This is a preprint of: “Darboux integrability and dynamics of the Basener-Ross population model”, Faruk Güngör, Jaume Llibre, Chara Pantazi, Rend. Circ. Mat. Palermo (2), 1–13, 2020. DOI: [10.1007/s12215-020-00507-4] DARBOUX INTEGRABILITY AND DYNAMICS OF THE BASENER-ROSS POPULATION MODEL FARUK GÜNGÖR1 , JAUME LLIBRE2 AND CHARA PANTAZI3 Abstract. We deal with the Basener and Ross model for the evolution of human population in Easter island. We study the Darboux integrability of this model and characterize all its global dynamics in the Poincaré disc, obtaining 15 different topological phase portraits. 1. Introduction and statement of the main results In order to explore the evolution of ecosystems an isolated island is a good labo- ratory due to the total absence of external distortion factors like migration. Basener and Ross, see equation (5) in [3], proposed the following model for the evolution of human population in Easter island c (1) ẋ = x(1 − y), ẏ = (h − 1)y 2 + (1 − c)y + x, k where c, k, h are positive constants, c is the growth rate of the recourses, k the carrying capacity of the population, h is the harvesting constant, y(t) is the quotient between the amount of resources x(t) and the human population in the island at time t. It is known that the population of Easter Island grew regularly for some time and then diminished very rapidly; to the extent that humans almost disappeared from the island. Basener and Ross [3] provided the mathematical model (1) that allowed to explain this type of behavior. With respect to other predator-prey models the Basener–Ross model shows a rich variety of dynamical behaviors, allowing the extinction in finite time. Therefore it has been considered as an acceptable model for the evolution of population in ancient civilizations and some generalizations have been done and studied, for instance see [2, 4, 5, 14] and the references therein. On the other hand, the Basener–Ross model has been extended considering that its ecological parameters can change with the time, see for instance [1, 11, 12, 13, 24, 25]. Nucci and Sanchini [19] applied the Lie group theory to system (1) and proved that this system can be integrated by quadrature for some values of the parameters. They also provided a comparison analysis with the qualitative study given by Basener and Ross. We have two objectives, first to study the Darboux first integrals of system (1), and second to characterize all the phase portraits of the differential system (1), thus 2010 Mathematics Subject Classification. Primary: 34C05 34C23. Key words and phrases. Basener-Ross population model, quadratic system, Poincaré disc, Darboux integrability, Darboux invariant. 1 2 F. GÜNGÖR, J. LLIBRE AND C. PANTAZI completing the initial qualitative analysis done by Basener and Ross. More precisely, we will classify all the phase portraits of the differential system (1) in the Poincaré disc. Thus, in particular we control all the orbits which come or go to infinity, which never were studied previously for system (1). By scaling the time and the x coordinate we can reduce the study to the values of the parameters k = c, and the analysis of the differential system (1) is reduced to study the differential system ẋ = x(1 − y), ẏ = (h − 1)y 2 + (1 − c)y + x. Additionally, setting b = h − 1 we have (2) ẋ = x(1 − y) = P (x, y), ẏ = by 2 + (1 − c)y + x = Q(x, y), with b > −1 and c > 0. Darboux [38] showed how can be constructed the first integrals of planar polynomial vector fields possessing sufficient invariant algebraic curves. System (2) is integrable on an open subset U of R2 if there exists a nonconstant analytic function H : U → R, called a first integral of the system on U, which is constant on all solution curves (x(t), y(t)) of system (2) contained in U. We say that an analytic function