Limit Cycle Bifurcations in a Quartic Ecological Model
aa r X i v : . [ m a t h . D S ] F e b Limit Cycle Bifurcationsin a Quartic Ecological Model ⋆ Henk W. Broer
University of Groningen, Department of Mathematics,P.O. Box 407, 9700 AK Groningen, The Netherlands
Valery A. Gaiko
Belarusian State University of Informatics and Radioelectronics,Department of Mathematics, L. Beda Str. 6–4, Minsk 220040, Belarus
Abstract
In this paper we complete the global qualitative analysis of a quartic ecologicalmodel. In particular, studying global bifurcations of singular points and limit cycles,we prove that the corresponding dynamical system has at most two limit cycles.
Keywords : quartic ecological model; field rotation parameter; bifurcation; singularpoint; limit cycle; separatrix cycle; Wintner–Perko termination principle
The paper is based on the applications of Bifurcation Theory and can be usedfor modeling problems, where system parameters play a certain role in variousbifurcations. In this paper we consider a particular (quartic) family of planarvector fields, which models the dynamics of the populations of predators andtheir prey in a given ecological system and which is a variation on the classicalLotka–Volterra system. For the latter system the change of the prey densityper unit of time per predator called the response function is proportional tothe prey density. This means that there is no saturation of the predator whenthe amount of available prey is large. However, it is more realistic to considera nonlinear and bounded response function, and in fact different response ⋆ This work is supported by the Netherlands Organization for Scientific Research.
Email address: [email protected] (Valery A. Gaiko).
Preprint submitted to Elsevier 2 November 2018 unctions have been used in the literature to model the predator response, see[2]–[6], [11]–[14], [16].For instance, Zhu et al. [16] have studied recently the following predator-preymodel: ˙ x = x ( a − λx ) − yP ( x ) (prey) , ˙ y = − δy + yQ ( x ) (predator) . (1 . x > y > P ( x ) is a non-monotonic response functiongiven by P ( x ) = mxαx + βx + 1 , (1 . α, m are positive and where β > − √ α. Observe that in the absenceof predators, the number of prey increases according to a logistic growth law.The coefficient a represents the intrinsic growth rate of the prey, while λ > δ > . In Gause’s model the function Q ( x ) isgiven by Q ( x ) = cP ( x ) , where c > x = x − λx − yαx + βx + 1 ! (prey) , ˙ y = y − δ − µy + xαx + βx + 1 ! (predator) , (1 . α ≥ , δ > , λ > , µ ≥ β > − √ α are parameters. We note that(1.3) is obtained from (1.1) by adding the term − µy to the second equationand after scaling x and y, as well as the parameters and the time t. In thisway we take into account competition between predators for resources otherthan prey. The non-negative coefficient µ is the rate of competition amongstpredators. For examples of populations that use the group defense strategysee [2]–[6].System (1.3) can be written in the form˙ x = x ((1 − λx )( αx + βx + 1) − y ) ≡ P, ˙ y = − y (( δ + µy )( αx + βx + 1) − x ) ≡ Q. (1 . x = P − γQ, ˙ y = Q + γP, (1 . In this paper geometric aspects of Bifurcation Theory are used and developed.First of all, the two-isocline method which was developed by Erugin is used,see [7]. An isocline portrait is the most natural construction for a polyno-mial equation. It is sufficient to have only two isoclines (of zero and infinity)to obtain principal information on the original polynomial system, becausethese two isoclines are right-hand sides of the system. Geometric properties ofisoclines (conics, cubics, quartics, etc.) are well-known, and all isocline por-traits can be easily constructed. By means of them, all topologically differentqualitative pictures of integral curves to within a number of limit cycles anddistinguishing center and focus can be obtained. Thus, it is possible to carryout a rough topological classification of the phase portraits for the polyno-mial dynamical systems. It is the first application of Erugin’s method. Afterstudying contact and rotation properties of the isoclines, the simplest (canon-ical) systems containing limit cycles can be also constructed. Two groups ofparameters can be distinguished in such systems: static and dynamic. Staticparameters determine the behavior of phase trajectories in principle, sincethey control the number, position, and character of singular points in a finitepart of the plane (finite singularities). The parameters from the first groupdetermine also a possible behavior