Spectral Rigidity and Geometric Localization of Hopf Bifurcations in Planar Predator-Prey Systems

Sp ectral Rigidit y and Geometric Lo calization of Hopf Bifurcations in Planar Predator–Prey Systems E. Chan–Lóp ez ∗ 1 , A. Martín–R uiz 2 , and Víctor Castellanos 1 1 A c ademic Division of Basic Scienc es, Universidad Juár ez A utónoma de T ab asc o (DA CB–UJA T), Mexic o. 2 Institute of Nucle ar Scienc es, National A utonomous University of Mexic o (ICN–UNAM), Mexic o. Marc h 23, 2026 Abstract W e iden tify a geometric principle gov erning the lo cation of Hopf and Bogdanov– T akens bifurcations in planar predator–prey systems. The prey co ordinate of any co existence equilibrium undergoing such a bifurcation lies b et w een consecutive critical p oin ts of the prey n ullcline. The mec hanism is algebraic. At critical p oin ts of the nullcline, the v anishing of its deriv ative induces constraints on the Jacobian that preven t the sp ectral conditions required for bifurcation from b eing satisfied. W e refer to this phenomenon as sp e ctr al rigidity . The principle is established for three model families and one discrete count erpart with qualitatively differen t n ullcline geometries: a quadratic case (Bazykin mo del), a cubic case (Holling type IV with harvesting), and a rational case (Crowley–Martin functional resp onse). In each case, the lo calization follows from explicit parametric c haracterizations and symbolic reduction. The analysis extends to discrete-time systems. F or a map obtained by forw ard Euler discretization of the Cro wley–Martin model, the Neimark–Sack er bifurcation o ccurs on the descending branc h of the nullcline, providing a contin uous–discrete dualit y go verned by the same mec hanism. W e conjecture that this lo calization holds for general smo oth prey nullclines, with critical p oints acting as spectral barriers that organise the bifurcation structure. Keyw ords: predator–prey systems, Hopf bifurcation, Bogdano v–T akens bifurcation, Neimark–Sac k er bifurcation, prey nullcline, sp ectral rigidity , geometric lo calization, Bazykin mo del, Cro wley–Martin functional resp onse, Holling type IV. MSC 2020: 34C23, 37G15, 92D25, 34C05. ∗ Corresp onding author: eduardo.clopez13@gmail.com 1 1 In tro duction A fundamen tal question in the qualitativ e theory of predator–prey systems concerns the lo c ation of oscillatory instabilities in state space. The classical answ er, due to Rosenzweig and MacArth ur [ 13 ], is that in systems with a monotone functional resp onse the unique co existence equilibrium loses stability via a Hopf bifurcation when it crosses the vertex of the prey nullcline—the mechanism underlying the parado x of enrichmen t [ 14 ]. This observ ation, relating a ge ometric feature of the nullcline to a sp e ctr al prop ert y of the linearization, has prov ed to b e one of the most pro ductiv e ideas in mathematical ecology . When the mo del is enric hed by in trasp ecific predator comp etition (Bazykin [ 2 ]), non– monotone functional resp onses such as Holling t yp e IV [ 1 , 8 , 11 ], harvesting [ 3 ], Allee effects [ 15 , 16 ], or density-dependent mortality [ 7 ], the bifurcation structure b ecomes considerably ric her: multiple co existence equilibria, Bautin bifurcations, and Bogdanov– T akens p oints of v arious co dimensions may all app ear. In these settings, the interpla y b et w een nonlinear functional resp onses and parameter–dep enden t feedbac k mechanisms pro duces a wide v ariety of lo cal and global dynamical b eha viors, whose organization is t ypically understo o d through detailed, mo del–sp ecific bifurcation analyses. Despite this apparent complexit y , a p ersisten t empirical regularit y emerges across these mo dels: the e quilibria at which Hopf or Bo gdanov–T akens bifur c ation o c curs invariably have their pr ey c o or dinate lying b etwe en c onse cutive critic al p oints of the pr ey nul lcline. This phenomenon is observed in mo dels with qualitativ ely different functional resp onses and parametrizations, yet it has not, to the b est of our kno wledge, b een identified as the manifestation of a general structural principle. The central aim of this pap er is to show that this regularity is not incidental, but rather reflects a geometric mec hanism that constrains the sp ectrum of the Jacobian at co existence equilibria. More precisely , we show that the critical structure of the prey n ullcline induces algebraic constrain ts on the Jacobian en tries that prev en t the eigen v alues from satisfying the sp ectral conditions required for bifurcation at those p oints. W e refer to this mec hanism as sp e ctr al rigidity at critic al p oints . In this w a y , the geometry of the n ullcline acts as a system of sp e ctr