H(x, y, t) : U × R → R is an invariant of system (2) on U, if H(x, y, t) = constant for all values of t for which the solution (x(t), y(t)) is defined and contained in U. If an invariant H is independent of t then, of course, it is a first integral. The knowledge provided by an invariant is weaker than the one provided by a first integral. The invariant, in general, only gives information about either the α– or the ω–limit set of the orbits of the system (see for instance [16]), while the level curves of a first integral contain the orbits of the system. Let f = f (x, y) be a real polynomial in the variables x and y. The algebraic curve f (x, y) = 0 is an invariant algebraic curve of system (2) if for some polynomial K = K(x, y) we have ∂f ∂f (3) P +Q = Kf. ∂x ∂y The polynomial K is called the cofactor of the invariant algebraic curve f = 0. We note that since the polynomial system has degree m, then any cofactor has at most degree m − 1. Since on the points of the algebraic curve f = 0 the gradient (∂f /∂x, ∂f /∂y) of the curve is orthogonal to the vector field (P, Q) associated to system (2), the vector field (P, Q) is tangent to the curve f = 0 at every point of this curve. Hence, the curve f = 0 is formed by orbits of system (2). This justifies the name of invariant algebraic curve given to the algebraic curve f = 0 satisfying (3) for some polynomial K, because it is invariant under the flow defined by system (2). In the next theorem we summarize the basic results on the Darboux theory of integrability that we shall use in this paper, for a proof see [9] or Chapter 8 of [10]. Theorem 1. Suppose that a polynomial system (4) ẋ = P (x, y), ẏ = Q(x, y), INTEGRABILITY AND DYNAMICS OF THE BASENER-ROSS POPULATION MODEL 3 admits p irreducible invariant algebraic curves fi = 0 with cofactors Ki for i = 1, . . . , p. P p (a) There exist λi ’s in R not all zero such that λi Ki = 0, if and only if the i=1 function (5) f1λ1 · · · fpλp is a first integral of system (4). P p (b) There exist λi ’s in R not all zero such that λi Ki = −s for some s ∈ R\{0}, i=1 if and only if the function (6) f1λ1 · · · fpλp est is an invariant of system (4). The first integrals of the form (5) are called the Darboux first integrals, and the invariants of the form (6) are called the Darboux invariants. Our main result on the Darboux integrability of the Basener–Ross differential sys- tem (2) is the following. Theorem 2. The following statements hold. (a) For c = 2b + 1 system (2) has the Darboux first integral  (7) H(x, y) = x2b (2 b + 1) y 2 − 2 (2 b + 1) y + 2 x . (b) For c = 1/2 and b = −1/4 (note that c = 2b + 1) system (2) has the previous Darboux first integral and the Darboux invariants (8) I1 = x−1/2 (y 2 + 4x) exp(−t/2), I2 = x−1/2 [(y − 2)2 + 4x] exp(t/2). (c) For b = (1 − c)/(2c − 3) system (2) has the Darboux invariant   2(1−c)/(2c−3)  2  2(1 − c) (9) I3 = x (y − (3 − 2c)) + 2(3 − 2c)x exp t . 3 − 2c Theorem 2 is proved in Section 2. In this paper all the phase portraits are drawn in the so called Poincaré disc, which roughly speaking, is the closed disc centered at the origin of coordinates and of radius one. The interior of this disc is identified with R2 . Its boundary, the circle S1 , is identified with the infinity of R2 . In the plane R2 we can go to infinity in as many directions as points in the circle S1 . For more details about the Poincaré disc and the coordinates for studying the polynomial differential system (2) in it, see Appendix 1. Theorem 3. The Basener–Ross differential system (2) has 15 non-topological equiv- alent phase portraits in the Poincaré disc (see Figures 1, 2 and 3 and the proof of this theorem). Theorem 3 is proved in Section 3. 