Hopf bifurcation in the Holling-Tanner model

Transcription

1 Hopf bifurcation in the Holling-Tanner model MANUEL FALCONI UNAM Departamento de Matemáticas Ciudad Universitaria D.F. MEXICO MARTHA GARCIA UNAM Departamento de Matemáticas Ciudad Universitaria D.F. MEXICO JAUME LLIBRE UAB Departament de Matemàtiques Bellaterra Barcelona SPAIN Abstract: We provide the explicit conditions for the existence of a Hopf bifurcation in the complete Holling-Tanner model with its six parameters. Key Words: Hopf bifurcation predator-prey model Holling Tanner model 1 Introduction The existence and the number of limit cycles are important topics in the qualitative theory of differential equations and in the study of many applied mathematical models. These studies allow a better understanding of many oscillatory phenomena which appear in the nature [ ]. For predator-prey systems usually the existence of limit cycles come from the change of stability of one equilibrium point i.e. from a Hopf bifurcation. Hopf bifurcation theory is a powerful tool for studying the existence number and properties of the limit cycles in mathematical biology. In this paper we study the Hopf bifurcation for the Holling-Tanner model. This model developed by Robert May incorporates the Holling s rate [5 6] at which predators consume prey and the Leslie s rate for modeling the dynamics of the predator [8 11]. This model has been studied first for its mathematical properties and second for its efficacy for describing real ecological systems such as mite/spider mite lynx/hare sparrow/sparrow hawk etc. by Tanner [16] and Wollkind Collings and Logan [17]. The Holling-Tanner model for predator-prey interaction is ẋ = rx 1 x qxy k x + a ẏ = sy 1 y 1. γx The variables xt and yt denote the number of preys and predators respectively. The dot denote the derivative with respect to the time t. The parameters r and s are the intrinsic growth rates of prey and predator respectively. The value k is the carrying capacity of the prey and γx takes on the role of a prey-dependent carrying capacity for the predator. The parameter γ is a measure of the quality of the prey as food for the predator. The rate at which predators consume the prey qx/x + a is known as a Holling type 2 predator response [ ]. The parameter q is the maximum number of preys that can be eaten per predator per unit of time and the parameter a is a saturation value; it corresponds to the number of preys necessary to achieve one half of the maximum rate q. We note that for a biological meaning all the parameters of system 1 are positive. Our objective is to provide the explicit conditions for the existence of a Hopf bifurcation in the complete Holling-Tanner model with its six parameters. Up to now all the studies of the Hopf bifurcation for this model have been done in the so-called dimensionless models with only three parameters see for instance [ ]. These dimensionless models are obtained doing convenient rescaling of the variables x y and the time. From a mathematical point of view to work with the six parameters is not relevant it is sufficient to work with the essential three parameters of the dimensionless system. But from a biological point of view where every one of the six parameters has his meaning it is important to characterize completely the existence of a Hopf bifurcation in the six parameter space. From our point of view the best published paper on the Hopf bifurcation of the Holling-Tanner model is the one of Gasull et al. [4] in These authors show that the Holling-Tanner systems can exhibit at least two limit cycles in a Hopf bifurcation. Later on in 1999 Sáez and González Olivares [14] described curves in 3 parameter space that delimit regions of two limit cycles semistable limit cycle etc. Hsu and Hwang [15] studied the existence of at least one limit cycle in a Hopf bifurcation but without quoting the ISBN:

2 results of [4] and in 2003 Braza again studies the existence of at least one limit cycle in a Hopf bifurcation without mention the results of [4 15]. The mathematical conditions for the characterization of the existence of a Hopf bifurcation for the Holling-Tanner model in the six parameter space are well known the problem is how to organize the huge computations with the six parameters that such characterization needs. In section 4 we describe the algorithm that following it allows to characterize if system 1 in the set of parameters r [r 1 r 2 ]k [k 1 k 2 ]q [q 1 q 2 ] a [a 1 a 2 ]s [s 1 s 2 ] γ [γ 1 γ 2 ] exhibits or not a Hopf bifurcation. 2 2 Two-dimensional Hopf bifurcation In this section we summarize for the differential systems in IR 2 some basic results on the Hopf bifurcation that will be used for characterizing the Hopf bifurcation in the six parameter space of the Holling-Tanner model. For more details of all the results state in this section see Marsden and McCracken [9]. Consider an autonomous system of ordinary differential equations du dt = F u μ 3 where F :IR 2 IR IR 2 is C and μ is the bifurcation parameter. Suppose that aμ is a equilibrium point of 3 for every μ in a neighborhood U of μ =0 i.e. F aμμ=0if μ U. Assume that DF aμμ has eigenvalues of the form αμ±iβμ. Poincaré [12] Andronov and Witt [2] and Hopf [7] a translation to the English of the Hopf s original paper can be found in section 5 of [9] showed that an one parameter family of periodic orbits of 3 arises from u μ =0 0 if i DF 00 has eigenvalues ±iβ0 0 ii dα/dμ μ=0 0 and iii V 0 see 5. We say that μ =0is the value of the Hopf bifurcation. In order to describe how to compute the kind of stability of the periodic orbit Hopf periodic orbit which appears at a Hopf bifurcation we write the differential system 3 in a neighborhood of the origin of IR 2 and of μ =0as ẋ = ẏ = P x y μ =αμx + βμy+ a ij x i y j i+j=2 Qx y μ = βμx + αμy+ b ij x i y j. i+j=2 We define the number 3π 3 P V = 4 β0 x P x y Q y 3 + 3π 4β0 2 2 P x 2 3 Q x 2 y + 2 P x y + 2 Q 2 Q y 2 x y + 2 Q 2 Q x 2 x y 2 P 2 P y 2 x y + 2 P 2 Q x 2 y 2 2 P y 2 2 Q y xyμ= If V < 0 then an attracting Hopf periodic orbit exists for μ>0 sufficiently small. If V>0 then an unstable Hopf periodic orbit exists for μ<0 sufficiently small. This V is the first Liapunov constant of the weak focus of system 1 when α =0. 3 Statement of the main result In order to simplify the computations we rename the positive parameters a and k of system 1 by a 2 and k 2 with the news a and k positive. There are two equilibrium solutions in the first closed quadrant. One equilibrium point given by x = k 2 and y =0 is not of interest because it represents the extinction of the predator population and the density of the prey population equilibrating at the carrying capacity. The other equilibrium point x y given by x = a2 r k 2 r+k 2 qγ 4a 2 k 2 r 2 +a 2 r k 2 r+k 2 qγ 2 y = γ 2r a 2 r k 2 r+k 2 qγ 4a 2 k 2 r 2 +a 2 r k 2 r+k 2 qγ 2 2r depends on the parameters of system 1 except the parameter s. We change the parameter q by a new parameter d by doing q = a2 r +2adkr + k 2 r k 2. 6 γ ISBN:

3 This change simplifies the expression of the equilibrium point. Now the equilibrium point is x y = ak 1+d 2 dakγ 1+d 2 d. We translate this equilibrium point with positive coordinates to the origin by the translation 1+d x = X + ak 2 d 1+d y = Y + akγ 2 d under which system 1 becomes r Ẋ = γk 2 a 2 + adk akr X akγ a 2 d +2ak +4ad 2 k + dk 2 a 2 R +4adkR + k 2 RX + aka 2 2adk k 2 d RY + γa 2 3adk k 2 +3akRX 2 +k 2 a 2 +2adkXY + γx 3 s Ẏ = akγ 2 d RX γadk akr X akγd RY γxy + Y 2 7 where R = 1+d 2. We prefer to work with polynomial differential equations instead of rational ones so we change the old time t by the new time s doing dt = γk 2 adk akr X a 2 + adk akr X ds. The expression of differential system 7 in the new time s is X = r a 2 k 2 γd Rk 2 a 2 d R +2ak1 + 2d 2 2dRX+ a 2 k 2 d R 2 a 2 2adk k 2 Y +akγ2a 2 k 2 d R ak2+ 7d 2 10dR +3R 2 X 2 2akd Ra 2 2adk k 2 XY + a 2 2adk k 2 X 2 Y + γk 2 a 2 +4akd RX 3 γx 4 Y = a 2 k 3 sγ 2 d R a + dk krx a 2 k 3 sγd R a + dk kry ak 3 sγ 2 d RX 2 + a 2 k 2 sγxy +ak 2 a + dk krsy 2 + k 2 sγx 2 Y k 2 sxy 2 8 where the prime denotes derivative with respect to the time s. In order to have a Hopf bifurcation we need that the characteristic polynomial of the Jacobian matrix at the origin of system 8 have complex eigenvalues. We control these eigenvalues by changing conveniently the parameters r and s by the new parameters α and β as follows α = β = a 2 k 2 dr 2akr 4ad 2 kr aks + dk 2 s + 1+d 2 4adkr+ k 2 a 2 r k 2 s 1 + 2d 2 k 4 + a 4 +4adk3+ 4d 2 k 2 a 2 +2a d d 4 k 2 r 2 2ka 3 d + a 2 k 2a 2 d 2 k 3adk 2 + k 3 +2d 2 k 3 rs +k 2 s 2 a 2 2adk + k 2 + 2d 2 k 2 2a 2 4adk k 2 a 2 k 2 d 2ak 4ad 2 kr 2 ka 3 2a 2 dk 3ak 2 +2dk 3 rs k 3 a dks 2 1+d 2. 9 Theorem 1. Consider differential system 8 defined in IR + 2 with the parameter values a>0 d>0γ > 0 k>0 β>0 and α. Then in a neighborhood U sufficiently small of α =0the following statements hold. a All the initial parameters of system 1 are positive. b The origin is an equilibrium point of system 8 for all α U and this system has a Hopf bifurcation at this equilibrium point when α =0. Proof. The eigenvalues of the Jacobian matrix of system 8 at the origin are given by λ 12 = 1 2 a2 k 2 R dγα ± βi. We solve 9 for r and s and we take the positive solution akαρ + σ r = 16a1 + d 2 k 1 a 4 4a 3 dk 6a 2 k 2 +4adk 3 + k 4 s = d Ra 3 4a 2 dk 3ak 2 a 2 ka 2 k k 3 σ akα161+ d 2 a dka 4 4a 3 dk 6a 2 k 2 + 4adk 3 + k 4 +a 2 k a 3 d +4a 2 d 2 k +3adk 2 + k 3 ρ 16k1 + d 2 a 4 4a 3 dk 6a 2 k 2 +4adk 3 + k 4 R+ aρra 2 4adk 3k 2 / 16ak d 2 a 2 2adk k 2 a 4 4a 3 dk 6a 2 k 2 +4adk 3 + k 4 10 ISBN:

4 where ρ = σ = 81+d 2 k 2 a 2 d +2ak +4ad 2 k+ k 2 a 2 +4adkR 64ak1 + d 2 2 a 2 4adk k 2 a 2 2adk k 2 a 2 k 2 d 2ak +2d 2 k α 2 a 4 4a 3 dk 6a 2 k 2 +4adk 3 + k 4 a 2 d dk 2 ak +4d 2 kβ 2 32akR 1 + d 2 a 2 2adk k 2 a 4 +2a 4 d 2 12a 3 dk 16a 3 d 3 k +2a 2 k 2 +28a 2 d 2 k a 2 d 4 k 2 +12adk 3 +16ad 3 k 3 + k 4 + 2d 2 k 4 α 2 +a 2 +2a 2 d 2 6adk 8ad 3 k k 2 2d 2 k 2 a 4 4a 3 dk 6a 2 k 2 +4adk 3 + k 4 β 2. We remark that the change of parameter from r and s to α and β is well defined if the radical inside the square root that appears in the change is non negative. By construction if such a change is not well defined then the conditions for having a Hopf bifurcation are not satisfied. So statement a is proved. Now we do a linear change of variables which writes the linear part of the differential system 8 in its real Jordan normal form. Under the transformation where with u v A = = A X Y a1 1 a 2 0 a 1 = 2γd1 + d 2 a 4 +4a 3 dk +6a 2 k 2 4adk 3 k 4 a 2 +2adk + k 2 α 2 + ak +4ad 2 k + dk 2 a 2 β 2 +γa 4 4a 3 dk 6a 2 k 2 +4adk 3 + k d 2 a 2 2adk k 2 α d 2 a 2 k 2 2ad3 + 4d 2 kβ 2 R + αγ 2d3+ 4d 2 a 2 k 2 2ak1 + 8d 2 +8d 4 Δ αγ d 2 a 2 k 2 8adk 1 + 2d 2 Rxx/ a 2 2adk k 2 d Ra 4 6a 2 k 2 +4adkk 2 a 2 R +2d1 + d 2 drα 2 + β 2 a 2 = 2aαβγdk1 + d d 2 a 4 6a 2 k 2 + 4adkk 2 a 2 +k 4 + 2βγ1 + 8d 2 + 8d 4 a 2 k 2 2ad5 + 20d 2 +16d 4 kδ +2aαβγk1 + d d 2 a 4 4a 3 dk 6a 2 k 2 +4adk 3 + k 4 R 2 2βγ 2d +2d 3 a 2 k 2 ak1 + 12d 2 +16d 4 ΔR and Δ= a1 + d 2 2 k2α 2 a 2 4adk k 2 a 2 2adk k 2 a 2 k 2 d 2ak 4ad 2 k+a 2 d ak 4ad 2 k dk 2 a 4 4a 3 dk 6a 2 k 2 +4adk 3 + k 4 β 2 Ra 2 2adk k d 2 k 4 + a 4 + 4adk3 + 4d 2 k 2 a 2 +2a 2 k d 2 +16d 4 