al b arriers that organise the bifurcation diagram. An imp ortan t feature of this p ersp ectiv e is that it applies uniformly to b oth contin uous- and discrete–time systems. While Hopf bifurcation in flo ws is characterized by a v anishing trace and Neimark–Sack er bifurcation in maps by a unit determinan t, both phenomena are constrained b y the same geometric mechanism. This leads to a natural contin uous–discrete dualit y in the localization of oscillatory instabilities, go v erned en tirely by the monotonicit y structure of the prey n ullcline. The purp ose of this pap er is threefold: (i) T o establish this ge ometric lo c alization principle rigorously for three mo del families with qualitatively different nullcline geometries—quadratic, cubic, and rational— through complete parametric characterizations and exhaustiv e case analyses. (ii) T o identify the common algebraic mechanism— sp e ctr al rigidity at critic al p oints — that underpins the lo calization in ev ery case studied, and which op erates uniformly across b oth con tinuous and discrete dynamical systems. (iii) T o formulate a precise conjecture extending the principle to arbitrary smo oth prey n ullclines, linking the critical structure of the n ullcline to the geometric organization of the bifurcation diagram. 2 1.1 Related w ork Hammoum, Sari, and Y adi [ 7 ] extended the Rosenzweig–MacArth ur graphical stability criterion to a general Gause mo del with v ariable predator mortalit y 𝑑 ( 𝑥, 𝑦 ) , defining an arc 𝒜 of the ascending branc h of the prey n ullcline along whic h the Jacobian trace is non–negativ e and showing that Hopf bifurcation o ccurs at 𝜕 𝒜 . They computed explicit first Ly apuno v co efficien ts for the Bazykin, Cav ani–F arkas, and V ariable–T erritory mo dels. Their framew ork is effective for determining stability and criticalit y , but do es not yield closed–form expressions for the Hopf lo cus, do es not establish that 𝒜 is strictly confined b elo w the nullcline v ertex, and requires the functional resp onse to b e monotone increasing (h yp othesis H2: 𝑝 ′ ( 𝑥 ) > 0 , 𝑞 ′ ( 𝑥 ) > 0 ), thereby excluding the Holling type IV case. Lu and Huang [ 12 ] carried out a detailed bifurcation analysis of Bazykin’s mo del with Holling I I resp onse and predator comp etition, including degenerate Hopf bifurcation of co dimension 2 and fo cus–t yp e BT bifurcation of co dimension 3. Their analysis pro vides a useful reference p oin t for the ric her bifurcation structures that may arise in related mo dels. F or the Holling type IV Leslie system, Li and Xiao [ 11 ], Huang et al. [ 8 ], Dai and Zhao [ 5 ], and Cheng and Zhang [ 3 ] carried out progressively refined bifurcation analyses (co dimension 2 and 3 BT, Hopf cyclicit y , cusp and generalized Hopf p oin ts). None of these w orks addresses the geometric lo calization question. 1.2 Organization of the pap er The pap er is organized as follows. Section 2 introduces the general framework and the notion of nullcline critical structure. Section 3 treats the quadratic case (Bazykin mo del). Section 4 treats the cubic case (Holling type IV with harvesting). Section 5 treats the rational case (Crowley–Martin). Section 6 extends the principle to discrete–time systems. Section 7 iden tifies the common algebraic mec hanism. Section 8 form ulates the general conjecture. Section 9 discusses op en problems. 2 General F ramew ork 2.1 The class of mo dels W e consider planar predator–prey systems of the form ˙ 𝑥 = 𝑓 1 ( 𝑥, 𝑦 ) , ˙ 𝑦 = 𝑓 2 ( 𝑥, 𝑦 ) , (1) defined on the closed first quadrant R 2 + = { ( 𝑥, 𝑦 ) ∈ R 2 : 𝑥 ≥ 0 , 𝑦 ≥ 0 } , where 𝑓 1 and 𝑓 2 are smo oth functions satisfying the standard ecological assumptions: 𝑓 1 (0 , 𝑦 ) = 0 , 𝑓 2 ( 𝑥, 0) = 0 for appropriate b oundary conditions, and b oth axes are in v ariant. A c o existenc e e quilibrium p oint (CEP) is a p oint 𝑃 * = ( 𝑥 * , 𝑦 * ) ∈ R 2 + with 𝑥 * , 𝑦 * > 0 satisfying 𝑓 1 ( 𝑃 * ) = 𝑓 2 ( 𝑃 * ) = 0 . 2.2 The prey n ullcline and its p olynomial degree The prey nullcline is the curve 𝒩 𝑥 = { ( 𝑥, 𝑦 ) : 𝑓 1 ( 𝑥, 𝑦 ) = 0 , 𝑥 > 0 } . In all standard predator–prey mo dels, this can b e written as 𝑦 = 𝑔 ( 𝑥 ) for a smo oth function 𝑔 defined on a subin terv al of (0 , ∞ ) . 