of separatrices and singular points at infin-ity (infinite singularities) under variation of the parameters from the secondgroup. The dynamic parameters are field rotation parameters, see [1], [7], [15].They do not change the number, position and index of the finite singularities,but only involve the vector field in a directional rotation. The rotation pa-rameters allow to control the infinite singularities, the behavior of limit cyclesand separatrices. The cyclicity of singular points and separatrix cycles, thebehavior of semi-stable and other multiple limit cycles are controlled by theseparameters as well. Therefore, by means of the rotation parameters, it is pos-sible to control all limit cycle bifurcations and to solve the most complicatedproblems of the qualitative theory of dynamical systems.In [7], [8] some complete results on quadratic systems have been presented.3n particular, it has been proved that for quadratic systems four is really themaximum number of limit cycles and (3 : 1) , i. e., three limit cycles aroundone focus and the only limit cycle around another focus, is their only possibledistribution (this is a solution of Hilbert’s Sixteenth Problem in the quadraticcase of polynomial dynamical systems). In [10] some preliminary results ongeneralizing new ideas and methods of [7] to cubic dynamical systems havealready been established. In particular, a canonical cubic system of Kuklestype has been constructed and the global qualitative analysis of its specialcase corresponding to a generalized Li´enard equation has been carried out. Ithas been proved also that the foci of such a Li´enard system can be at most ofsecond order and that such system can have at most three limit cycles on thewhole phase plane. Moreover, unlike all previous works on the Kukles-typesystems, global bifurcations of limit and separatrix cycles using arbitrary (in-cluding as large as possible) field rotation parameters of the canonical systemhave been studied in [10]. As a result, the classification of all possible typesof separatrix cycles for the generalized Li´enard system has been obtained andall possible distributions of its limit cycles have been found. In [9] a solutionof Smale’s Thirteenth Problem proving that the Li´enard system with a poly-nomial of degree 2 k + 1 can have at most k limit cycles has been presented.All of these methods and results can be applied to quartic dynamical systemsas well. In this paper, using [7]–[10], we will complete the global qualitativeanalysis of quartic ecological model (1.4). In particular, studying global bifur-cations of singular points and limit cycles, we will prove that the correspondingdynamical system has at most two limit cycles. The study of singular point of system (1.4) will use two index theorems byH. Poincar´e, see [1]. But first let us define the Poincar´e index [1].
Definition 3.1.
Let S be a simple closed curve in the phase plane not passingthrough a singular point of the system˙ x = P ( x, y ) , ˙ y = Q ( x, y ) , (3 . P ( x, y ) and Q ( x, y ) are continuous functions (for example, polynomials),and M be some point on S. If the point M goes around the curve S in positivedirection (counterclockwise) one time, then the vector coinciding with thedirection of a tangent to the trajectory passing through the point M is rotatedthrough the angle 2 πj ( j = 0 , ± , ± , . . . ) . The integer j is called the Poincar´eindex of the closed curve S relative to the vector field of system (3.1) and has4he expression j = 12 π I S P dQ − Q dPP + Q . According to this definition, the index of a node or a focus, or a center is equalto +1 and the index of a saddle is − . Theorem 3.1 (First Poincar´e Index Theorem). If N, N f , N c , and C arerespectively the number of nodes, foci, centers, and saddles in a finite part ofthe phase plane and N ′ and C ′ are the number of nodes and saddles at infinity,then it is valid the formula N + N f + N c + N ′ = C + C ′ + 1 . Theorem 3.2 (Second Poincar´e Index Theorem).
If all singular pointsare simple, then along an isocline without multiple points lying in a Poincar´ehemisphere which is obtained by a stereographic projection of the phase plane,the singular points are distributed so that a saddle is followed by a node or afocus, or a center and vice versa. If two points are separated by the equatorof the Poincar´e sphere, then a saddle will be followed by a saddle again and anode or a focus, or a center will be followed by a node or a focus, or a center.
We will use also the following theorem by A. N. Berlinskii, see [7].