4 F. GÜNGÖR, J. LLIBRE AND C. PANTAZI . r1 . L1 . r2 . . .. .. . . .. .. . . . . . . S=19, R=6 L2 . S=17, R=6 r3 . S=19, R=6 L3 . .. .. . . .. . .. . . . . . . S=16, R=5 r4 . S=18, R=5 L4 . S=18, R=5 r5 .. .. .. . . .. . .. . . . S=18, R=5 . S=18, R=5 . S=18, R=5 Figure 1. Global phase portraits of system (2). 2. Darboux integrability In this section we prove Theorem 2. We denote by ∂ ∂ (10) X = x(1 − y) + (by 2 + (1 − c)y + x) ∂x ∂y the vector field defined by system (2) . The straight line f1 = x = 0 is an invariant algebraic curve of the vector field (10) with cofactor K1 = 1 − y. First consider c = 2b + 1 6= 0. Then system (2) admits the invariant algebraic curve 2b + 1 2 f2 = y − (2b + 1)y + x = 0, 2 INTEGRABILITY AND DYNAMICS OF THE BASENER-ROSS POPULATION MODEL 5 . L5 . r6 . L6 .. .. . . . .. . .. . . . . . . . S=18, R=5 S=18, R=5 S=14,R=3 r7 . q 2 L7 . . . . . .. . .. . . .. . .. . S=16, R=3 r8 . S=14, R=3 L8 . S=18,R=5 L 13 . . . . . .. . .. . . . . S=18, R=5 . S=16, R=5 . S=14, R=3 Figure 2. Global phase portraits of systems (2). with cofactor K2 = 2b(y − 1). Note that 2bK1 + K2 = 0, so from statement (a) of Theorem 1 we have that for c = 2b + 1 system (2) admits the Darboux first integral H = f12b f2 given in (7). Therefore statement (a) of Theorem 2 is proved. Now we consider the special choice of the parameters c = 1/2 and b = −1/4. Note that still relation c = 2b + 1 holds, and consequently system (2) has the Darboux first integral (7) for these values of the parameters. Furthermore, f3 = y 2 + 4x = 0 and f4 = (y − 2)2 + 4x = 0 are invariant algebraic curves of system (2) with cofactors K3 = (2 − y)/2 and K4 = −y/2, respectively. Since 1 1 1 1 − K 1 + K3 = and − K1 + K4 = − , 2 2 2 2 from statement (b) of Theorem 1 it follows that I1 and I2 are Darboux invariants of system (2). This completes the proof of statement (b) of Theorem 2. 6 F. GÜNGÖR, J. LLIBRE AND C. PANTAZI . . r12 . L12 . . L11 . . . . . . . .. . . . . . . . S=15, R=4 r11 . S=17, R=4 L10 . S=17, R=4 r10 . . .. . . . .. . .. . . . . . . S=17, R=4 L9 . S=16, R=5 r9 .. S=19, R=6 q1 .. .. .. . . .. . .. .. . .. . S=18,R=5 . S=18,R=5 .. Figure 3. Global phase portraits of system (2). Finally we consider system (2) with b = (1 − c)/(2c − 3). Then f5 = 2(3 − 2c)x + (−3 + 2c + y)2 = 0 is an invariant algebraic curve of this system with cofactor K5 = 2(c − 1)y/(3 − 2c). Since 2(1 − c) 2(c − 1) K1 + K5 = . 2c − 3 3 − 2c Again from statement (b) of Theorem 1 we get that I3 is a Darboux invariant of system (2). So statement (c) of Theorem 2 is proved. 3. The global phase portraits in the Poincaré disc of system (2) In order to present the global phase portraits of the two-parametric family (2) we first study its finite singular points, in Subsection 3.1 and after the infinite singular points in Subsection 3.2. INTEGRABILITY AND DYNAMICS OF THE BASENER-ROSS POPULATION MODEL 7 3.1. The Finite singular points of system (2). System (2) has the following finite singular points whenever they are defined (i.e. if b 6= 0) (11) P0 = (0, 0), P1 = (0, (c − 1)/b), P2 = (c − b − 1, 1). The origin P0 has eigenvalues 1 and 1 − c. So P0 is a hyperbolic unstable node for c < 1 and a hyperbolic saddle for c > 1. For c = 1 the origin P0 collapses to the point P1 and becomes a semi–hyperbolic saddle node, see Theorem 2.19 of [10]. Hence we consider the bifurcation curve g1 = c − 1 = 0. The point P1 is defined for b 6= 0 and has eigenvalues c − 1 and (b − c + 1)/b. We consider the bifurcation curves g0 = b = 0, and g2 = b − c + 1 = 0. Note that for our study we always have g0 > −1. Then for b > 0, g1 > 0 and g2 > 0 the point P1 is a hyperbolic unstable node, and for b > 0, g1 < 0 and g2 > 0 it is a hyperbolic saddle. Then for b < 0, g1 > 0 and g2 > 0 it is a hyperbolic saddle, and for b < 0, g1 < 0 and g2 > 0 it is a hyperbolic stable node. Then for b > 0, g1 > 0 and g2 < 0 it is a hyperbolic saddle, and for b > 0, g1 < 0 and g2 < 0 it is