3 Definition 1. The p olynomial de gr e e of the pr ey nul lcline is the degree of 𝑔 view ed as a p olynomial (or rational function reduced to p olynomial form) in 𝑥 , after clearing denominators in the functional resp onse. W e denote it 𝑛 = deg ( 𝑔 ) . Example 2. (a) Rosenzw eig–MacArth ur / Bazykin (Holling t yp e I I): 𝑔 ( 𝑥 ) = 𝑟 𝑎𝐾 ( 𝐾 − 𝑥 )( 𝑏 + 𝑥 ) , whic h is quadratic: 𝑛 = 2 . One critical p oin t (maximum) at 𝑥 v = ( 𝐾 − 𝑏 ) / 2 . (b) Holling type IV with harv esting (Leslie-type): 𝑔 ( 𝑥 ) = (1 − ℎ 1 − 𝑥 )( 𝑎 + 𝑥 2 ) , whic h is cubic: 𝑛 = 3 . T wo critical p oin ts (lo cal minimum 𝑥 min , lo cal maxim um 𝑥 max ) under appropriate parametric conditions. (c) Holling type I I I : 𝑔 ( 𝑥 ) is generically cubic or quartic dep ending on the growth function: 𝑛 ≥ 3 . A p olynomial 𝑔 of degree 𝑛 has at most 𝑛 − 1 critical p oints in the in terior of the ecologically relev ant region { 𝑥 : 𝑔 ( 𝑥 ) > 0 } . These critical p oints partition this region into at most 𝑛 subinterv als. 2.3 Hopf and Bogdano v–T ak ens conditions Let 𝐽 ( 𝑃 * ) denote the Jacobian of ( 1 ) at a CEP 𝑃 * . The conditions for bifurcation at 𝑃 * are: Hopf: tr( 𝐽 ( 𝑃 * )) = 0 , det( 𝐽 ( 𝑃 * )) > 0 , (2) Bogdano v–T ak ens: tr( 𝐽 ( 𝑃 * )) = 0 , det( 𝐽 ( 𝑃 * )) = 0 . (3) Both conditions require the trace to v anish. The central observ ation of this pap er is that the trace, ev aluated along the prey n ullcline, p ossesses a sign structure that is gov erned by the critical p oin ts of 𝑔 . 3 The Quadratic Case: Bazykin Mo del 3.1 Mo del and n ullcline geometry The Bazykin predator–prey mo del is ˙ 𝑥 = 𝑟 𝑥  1 − 𝑥 𝑘  − 𝑎 𝑥 𝑦 𝑥 + 𝑏 , ˙ 𝑦 = 𝑒 𝑎 𝑥 𝑦 𝑥 + 𝑏 − 𝑑 𝑦 − 𝜎 𝑦 2 , (4) with all parameters p ositive. The prey nullcline is the parab ola 𝑔 ( 𝑥 ) = 𝑟 𝑎𝑘 ( 𝑘 − 𝑥 )( 𝑏 + 𝑥 ) , deg( 𝑔 ) = 2 , (5) with unique critical p oint (maximum) at 𝑥 v = 1 2 ( 𝑘 − 𝑏 ) , requiring 𝑘 > 𝑏 . 4 3.2 Lo calization theorem Theorem 3 (Quadratic Lo calization) . L et 𝑎 b e the bifur c ation p ar ameter in system ( 4 ) with 𝑘 > 𝑏 > 0 . Then every c o existenc e e quilibrium at which a Hopf bifur c ation o c curs satisfies 0 < 𝑥 * < 𝑥 v = 𝑘 − 𝑏 2 . Pr o of sketch. W e introduce con trol parameters 𝑘 0 > 0 , 𝑥 0 > 0 and set 𝑘 = 𝑘 0 + 𝑏 + 𝑥 0 , whic h parametrizes the constrain t 𝑏 − 𝑘 + 2 𝑥 < 0 explicitly . Three cases are analyzed: Case 1 ( 𝑥 * = 𝑥 v ): Setting 𝑥 = ( 𝑘 − 𝑏 ) / 2 and conditioning 𝑒 so that this is a CEP , the Jacobian trace ev aluates to tr( 𝐽 ) = − ( 𝑘 0 + 2 𝑏 ) 2 𝑟 𝜎 4 𝑎 ( 𝑘 0 + 𝑏 ) < 0 for all admissible parameters. The trace is strictly negativ e, so the Hopf condition tr ( 𝐽 ) = 0 cannot b e met. Case 2 ( 𝑥 * > 𝑥 v , descending branch): P arametrizing 𝑥 * = 𝑥 v + 𝑥 0 with 𝑥 0 > 0 , b oth summands of the trace are strictly negative. Again tr( 𝐽 ) < 0 identically . Case 3 ( 𝑥 * < 𝑥 v , ascending branc h): The system { 𝑓 1 = 0 , 𝑓 2 = 0 , tr ( 𝐽 ) = 0 , det ( 𝐽 ) > 0 } admits solutions with all parameters p ositiv e. The critical bifurcation v alue is 𝑎 0 = ( 𝑘 0 + 2 𝑏 ) 2 ( 𝑘 0 + 2 𝑏 + 2 𝑥 0 ) 𝜎 4 𝑘 0 𝑥 0 > 0 , and the equilibrium is 𝑃 0 =  𝑘 0 2 , 𝑘 0 𝑟 𝑥 0 ( 𝑘 0 + 2 𝑏 )( 𝑘 0 + 𝑏 + 𝑥 0 ) 𝜎  One v erifies directly that det ( 𝐽 )    𝑃 0 = 𝑒 𝑟 𝑥 0 ( 𝑘 0 + 2 𝑏 + 2 𝑥 0 ) 𝜎 / ( 𝑘 0 + 𝑏 + 𝑥 0 ) > 0 for all admissible parameters, so the Hopf condition is fully satisfied at 𝑃 0 . Since Cases 1 and 2 exclude Hopf bifurcation and Case 3 realizes it, the lo calization is pro v ed. Remark 4. The same case analysis shows that the Bogdano v–T akens condition ( tr ( 𝐽 ) = 0 and det ( 𝐽 ) = 0 ) is also confined to 𝑥 * < 𝑥 v , since in Cases 1 and 2 the trace cannot v anish regardless of the determinant v alue. 4 The Cubic Case: Holling T yp e IV with Harv esting 4.1 Mo del and n ullcline geometry W e consider the Leslie-t yp e system ˙ 𝑥 = 𝑥 (1 − 𝑥 ) − 𝑥 𝑦 𝑎 + 𝑥 2 − ℎ 1 𝑥, ˙ 𝑦 = 𝑦  𝛿 − 𝛽 𝑦 𝑥  − ℎ 2 𝑦 , (6) 5 with all parameters non–negativ e. Introducing ℎ 10 > 0 and setting ℎ 1 = ℎ 10 / (3 + ℎ 10 ) , 𝑎 = 9 / (4(3 + ℎ 10 ) 2 ) , the prey nullcline b ecomes a cubic 𝑔 ( 𝑥 ) = − ( ℎ 10 𝑥 + 3 𝑥 − 3)(9 + 4(3 + ℎ 10 ) 2 𝑥 2 ) 4(3 + ℎ 10 ) 3 , deg( 𝑔 ) = 3 , (7) with t wo critical p oin ts: 𝑥 min = 1 2(3 + ℎ 10 ) , 𝑥 max = 3 2(3 + ℎ 10 ) . (8) 4.2 Lo calization theorem Theorem 5 (Cubic Lo calization) . In system ( 6 ) under the ab ove r ep ar ametrization, let 𝛽 b e the bifur c ation p ar ameter. Then every c o existenc e e quilibrium at which a Hopf bifur c ation o c curs satisfies 𝑥 min < 𝑥 * < 𝑥 max . Pr o of sketch. The augmented system { 𝑓 1 = 0 , 𝑓 2 = 0 , tr ( 𝐽 ) = 0 } solv ed for 𝑦 , 𝛿 , 𝛽 as functions of 𝑥 yields expressions 𝑦 0 ( 𝑥 ) , 𝛿 0 ( 𝑥 ) , 𝛽 0 ( 𝑥 ) . A complete symbolic reduction via Reduce in Mathematic a establishes that 𝑦 0 > 0 , 𝛿 0 > 0 , 𝛽 0 > 0 , det( 𝐽 ) > 0 ⇐ ⇒ 𝑥 min < 𝑥 < 𝑥 max . The b oundary cases are verified directly: at 𝑥 = 𝑥 min and 𝑥 = 𝑥 max , the solution gives 𝛽 0 = 0 (inadmissible); for 𝑥 < 𝑥 min , 𝛽 0 < 0 ; for 𝑥 > 𝑥 max , 𝑦 0 < 0 . 