α 2 + a 2 k d 2 6adk 8ad 3 k a 4 4a 3 dk 6a 2 k 2 +4adk 3 + k 4 β 2 system 8 becomes u = AMA 1 u v v p2 u v + q 2 u v 11 where M is the matrix of the linear part of system 8 at the origin and p 2 and q 2 start with second order terms in u and v. We do not write this system explicitly because its complete expression should need several pages. Since AMA 1 = 1 2 a2 k 2 α β R d β α system 11 has the form of system 4 of section 2. Now we need to compute the first Liapunov constant V at the origin for the differential system 11 when α = 0 and to consider that it is not zero. After a tedious computation we obtain the number V. We do not provide the expression of V because it is very large. From the summary of section 2 when V is different from zero we have a Hopf bifurcation at the origin. V = 0 is 4 dimensional hypersurface in the 5 dimensional parameter space P = {a d k γ β a > 0k > 0γ > 0 } wich separates the parameter space in two regions according to the kind of stability of the Hopf periodic orbit. Consequently statement b is proved. 4 The algorithm In this section we describe an algorithm for characterize the existence or not of a Hopf bifurcation in a given system 1 with parameters a k q r s γ satisfying 2. First using the expression 6 we compute d = da k q r γ. After from 9 we can determine α = αa d k r s and β = βa d k r s and finally we can compute V = V a d k β γ. Summarizing the results of the previous section we have ISBN:

5 Theorem 2. System 1 satisfying 2 has a Hopf bifurcation if the following four conditions hold: σ 0 see 10 α 0 V 0and α =0for some values a d k r s satisfying 2. Acknowledgements: The second author is partially supported by the grants CONACYT and SMM-Sofía Kovalévskaia foundation. The third author is partially supported by the grants MEC/FEDER MTM CIRIT 2009SGR 410 and ICREA Academia. References: [1] F. Albrecht H. Gatzke and N. Wax Stable limit cycles in prey predator populations Science pp [2] A.A. Andronov and A. Witt Sur la théorie mathematiques des autooscillations.c. R. Acad. Sci. Paris pp [3] P.A. Braza The bifurcation structure of the Holling-Tanner model for predator-prey interactions using two-timing SIAM J. Appl. Math pp [4] A. Gasull R.E. Kooij and J. Torregrosa Limit cycles in the Holling-Tanner model Publicacions Matemàtiques pp [5] M.P. Hassell The dynamics of Arthropod Predator-Prey Systems Princeton University Press Princenton NJ [6] C.S. Holling The functional response of invertebrate predators to prey density Mem. Ent. Soc. Can pp [7] E. Hopf Abzwigung einer periodischen Losung von einer stationaren Losung eines Differentialsystems Ber. Math. Phys. Sachsische Adademie der Wissenschaften Leipzig pp [8] P.H. Leslie Some Further Notes on the Use of Matrices in Population Mathematics Biometrica pp [9] J.E. Marsden and M. McCracken The Hopf bifurcation and its applications Springer-Verlag New York [10] R.M. May Limit cycles in predator-prey communities Science pp [11] R.M. May Stability and Complexity in Model Ecosystems Princeton University Press Princenton NJ [12] H. Poincaré Les Méthodes Nouvelles de la Mécanique Céleste 1 Paris [13] M.L. Rosenzweig Paradox of enrichement: Destabilization of exploitation ecosystems in ecological time Science pp [14] E. Sáez and E. González Olivares Dynamics of a predator-prey model SIAM J. Appl. Math pp [15] Sze Bi Hsu and Tzy Wei Hwang Hopf bifurcation analysis for a predator-prey system of Holling and Leslie type Taiwanese J. Math pp [16] J.T. Tanner The stability and the intrinsic growth rates of prey and predator populations Ecology pp [17] D.J. Wollkind J.B. Collings and J.A. Logan Metastability in a temperature-dependent model system for predator prey mite outbreak interactions on fruit flies Bull. Math. Biol pp ISBN: