Abstract. We consider a generalised Gause predator-prey system with a generalised Holling. ax 2. We study the cases where b is positive or negative.

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1 BIFURCATION ANALYSIS OF A PREDATOR-PREY SYSTEM WIT GENERALISED OLLING TYPE III FUNCTIONAL RESPONSE YANN LAMONTAGNE, CAROLINE COUTU, AND CRISTIANE ROUSSEAU Abstract. We consider a generalised Gause predator-prey system with a generalised olling m response function of type III: p() = 2 a 2. We study the cases where b is positive or negative. +b+ We make a complete study of the bifurcation of the singular points including: the opf bifurcation of codimension and 2, the Bogdanov-Takens bifurcation of codimension 2 and 3. Numerical simulations are given to calculate the homoclinic orbit of the system. Based on the results obtained, a bifurcation diagram is conjectured and a biological interpretation is given. Key words. Gause predator-prey system, response function, Bogdanov-Takens bifurcation, opf bifurcation, limit cycles, homoclinic orbit, homoclinic bifurcation, generalised olling response function of type III AMS subject classifications. 34C5, 34C23, 34C25, 34C26, 34C37, 37G, 37G5, 92D25. Introduction. We consider a generalised Gause predator-prey system [2, ] of the following form ẋ = r ( k ) yp() ẏ = y( d + cp()) () >, y() > with a generalised olling response function of type III p() = (.) m 2 a 2 + b +. (.2) The system has seven parameters, the parameters a, c, d, k, m and r are positive. We study the cases where b is positive or negative. The parameters a, b, m are fitting parameters of the response function. The parameter d is the death rate of the predator while c is the efficiency of the predator to convert prey into predators. The prey follows a logistic growth with a rate r in the absence of predator. The environment has a prey capacity determined by k. The case b has been studied biologically by Jost and al. in 973 [6, 7]. The generalised olling functional response of type III was used to model the predation of Tetrahymena pyriformis on Escherichia coli or Azotobacter vinelandii in a chemostat. The chemostat system with the generalised olling response function of type III was used instead of the olling response function of type II (p() = b +a ) since the former response function gave better results. This is partly due to the rate of predation which does not diminish sufficiently fast in type II when the prey population tends to zero. Indeed, for type II (resp. III) the response function has a nonzero (resp. zero) slope at = (see Fig..(a)). We show that the case b > is qualitatively equivalent to the case b = (i.e. p() = m2 a 2 + also known as olling response function of type III [5, 9]). This work was supported by NSERC in Canada. yann@daume.org. Collège Édouard-Montpetit, caroline.coutu@college-em.qc.ca. Département de Mathématiques, Université de Montréal, C.P. 628, Succursale Centre-ville, Montréal, Québec, 3C 3J7, CANADA, rousseac@dms.umontreal.ca.

2 2 Y. Lamontagne, C. Coutu & C. Rousseau m a M L (a) b M (b) b < Fig... Generalised olling type III functional response. The case b < provides a model for a functional response with limited group defence. In opposition to the generalised olling function of type IV (p() = m a 2 +b+ ) studied in [,, 2, 9, 2] where the response function tends to zero as the prey population tends to infinity, the generalised function of type III tends to a nonzero value L as the prey population tends to infinity (see Fig..(b)). The functional response of type III with b < has a maimum M at M. It is possible to obtain a, b, and m as functions of L, M and M. When studying the case b < we will find a Bogdanov-Takens bifurcation of codimension 3 which is an organizing center for the bifurcation diagram. The system (.) has 7 parameters but, with the following scaling (X, Y, T ) = ( k, ck y, cmk2 t) of the variables, we are able to reduce the number of parameters to 4. The system which we consider is where ẋ = ρ( ) yp() ẏ = y( δ + p()) () >, y() > (.3) p() = β + (.4) and the parameters are (, β, δ, ρ) = (ak 2 d r, bk, cmk, 2 cmk ). The parameter space is 2 of dimension 4 but we noticed that all but the homoclinic orbit bifurcation and the bifurcation of the double limit cycle do not depend on the ρ parameter. Therefore the bifurcation diagrams are given in the (, δ) plane for values of β. The values of β are either generic or bifurcation values. The article is organized as follows; in section 2 we study the number of singular points. In section 3 we study the saddle/anti-saddle character of the singular points. In section 4 we discuss the Bogdanov-Takens bifurcation of codimension 2 and 3. In section 5 we study the opf bifurcation of codimension and 2. In section 6 a bifurcation diagram is conjectured, and in section 7 we study the homoclinic bifurcation by numerical simulation and find necessary conditions for the eistence of a homoclinic orbit. We discuss in section 8 a biological interpretation. 2. Bifurcation of Number of the Singular Points. We give different surfaces in the parameter space which describe the bifurcation of the number of singular points in the first quadrant. There are always two singular points on the -ais: (, ) and (, ), but (, ) can change multiplicity when singular points enter or eit the first quadrant. In addition, a singular point of multiplicity 2 may appear in the first

3 BIFURCATION ANALYSIS OF A PREDATOR-PREY SYSTEM 3 quadrant and bifurcate into 2 singular points. In the case β (resp. β < ) there is a possibility of up to one singular point (resp. two singular points) in the open first quadrant. We want to determine the singular points of the system (.3) apart from those on the -ais. There is the possibility of up to two singular points in the open first quadrant which are the solutions of p() = δ with y = ρ δ ( ). We must verify the number of positive solutions of p() = δ such that ρ δ ( ). The number of real solutions of p() = δ is determined by the sign of: = δ(δβ 2 4δ + 4). (2.) The study of the sign of y is more difficult, but y can only change sign by vanishing or going to infinity. We will show that the latter is ecluded when we are in the first quadrant. ence we will conclude that the number of singular points in the first quadrant can only change in two ways. The first when a singular point passes through the singular point (, ). The second when two singular points coalesce and then vanish. We will study both bifurcations. For the case β we note that p () > for all >, therefore p() = δ has at most one positive solution. It is possible that a singular point (, y ) is not in the first quadrant and, in consequence, has no biological significance. The number of points in the first quadrant is given by the following theorem. Theorem 2.. For all parameter values, (, ) and (, ) are singular points of the system (.3). The number of singular points in the open first quadrant is given in Fig. 2.. The surfaces appearing in Fig. 2. are defined below. To help define the surfaces we consider the Jacobian of the system (.3): [ ] ρ 2ρ y(β+2) 2 ( A = 2 +β+) 2 2 +β+ y(β+2). (2.2) ( 2 +β+) δ β+ S : The Jacobian has a zero eigenvalue at (, ) when the parameters are on S : δ = + β +. This corresponds to (, ) being a saddle-node for β 2 or a singular point of multiplicity 3 if β = 2. When crossing S a singular point passes through (, ) and changes quadrant. In the case where β > 2: if δ < +β+ (resp. δ > +β+ ) then the singular point is in the first (resp. fourth) quadrant, and we have the inverse situation when β < 2. : The surface represents when a singular point in the open first quadrant is of multiplicity 2. The surface has a biological significance when β 2. We observe that p() = δ if and only if f() = where f() = (δ ) 2 + βδ + δ. (2.3) The surface corresponds to the discriminant (see (2.)) of f() being zero. Therefore the surface is given by : δ = 4 4 β 2. We observe that the surface and S are identical when β = 2, in which case (, ) is a singular point of multiplicity 3.