a hyperbolic stable node. Then for b < 0, g1 > 0 and g2 < 0 it is a hyperbolic unstable node, and for b < 0, g1 < 0 and g2 < 0 it is a hyperbolic saddle. For c = b + 1 we have that g2 = 0 and the point P1 collapses to the point P2 and it becomes a semi–hyperbolic saddle–node, see again Theorem 2.19 of [10]. The point P2 has the eigenvalues √ 2b − c + 1 D1 λ± = ± , 2 2 with D1 = c2 − 4 cb + 4 b2 − 6 c + 8 b + 5 = 0 a parabola in the (b, c)-plane and it is a bifurcation curve. Note that λ− λ+ = c − b − 1 = −g2 . First consider D1 ≥ 0. Then for g2 < 0 the point P2 is a hyperbolic node, whereas for g2 > 0 is a hyperbolic saddle. For g2 = 0 we recall that the point P2 collapses to the point P1 and it is a semi–hyperbolic saddle node. Now we consider D1 < 0. Additionally we must consider the curve g3 = 2b−c + 1 = 0. For g3 > 0 the point P2 is a hyperbolic unstable focus, whereas for g3 < 0 we have that P2 is a hyperbolic stable focus. For g3 = 0 the point P2 could be a focus or a center. Since for c = 2b + 1 6= 0 there is the first integral (7) defined at P2 , it follows that P2 is a center. 3.2. The infinite singular points of system (2). In order to describe the global phase portraits of system (2) in the Poincaré disc we must study the infinite singular points, see Appendix 1. System (2) in the chart (U1 , F1 ) is written z˙1 = z2 + (b + 1)z12 − cz1 z2 , (12) z˙2 = z2 (z1 − z2 ) , and the origin (0, 0) of (U1 , F1 ) is a singular point. Note that the linear part of system (12) at the origin is not identically zero but its two eigenvalues are equal to zero. So the origin is a nilpotent singular point. Applying Theorem 3.5 of [10] we have that 8 F. GÜNGÖR, J. LLIBRE AND C. PANTAZI b g1=0 g 2=0 g 3 =0 3 4 2 5 L3 L2 1 L4 L1 D1 =0 L5 6 g 0=0 L12 L13 L6 q1 q2 12 L11 8 L8 7 11 9 L7 L10 L9 10 g4=0 Figure 4. The bifurcation diagram of system (2) with c > 0 and b > −1. for b > −1 the origin (0, 0) of the chart (U1 , F1 ) is the union of one hyperbolic and one elliptic sector. System (2) in the chart (U2 , F2 ) is z˙1 = z1 (cz2 − z1 z2 − b − 1) , (13) z˙2 = z2 (cz2 − z1 z2 − z2 − b) , and the origin is the only singular point with eigenvalues −b − 1, −b. We consider the bifurcation curve g4 = b + 1 = 0. Since b > −1 we have that the origin of the chart (U2 , F2 ) is a hyperbolic saddle for −1 < b < 0, and for b > 0 is a hyperbolic stable node. For b = 0 it is a semi–hyperbolic saddle–node. 3.3. The topological classification of the global phase portraits of system (2). From Subsections 3.1 and 3.2 we obtain the bifurcation diagram of system (2) given in Figure 4. In the bifurcation diagram of Figure 4 we have twelve regions r1 , . . . , r12 , the thir- teen lines L1 , . . . , L13 , and two points q1 = (1, 0) and q2 = (5, 0). INTEGRABILITY AND DYNAMICS OF THE BASENER-ROSS POPULATION MODEL 9 Using that the straight line x = 0 of system (2) is invariant, how is the flow of system (2) on y = 0, and the local phase portraits of the finite and infinite singular points of system (2) we obtain the phase portraits of system (2) described in the Figures 1, 2 and 3 according with the twelve regions, the thirteen lines and the two points. In these figures we denote by S the number of separatrices and by R the number of the canonical regions, see for more details Appendix 2. In Figures 1, 2 and 3 are claimed that the phase portraits of system (2) has no limit cycles. Now we prove the claim. First it is known that if a quadratic polyno- mial differential system has a limit cycle this limit cycle must surround a focus, see Theorem 6 of [7]. Also it is known that if a quadratic polynomial differential system has an invariant straight line it has at most one limit cycle, which must be stable or unstable. The first proof of this result is due to Ryckov [23], a more clear proofs appear later on in [6, 8]. From the phase portraits of Figures 1, 2 and 3 it follows that if there is a focus surrounded by a limit cycle, this limit cycle must be semistable, but since system (2) has the invariant straight line x = 0 if it has a limit cycle this must be unique and either stable or unstable, hence system (2) cannot have limit cycles. The claim is proved. Proof of Theorem 3. We recall that two global phase portraits are topological equiv- alent if and only if does exist a homeomorphism to bring the separatrix configuration of one phase portrait into the separatrix configuration of the other, see Theorem 5 in Appendix 2. We distinguish the following 15 topologically different phase portraits in the Poincaré disc using the mentioned Theorem 5. Case S = 14, R = 3. We have that the phase portraits of the lines L6 and L13 , and of the point q2 are topological equivalents, and we simply write L6 = L13 = q2 . Case S = 15, R = 4. We obtain the unique phase portrait L12 . Case S = 16, R = 3. There is only the phase portrait r7 . Case S = 16, R = 5. We have three different topological phase portraits L2 , L8 and L10 . Case S = 17, R = 4. We obtain a unique phase portrait r12 = L11 = r11 . Case S = 17, R = 6. There is only the phase portrait L1 . Case S = 18, R = 5. We have four different topological phase portraits r3 = r4 = L3 , r5 = r6 = L5 , r8 = r9 = L9 = L7 and L4 . Case S = 19, R = 6. We obtain two different topological phase portraits r1 = r2 and r10 . Case q1 . This completes the proof of the theorem.  10 F. GÜNGÖR, J. LLIBRE AND C. PANTAZI 4. Appendix 1: Poincaré compactification We consider the quadratic polynomial differential system (2) and its corresponding vector field X = (P, Q). We want to obtain the global phase portrait of system (2), and consequently we need to control the orbits that come from or escape to infinity. For this reason we use the so called Poincaré compactification, see Chapter 5 of [10]. Let R2 be the plane in R3 defined by (y1 , y2 , y3) = (x1 , x2 , 1). Consider the Poincaré sphere S2 = {y = (y1 , y2, y3 ) ∈ R3 : y12 + y22 + y32 = 1} and we denote by T(0,0,1) S2 the tangent space to S2 at the point (0, 0, 1) (see also [21]). We consider the central projection f : T(0,0,1) : R2 → S2 . Note that f defines two copies of X , one in the northern hemisphere {y ∈ S2 : y3 > 0} and the other in the southern hemisphere. Now set Xˆ = Df ◦ X . We observe that X̂ is defined on S2 except on its equator S1 . Hence, the points at infinity of R2 are in bijective correspondence with S1 = {y ∈ S2 : y3 = 0}, (the equator of S2 ). So according to this construction S1 is identified to be the infinity of R2 . The resulting Poincaré compactified vector field p(X ) of X will be an analytic vector field induced on S2 as follows: First we multiply X̂ by the factor y32, and so the vector field y32Xˆ is defined in the whole S2 . Additionally, on S2 \ S1 there are two symmetric copies of X and note that the behavior of p(X ) around S1 gives the behavior of X near the infinity. Then the Poincaré disc D2 is the projection of the closed northern hemisphere of S2 on y3 = 0 under (y1 , y2 , y3 ) 7−→ (y1, y2 ). Since S2 is a differentiable manifold, we can consider the six local charts Ui = {y ∈ S2 : yi > 0}, and Vi = {y ∈ S2 : yi < 0} for i = 1, 2, 3 with the diffeomorphisms Fi : Ui −→ R2 and Gi : Vi −→ R2 , which are the inverses of the central projections from the planes tangent at the points (1, 0, 0), (−1, 0, 0), (0, 1, 0), (0, −1, 0), (0, 0, 1) and (0, 0, −1) respectively. We set z = (z1 , z2 ) to be the value of Fi (y) or Gi (y) for any i = 1, 2, 3. Hence, the expressions of the compactified vector field p(X ) of X are ! 