27) x x 3 x x 2 > y y!g!a y3 y 2 = y y!g;a y y 3 y y 2!A 3

31 From Theorem 8.. follows that y < 6 and because the point (x 3 y 3 95) is in s < we get y 3 > Using these facts and Lemma.4 and.5 in inequality (.27) we get inequality (.23). Inequalities (.24) and (.26) follow from Lemmas.5 and.8, respectively. Because x > for <s<95 we get inequality (.25). The proof of the lemma is nished. We now use the lemmas to prove the theorem. Combining the inequalities in Lemma.9 we get x 5 =x > a 34 2! =(3a ) e 69=a 2 2A;G where A and G are from Lemmas.7 and.6. After some calculations we see that x 5 =x > follows from > 2=9 ; ( 2 ln( 2 ) ; 2 ln( ))=(3 69 ; 2"=95 + a 2 (ln( =(6 5)) + 34 ln a 2 )=69 which follows from > 4=3 ; 2 ln 2 ; 2" +2a 2 ln a 2 Combining Lemmas.-.5 we also get that y 5 > 83 and the limit cycle in the ys-plane intersects s = 2 y>83. Thus the x-value increases when the trajectory winds near the ys-plane and the limit cycle must be unstable. The statement of the theorem now follows. References [] A.D. Bazykin The mathematical biophysics of interacting populations. Nauka, Moscow, 985, (in Russian). [2] G.J. Butler and P. Waltman Bifurcation from a limit cycle in a two predator-one prey ecosystem modeled on a chemostat.j. Math. Biol. 2 (98), [3] G.J. Butler, S.B. Hsu, and P. Waltman A mathematical model of the chemostat with periodic washout rate. SIAM J. Appl. Math. 45 (985), [4] K.S. Cheng Uniqueness of limit cycle for a predator-prey system. SIAM J Appl Anal 2 (98), [5] J. Coste Dynamical regression in many species ecosystems The case of many predators competing for several preys. SIAM J Appl Math 45 (985), [6] J.M. Cushing Periodic two-predator, one-prey interactions and the time sharing of a resource niche. SIAM J. Appl. Math. 44 (984),

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