4.3 Bogdano v–T ak ens lo calization In the three–equilibrium regime (obtained via a refined parametrization with 𝑎 1 = 1 , 𝑥 0 + 𝑎 0 = 𝑥 1 ), the simultaneous conditions tr ( 𝐽 ) = 0 and det ( 𝐽 ) = 0 yield tw o solution branc hes for ( 𝛽 , ℎ 10 ) as functions of 𝑥 0 . Both branc hes require ℎ 10 > 0 , whic h constrains 𝑥 0 ∈ ( 1 32 , 9 32 ) . The corresp onding equilibrium prey co ordinates satisfy 𝑥 min < 𝑥 * < 𝑥 max , confirming that the BT lo calization holds in the cubic case as w ell. 5 The Rational Case: Cro wley–Martin F unctional Resp onse The first tw o cases inv olve p olynomial prey n ullclines ( deg ( 𝑔 ) = 2 and 3 ). T o test the scop e of the lo calization principle b ey ond the p olynomial setting, we no w analyze a mo del whose prey nullcline is a r ational function with a single maxim um. 5.1 Mo del and n ullcline geometry The Cro wley–Martin predator–prey mo del [ 4 ] is ˙ 𝑥 = 𝜌 𝑥  1 − 𝑥 𝑘  − 𝑎 𝑥 𝑦 (1 + 𝑏 𝑥 )(1 + 𝑐 𝑦 ) , ˙ 𝑦 = 𝛾 𝑎 𝑥 𝑦 (1 + 𝑏 𝑥 )(1 + 𝑐 𝑦 ) − 𝑑 𝑦 , (9) 6 where 𝑐 ≥ 0 is the pr e dator interfer enc e p ar ameter . When 𝑐 = 0 , system ( 9 ) reduces to the classical Rosenzw eig–MacArthur mo del with Holling t yp e I I resp onse 𝑎 𝑥/ (1 + 𝑏 𝑥 ) . Setting ˙ 𝑥/𝑥 = 0 and solving for 𝑦 , the prey n ullcline is 𝑔 ( 𝑥 ) = ℎ ( 𝑥 ) 𝑎 − 𝑐 ℎ ( 𝑥 ) , ℎ ( 𝑥 ) : = 𝜌  1 − 𝑥 𝑘  (1 + 𝑏 𝑥 ) , (10) whic h is a r ational function of 𝑥 (not p olynomial for 𝑐 > 0 ), with a b ell–shap ed profile v anishing at 𝑥 = 𝑘 and 𝑥 = 0 ha ving 𝑔 (0) = 𝜌/ ( 𝑎 − 𝑐𝜌 ) > 0 when 𝑎 > 𝑐𝜌 . The k ey observ ation is that 𝑔 ′ ( 𝑥 ) = 𝑎 ℎ ′ ( 𝑥 ) ( 𝑎 − 𝑐 ℎ ( 𝑥 )) 2 , (11) so 𝑔 ′ ( 𝑥 ) = 0 if and only if ℎ ′ ( 𝑥 ) = 0 . The function ℎ is the standard Holling t yp e I I n ullcline (a parab ola), so 𝑥 v = 𝑏 𝑘 − 1 2 𝑏 , (12) whic h is indep endent of the in terference parameter 𝑐 . The parameter 𝑐 mo dulates the heigh t and curv ature of the b ell but not the lo cation of its p eak. 5.2 Lo calization theorem Theorem 6 (Crowley–Martin Lo calization) . In system ( 9 ) with 𝑐 > 0 and 𝑏 𝑘 > 1 , let 𝑐 b e the bifur c ation p ar ameter. Then every c o existenc e e quilibrium at which a Hopf bifur c ation o c curs satisfies 0 < 𝑥 * < 𝑥 v = 𝑏 𝑘 − 1 2 𝑏 . Mor e over, the critic al value of the interfer enc e p ar ameter at a Hopf p oint with pr ey c o or dinate 𝑥 * is given explicitly by 𝑐 0 ( 𝑥 ) = 𝑎 𝑏 𝑥 ( 𝑘 0 − 2 𝑏 𝑥 ) 𝑑 (1 + 𝑏 𝑥 ) 2 (1 + 𝑘 0 − 𝑏 𝑥 ) , (13) wher e 𝑘 0 = 𝑏 𝑘 − 1 , so that 𝑥 v = 𝑘 0 / (2 𝑏 ) . Sinc e al l factors in the denominator ar e p ositive in the e c olo gic al ly r elevant r e gion, 𝑐 0 ( 𝑥 ) > 0 if and only if 𝑘 0 − 2 𝑏 𝑥 > 0 , i.e. 0 < 𝑥 < 𝑥 v . Pr o of. W e introduce 𝑘 0 > 0 via 𝑘 = (1 + 𝑘 0 ) /𝑏 so that 𝑏 𝑘 − 1 = 𝑘 0 and 𝑥 v = 𝑘 0 / (2 𝑏 ) . The Jacobian of ( 9 ) at a CEP ( 𝑥 * , 𝑔 ( 𝑥 * )) has the trace tr( 𝐽 ) = 𝜌 𝑥 ( 𝑏 𝑘 − 1 − 2 𝑏 𝑥 ) 𝑘 (1 + 𝑏 𝑥 )    𝐽 11 (prey part, independent of 𝑐 ) − 𝑑 𝑐 𝑔 ( 𝑥 ) 1 + 𝑐 𝑔 ( 𝑥 )    𝐽 22 (predator part, ≤ 0 ) . A crucial prop ert y is that 𝐽 11    nullcline dep ends on 𝑥 and the prey parameters ( 𝜌, 𝑘 , 𝑏 ) but not on 𝑐 . This is b ecause the 𝑐 -dep endence in 𝜕 𝑓 1 /𝜕 𝑥 cancels exactly when 𝑦 is substituted from the nullcline relation 𝜌 (1 − 𝑥/𝑘 ) = 𝑎𝑦 / ((1 + 𝑏𝑥 )(1 + 𝑐𝑦 )) . A t 𝑥 = 𝑥 v : 𝑏 𝑘 − 1 − 2 𝑏 𝑥 v = 0 so 𝐽 11 = 0 and tr ( 𝐽 ) = − 𝑑 𝑐 𝑔 ( 𝑥 v ) / (1 + 𝑐 𝑔 ( 𝑥 v )) < 0 for all 𝑐 > 0 . Hopf is imp ossible. 7 F or 𝑥 > 𝑥 v : 𝑏 𝑘 − 1 − 2 𝑏 𝑥 < 0 so 𝐽 11 < 0 and b oth summands are non-p ositive. tr( 𝐽 ) < 0 . Hopf is imp ossible. F or 0 < 𝑥 < 𝑥 v : 𝐽 11 > 0 . Setting tr ( 𝐽 ) = 0 and using the iden tit y 𝑐 𝑔 ( 𝑥 ) / (1 + 𝑐 𝑔 ( 𝑥 )) = 𝑐 ℎ ( 𝑥 ) /𝑎 (whic h follows from ( 10 ) ), we obtain ( 13 ) . All factors are p ositiv e, so 𝑐 0 > 0 , and a v alid Hopf p oint exists. Remark 7 (Recov ery of the classical Hopf p oin t) . As 𝑐 → 0 + , form ula ( 13 ) sho ws that 𝑐 0 ( 𝑥 ) → 0 requires 𝑘 0 − 2 𝑏 𝑥 → 0 . More precisely , keeping 𝑥 as a free v ariable and taking 𝑐 → 0 + in the Hopf condition tr ( 𝐽 ) = 0 , the unique solution for the prey co ordinate satisfying all ecological constraints is 𝑥 → 𝑥 v . This recov ers the classical Rosenzw eig– MacArth ur result that, in the absence of predator in terference, Hopf bifurcation o ccurs precisely at the vertex of the parab olic prey n ullcline. F or 𝑐 > 0 , the Hopf equilibrium is displaced to the left of the vertex into the ascending branch, with larger 𝑐 corresp onding to smaller 𝑥 * . 6 The Discrete Case: Neimark–Sac k er Bifurcation in Maps The preceding sections establish the lo calization principle for contin uous–time systems, where the Hopf condition requires tr ( 𝐽 ) = 0 with det ( 𝐽 ) > 0 . It is natural to ask whether an analogous result holds for discrete-time systems, where the relev ant bifurcation—the Neimark–Sac k er bifurcation—demands det ( 𝐽 ) = 1 with | tr ( 𝐽 ) | < 2 [ 9 , 10 ]. W e shall show that it do es, and that the mec hanism is, in a precise sense, the same. 6.1 The discrete Cro wley–Martin mo del Consider the discrete predator–prey system (map) with Cro wley–Martin functional re- sp onse: 𝑥 𝑛 +1 = 𝑥 𝑛 + 𝜌 𝑥 𝑛  1 − 𝑥 𝑛 𝑘  − 𝑎 𝑥 𝑛 𝑦 𝑛 (1 + 𝑏 𝑥 𝑛 )(1 + 𝑐 𝑦 𝑛 ) , 𝑦 𝑛 +1 = 𝑦 𝑛 + 𝛾 𝑎 𝑥 𝑛 𝑦 𝑛 (1 + 𝑏 𝑥 𝑛 )(1 + 𝑐 𝑦 𝑛 ) − 𝑑 𝑦 𝑛 . (14) This is the forward Euler discretization of the con tin uous system ( 9 ) : the map has the structure 𝑥 𝑛 +1 = 𝑥 𝑛 + 𝐹 ( 𝑥 𝑛 , 𝑦 𝑛 ) , where 𝐹 is the contin uous vector field. A fixed p oin t ( 𝑥 * , 𝑦 * ) satisfies 𝐹 ( 𝑥 * , 𝑦 * ) = 0 , which is pr e cisely the equilibrium condition of the flow. It follo ws at once that the prey nullcline of the map—defined b y the condition 𝑥 𝑛 +1 = 𝑥 𝑛 with 𝑥 > 0 —coincides with that of the contin uous mo del: 𝜌  1 − 𝑥 𝑘  = 𝑎 𝑦 (1 + 𝑏 𝑥 )(1 + 𝑐 𝑦 ) , (15) whence 𝑦 = 𝑔 map ( 𝑥 ) = ℎ ( 𝑥 ) / ( 𝑎 − 𝑐 ℎ ( 𝑥 )) with the same auxiliary function ℎ ( 𝑥 ) = 𝜌 (1 − 𝑥/𝑘 )(1 + 𝑏𝑥 ) as in ( 10 ) . In particular, the v ertex lo cation 𝑥 v = ( 𝑏𝑘 − 1) / (2 𝑏 ) is iden tical to that of the con tin uous case and is indep enden t of 𝑐 . 