PHASE PORTRAITS OF QUADRATIC LOTKA–VOLTERRA SYSTEMS WITH A DARBOUX INVARIANT IN THE POINCARÉ DISC YUDY BOLAÑOS 1 , JAUME LLIBRE 1 AND CLAUDIA VALLS 2 Abstract. We characterize the global phase portraits in the Poincaré disc of all the planar Lotka–Volterra quadratic polynomial differential systems having a Darboux invariant. 1. Introduction and statement of main results Let R[x, y ] (resp. C[x, y ]) be the ring of the polynomials in the variables x and y with coefficients in R (resp. C). We consider a system of polynomial differential equations or simply a polynomial differential system in R 2 defined by (1) ˙ x = P (x, y ), ˙ y = Q(x, y ), where P,Q ∈ R[x, y ] and the dot denotes derivative with respect to the independent variable t usually called the time. We say that the maximum of the degrees of the polynomials P and Q is the degree of system (1). Usually a quadratic polynomial differential system of degree 2 is called simply a quadratic system. Sometimes we shall talk about the quadratic vector field (2) X = P (x, y ) ∂ ∂x + Q(x, y ) ∂ ∂y . associated to the quadratic system (1). A non–locally constant real function H of class C 1 defined on an open set U is a first integral of the polynomial vector field X if H (x(t),y (t)) is constant for all value of t for which the solution (x(t),y (t)) of X is defined on U . We observe that H is a first integral of X if and only if X H =0 on U . Let f ∈ C[x, y ] \{0}. The algebraic curve f (x, y )=0 is an invariant algebraic curve of the polynomial system (1) if for some polynomial k ∈ C[x, y ] we have X f = P ∂f ∂x + Q ∂f ∂y = kf. The polynomial k is called the cofactor of the invariant algebraic curve f =0. Let g,h ∈ C[x, y ] and assume that g and h are relatively prime in the ring C[x, y ] or that h =1. Then the function exp(g/h) is called an exponential factor of system (1) if for some polynomial k ∈ C[x, y ] of degree at most 1 we have that X  exp  g h  = k exp  g h  . 2010 Mathematics Subject Classification. 34A34, 34C05, 34C14. Key words and phrases. Phase portrait, quadratic Lotka–Volterra systems, Darboux invariant, Poincaré compactification, Poincaré disc. 1 This is a preprint of: “Phase portraits of quadratic Lotka-Volterra systems with a Darboux invari- ant in the Poincar´ e disc”, Yudi Marcela Bola˜ nos Rivera, Jaume Llibre, Cl` audia Valls, Commun. Contemp. Math., vol. 16(6), 1350041 (23 pages), 2014. DOI: [10.1142/S0219199713500417]