4 4 Y. Lamontagne, C. Coutu & C. Rousseau δ S S δ S S (a) β (b) < β < δ S δ S S (c) β =, where S = S δ S S (d) 2 < β < (e) β = 2, where = S δ S S 2 (f) β < 2 Fig. 2.. Surfaces of parameters and number of singular points. : The surface represents the boundary of the parameter region where the model has biological significance in the first quadrant. The model has a biological significance when the generalised olling function of type III is well defined (i.e. when the denominator of p() does not vanish). The condition is β 2

5 BIFURCATION ANALYSIS OF A PREDATOR-PREY SYSTEM 5 (a) β = 2 + ɛ (b) β = 2 (c) β = 2 ɛ Fig. 3.. Behavior of solutions in a neighborhood of the -ais when the paramenters are on the surface S and in a neighborhood of β = 2. 4 < when β < and the surface is given by : β 2 = 4. S : This surface represents a singular point which passes to infinity as its -coordinate becomes infinite. The surface is given by S : δ = Note that the surfaces S and S are identical when β =. In addition we notice that the surface S is not relevant for the biological system since it does not change the number of singular points in the first quadrant. This comes from the fact that a singular point can only tend to infinity while it is in the region y < and hence it moves from the fourth to the third quadrant. 3. Type of Singular Points. 3.. Statements of Results. Theorem 3.. The singular point at the origin is a saddle. The type of (, ) is: δ > +β+ : an attracting node, δ < +β+ : a saddle, δ = +β+ and β 2: an attracting saddle-node (see Fig. 3.), β = 2: an attracting semi-hyperbolic node (see Fig. 3.). The unfolding of the semi-hyperbolic node (, ) of codimension 2 is a cylinder over the bifurcation diagram given in Fig Theorem 3.2. If the parameters of the system (.3) are on the surface, β < 2 and β 4 then there is a saddle-node in the open first quadrant. If β ( 4, 2) then the saddle-node is attracting. If β < 4 then the saddle-node is repelling. Definition 3.3. A singular point is simple if the determinant det(a (,y )) of the Jacobian A (,y ) is not zero. A singular point is of saddle (resp. anti-saddle) type if the determinant, det(a (,y )), is negative (resp. positive). It follows that a singular point of anti-saddle type is either a center, a node, a focus, or a weak focus. Theorem 3.4. If there eists eactly one singular point in the open first quadrant then it is an anti-saddle. If there eists eactly two singular points in the open first quadrant then the singular point on the left is an anti-saddle and the singular point on the right is a saddle.

6 6 Y. Lamontagne, C. Coutu & C. Rousseau IV III S II I II III V I IV V VI VI VII VIII VII VIII S Fig Bifurcation diagram of the semi-hyperbolic point (, ) of codimension Proofs of Results. Proof of Theorem 3.. The Jacobian A (see (2.2)) evaluated at (, ) is [ ] ρ A (,) =. δ It clearly follows that the origin is a saddle since it has eigenvalues ρ and δ with eigenvectors (, ) and (, ), respectively. We observe directly from the system (.3) that the and y aes are invariant. If A is evaluated at (, ) we obtain [ ρ ] +β+ A (,) = δ +. +β+ We notice that ρ is always an eigenvalue and its associated eigenvector is (,), hence there is an heteroclinic connection between the singular points, on the -ais. The second eigenvalue is δ + +β+, its sign depends on the parameters. If the parameters are above the surface S, i.e. δ > +β+, then (, ) is an attracting node. If the parameters are below the surface S, i.e. δ < +β+, then (, ) is a saddle. If the parameters are on the surface S (i.e. satisfy the relation δ = +β+ ), then the second eigenvalue is zero. To determine the type of (, ), we first localise the system at (, ) and diagonalise the linear part. We obtain the following system Ẋ = ρx + o( X, Y ) Ẏ = BY 2 + o( X, Y 2 ) (3.) where 2 + β B = ρ( + β + ) 3. If a change of coordinates is performed to bring the system onto the center manifold, the term BY 2 does not change. ence the local behavior of the solutions is determined

7 BIFURCATION ANALYSIS OF A PREDATOR-PREY SYSTEM 7 by B when B. The singular point (, ) is an attracting saddle-node when the parameters are on the surface S and β 2. The surfaces and S intersect on β = 2, in which case (, ) is of multiplicity 3. Then (3.) has the following form Ẋ = ρx + o( X, Y ) Ẏ = CY 3 + XO( X, Y 2 ) + o( X, Y 3 ) where C = ρ( ) 4 ρ 2. The term CY 3 is sufficient to determine the local behavior of the solutions since it remains unchanged when reducing the flow on the center manifold. ence (, ) is a semi-hyperbolic attracting node since C is always negative. Proof of Theorem 3.2. The Jacobian A evaluated at the singular point (, y ) = ( 2 β, 2ρ(β+2) δβ 2 ) of multiplicity 2 is A (,y ) = [ ρ(β+4) β 4 β 2 4 and its trace is equal to ρ(β+4) β. ence (, y ) is an attracting (resp. repelling) saddle-node for β < 4 (resp. β > 4). We introduce some relations and a function which will be used both for the proof of Theorem 3.4 and for the study of the opf bifurcation. The singular point (, y ) is not necessarily on the -ais if p( ) = δ (see (.4)), in which case the following two relations are satisfied ], and I : δ = β +, (3.2) II : y = ρ δ ( ). (3.3) To determine the type of the singular point we take the determinant of A (Jacobian of the system, see (2.2)) and evaluate it at a singular point not necessarily on the -ais, det(a (,y )) = 3 y (β + 2) ( 2 + β + ) 3. It is possible to simplify det(a (,y )) by substituting II and I, yielding for h defined by det(a (,y )) = ( + )( β + 2)ρ 2 ( 2 + β + ) 2 = δ 2 ( + )( β + 2)ρ = h( ) h() = δ 2 ( )(β + 2)ρ. (3.4)