Theorem 3.3 (Berlinskii Theorem) . If a quadratic system (3 . has foursingular points in a finite part of the phase plane, then only one of the fol-lowing cases is possible : 1) these points are vertices of a convex quadrangular,where two opposite vertices are saddles ( antisaddles ) and two others are anti-saddles ( saddles ); 2) three singular points are vertices of a triangle containingthe fourth point inside, and if this point is a saddle ( antisaddle ) , then the oth-ers are antisaddles ( saddles ) , where antisaddles are singularities that are nota saddle. Consider system (1.4) which has two invariant straight lines: x = 0 and y = 0 . Its finite singularities are determined by the algebraic system x ((1 − λx )( αx + βx + 1) − y ) = 0 ,y (( δ + µy )( αx + βx + 1) − x ) = 0 . (3 . ,
0) and (0 , − δ/µ ) , at most twopoints defined by the condition αx + βx + 1 = 0 , y = 0 , (3 . y = (1 − λx )( αx + βx + 1) ,y ( δ + µy ) − x (1 − λx ) = 0 , (3 . /λ, . To investigate the character and distribution of the singular points in the phaseplane, we will use the method developed in [7]–[10]. The sense of this methodis to obtain the simplest (well-known) system by vanishing some parameters(usually field rotation parameters) of the original system and then to inputthese parameters successively one by one studying the dynamics of the singularpoints (both finite and infinite) in the phase plane.Let the parameters α, β vanish and consider first the quadratic system˙ x = x (1 − λx − y ) , ˙ y = − y ( δ + µy − x ) . (3 . δ = 1 /λ. Studying isocline portraitsof the equation corresponding to system (3.5) and applying theorems 3.1–3.3,we can see that for the case, when δ > /λ, system (3.5) has two saddles:(0 ,
0) and a point of intersection of two straight line-isoclines:1 − λx − y = 0 , δ + µy − x = 0 , (3 . , − δ/µ ) and (1 /λ, . For the case, when δ < /λ, system(3.5) has two saddles: (0 ,
0) and (1 /λ, , — and two nodes: (0 , − δ/µ ) and(3.6). If δ = 1 /λ, it has three singularities: a saddle (0 , , a node (0 , − δ/µ ) , and a saddle-node (1 /λ, . Since we consider the first coordinate quadrantwith respect to the variables x and y, we will be interested basically in the caseof δ < /λ, when the singular point defined by (3.6) is in the first quadrant.To study singular points at infinity, consider the corresponding differentialequation dydx = − y ( δ + µy − x ) x (1 − λx − y ) . (3 . x ( x = 0) and denoting y/x by u, we will get the algebraic equation(1 − µ ) u + (1 + λ ) u = 0 , where u = y/x, (3 . x = 0 (the “ends” of the y -axis), see [1], [7]. For this special case we can divide the numerator and6enominator of the right-hand side of (3.7) by y ( y = 0) denoting x/y by v and consider the algebraic equation(1 + λ ) v + (1 − µ ) v = 0 , where v = x/y. (3 . x and y axes and a saddle in the direction of u = ( λ + 1) / ( µ − . Fix the parameters δ, λ, µ and take β < β, we will have a cubic system:˙ x = x ((1 − λx )( βx + 1) − y ) , ˙ y = − y (( δ + µy )( βx + 1) − x ) . (3 . δ < /λ and β < , system (3.10) has five finite singularities: two sad-dles — (0 ,
0) and (1 /λ, , two nodes — (0 , − δ/µ ) and ( − /β, , and anantisaddle (a node, a focus, or a center) defined as a point of intersection oftwo isoclines: (1 − λx )( βx + 1) − y = 0 , ( δ + µy )( βx + 1) − x = 0 . (3 . dydx = − y (( δ + µy )( βx + 1) − x ) x ((1 − λx )( βx + 1) − y ) (3 . µu − λu = 0 , where u = y/x, (3 . λv − µv = 0 , where v = x/y, (3 . x -axis,a saddle-node on the “ends” of the y -axis, and a saddle in the direction of u = λ/µ. Fix the parameters β, δ, λ, µ and take α > . Studying the bundle of cubiccurves y = (1 − λx )( αx + βx + 1) (3 . /λ,
0) and contact at the point (0 , , we cansee that system (1.4) obtained after inputting α has first six finite singularpoints: three saddles — (0 , , (1 /λ, , and (( − β + q β − α ) / (2 α ) , , two7odes — (0 , − δ/µ ) and (( − β − q β − α ) / (2 α ) , , and an antisaddle definedas a point of intersection of isoclines (3.4).On increasing the parameter α, the points (( − β − q β − α ) / (2 α ) ,
0) and(( − β + q β − α ) / (2 α ) ,
0) combine a saddle-node which then disappears. Onfurther increasing α, the point (1 /λ,
0) becomes a triple saddle from which asaddle and a node (or a saddle-node) will appear. Thus, we will have threesingular points in the first quadrant: a saddle S and two antisaddles — A and A which are defined as points of intersection of isoclines (3.4). Supposethat with respect to the x -axis they have the following sequence: A , S, A . To study singular points of (1.4) at infinity, consider the corresponding diffe-rential equation dydx = − y (( δ + µy )( αx + βx + 1) − x ) x ((1 − λx )( αx + βx + 1) − y ) (3 . µu − λu = 0 , where u = y/x, (3 . λv − µv = 0 , where v = x/y, (3 . x -axis, a triple node on the “ends” of the y -axis, and a simple saddle in thedirection of u = λ/µ. Note that all results on finite singularities of system (1.4) agree with the resultsof [3]–[6], [14], [16], but where infinite singularities have not been investigatedat all. Using the obtained information and applying the approach developedin [7]–[10], we can study limit cycle bifurcations of system (1.4) now. Thisstudy will use also some results obtained in [3]–[6], [14], [16]. In particular,the results on the cyclicity of singular points of (1.4) will be used. However,it is surely not enough to have only these results to prove the main theoremof this paper: on the maximum number of limit cycles of (1.4).