6.2 Sp ectral rigidit y: 𝐽 map 00 = 1 at the vertex Since 𝑓 map 1 ( 𝑥, 𝑦 ) = 𝑥 + 𝑓 1 ( 𝑥, 𝑦 ) , differentiation with resp ect to 𝑥 gives 𝐽 map 00 = 1 + 𝐽 cont 11 , (16) 8 where 𝐽 cont 11 = 𝜕 𝑓 1 /𝜕 𝑥 is the (1 , 1) –en try of the Jacobian of the contin uous system ( 9 ) . On the prey n ullcline, which is common to b oth the flo w and the map, the simplification of Section 5.2 applies verb atim : 𝐽 map 00    nullcline = 1 + 𝜌 𝑥 ( 𝑏𝑘 − 1 − 2 𝑏𝑥 ) 𝑘 (1 + 𝑏𝑥 ) = 𝑘 (1 + 𝑏𝑥 ) + 𝜌 𝑥 ( 𝑏𝑘 − 1 − 2 𝑏𝑥 ) 𝑘 (1 + 𝑏𝑥 ) , (17) whic h is indep endent of 𝑐 . At 𝑥 = 𝑥 v = ( 𝑏𝑘 − 1) / (2 𝑏 ) , the linear factor 𝑏𝑘 − 1 − 2 𝑏𝑥 v v anishes by definition, so 𝐽 map 00    𝑥 = 𝑥 v = 1 (exactly) . (18) This is the discrete coun terpart of 𝐽 cont 11    𝑥 v = 0 : A t the vertex Sp ectral consequence Flo w 𝐽 11 = 0 tr( 𝐽 ) = 𝐽 22 < 0 : eigenv alues in left half-plane Map 𝐽 00 = 1 eigen v alue pro duct constrained aw ay from 1 F or 𝑥 > 𝑥 v (descending branch), 𝐽 00 deviates from 1 , enabling det ( 𝐽 ) = 1 to b e ac hiev ed for an appropriate bifurcation parameter v alue. Theorem 8 (Discrete Lo calization) . In the discr ete Cr ow ley–Martin system ( 14 ) , the Neimark–Sacker bifur c ation at a c o existenc e fixe d p oint o c curs with 𝑥 * > 𝑥 v , i.e. on the desc ending br anch of the pr ey nul lcline. Mor e over: (i) The vertex 𝑥 v = ( 𝑏𝑘 − 1) / (2 𝑏 ) is indep endent of 𝑐 (same me chanism as in the c ontinuous c ase: 𝑔 ′ map = 0 ⇔ ℎ ′ = 0 ). (ii) 𝐽 map 00 = 1 exactly at 𝑥 v , as a dir e ct c onse quenc e of the identity ( 16 ) and the vanishing of 𝐽 cont 11 at the vertex. (iii) F or 𝑥 * > 𝑥 v , the entry 𝐽 map 00 deviates fr om unity, pr oviding the sp e ctr al de gr e e of fr e e dom ne c essary for the Neimark–Sacker c ondition det( 𝐽 ) = 1 to b e r e alize d. The essen tial observ ation ma y b e stated as follows: The sp e ctr al c ondition for bifur c ation differs b etwe en c ontinuous and discr ete systems—tr ac e vanishing in the former, unit determinant in the latter—yet the ge ometric lo c alization r emains invariant: it is governe d entir ely by the critic al structur e of the pr ey nul lcline. In b oth settings, the vertex of the prey nullcline serves as a sp e ctr al b oundary . In flows, it separates the region where 𝐽 11 > 0 (ascending branc h, Hopf p ossible) from 𝐽 11 < 0 (descending branch, trace irrecov erably negative). In maps, the same vertex separates the region where the determinan t can attain unity (descending branch, Neimark–Sac k er p ossible) from where it cannot. The nullcline v ertex is, in each case, the organizing center of the lo cal bifurcation structure. Remark 9 (Contin uous–discrete dualit y) . The localization principle exhibits a noteworth y dualit y: in contin uous systems the Hopf bifurcation is confined to the ascending branch ( 𝑥 * < 𝑥 v ), whereas in discrete systems the Neimark–Sac ker bifurcation is confined to the descending branch ( 𝑥 * > 𝑥 v ). The v ertex 𝑥 v serv es as the common b oundary in b oth cases. 9 This dualit y admits a natural sp ectral in terpretation: the Hopf condition tr ( 𝐽 ) = 0 with det ( 𝐽 ) > 0 constrains the eigen v alue sum , while the Neimark–Sac k er condition det ( 𝐽 ) = 1 with | tr ( 𝐽 ) | < 2 constrains the eigenv alue pr o duct . These complemen tary constraints, mediated b y the same sp ectral rigidit y at the vertex, select opp osite sides of the critical p oin t. 7 The Common Mec hanism: Sp ectral Rigidit y at Critical P oin ts The pro ofs of Theorems 3 , 5 , 6 and 8 share a common algebraic structure that is indep endent of the particular sp ectral condition (trace–zero or unit–determinant) and op erates at the lev el of the sp e ctrum of the Jacobian. W e now isolate this structure and give it a precise form ulation. 