8 8 Y. Lamontagne, C. Coutu & C. Rousseau Proof of Theorem 3.4. By continuity the determinant of a singular point cannot change sign without vanishing. There are only two posibilities for h() =, namely = or = 2 β. We obtain a surface of parameters where the determinant is zero at a singular point by evaluating f() (see (2.3)) for each case. The relation f() = and f( 2 β ) = are in fact the surfaces S and, respectively. Therefore the sign of the determinant of a singular point in the first quadrant will only change on the bifurcation surfaces S and. ence the sign of the determinant of the Jacobian at a singular point in the open first quadrant is determined by the local analysis of the bifurcations. As we have studied the bifurcations on S and, this completes the proof of the theorem. 4. Bogdanov-Takens Bifurcation. Theorem 4.. If the parameters are on, β = 4 and 8 then there eists a Bogdanov-Takens bifurcation: the system localised at the singular point of multiplicity 2 is conjugate to ẋ 2 = y 2 ẏ 2 = D 2 y 2 + o( 2, y 2 2 ) (4.) where D = 2ρ(8 ). (4.2) 4 Theorem 4.2. If the parameters are on and (, β) = (8, 4) then there eists an attracting Bogdanov-Takens bifurcation of codimension 3: the system localised at the singular point of multiplicity 2 is equivalent to X = Y Y = X 2 + EX 3 Y + o( X, Y 4 ) (4.3) where E = 24 ρ 2. (4.4) Proof of Theorem 4.. If the parameters are on the surface and β = 4 then the Jacobian evaluated at the singular point (, y ) = ( 2 β, 2ρ(β+2) δβ 2 ) of multiplicity 2, A (,y ) = [ ] 4, is nilpotent. Therefore it has two zero eigenvalues and we observe that 4 < since β 2 4 < and β = 4 which implies > 4 (restriction of the parameters by the surface ). This implies that the linear part of the system is non-diagonalisable, hence there is a Bogdanov-Takens bifurcation. To study the Bogdanov-Takens bifurcation, we want to put the system into the form (4.). We perform two changes of coordinates. The first step is to localise the singular point (, y ) to the origin by means of (, y ) = (, y y ) and to

9 BIFURCATION ANALYSIS OF A PREDATOR-PREY SYSTEM 9 O II III µ 2 I IV I O O µ II III IV O (, µ 2 ), µ 2 > (, µ 2 ), µ 2 < Fig. 4.. Bogdanov-Takens bifurcation diagram when D >. perform a second changes of coordinates, 4ρ 2 = ( 4) 2 y 2 = + 4ρ ( 4) 2 ( 4 y ρ (3 6)ρ ( 4) ( 4) 2 3 y + o(, y 4 ) ( 4) 2 2 y ). Simplifying we obtain a system of the form (4.) with D given in (4.2). D is well defined since > 4. The coefficient D can change sign, therefore we continue the analysis of the Bogdanov-Takens bifurcation. If D and there eists an unfolding of (4.) then we have a system of the form ẋ 2 = y 2 ẏ 2 = µ + µ 2 y D 2y 2 + o( 2, y 2 2 ), and in the case D > the system will have the bifurcation diagram given in Fig. 4.. In the case D < the bifurcation diagram can be obtained by applying the transformation ( 2, y 2, t, µ, µ 2 ) ( 2, y 2, t, µ, µ 2 ) to the bifurcation diagram of Fig. 4.. The Bogdanov-Takens bifurcation is studied in [2, 7, 4]. If = 8 then D = and we have the following system ẋ 2 = y 2 ẏ 2 = ρ ( 6 + y 2 ( ρ ρ + y 2 2 ρ ( 28 ρ ρ ) ( 256 ρ ρ ) 2 + o( 2, y 2 4 ). ) ) 3 2 (4.5) We want to bring it to the normal form (4.3), for that purpose we introduce the following result.

10 Y. Lamontagne, C. Coutu & C. Rousseau Proposition 4.3. The system ẋ = y ẏ = 2 + a a y(a a 3 3 ) + y 2 (a 2 + a 22 2 ) + o(, y 4 ) (4.6) is equivalent to the system X = Y Y = X 2 + EX 3 Y + o( X, Y 4 ) (4.7) where E = a 3 a 3 a 2 (4.8) after changes of coordinates and a rescaling of time. Proof. The first change of coordinates removes the terms 2 y, y 2, 2 y 2. = 3 + N 3 y 3 + N N N y 3 y = y 3 + N y N N 2 2 3y 3 + a 3 N N 3 3 3y 3 + 3N 4 2 3y 2 3 where (N, N 2, N 3, N 4 ) = ( a2 coordinate we obtain the system 3, a2 6, a2 2 +9a22 8, a2a2 ẋ 3 = y 3 + o( 3, y 3 4 ) = h( 3, y 3 ) ẏ 3 = k( 3 ) + E 3 3y 3 + o( 3, y 3 4 ) 6 ). Using the above change of where k() = 2 + b b 4 4 and (b 3, b 4, E) = (a 3, a 4 a2 6, a 3 a 3 a 2 ). We perform a second change of coordinates, 4 = 3 y 4 = h( 3, y 3 ) = y 3 + n( 3, y 3 ). with h(, y) = o(, y 4 ) which results in the system brings the system to ẋ 4 = y 4 + o( 4, y 4 4 ) ẏ 4 = k( 4 ) + E 3 4 y 4 + o( 4, y 4 4 ). For the third change, let K() = k(z)dz and X = (3K( 4 )) 3 = 4 + o( 4 ) Y = y 4 be a change of coordinates tangent to the identity. coordinates we obtain If we perform the change of Ẋ = (3K( 4 )) 2 3 k(4 )Y Ẏ = k( 4 ) + E 3 4Y + o( 4, Y 4 ). We then rescale the time by dividing the system by (3K( 4 )) 2 3 k( 4 ) = + O( 4 ) this will not change the orientation nor the qualitative behavior of the solution since

11 BIFURCATION ANALYSIS OF A PREDATOR-PREY SYSTEM O µ 2 BT C + O 2 O + 2 DC BT + SN + µ SN 3 µ Fig Surfaces of the bifurcation Bogdanov-Takens of codimension 3. SN III O 2 II IV DC 2 BT + O + + I I II III C VI V O BT SN + IV V VI Fig Bifurcation diagram of generic region of the Bogdanov-Takens bifurcation of codimension 3. the dividing factor is positive in a neighborhood of the bifurcation. With proper substitution we obtain the system (4.7). Proof of Theorem 4.2. Using Proposition 4.3 the system (4.5) is equivalent to a system of the form (4.3) where E is given in (4.4). The coefficient E is always negative and a system of this form has been studied in [8, 7]. The bifurcation diagram for such a system is known and, if there eists an unfolding, then the unfolding of the system (4.3) is equivalent to ẋ = y ẏ = µ + µ 2 y + µ 3 y y + o(, y 4 ). We give a bifurcation diagram in Fig. 4.2 and Tab. 4 gives a description of the bifurcation surfaces. The bifurcations BT and BT + occur on the µ 3 -ais, separated by the Bogdanov-Takens bifurcation of codimension 3. To visualize the generic regions of the bifurcation diagram we notice that, when µ >, there is no singular point, hence the bifurcations are in the half-space where µ <. We take a sphere S ɛ = {(µ, µ 2, µ 3 ) µ 2 + µ2 2 + µ2 3 = ɛ2 }, which we intersect with the bifurcation diagram of Fig. 4.2 and we then remove a point on the sphere in