1 z  1 z  1 z  1 1 1 z22 ∆(z) Q , − z1 P , , −z2 P , in U1 , z2 z2 z2 z2 z2 z2 ! z 1  z 1  z 1  1 1 1 z22 ∆(z) P , − z1 Q , , −z2 Q , in U2 , z2 z2 z2 z2 z2 z2  ∆(z) P (z1 , z2 ), Q(z1 , z2 ) in U3 , 1 where ∆(z) = (z12 + z22 + 1)− 2 . The expressions of the vector field p(X ) in the local chart Vi is the same as in the chart Ui multiplied by the factor −1. In these coordinates z2 = 0 denotes the points of S1 . Usually, we omit the factor ∆(z) by rescaling the vector field p(X ), and so we obtain a polynomial vector field in each local chart. Also note that the infinity S1 is invariant with respect to p(X ). Two polynomial vector fields X and Y on R2 are topologically equivalent if there exists a homeomorphism on S2 preserving the infinity S1 carrying orbits of the flow induced by p(X ) into orbits of the flow induced by p(Y). Note that the homeomor- phism should preserve or reverse simultaneously the sense of all orbits of the two compactified vector fields p(X ) and p(Y). INTEGRABILITY AND DYNAMICS OF THE BASENER-ROSS POPULATION MODEL 11 5. Appendix 2: Separatrix configuration In order to proceed with the topological classification of the global phase portraits of system (2) we need to consider the definition of parallel flows. We use the definition given by Markus [17] and Neumann in [18]. Let φ be a C k local flow on the two dimensional manifold R2 or R2 \ {0}. Here k is either a positive integer, or ∞ (smooth), or ω (analytic). The flow (M, φ) is C k parallel if it is C k -equivalent to one of the following ones: strip: (R2 , φ) with the flow φ defined by ẋ = 1, ẏ = 0; annular: (R2 \ {0}, φ) with the flow φ defined (in polar coordinates) by ṙ = 0, θ̇ = 1; spiral: (R2 \ {0}, φ) with the flow φ defined by ṙ = r, θ̇ = 1. The separatrices of the vector field p(X ) in the Poincaré disc D are (i) all the orbits of p(X ) which are in the boundary S1 of the Poincaré disc (recall that S1 is the infinity of R2 ); (ii) all the finite singular points of p(X ); (iii) all the limit cycles of p(X ); and (iv) all the separatrices of the hyperbolic sectors of the finite and infinite singular points of p(X ). We denote by Σ the union of all separatrices of the flow (D, φ) defined by the compactified vector field p(X ) in the Poincaré disc D. Note that Σ is a closed invariant subset of D. Every open connected component of D \ Σ, with the restricted flow, is called a canonical region of φ. For a proof of the following result see [15, 18]. Theorem 4. Let φ be a C k flow in the Poincaré disc with finitely many separatri- ces, and let Σ be the union of all its separatrices. Then the flow restricted to every canonical region is C k parallel. The separatrix configuration Σc of a flow (D, φ) is the union of all the separatrices Σ of the flow together with an orbit belonging to each canonical region. The sepa- ratrix configuration Σc of the flow (D, φ) is said to be topologically equivalent to the separatrix configuration Σ̃c of the flow (D, φ̃) if there exists a homeomorphism from Σc to Σ̃c which transforms orbits of Σc into orbits of Σ̃c , and orbits of Σ into orbits of Σ̃. For a proof of the next Theorem see [17, 18, 20]. Theorem 5. Let (D, φ) and (D, φ̃) be two compactified Poincaré flows with finitely many separatrices coming from two polynomial vector fields (2). Then they are topo- logically equivalent if and only if their separatrix configurations are topologically equiv- alent. Note that from Theorem 5 in order to classify the phase portraits in the Poincaré disc of a planar polynomial differential system having finitely many separatrices, it is enough to describe their separatrix configuration. 