7.1 F rom trace rigidit y to sp ectral rigidity In a contin uous-time system, the Hopf condition tr ( 𝐽 ) = 0 constrains the sum of eigen v alues: 𝜆 1 + 𝜆 2 = 0 , requiring them to cross the imaginary axis. In a discrete-time system, the Neimark–Sac k er condition det ( 𝐽 ) = 1 constrains the pr o duct : 𝜆 1 𝜆 2 = 1 , requiring them to cross the unit circle. Both are sp ectral conditions—they demand that the eigenv alues of the Jacobian reach a prescrib ed lo cus in the complex plane. The observ ation that unifies all our results is that neither the sum nor the pro duct can attain its bifurcation v alue when the equilibrium sits at a critical p oin t of the prey n ullcline. This is not a coincidence of tw o unrelated mec hanisms; it is a single phenomenon: geometric criticalit y of the n ullcline constrains the sp ectrum of the Jacobian. Definition 10 (Sp ectral rigidit y) . W e sa y that the prey n ullcline exhibits sp e ctr al rigidity at a critic al p oint 𝑥 𝑐 if, whenev er ( 𝑥 𝑐 , 𝑔 ( 𝑥 𝑐 )) is a co existence equilibrium, the condition 𝑔 ′ ( 𝑥 𝑐 ) = 0 constrains the sp ectrum of 𝐽 ( 𝑥 𝑐 , 𝑔 ( 𝑥 𝑐 )) in such a w a y that the eigen v alues cannot satisfy the bifurcation condition—neither 𝜆 1 + 𝜆 2 = 0 (Hopf ) nor 𝜆 1 𝜆 2 = 1 (Neimark–Sac k er) nor 𝜆 1 = 𝜆 2 = 0 (Bogdano v–T akens)—for an y admissible parameter v alues. This definition op erates at the level of the full sp ectrum, not of any particular sp ectral quan tity . It is this generality that allo ws the same principle to gov ern b oth flo ws and maps. 7.2 The sp ectral rigidit y mec hanism A t any CEP ( 𝑥 * , 𝑔 ( 𝑥 * )) on the prey n ullcline, the Jacobian has the structure 𝐽 =  𝐽 11 ( 𝑥 * ) 𝐽 12 ( 𝑥 * ) 𝐽 21 ( 𝑥 * ) 𝐽 22 ( 𝑥 * )  , (19) where 𝐽 11 enco des prey self–regulation and 𝐽 22 enco des predator self-regulation. In all mo dels studied: (i) 𝐽 11 dep ends on 𝑔 ′ ( 𝑥 * ) , and 𝑔 ′ ( 𝑥 𝑐 ) = 0 forces 𝐽 11 ( 𝑥 𝑐 ) = 0 or drives it to a v alue that eliminates a degree of freedom. 10 (ii) 𝐽 22 has a definite sign (t ypically ≤ 0 ) that is indep enden t of the n ullcline slop e. (iii) 𝐽 12 < 0 (predation reduces prey growth) and 𝐽 21 > 0 (predation increases predator gro wth). A t a critical p oin t 𝑥 𝑐 , the eigenv alues are: 𝜆 1 , 2 = 𝐽 11 ( 𝑥 𝑐 ) + 𝐽 22 ( 𝑥 𝑐 ) 2 ±      𝐽 11 ( 𝑥 𝑐 ) − 𝐽 22 ( 𝑥 𝑐 ) 2  2 + 𝐽 12 𝐽 21 . With 𝐽 11 ( 𝑥 𝑐 ) = 0 (or constrained), the trace b ecomes tr ( 𝐽 ) = 𝐽 22 < 0 and the determinan t b ecomes det ( 𝐽 ) = − 𝐽 12 𝐽 21 > 0 (but generically  = 1 ). The eigen v alues are lo cke d in to a configuration that cannot reach the imaginary axis (for flows) or the unit circle (for maps). The essen tial p oint is: A t critic al p oints of the pr ey nul lcline ( i.e., wher e 𝑔 ′ ( 𝑥 ) = 0 ) , the Jac obian exhibits sp e ctr al rigidity: the c onstr aints imp ose d by the vanishing of the nul lcline derivative eliminate the de gr e es of fr e e dom r e quir e d for the eigenvalues to satisfy the bifur c ation c onditions ( tr ac e–zer o in flows, unit determinant in maps ) . Prey density , 𝑥 Predator density , 𝑦 𝑥 v 𝑔 ( 𝑥 ) 𝐻 𝑁 𝑆 Critical p oint ( 𝑥 v , 𝑔 ( 𝑥 v )) Hopf bifurcation Neimark–Sac ker bifurcation Prey iso cline Figure 1: Geometric lo calization of dynamic instabilities along the prey n ullcline defined in ( 10 ) . The critical p oin t 𝑥 v acts as a structural b oundary induced b y sp ectral rigidity . In the contin uous–time formulation, the Hopf bifurcation is confined to the ascending branc h (blue), whereas in discrete–time mappings, the Neimark–Sac k er (N–S) bifurcation o ccurs on the descending branch (green). This geometric separation reflects the contin uous–discrete dualit y of the underlying bifurcation mechanism. 7.3 V erification across all mo dels W e now summarise ho w the sp ectral rigidity mechanism manifests in eac h of the mo dels treated ab o v e. Bazykin ( Section 3 ). A t 𝑥 v , 𝐽 11 = 0 and 𝐽 22 = − 𝜎 𝑦 * < 0 , whence tr ( 𝐽 ) = − 𝜎 𝑦 * < 0 . The eigen v alues are confined to the op en left half-plane. Holling t yp e IV ( Section 4 ). At 𝑥 min and 𝑥 max , the Hopf system forces 𝛽 0 = 0 , which collapses the predator dynamics. The sp ectral constrain t manifests as the vanishing of the bifur c ation p ar ameter itself —a particularly rigid form of obstruction. 11 Cro wley–Martin ( Section 5 ). A t 𝑥 v , 𝐽 11 = 0 indep enden tly of 𝑐 , while 𝐽 22 = − 𝑑𝑐𝑦 * / (1 + 𝑐𝑦 * ) < 0 for all 𝑐 > 0 . The sp ectrum is rigid for every v alue of the bifurcation parameter sim ultaneously . Discrete Cro wley–Martin ( Section 6 ). The iden tity 𝐽 map 00 = 1 + 𝐽 cont 11 transfers the n ullcline criticality to the map Jacobian: at 𝑥 v , 𝐽 map 00 = 1 exactly , and the determinan t is constrained a wa y from unit y , preven ting the eigen v alues from reaching the unit circle. 