Let us first formulate the Wintner–Perko termination principle [15] for thepolynomial system ˙ x = f ( x , µ ) , (4 . µ )where x ∈ R ; µ ∈ R n ; f ∈ R ( f is a polynomial vector function).8 heorem 4.1 (Wintner–Perko termination principle). Any one-para-meter family of multiplicity- m limit cycles of relatively prime polynomial sys-tem (4 . µ ) can be extended in a unique way to a maximal one-parameter familyof multiplicity- m limit cycles of (4 . µ ) which is either open or cyclic.If it is open, then it terminates either as the parameter or the limit cycles be-come unbounded; or, the family terminates either at a singular point of (4 . µ ) , which is typically a fine focus of multiplicity m, or on a ( compound ) separatrixcycle of (4 . µ ) , which is also typically of multiplicity m. The proof of this principle for general polynomial system (4 . µ ) with a vectorparameter µ ∈ R n parallels the proof of the planar termination principle forthe system ˙ x = P ( x, y, λ ) , ˙ y = Q ( x, y, λ ) (4 . λ )with a single parameter λ ∈ R (see [7], [15]), since there is no loss of generalityin assuming that system (4 . µ ) is parameterized by a single parameter λ ; i. e.,we can assume that there exists an analytic mapping µ ( λ ) of R into R n suchthat (4 . µ ) can be written as (4 . µ ( λ ) ) or even (4 . λ ) and then we can repeateverything, what had been done for system (4 . λ ) in [15]. In particular, if λ is afield rotation parameter of (4 . λ ) , the following Perko’s theorem on monotonicfamilies of limit cycles is valid (see [15]). Theorem 4.2. If L is a nonsingular multiple limit cycle of (4 . ) , then L belongs to a one-parameter family of limit cycles of (4 . λ ); furthermore :1) if the multiplicity of L is odd, then the family either expands or contractsmonotonically as λ increases through λ ;2) if the multiplicity of L is even, then L bifurcates into a stable and anunstable limit cycle as λ varies from λ in one sense and L disappears as λ varies from λ in the opposite sense; i. e., there is a fold bifurcation at λ . Applying the definition of a field rotation parameter [1], [7], [15], i. e., a param-eter which rotates the field in one direction, to system (1.4), let us calculatethe corresponding determinants for the parameters α and β, respectively:∆ α = P Q ′ α − QP ′ α = x y ( y ( δ + µy ) − x (1 − λx )) , (4 . β = P Q ′ β − QP ′ β = x y ( y ( δ + µy ) − x (1 − λx )) . (4 . α or β the vector field of(1.4) in the first quadrant is rotated in positive direction (counterclockwise)9nly on the outside of the ellipse y ( δ + µy ) − x (1 − λx ) = 0 . (4 . γ : ∆ γ = P + Q ≥ . (4 . Theorem 4.3.
System (1 . has at most two limit cycles. Proof.