12 2 Y. Lamontagne, C. Coutu & C. Rousseau + O + O 2 O 2 DC C BT + BT SN + SN repelling opf bifurcation attracting opf bifurcation repelling homoclinic bifurcation attracting homoclinic bifurcation opf bifurcation of codimension two homoclinic bifurcation of codimension two double limit cycle intersection of and O Bogdanov-Takens bifurcation with positive y coefficient Bogdanov-Takens bifurcation with negative y coefficient repelling saddle-node attracting saddle-node Table 4. δ S Fig. 5.. opf bifurcation diagram when β. the region with no singular point. The sphere minus the point can be represented on a plane and the bifurcation diagram appears in Fig opf Bifurcation. In this section we give a full analysis of the opf Bifurcation which is summarised in the following theorem. Theorem 5.. The system (.3) has a generic opf bifurcation of codimension when β (bifurcation diagram given in Fig. 5.). It has a generic opf bifurcation of codimension (bifurcation diagram given in Fig. 5.2) or 2 (bifurcation diagram given in Fig. 5.3) when β <. In all cases it has a complete unfolding. The proof of the theorem is spread throughout sections 5. to 5.4. In section 5. we study some of the necessary conditions for the opf bifurcation. In the sections 5.2 and 5.3 the method of Lyapunov coeffficients is used to determine the codimension of the opf bifurcation. We then proceed in sections 5.3 and 5.4 to show that the bifurcation has a complete unfolding. 5.. Necessary Conditions for the opf Bifurcation. We deduce two necessary conditions for the opf bifurcation to eist. The trace (resp. determinant) of the Jacobian A (,y ) evaluated at a singular point in the open first quadrant is zero (resp. positive). The sign of the determinant of the Jacobian has already been studied in Theorem 3.4. The trace of A (see (2.2)) at a singular point (, y ) not necesserily on the -ais vanishes when ρ 2ρ y(β+2) =. With the use of relation I and II (see ( 2 +β+)2 (3.2) and (3.3)) we obtain that the trace of A (,y ) is zero if and only if g( ) =

13 BIFURCATION ANALYSIS OF A PREDATOR-PREY SYSTEM 3 δ S δ S (a) 3 β < (b) 4 < β < 3 δ S δ S (c) β = 4 (d) β < 4 Fig opf bifurcation of codimension when β <. δ S δ S BT (a) 3 β = 4 2 (β) (b) β 2 S < β < 4 δ S δ S 2 S (c) β = β 2 S 2 (β) (d) 6 < β < β 2 S Fig opf bifurcation diagram of codimension 2 when β <. where g() = (βδ + ) 2 + δ(2 β) 2δ. (5.) To obtain a surface in the parameter space which represents the trace of A (,y ) vanishing for (, y ) in the first quadrant, we take the resultant of f() (see (2.3)) and g() with respect to. This yields P (, β, δ) = where resultant(f(), g(), ) P (, β, δ) = δ 2 = ( + β + )(4 β 2 )δ 3 + (β 2 β 2 + β )δ 2 + ( 5 + 2β)δ +. (5.2)

14 4 Y. Lamontagne, C. Coutu & C. Rousseau δ SP δ (β) (β) Fig Diagram of the curves SP and for β < 4. We note that P (, β, δ) is a cubic with respect to δ. ence its discriminant gives information on the number of roots. The discriminant of P (, β, δ) with respect to δ is Q(, β) = Q (, β)q 2 (, β) (5.3) where Q (, β) = 4( β 3 + 3β β ) Q 2 (, β) = ( 2 + β β 2 β 3 ) 2. (5.4) When β is fied we note that, apart from a double root = (β), Q 2 (, β) is always strictly positive, (β) is positive if and only if β R\[ 2, 2]. We analyse Q (, β) and, since it is cubic in, we take its discriminant with respect to which is equal to 9683β 2 (β + 4). ence if β > 4, (resp. β = 4, β < 4) is fied, then Q (, β) = has three roots, (resp. one root of multiplicity 2, one root). For β < there is always an even number of positive roots of Q (, β) =. They are denoted and 2 when they eist. When β >, Q (, β) has eactly one positive root which is denoted. Proposition 5.2. If β = 3± 3 2, 2, 4 then Q (, β ) = and Q 2 (, β ) = have a common root. Proof. The proof is trivial since the root of Q 2 (, β) is eplicitly known. Proposition 5.3. For β fied, the graph of the function Q(, β) and the algebraic curve P (, β, δ) = are given (qualitatively) for generic or bifurcation values of β in Fig. 5.5, 5.6, 5.7, 5.8, 5.9, 5., 5., 5.2, 5.3, 5.4, 5.5. Proof. The proof is lenghty but relatively straightforward and consists of a study of Q(, β) and P (, β, δ) = with algebraic tools. In particular a solution branch of P (, β, δ) = tends to infinity with respect to δ for = β. Full details appear in [8]. Not all solution branches given above satisfy the necessary condition for the opf bifurcation. For eample one branch represents a singular point that is not in the first quadrant, while another branch represents a trace zero saddle in the open first quadrant. Proposition 5.4. Among the solution branches of P (, β, δ) =, only the ones in Fig. 5. for β and in Fig. 5.2 for β < satisfy the necessary conditions for the opf bifurcation.