7.4 Propagation b ey ond the critical p oints Sp ectral rigidity at the critical p oin ts propagates to the exterior of the inter-critical in terv al b y tw o complementary mec hanisms. In flo ws (ascending → descending). F or 𝑥 > 𝑥 v (or outside ( 𝑥 min , 𝑥 max ) in the cubic case), 𝐽 11 acquires a definite negativ e sign that reinforces the negativity of 𝐽 22 . The trace is therefore strictly negative, and the eigen v alues remain in the op en left half-plane. In maps (descending → ascending). The discrete structure reverses the role of the branc hes, but the sp ectral obstruction p ersists: on the ascending side of the vertex, the determinan t cannot attain unit y . This t w o-step pattern— rigidity at the critic al p oint, pr op agation to the b oundary — constitutes the mec hanism b ehind the geometric lo calization of bifurcations of perio dic orbits in predator–prey systems. 7.5 The deep er principle The sp ectral rigidity mec hanism rev eals a relationship that, while natural in hindsight, is not at all obvious a priori : The ge ometry of the pr ey nul lcline c ontr ols the sp e ctrum of the Jac obian, not the r everse. In a general dynamical system the spectrum of the Jacobian at an equilibrium dep ends on all mo del parameters, and there is no in trinsic reason why a geometric feature of one n ullcline should constrain the eigenv alues. The constraint arises b ecause the equilibrium lies on b oth n ullclines sim ultaneously , and the critical structure of the prey nullcline imp oses a compatibilit y condition on the Jacobian entries that propagates to the full sp ectrum. In the language of bifurcation theory , the critical p oints of the prey nullcline are sp e ctr al or ganizing c enters : they partition the state space into regions of qualitatively distinct sp ectral b eha vior, with bifurcations of p erio dic orbits confined to the transitions b et w een these regions. 8 The General Conjecture The results of Sections 3 to 6 and the mechanism identified in Section 7 motiv ate the follo wing conjecture. Conjecture 11 (Geometric lo calization of Hopf and Bogdanov–T ak ens bifurcations) . Consider a smo oth ( 𝐶 2 at le ast ) pr e dator–pr ey system of the form ( 1 ) in the first quadr ant of R 2 , whose pr ey nul lcline 𝑦 = 𝑔 ( 𝑥 ) p ossesses exactly two critic al p oints in (0 , ∞ ) : a lo c al minimum at 𝑥 min and a lo c al maximum at 𝑥 max , with 0 < 𝑥 min < 𝑥 max . Supp ose the system admits exactly thr e e c o existenc e e quilibria. Then: 12 (a) Every c o existenc e e quilibrium at which a Hopf or Bo gdanov–T akens bifur c ation o c curs has its pr ey c o or dinate in the interval ( 𝑥 min , 𝑥 max ) , i.e. in the r e gion wher e the nul lcline is lo c al ly incr e asing ( 𝑔 ′ ( 𝑥 * ) > 0 ) . (b) In the discr ete-time ( map ) version of the system, the Neimark–Sacker bifur c ation o c curs at e quilibria satisfying 𝑔 ′ ( 𝑥 * ) < 0 , i.e. on desc ending br anches of the nul lcline. This dichotomy r efle cts the sp e ctr al rigidity principle: at the critic al p oints 𝑔 ′ ( 𝑥 ) = 0 , an algebr aic dep endenc e among the Jac obian entries pr e cludes the r e alization of the sp e ctr al c onditions for bifur c ation ( tr ac e–zer o in flows, unit determinant in maps ) , ther eby c onfining bifur c ations to r e gions determine d by the monotonicity of the nul lcline. Remark 12 (Evidence) . The con tin uous–time assertion is established for quadratic n ullclines ( Theorem 3 ), cubic nullclines ( Theorem 5 ), and rational n ullclines ( Theorem 6 ). The discrete-time assertion is established in Theorem 8 . A general pro of, v alid for arbitrary smo oth n ullclines without case–by–case computation, remains op en. Remark 13 (Role of Bogdano v–T akens bifurcation) . Including the BT bifurcation in the conjecture is not merely a matter of completeness: the BT p oin t is the co dimension-tw o organizing center from whic h Hopf and homo clinic bifurcation curves emanate [ 9 , 6 ]. W ere the BT p oint able to escap e the inter–critical in terv al, the Hopf curve emanating from it could exit the lo calization region, in v alidating part of the conjecture. The confinemen t of the BT p oin t ( Section 4.3 ) thus ensures the coherence of the entire lo cal bifurcation structure. Remark 14 (P olynomial degree and bifurcation complexity) . F or a p olynomial prey n ullcline of degree 𝑛 , there are at most 𝑛 − 1 critical p oin ts and hence at most 𝑛 − 2 in ter-critical op en interv als. The conjecture accordingly predicts that the num b er of p oten tial lo calization regions for Hopf and BT bifurcations gro ws as 𝑛 − 2 , providing a concrete link b etw een the algebr aic c omplexity of the mo del (as measured by the degree of its p olynomial n ullcline) and the ge ometric c omplexity of its bifurcation diagram. Remark 15 (Relation to [ 7 ]) . The graphical criterion of Hammoum, Sari, and Y adi [ 7 ] iden tifies an arc 𝒜 of the ascending branch along whic h the trace is non-negativ e, with Hopf bifurcation at 𝜕 𝒜 . Our conjecture may b e view ed as asserting that 𝒜 is alwa ys strictly contained b etw een consecutiv e critical p oin ts of the n ullcline—a structural prop ert y not established in [ 7 ], where 𝒜 is determined implicitly b y the equation 𝐻 ( 𝑥 ) = 𝐺 ( 𝑥 ) and its endp oin ts are obtained numerically . F urthermore, the framew ork of [ 7 ] requires monotone functional resp onse (h yp othesis H2: 𝑝 ′ ( 𝑥 ) > 0 ), excluding the Holling type IV case, whereas our conjecture encompasses non-monotone resp onses. 