First let us prove that system (1.4) can have at least two limit cycles.Begin with quadratic system (3.5). It is clear that such a system, with twoinvariant straight lines, cannot have limit cycles at all [7]. Inputting a nega-tive parameter β into this system, the vector field of cubic system (3.10) willbe rotated in negative direction (clockwise) at infinity, the structure and thecharacter of stability of infinite singularities will be changed, and an unstablelimit, Γ , will appear immediately from infinity in this case. This cycle willsurround a stable antisaddle (a node or a focus) A which is in the first quad-rant of system (3.10). Inputting a positive parameter α into system (3.10),the vector field of quartic system (1.4) will be rotated in positive direction(counterclockwise) at infinity, the structure and the character of stability ofinfinite singularities will be changed again, and a stable limit, Γ , surroundingΓ will appear immediately from infinity in this case. On further increasingthe parameter α, the limit cycles Γ and Γ combine a semi-stable limit, Γ , which then disappears in a “trajectory concentration” [1], [7].As we saw above, on further increasing α, two other singular points, a saddle S and an antisaddle A , will appear in the first quadrant in system (1.4).We can fix the parameter α, fixing simultaneously the positions of the finitesingularities A , S, A , and consider system (1.5) with a positive parameter γ which acts like a positive parameter α of system (1.4), but on the whole phaseplane.So, consider system (1.5) with a positive parameter γ. On increasing this pa-rameter, the stable nodes A and A becomes first stable foci, then they changethe character of their stability, becoming unstable foci. At these Andronov–Hopf bifurcations [1], [7], stable limit cycles will appear from the foci A and A . On further increasing γ, the limit cycles will expand and will disappearin small separatrix loops of the saddle S. If these loops are formed simultane-ously, we will have a so-called eight-loop separatrix cycle. In this case, a bigstable limit surrounding three singular points, A , S, and A , will appear from10he eight-loop separatrix cycle after its destruction, expanding to infinity onincreasing γ. If a small loop is formed earlier, for example, around the point A ( A ) , then, on increasing γ, a big loop formed by two lower (upper) adjoiningseparatrices of the saddle S and surrounding the points A and A will appear.After its destruction, we will have simultaneously a big limit cycle surroundingthree singular points, A , S, A , and a small limit cycle surrounding the point A ( A ) . Thus, we have proved that system (1.4) can have at least two limitcycles, see also [3]–[6], [14], [16].Let us prove now that this system has at most two limit cycles. The proofis carried out by contradiction applying Catastrophe Theory, see [7], [15].Consider system (1.5) with three parameters: α, β, and γ (the parameters δ, λ, and µ can be fixed, since they do not generate limit cycles). Supposethat (1.5) has three limit cycles surrounding the only point, A , in the firstquadrant. Then we get into some domain of the parameters α, β, and γ beingrestricted by definite conditions on three other parameters, δ, λ, and µ . Thisdomain is bounded by two fold bifurcation surfaces forming a cusp bifurcationsurface of multiplicity-three limit cycles in the space of the parameters α, β, and γ [7], [15].The corresponding maximal one-parameter family of multiplicity-three limitcycles cannot be cyclic, otherwise there will be at least one point correspondingto the limit cycle of multiplicity four (or even higher) in the parameter space.Extending the bifurcation curve of multiplicity-four limit cycles through thispoint and parameterizing the corresponding maximal one-parameter family ofmultiplicity-four limit cycles by the field rotation parameter, γ, according toTheorem 4.2, we will obtain two monotonic curves of multiplicity-three andone, respectively, which, by the Wintner–Perko termination principle (Theo-rem 4.1), terminate either at the point A or on a separatrix cycle surroundingthis point. Since we know at least the cyclicity of the singular point which isequal to two (see [3]–[6], [14], [16]), we have got a contradiction with the termi-nation principle stating that the multiplicity of limit cycles cannot be higherthan the multiplicity (cyclicity) of the singular point in which they terminate.If the maximal one-parameter family of multiplicity-three limit cycles is notcyclic, using the same principle (Theorem 4.1), this again contradicts thecyclicity of A (see [3]–[6], [14], [16]) not admitting the multiplicity of limitcycles to be higher than two. This contradiction completes the proof in thecase of one singular point in the first quadrant.Suppose that system (1.5) with three finite singularities, A , S, and A , hastwo small limit cycles around, for example, the point A (the case when limitcycles surround the point A is considered in a similar way). Then we get intosome domain in the space of the parameters α, β, and γ which is bounded bya fold bifurcation surface of multiplicity-two limit cycles [7], [15].11he corresponding maximal one-parameter family of multiplicity-two limitcycles cannot be cyclic, otherwise there will be at least one point correspondingto the limit cycle of multiplicity three (or even higher) in the parameter space.Extending the bifurcation curve of multiplicity-three limit cycles through thispoint and parameterizing the corresponding maximal one-parameter familyof multiplicity-three limit cycles by the field rotation parameter, γ, accordingto Theorem 4.2, we will obtain a monotonic curve which, by the Wintner–Perko termination principle (Theorem 4.1), terminates either at the point A or on some separatrix cycle surrounding this point. Since we know at least thecyclicity of the singular point which is equal to one in this case [3]–[6], [14],[16], we have got a contradiction with the termination principle (Theorem 4.1).If the maximal one-parameter family of multiplicity-two limit cycles is notcyclic, using the same principle (Theorem 4.1), this again contradicts thecyclicity of A (see [3]–[6], [14], [16]) not admitting the multiplicity of limitcycles higher than one. Moreover, it also follows from the termination principlethat either an ordinary (small) separatrix loop or a big loop, or an eight-loopcannot have the multiplicity (cyclicity) higher than one in this case. Therefore,according to the same principle, there are no more than one limit cycle in theexterior domain surrounding all three finite singularities, A , S, and A . 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