15 BIFURCATION ANALYSIS OF A PREDATOR-PREY SYSTEM 5 δ P (, β, δ) = Q(, β) Fig Q(, β) and P (, β, δ) = when β > is fied. δ P (, β, δ) = Q(, β) Fig Q(, β) and P (, β, δ) = when β = δ P (, β, δ) = Q(, β) Fig Q(, β) and P (, β, δ) = when 2 < β < is fied. Proof. We eliminate the solution branches of P (, β, δ) = which either do not represent a singular point (, y ) in the open first quadrant, or are attached to a singular point with zero trace at which the determinant of the Jacobian A (,y ) is negative. In the case β a branch is easily discarded as part of it is located in a region where the system has no singular point in the open first quadrant. The same reasoning can be applied in the case β ( 4, ) and the upper left solution branch is discarded. In the case β < 4, the branch which intersects the valid opf solution branch,, is discarded since it represents a trace zero saddle, SP, see Fig owever this branch will be of interest in section 7.4 where we consider the homoclinic bifurcation of codimension 2, O 2. In the case β < the lowest solution branch (the one closest to the -ais) represents a singular point not in the first quadrant, hence it is discarded. Full details appear in [8, 6] opf Bifurcation of Codimension 2. We study the opf bifurcation with the method of Lyapunov constants [3, 5, 4, 3]. We first localise the system at a singular point (, y ) in the open first quadrant. We then rescale the time by dividing the system by p( + ) (see (.4)). Since p() > for all >, neither the orientation of trajectories, nor the number of periodic solutions will change. The

16 6 Y. Lamontagne, C. Coutu & C. Rousseau δ P (, β, δ) = Q(, β) Fig Q(, β) and P (, β, δ) = when β 2 is fied. δ P (, β, δ) = 2 Q(, β) 2 Fig Q(, β) and P (, β, δ) = when < β < is fied. δ P (, β, δ) = 2 Q(, β) 2 Fig. 5.. Q(, β) and P (, β, δ) = when β =. system has the form where = h( ) y (5.5) ( ) y = (y δ + y ) p( + ) + (5.6) h() = ρ( ( + ) 3 + ( β)( + ) 2 + (β )( + ) + ) + y. We proceed with a second change of coordinates ( 2, y 2 ) = (, y h( )) and we obtain a system of the form 2 = y 2 ( ) ( ) y 2 δ = (h( 2 ) + y ) p( 2 + ) + dh(2 ) δ + y 2 + d 2 p( 2 + ) ( 5 4 ) = a i (λ) i 2 + y 2 b i (λ) i 2 + o( 2 4 ) + o( 2 5 ). i= i= (5.7)

17 BIFURCATION ANALYSIS OF A PREDATOR-PREY SYSTEM 7 δ P (, β, δ) = 2 Q(, β) 2 Fig. 5.. Q(, β) and P (, β, δ) = when 2 < β < is fied. δ P (, β, δ) = 2 Q(, β) 2 Fig Q(, β) and P (, β, δ) = when β = 2. δ P (, β, δ) = 2 P Q(, β) 2 Fig Q(, β) and P (, β, δ) = when 4 < β < 2 is fied. The method of Lyapunov constants to study the opf bifurcation consists of looking for a Lyapunov function for the system. The method is usually applied to a system of the form Ẋ = c(λ)x Y + G (X, Y, λ) Ẏ = X + d(λ)y + G 2 (X, Y, λ) (5.8) where c, d, G, G 2 are smooth functions with G i (X, Y ) = o( X, Y ) and, either c or c d and λ a multi-parameter. Proposition 5.5. Consider a family of Liénard vector fields (5.7) with λ a multi-parameter, to which we apply the change of coordinates ] [ = y 2 [ 2 b (λ) 2 b(λ) 2 +4a (λ) 2 ] [X Y ]. (5.9)

18 8 Y. Lamontagne, C. Coutu & C. Rousseau δ P (, β, δ) = Q(, β) Fig Q(, β) and P (, β, δ) = when β = 4. δ P (, β, δ) = Q(, β) Fig Q(, β) and P (, β, δ) = when β < 4 is fied. Then there eists a polynomial F λ, F λ (X, Y ) = 2 (X 2 + Y 2 ) + 6 F j (X, Y, λ), with F j a homogeneous polynomial of degree j with respect to X and Y. Then there eists L (λ) such that j=3 F λ = L (λ)(x 2 + Y 2 ) + o( X, Y 2 ), and for all values of λ such that L (λ) =, there eist L (λ) and L 2 (λ) such that F λ = L (λ)(x 2 + Y 2 ) 2 + L 2 (λ)(x 2 + Y 2 ) 3 + o( X, Y 6 ). The Lyapunov constants are where L (λ) = b (λ) 2, L (λ) = b 2(λ)a (λ) b (λ)a 2 (λ), L= 8a (λ) L 2 (λ) = L= 92a (λ) 3 (L() 2 (λ)(b 2(λ)a (λ) b (λ)a 2 (λ)) + L (2) 2 (λ)) L () 2 (λ) = 2a 3(λ)a (λ) + b (λ) 2 a (λ) a 2 (λ) 2 L (2) 2 (λ) = 2a (λ) 3 b 4 (λ) 2a (λ) 2 b 3 (λ)a 2 (λ) 2a (λ) 2 b (λ)a 4 (λ) + 2a 2 (λ)b (λ)a 3 (λ)a (λ). (5.) (5.) In addition, L, L, L 2 are smooth with respect to λ.

19 BIFURCATION ANALYSIS OF A PREDATOR-PREY SYSTEM 9 Proof. The Lyapunov method guarantees the eistence and differentiability of the Lyapunov constants of a system of the form (5.8) shown in [4]. After the change of coordinates given by (5.9) is applied to the system (5.7) we determine the Lyapunov constants. This is an algebraic procedure, therefore it is possible to program it with a symbolic manipulation program. The program used in this case was MAPLE. The calculations have been made independently by two authors Analysis of the Lyapunov Constants. With the analysis of the Lyapunov constants we determine the codimension of the opf Bifurcation depending on the parameters. We proceed with the L Lyapunov constant (see (5.)), once the coefficient b is substituted, L depends on, β, δ, ρ,, y. The use of two successive substitutions: substituting y with the relation II (see (3.3)), substituting δ with the relation I (see (3.2)), removes the dependence on y and δ. ence, we obtain L = ( + (β ) )ρ 2 2. (5.2) As epected it follows that L = if and only if the trace of the Jacobian A (,y ) is zero, which we had shown to be equivalent to g() = (see (5.)). The locus in the parameter space where L = has already been studied in the analysis of P (, β, δ) =. We continue with the analysis of the first Lyapunov constant L under the hypothesis L =. ence g( ) = (see (5.)), and we can substitute δ with the relation I and isolate from the factor of the numerator that can vanish in the open first quadrant. We obtain III : = + β2 2 ( 2 ). (5.3) The constant L (see (5.)) depends on, β, δ, ρ,, y. The substitution of y with the relation II (see (3.3)), and of δ with the relation I (see (3.2)) and with the relation III, removes the dependence on y, δ and. ence, this yields L = (23 β2 + (6 2 + )β + 6)ρ 8 3 (β. (5.4) + 2)(2 ) The following Lemma will be useful for the analysis of L and L 2. Lemma 5.6. For β (resp. β < ), if the system (.3) has a opf bifurcation at a point (, y ), then necessarily < 2 (resp. 2 ). Proof. If we suppose 2 then it implies that, if the numerator of L is zero and positive, then β is negative. In the case β <, if = 2 and the Jacobian trace is zero, then β = 4. It is then the case of the Bogdanov-Takens bifurcation. Proposition 5.7. If β, the opf bifurcation is of codimension and the sign of the first Lyapunov constant, L, is negative. Proof. We observe that the numerator of L (see (5.4)) is always positive which implies that the opf bifurcation is at most of codimension. By Lemma 5.6 the denominator is negative. ence L is negative. The net proposition eplores the possibility of having L and L vanishing. Proposition 5.8. If β <, L (λ) = and is equal to 2 (β) = β2 β + 6 (5.5)