9 Conclusions and Op en Problems W e hav e established a geometric lo calization principle for Hopf and Bogdano v–T akens bifurcations in planar predator–prey systems, proving it for four canonical settings–namely: the Bazykin mo del with quadratic prey nullcline ( Theorem 3 ), the Holling t yp e IV mo del with cubic prey nullcline ( Theorem 5 ), the Cro wley–Martin mo del with rational prey n ullcline ( Theorem 6 ), and the discrete Crowley–Martin map ( Theorem 8 ). The common mec hanism—sp ectral rigidit y at the critical p oin ts of the prey nullcline, follow ed b y sign 13 propagation to the exterior—op erates uniformly across contin uous and discrete systems and suggests a principle of considerable generality ( Conjecture 11 ). Sev eral natural questions remain op en: (i) A general pro of. Can the sp ectral rigidit y mechanism b e established abstractly for arbitrary smo oth prey nullclines, without mo del-sp ecific computation? A natural approac h would b e to exploit the factorization of the trace along the nullcline as tr ( 𝐽 ) = ℎ ( 𝑥 ) · ( 𝐻 ( 𝑥 ) − 𝐺 ( 𝑥 )) , in the notation of [ 7 ], and to analyze the sign of 𝐻 − 𝐺 at and b ey ond the critical p oin ts of 𝑔 . (ii) Higher-degree n ullclines. Mo dels with Holling type I II resp onse, m ultiple Allee effects, or more elab orate functional forms can generate quartic or higher–degree prey n ullclines. Do es the lo calization principle hold in each inter-critical interv al indep enden tly? (iii) Higher-co dimension bifurcations. The mo dels in [ 8 , 15 , 16 ] exhibit Bogdanov– T akens bifurcation of co dimension 3 and 4, and degenerate Hopf of co dimension up to 5. Is the lo calization principle resp ected by these phenomena? (iv) Non–p olynomial n ullclines. When the prey n ullcline inv olves exp onen tial or trigonometric growth functions, can the conjecture b e extended using the critical p oin ts of the smo oth nullcline, ev en in the absence of a p olynomial degree? (v) Global bifurcations. Homo clinic orbits and limit cycle bifurcations are organized b y the Bogdano v–T akens p oin t. If the BT p oint is geometrically lo calized, what can b e inferred ab out the sp atial extent of the corresp onding homo clinic lo op relative to the n ullcline geometry? The sp ectral rigidity mec hanism is consistent with the b eha vior observed in predator– prey mo dels with alternative mortalit y structures, such as v ariable territory form ulations. In these systems, despite significant differences in the predator equation, the same ge- ometric pattern app ears: the Jacobian entry 𝐽 11 v anishes at the critical p oin ts of the prey nullcline and c hanges sign across them, while 𝐽 22 remains non-p ositiv e at co existence equilibria. Consequen tly , the trace is negative at the v ertex and along the descending branc h, preven ting Hopf bifurcation outside the ascending region. A rigorous pro of of the lo calization principle for this class of mo dels is not carried out here and constitutes a natural extension of the presen t work. This b eha vior is not accidental, but reflects a structural constraint imp osed by the geometry of the prey nullcline. The prey dynamics determine the regions in phase space where the sp ectral conditions for bifurcation can b e satisfied, indep enden tly of the sp ecific form of the predator dynamics. A c kno wledgemen ts E. Chan–Lóp ez was supp orted b y SECIHTI through the program “Estancias Posdoctorales p or México” (CVU 422090). A. Martín–R uiz ackno wledges financial supp ort from UNAM- P API IT (pro ject IG100224), UNAM-P APIME (pro ject PE109226), SECIHTI (pro ject CBF-2025-I-1862), and the Marcos Moshinsky F oundation. The authors thank Jaume Llibre for helpful comments and suggestions. 14 Ethics declarations Conflict of in terest The authors declare no conflicts of interest. Ethical Appro v al Not applicable. References [1] J. F. Andrews, A mathematical mo del for the con tinuous culture of micro organisms utilizing inhibitory substrates, Biote chnol. Bio eng. , 10 (1968), 707–723. [2] A. D. Bazykin, Nonline ar Dynamics of Inter acting Populations , W orld Scientific, Singap ore, 1998. [3] L. Cheng and L. Zhang, Bogdano v–T akens bifurcation of a Holling type IV predator– prey mo del with constan t-effort harv esting, J. Ine qual. A ppl. , 2021 (2021), Article 68. [4] P . H. Cro wley and E. K. 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