12 F. GÜNGÖR, J. LLIBRE AND C. PANTAZI References [1] P. Amarasekare, Effects of temperature on consumer-resource interactions, J. Anim. Ecol. 84 (2015), 665–679. [2] W. Basener, B. Brooks, M. Radin and T. Wiandt, Rat instigated human population collapse on Easter Island, Nonlinear Dyn. Psychol. Life Sci. 12(3) (2008), 227–240. [3] B. Basener and D.S. Ross, Booming and crashing populations and Easter Island, SIAM J. Appl. Math. 65 (2004), 684–701. [4] M. Bologna and J.C. Flores, A simple mathematical model of society collapse applied to Easter Island, Europhys. Lett. EPL 81 (2008), no. 4, Art. 48006, 4 pp. [5] G. Brandt and A. Merico, The slow demise of Easter Island: insights from a modeling investigation, Front. Ecol. Evol. 3 (2015), 1–13. [6] B. Coll and J. Llibre, Limit cycles for a quadratic system with an invariant straight line and some evolution of phase portraits, in “Qualitative Theory of Differential Equations”, Colloquia Mathematica Societatis János Bolyai, Bolyai Institut, Szeged, Hungria 53 (1988), 111–123. [7] W.A. Coppel, A survey of quadratic systems, J. Diff. Eqns. 2 (1966), 293–304. [8] W.A. Coppel, Some quadratic systems with at most one limit cycle, Dynamics reported, Vol. 2 (1989), 61–88. [9] G. Darboux, Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré (Mélanges), Bull. Sci. Math. 2ème Série 2 (1878), 60–96, 123–144, 151–200. [10] F. Dumortier, J. Llibre and J. C. Artés, Qualitative theory of planar polynomial systems, Springer, 2006. [11] M. Feng, M. Fan, X. Yuan and H. Zhu, Effect of seasonal changing temperature on the growth of phytoplankton, Math. Biosci. Eng. 14(56), (2017), 1091–1117. [12] F. Güngör and P.J. Torres, Integrability of the BasenerRoss model with time-dependent coefficients, SeMA Journal 76 (2019), 485–493. [13] A. Huppert, B. Blasius R. Olinky and L. Stone, A model for seasonal phytoplankton blooms, J. Theor. Biol. 236, (2005), 276–290. [14] L. Koss, Ordinary differential equations and Easter Island: a survey of recent research devel- opments on the relationship between humans, trees, and rats, Eur. J. Math. 5 (2019), 929–936. [15] W. Li, J. Llibre, M. Nicolau and X. Zhang, On the differentiability of first integrals of two dimensional flows, Proc. Amer. Math. Soc. 130 (2002), 2079–2088. [16] J. Llibre and R. Oliveira, Quadratic systems with an invariant conic having Darboux invariants, Commun. Contemp. Math. 20 (2018), no. 4, 1750033, 15 pp. [17] L. Markus, Global structure of ordinary differential equations in the plane, Trans. Amer. Math Soc. 76 (1954), 127–148. [18] D.A. Neumann, Classification of continuous flows on 2–manifolds, Proc. Amer. Math. Soc. 48 (1975), 73–81. [19] M.c. Nucci and G. Sanchini, Symmetries, Lagrangians and conservation laws of an Easter island population model, Symmetry 7(3) (2015), 1613–1632. [20] M.M. Peixoto, Dynamical Systems. Proccedings of a Symposium held at the University of Bahia, 389–420, Acad. Press, New York, 1973. [21] H. Poincaré, Sur l’intégration des équations différentielles du premier ordre et du premier degré I, Rend. Circ. Mat. di Palermo, 5 (1891), 161–191. [22] H. Poincaré, Sur les courbes définies par une équation différentielle, Oevres compltes, Vol.1, 1928. [23] G.S. Ryckov, The limit cycles of the equation u(x + 1)du = (−x + ax2 + bxu + cu + du2 )dx Differential Equations 8 (1972), 1748–1750. [24] M. Scheffer, S. Rinaldi, Y.A. Kuznetsov and E.H. van Nes, Seasonal dynamics of Daphnia and Algae explained as a periodically forced predator-prey system, Oikos 80(3), (1997), 519–532. [25] M. Zhien and T.G. Hallam, Effects of parameter fluctuations on community survival, Math. Biosci. 86(1) (1987), 35–49. INTEGRABILITY AND DYNAMICS OF THE BASENER-ROSS POPULATION MODEL 13 1 Department of Mathematics, Faculty of Science and Letters, Istanbul Technical University, 34469 Istanbul, Turkey E-mail address: gungorf@itu.edu.tr 2 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bel- laterra, Barcelona, Catalonia, Spain E-mail address: jllibre@mat.uab.cat 3 Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, (EPSEB), Av. Doctor Marañón, 44–50, 08028 Barcelona, Spain E-mail address: chara.pantazi@upc.edu