20 2 Y. Lamontagne, C. Coutu & C. Rousseau then L (λ) =. Moreover, if β ( 6, 4) and = 2 (β) then there eists λ such that for λ = ( 2 (β), β, δ 2 (β), ρ) the system has a opf bifurcation of codimension at least 2. See Fig Proof. It is sufficient to determine when the numerator of L and L have a root in common. We take the resultant of the numerator of L and L with respect to, resultant(num(l ), num(l ), ) = 2(β 2 β 6) 3 (β + 4)ρ 6. If L (λ) = and is equal to β2 β+6 then L (λ) =. The case β = 4 is not a possibility since it is the Bogdanov-Takens bifurcation of codimension 3. The above proposition indicates that there eists values of the parameters such that L and L are zero simultaneously. A corollary which follows directly is that the maimum codimension of the opf bifurcation is at least 2. We analyse the second Lyapunov constant L 2 under the condition L = L =, it suffices to analyse L (2) 2 (see (5.)). Using the same simplifications as for L and using that L = we get, 2 = ((52 )β 2 + ( )β 48 5 ( ) 3 (β + 2) 2 (2 ) L (2) )ρ 48 5 ( ) 3 (β + 2) 2 (2 ). (5.6) The net proposition yields that the codimension of the opf bifurcation is at most 2. Proposition 5.9. The Lyapunov constants L, L and L 2 never vanish simultaneously for a singular point in the first quadrant. Moreover the second Lyapunov constant, L 2, is negative when L = L =. Proof. Under the hypothesis L = it is sufficient to take the resultant of the numerator of L and L 2 with respect to β, resultant(num(l ), num(l 2 ), β) = 48 ( )(2 ) 2 ( ) 7 ρ 4. Since we only consider a singular point in the open first quadrant then the only possibility of a root in common is = 2. This is ecluded by Lemma 5.6. Therefore L and L 2 never vanish simultaneously. Secondly, it remains to show that L 2 is negative. Since L and L 2 are not zero simultaneously, L 2 does not change sign when L = (by the intermediate value theorem). If β = 9 2 ( 6, 4) then for = 2 (β) and = 3 we have L = L =. If we evaluate L (2) 2 at (,, β) = ( 3, ( ) 9 2 2, 9 2 ), we obtain L(2) 2 = ρ. Therefore L 2 is negative when L = L = Submersion of the Lyapunov Constants. We show that all opf Bifurcations have a complete unfolding by showing that L and (L, L ) are submersions with respect to the parameters and δ. For the proof we will need: and (, δ) (, δ) δ (, δ) 2 δ = 2 (, δ)δ 2 (, δ) + δβ = (, δ) 2 + β (, δ) + 2 (, δ)δ 2 (, δ) + δβ (5.7) (5.8)

21 BIFURCATION ANALYSIS OF A PREDATOR-PREY SYSTEM 2 DC I L = III II DC I II III Fig Bifurcation diagram of the opf bifurcation of codimension 2. which are obtained by differentiating I (see (3.2)) with respect to and δ, respectively. Proposition 5.. The map λ = (, β, δ, ρ) L is a submersion with respect to and δ when L (λ) =. Proof. It suffices to show that L L and δ never vanish simultaneously. Using (5.7), (5.8) and substituting by the relation III (see (5.3)) we obtain L = (23 β + (8 4β)2 + (β 4) + 4)ρ 2(β + 2)(2 ) L δ = ( ) 2 (β + 2)( 3 β + 3 )ρ 4 (2 ) 3. We must verify that [ L (λ) L δ (λ)] has rank for all λ such that L (λ) =. The resultant of the numerators of L L and δ with respect to β, ( resultant num ( ) L, num δ = 2 2 ( ) 4 (2 ) 4 ρ 3, ( ) ) L, β does not vanish since, 2,, using Lemma 5.6. Proposition 5.. The map λ (L (λ), L (λ)) is a submersion with respect to and δ when L = L =. Proof. We want to show that the two vectors V i = ( L i, ) Li δ for i =,, are linearly independent at all points in the parameter space λ where L = L =. We use the following idea coming from Lemma 6 in [4]: for fied β, let (, δ ) be a point where L = L =. Proposition 5. ensures that L = is a smooth curve near (, δ ) by the implicit function theorem and that V is normal to L =. We now restrict L to L (, δ) = and we show that this map is a submersion with respect to δ when L = L =. This guarantees that V is linearly independent from V. We use the relation III (see (5.3)) to simplify L and calculate L δ. After substituting (δ) δ by the relation (5.8), we obtain L δ = 4(2 ) 4 (2 + β ) 5 [(4 ) 4 β3 + ( ) β 2 + ( )β ]( ) 2 ρ.

22 22 Y. Lamontagne, C. Coutu & C. Rousseau δ S I δ S I II III II III (a) β δ S I δ (b) 2 < β < IV I S II III II III (c) β = 2, where S = (d) 3 β < 2 (e) I (f) II (g) III (h) IV Fig. 6.. Bifurcation diagram and phase portraits for generic regions of parameters. L and L δ do not vanish simultaneously as resultant(num ( ) L, num(l ), β) δ = 26 6 (2 ) 4 ( ) 6 ρ 5 does not vanish since, 2, using Lemma 5.6. We prove the main theorem of this section. Proof of Theorem 5.. If β there is a opf bifurcation of codimension. The bifurcation is called super-critical: the singular point is stable and, when a limit cycle appears from the singular point, then the singular point becomes unstable and the limit cycle is stable. This follows from the Propositions 5.4, 5.5, 5.7, 5. and the fact that the sign of L is negative when L =. If β < there is a opf bifurcation of codimension or 2. In the case of the codimension 2, the bifurcation will have the bifurcation diagram given in Fig This follows from the Propositions 5.4, 5.5, 5.8, 5.9, 5. and the fact that the sign of L 2 is negative when L = L =.

23 BIFURCATION ANALYSIS OF A PREDATOR-PREY SYSTEM 23 δ I S O IV II V III VI δ BT + BT 3 (a) 4 < β < 3 I S O IV II V VI III BT (b) β = 4 (c) I (d) II (e) III (f) IV (g) V (h) VI Fig Bifurcation diagram and phase portraits for generic regions of parameters. 6. Bifurcation Diagram. In this section we conjecture the global topological bifurcation diagram based on the local study of the singular points. In particular we use the local bifurcation diagram near the Bogdanov-Takens bifurcations of codimension 2 and 3 which yields bifurcation surfaces of the homoclinic orbit, O, and the double limit cycle, DC, for β close to 4. Essentially we conjecture the global etension of these surfaces for other values of β. For the bifurcation diagram we do not consider the parameter ρ since we have shown that it does not have a role in the bifurcation of singular points. We want to represent the bifurcation diagram in the 3-parameter space (, β, δ). This is accomplished by taking slices with β fied where, we give the bifurcation diagram in the (, δ) plane. The values of β for the different slices are either part of a generic interval or bifurcation values. The bifurcation values are those which produce a topological change of the bifurcation diagram in the (, δ)

24 24 Y. Lamontagne, C. Coutu & C. Rousseau δ O + VII IV O 2 VIII VI 2 + I S O DC II V O δ O + VII IV O 2 VIII (a) β S 2 < β < 4 I VI + S III O DC V O II (b) β = β S 2 2 III (c) I (d) II (e) III (f) IV (g) V (h) VI (i) VII (j) VIII Fig Bifurcation diagram and phase portraits for generic regions of parameters. plane. For each slice of the bifurcation we identify the open regions in the parameter space: I, II, III,..., for each region we give a qualitative phase portrait. For the case β 3 (see Fig. 6.(a), 6.(b), 6.(c), 6.(d)) there is no intersection between the opf bifurcation,, and the surface S. This follows from the result that the intersection between the two surfaces has -coordinate given by S (β) = (β+)(β 2 +3β+) β+3. ence is always located below S or, yielding that tends to the -ais as tends to infinity. Also, we conjecture for β [ 3, 2) (see Fig. 6.(d)) that there is no homoclinic orbit, nor any limit cycle, when there are two singular points in the open first quadrant. In the case β < 3 (see Fig. 6.2(a), 6.2(b), 6.3(a), 6.3(b), 6.4(a), 6.4(b)) there is an intersection between and S, hence a limit cycle may eist when there are

25 BIFURCATION ANALYSIS OF A PREDATOR-PREY SYSTEM 25 δ O + VII IV O 2 VIII VI I + S O DC V O δ II O + VII IV O 2 IX (a) 6 < β < β S 2 I VI 2 S III + O DC VIII V O II IX III (b) β 6 (c) I (d) II (e) III (f) IV (g) V (h) VI (i) VII (j) VIII (k) IX Fig Bifurcation diagram and phase portraits for generic regions of parameters. two singular points in the open first quadrant. In addition, from the unfolding of the Bogdanov-Takens bifurcation of codimension 2 and 3, we know that there is a curve of homoclinic bifurcation for fied β near 4. We do not know at which value of β such bifurcation curves starts to eist. Near (, β, δ) = (8, 4, 4 ) the shape of the homoclinic bifurcation curve, O, is given by taking slices of Fig. 4.2 where µ 2 is constant. This yields the conjectured homoclinic bifurcation curve of Fig. 6.2, 6.3 and 6.4. We conjecture that the homoclinic bifurcation curve tends to infinity when β 3 (i.e. its left etremum point goes to infinity at β = 3). This comes from the fact that the region in which this curve may eist disappears (the region is created by the intersection of and S ). This curve was confirmed numerically for β = 3.5 in section 7 (see Fig. 7.2(a)). The case β = 4 (see Fig. 6.2(b)) is partially obtained by taking the slice µ 2 = of the Fig. 4.2.

26 26 Y. Lamontagne, C. Coutu & C. Rousseau For the cases β < 4 the dynamics are much richer. In particular we conjecture the presence of a curve DC which represents the bifurcation of a double limit cycle. We have shown that it eists in a neighborhood of 2 and also in the unfolding of the Bogdanov-Takens bifurcation of codimension 3. The curve DC starts at 2 and ends at the homoclinic bifurcation of codimension 2, O 2. We conjecture that it is always the case. The O eists in the neighborhood of the Bogdanov-Takens, near β = 4. Its etension is necessarily, negatively further from β = 4 to obtain a coherent bifurcation diagram. The main difference in the cases when β < 4 (see Fig. 5.3) is the position of 2 determined by 2 (β) = β2 β+6 (see Proposition 5.8). The bifurcation point 2 will eventually cross S at β = β 2 S and tend to infinity as β Numerical Simulations and omoclinic Bifurcation. For interesting fied values of β, numerical bifurcation diagrams are given in the (, δ) plane of the parameter space. When β > we have taken the particular value β = 6 for which we have a generic numerical bifurcation diagram. When β < the values -3.5, -4, -5 of β are chosen to calculate the homoclinic orbit before, at the Bogdanov-Takens bifurcation of codimension 2 and 3 and after. Their choice allows to confirm numerically some of the conjectures of section 6. The opf bifurcation,, and the homoclinic bifurcation, O, are calculated using XPPAUTO in which some of the AUTO routines are integrated. The curve O is calculated by taking a periodic solution of very large period and following it in the (, δ) plane. The parameter ρ has an effect on the homoclinic bifurcation curve, O, but we conjecture that it has no effect on the topological bifurcation diagram. For ease of numerical computation we used ρ =. This choice is based on the fact that, when ρ is larger, for many parameter values the vector field looks like a singular perturbation in at least one region of the first quadrant. The curves S, and are directly computed with MATLAB. The phase portraits were calculated with ode45 of MATLAB. This is done by calculating the solutions forward and backward in time for intial values located on a equally spaced grid in the first quadrant. The grid size varies for best representation of the vector field. The hollow circles on the phase portrait represent the singular points. 7.. A Typical Numerical Bifurcation Diagram for β >. The case β = 6 is a generic value and we obtain the numerical bifurcation diagram of Fig. 7.. We also give the typical phase portraits at points a, b and c in Fig. 7.. In Fig. 7.(a) and Fig. 7.(b), (, ) is a saddle point and in Fig. 7.(c), (, ) is an attracting node. The phase portrait Fig. 7.(a) (resp. Fig. 7.(b)) has a singular point in the open first quadrant which is an attracting focus (resp. repelling focus), while Fig. 7.(c) has no singular point in the open first quadrant. In addition, in Fig. 7.(b) there is an attracting limit cycle around the singular point in the open first quadrant. We notice that the numerical bifurcation diagram and numerical phase portraits confirm the results established by the theory Numerical Simulations of the omoclinic Orbit Bifurcation. The case β = 3.5 is a generic value and we obtain the numerical bifurcation diagram of Fig This diagram confirms the c shape of O, conjectured in Fig. 6.2(a), with the help of Fig Therefore, numerically, it is observed that O etends at least to β = 3.5. We conjectured that the curve will disappear towards the right (the left etremal point tends to infinity) as β tends to 3. We give a representation of

27 y y y BIFURCATION ANALYSIS OF A PREDATOR-PREY SYSTEM 27.6 S.4.2 c "..8 b.6.4 a ! (a) no limit cycle (b) one limit cycle (c) no limit cycle Fig. 7.. Numerical bifurcation diagram and phase portraits (β = 6)..24 S S O d c O ".23 " b.23.3 a ! ! (a) c shape of O (b) zoom of the diagram Fig Numerical bifurcation diagram (β = 3.5). different phase portraits in Fig. 7.3 for values a, b, c and d appearing in Fig. 7.2(b). In all the eamples of phase portraits the singular point (, ) is an attracting node and the left singular point (resp. right singular point) is a repelling focus (resp. saddle point). The eamples show the change in the gobal behavior of the system from: attracting limit cycle to homoclinic orbit, to no limit cycle, and back to a limit cycle. We also notice that some of the phase portraits are similar to those observed

28 y y y y 28 Y. Lamontagne, C. Coutu & C. Rousseau (a) one limit cycle (b) close to a homoclinic orbit (c) no limit cycle (d) close to a homoclinic orbit Fig Phase portraits (β = 3.5). in singular perturbations: we eplore this phenomenon in section 7.3. We discuss the biological interpretation in section 8. The case β = 4 is where we find the Bogdanov-Takens bifurcation of codimension 3: indeed it is observed numerically that O and meet at (, δ) = (8, 4 ). This confirms the behavior of Fig. 6.2(b) predicted by Fig. 4.2 when µ 2 =. We give a representation of some phase portraits in Fig The singular point (, ) in Fig. 7.4(c) is a saddle point and for all other cases it is an attracting node. In Fig. 7.4(a) and 7.4(c) the left singular point is an attracting focus, while in the remaining cases the left singular point is a repelling focus when it eists. In all cases the right singular point is a saddle point when it eists. The case β = 5 is more comple since there is a opf bifurcation of codimension 2, 2, as predicted by the theory. While we could not numerically compute DC which represents a double limit cycle, we give evidence that it eists locally in a neighborhood of 2. Indeed we are able to find parameter values for which there are two limit cycles around the singular point which undergoes the opf bifurcation. The eternal limit cycle is attracting and the internal limit cycle is repelling, as predicted by the theory. See Fig. 7.5(c). We had conjectured that DC begins at 2 and ends at the homoclinic bifurcation of codimension 2, O 2. The eistence of O 2 is confirmed by the numerical eistence of both a repelling homoclinic orbit and an attracting homoclinic orbit. In section 7.4 we compute the location of O 2, in particular when β = 5, its location is given in Fig See Fig. 7.5(a), where the phase portrait shows a repelling homoclinic. In all three phase portraits of Fig. 7.5 the singular points have the same type: (, ) is an attracting node, the left singular point is an attracting focus, and the right singular point is a saddle.

29 y y y y y y BIFURCATION ANALYSIS OF A PREDATOR-PREY SYSTEM a b S O.4.3 ".2. c..9 d e f ! (a) no limit cycle (b) no limit cycle (c) no limit cycle (d) no limit cycle (e) one limit cycle (f) close to a homoclinic orbit Fig Zoom of the numerical bifurcation diagramme and phase portraits (β = 4) Condition of Eistence of an omoclinic Orbit in the Limit Case of a Singular Perturbation. The system is a singular perturbation if the family of vector fields has the following form y 2 ẋ = ρ( ) 2 + β + ẏ = ɛ(,, β, δ)y (7.) where ɛ(,, β, δ) is very small for (a γ, b + γ 2 ), with γ (, a) and γ 2 is sufficiently large, where a and b are defined in Fig. 7.6(a). This implies that the vector field is nearly horizontal far from the nullcline of ẋ. If the system (7.) has a limit cycle, it will have the appearance of the limit cycle shown in Fig. 7.6(b). If the

30 y y y 3 Y. Lamontagne, C. Coutu & C. Rousseau.4.3 a S B O 2.2 " b...9 c ! (a) close to a homoclinic orbit (b) a limit cycle (c) two limit cycles Fig Zoom of the numerical bifurcation diagram and phase portraits (β = 5). y ẋ = y ẋ = y ma y min a min ma b (a) Nullcline of ẋ (b) Limit cycle Fig Vector field in the limit case of a singular perturbation. vector field is of the form (7.) then there eists a homoclinic orbit at the limit case (i.e. ɛ(,, β, δ) vanishing) when the singular point of saddle type is approimately located at (b, y min ).

31 y y BIFURCATION ANALYSIS OF A PREDATOR-PREY SYSTEM 3 Isocline = 5 Isocline = (a) Not a singular perturbation vector field (b) Singular perturbation vector field Fig Eamples of vector fields with a homoclinic orbit. β δ ρ Fig (a) (b) Table 7. Parameter values for the eamples of Fig It is possible to compute in the parameter space, the surface which represents the homoclinic orbit in the limit case of a singular perturbation. Theorem 7.. If a family of vector fields of the form (7.) has a homoclinic orbit, then at the limit (when ɛ(,, β, δ) ) the parameters satisfy (, β, δ) = where (, β, δ) = ( b b β 2 )δ 3 + (2 2 β β 6 3 β + 2β 2 3 2β 2 2 )δ 2 + ( β 2β β 4 )δ 6 2. (7.2) Proof. Let y() be the nullcline of ẋ then min is a solution of y () =. We look for the condition for which a local minimum ( min, y min ) of the nullcline ẋ has the same y coordinate as the saddle (, y ), that is d( min, ) = y( min ) y( ) =. We want to obtain a condition in the parameter space, therefore we take the resultant R of the numerator of d( min, ) and f( ) (where f is given in (2.3)) with respect to. We proceed to take the resultant of R and the numerator of y ( min ) with respect to min. We obtain 2 ρ 2 P (, β, δ) 2 (, β, δ) where P (, β, δ) is given in (5.2) and (, β, δ) given in (7.2). The factor P (, β, δ) 2 has already been studied and its solution branch does not represent a homoclinic orbit: it corresponds to the case min =. Fig. 7.7 gives two eamples with the nullcline of ẋ superposed with solution(s). Fig. 7.7(a) gives a phase portrait which is not singular, while Fig. 7.7(b) is the phase portrait of a singular perturbation vector field whose parameters satisfy the function (, β, δ) =. The parameters of both eamples are given in Tab omoclinic Orbit of Codimension 2. The eistence of a Bogdanov- Takens bifurcation of codimension 3 implies that there eists, in a neighborhood of that bifurcation, a curve in the parameter space for which there is a homoclinic bifurca-