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2 Mathematical Biosciences 225 (21) Contents lists available at ScienceDirect Mathematical Biosciences journal homepage: Bifurcation analysis of a moel for hormonal regulation of the menstrual cycle q James F. Selgrae Department of Mathematics an Biomathematics Program, North Carolina State University, Raleigh, NC , USA article info abstract Article history: Receive 8 October 29 Receive in revise form 15 February 21 Accepte 22 February 21 Available online 26 February 21 Keywors: Estraiol Follicle Parameter Transcritical bifurcation A moel for hormonal control of the menstrual cycle with 13 orinary ifferential equations an 41 parameters is presente. Important changes in moel behavior result from variations in two of the most sensitive parameters. One parameter represents the level of estraiol sufficient for significant synthesis of luteinizing hormone, which causes ovulation. By stuying bifurcation iagrams in this parameter, an interval of parameter values is observe for which a unique stable perioic solution exists an it represent an ovulatory cycle. The other parameter measures mass transfer between the first two stages of ovarian evelopment an is inicative of healthy follicular growth. Changes in this parameter affect the uniqueness interval efine with respect to the first parameter. Hopf, sale-noe an transcritical bifurcations are examine. To attain a normal ovulatory menstrual cycle in this moel, a balance must be maintaine between healthy evelopment of the follicles an flexibility in estraiol levels neee to prouce the surge in luteinizing hormone. Ó 21 Elsevier Inc. All rights reserve. 1. Introuction Systems of ifferential equations which track crucial hormonal changes uring the month have been use to stuy hormonal regulation of the human menstrual cycle, e.g., see [2,3,7,8,13,18,19]. This is a ual control system where hormones secrete by the hypothalamus an the pituitary glans affect the ovaries an the ovarian hormones affect the brain. Follicle stimulating hormone (FSH) an luteinizing hormone (), which are synthesize by the pituitary, control ovarian activity an ovulation, see [1,9,26,27]. In turn, the ovaries prouce estraiol ðe 2 Þ, progesterone ðp 4 Þ an inhibin (INH) which influence the synthesis an release of FSH an, see [1,12,23]. A mechanistic moel evelope by Harris-Clark et al. [8], Pasteur [17], an Schlosser an Selgrae [2,22] captures this interplay with a 13-imensional system of elaye ifferential equations. After parameter ientifications were one using two ifferent clinical ata sets for normally cycling women (McLachlan et al. [14] an Welt et al. [24]), moel simulations closely approximate the ata in both cases, see [8,17]. The system with parameters ientifie using the McLachlan ata [14] is referre to as the McLachlan moel an the system using the Welt ata [24] is calle the Welt moel. Aitional stuies of both moels [8,17] reveale that each has ifferent ynamical behavior. The McLachlan moel has two stable perioic solutions [8] one fits the McLachlan ata for normally cycling women an the other, which is anovulatory because of no surge, has similarities to an abnormal cycle of a woman with q Research partially supporte by NSF Grant DMS aress: selgrae@math.ncsu.eu polycystic ovarian synrome (PCOS) [25]. On the other han, the Welt moel has only one stable perioic solution an it fits the Welt ata for normally cycling women. Selgrae et al. [21] explaine this apparent moel iscrepancy by illustrating that a change in only one sensitive parameter of the Welt system will result in the Welt moel exhibiting two stable cycles like the McLachlan moel. In aition, when each moel has two stable cycles, it was observe [21] that the E 2 profiles of the normal cycles are almost the same an at the high en of the normal range for E 2. This prompte the conjecture that high E 2 levels may inicate a greater risk of cycling abnormally. The rationale for this stuy is to illustrate how the choice of sensitive moel parameters may reuce the chance of cycling abnormally. We focus on two of the three most sensitive parameters an analyze bifurcation iagrams for the Welt moel. Selgrae et al. [21] constructe one iagram for this moel with three iscrete time-elays present in the system of ifferential equations. In orer to track the large amplitue normal cycle an the unstable cycle, an a hoc shooting metho was evelope [21] which is too time consuming to use for a thorough bifurcation stuy. Instea, here we set the time-elays equal to zero an use the features of AUTO [4] in XPPAUT [5]. Taking the elays equal to zero oes not seem to change the qualitative behavior of moel solutions but reuces some hormone peak levels an ecreases the cycle length slightly. In this paper, Section 2 iscusses moel equations an parameters. Section 3 examines bifurcation iagrams with respect to Km, a parameter in the synthesis term, an c 2, an ovarian mass transfer parameter. Hopf, sale-noe an transcritical bifurcations are observe. The notion of a cycle uniqueness interval is introuce, which is an interval of Km values for which a unique /$ - see front matter Ó 21 Elsevier Inc. All rights reserve. oi:1.116/j.mbs

3 J.F. Selgrae / Mathematical Biosciences 225 (21) stable perioic solution exists an it represent an ovulatory cycle. How variations in the parameter c 2 change the size of the cycle uniqueness interval are stuie. Biological implications are iscusse in Section Moel equations an parameters The menstrual cycle for an ault female consists of the follicular phase, ovulation an the luteal phase (e.g., see [15,16]). During the follicular phase, uner the influence of FSH, 6 12 follicles evelop an a ominant follicle is selecte to grow to maturity. These follicles prouce E 2 which primes the pituitary to secrete in large amounts. At mi-cycle, a rapi rise an fall of over a perio of 5 ays is referre to as the surge an is necessary for ovulation, which commences h after the surge. The ovum is release an the ominant follicle becomes the corpus luteum, which prouces hormones in preparation for pregnancy. If fertilization oes not occur, the corpus luteum atrophies which signals the en of one cycle an the beginning of the next. Selgrae an Schlosser [22] evelope a nine-imensional system of orinary ifferential equations for the ovarian hormones E 2 ; P 4 an total INH an, concurrently, Schlosser an Selgrae [2] constructe a four-imensional system for the pituitary hormones an FSH. Harris-Clark et al. [8] merge these two moels into a 13-imensional system of ifferential equations, system (S), escribing the concentrations of the five hormones, estimate parameters an showe that moel simulations agree with ata in McLachlan et al. [14], for normally cycling women. Pasteur [17] estimate parameters for the same system using the ata in Welt et al. [24], which iffer in units for some hormones from the McLachlan ata. Pasteur s simulations [17] were a goo approximation of the Welt ata. Special features of system (S) inclue four equations for the synthesis, release an clearance of the gonaotropin hormones (S1) (S4), where state variables RP an RP FSH represent the amounts of hormones in the pituitary an an FSH represent the bloo concentrations of these hormones. Schlosser an Selgrae [2] assume that E 2 inhibits the release of the gonaotropin hormones (see the enominators in the secon terms of (S1) an (S3)) but at high levels E 2 significantly promotes synthesis (see the Hill function in the numerator of the first term of (S1)). Also P 4 an INH inhibit an FSH synthesis, respectively, an hormone clearance is linear. Three iscrete timeelays (62 ays) were assume for the effects of E 2 ; P 4 an INH on gonaotropin synthesis but for this stuy we set these elays equal to zero to obtain system (S) below. System (S) V ; þ V 1;E 8 2 t RP Km ¼ þe8 2 k ½1 þ c ;P P 4 ŠRP ; 1 þ P 4 =Ki ;P 1 þ c ;E E 2 ðs1þ t ¼ 1 k ½1 þ c ;P P 4 ŠRP a v ; 1 þ c ;E E 2 ðs2þ t RP V FSH FSH ¼ k FSH½1 þ c FSH;P P 4 ŠRP FSH ; 1 þ INH=Ki FSH;INH 1 þ c FSH;E E 2 2 ðs3þ t FSH ¼ 1 k FSH ½1 þ c FSH;P P 4 ŠRP FSH a v 1 þ c FSH;E E 2 FSH FSH; 2 ðs4þ t MsF ¼ bfsh þ½c 1FSH c 2 a ŠMsF; ðs5þ t SeF ¼ c 2 a MsF þ½c 3 b c 4 ŠSeF; ðs6þ t PrF ¼ c 4SeF c 5 c PrF; ðs7þ t Sc 1 ¼ c 5 c PrF 1 Sc 1 ; ðs8þ t Sc 2 ¼ 1 Sc 1 2 Sc 2 ; ðs9þ t Lut 1 ¼ 2 Sc 2 k 1 Lut 1 ; ðs1þ t Lut 2 ¼ k 1 Lut 1 k 2 Lut 2 ; ðs11þ t Lut 3 ¼ k 2 Lut 2 k 3 Lut 3 ; ðs12þ t Lut 4 ¼ k 3 Lut 3 k 4 Lut 4 : ðs13þ To escribe the prouction of E 2 ; P 4, an INH in the ovary, Selgrae an Schlosser [22] use (S5) (S13) to represent nine istinct stages of the ovary base on the capacity of each stage to prouce hormones. This capacity was assume proportional to the mass of each stage, so the state variables represent the masses of the follicular or luteal tissues uring the corresponing stages of the cycle. The gonaotropins promote tissue growth within a stage an the transformation of tissue from one stage to the next. Since clearance from the bloo of the ovarian hormones is on a fast time scale, we assume that bloo levels of E 2 ; P 4, an INH are at quasi-steay state [11] as i Bogumil et al. [2]. Hence, we take these concentrations to be proportional to the tissue masses uring the appropriate stages of the cycle giving the following three auxiliary equations A1, A2 an A3 for the ovarian hormones which appear in (S). Table 1 Parameters an values for system (S) an auxiliary equation (A). Eqs. S1 S4 k 2.42 ay 1 a 14. ay 1 V ; 5 IU/ay V 1; 45 IU/ay Km 2 pg/ml Ki ;P 12.2 ng/ml c ;E.4 ml/pg c ;P.26 ml/ng V FSH 375 IU/ay a FSH 8.21 ay 1 k FSH 1.9 ay 1 c FSH;E.18 ml 2 /pg 2 Ki FSH;INH 3.5 IU/mL c FSH;P 12. ml/ng v 2.5 L Eqs. S5 S13 b c 1 c 2 c 3 c 4 c k 1 k 2 k 3 k 4 a.79 b.16 c.2 Eq. (A) e e 1 e 2 e 3 p p 1 p 2 h h 1 h 2 h 3.5 L lg/(iu ay).8 L/(IU ay).7 (L/IU) a /ay.13 (L/IU) b /ay.27 L/(IU ay).51 (L/IU) c /ay.5 ay 1.56 ay 1.55 ay 1.69 ay 1.85 ay 1.85 ay 1 3 pg/ml.11 L 1.21 L 1.45 L 1 ng/ml.48 kl 1.48 kl 1.4 IU/mL.9 IU/(lg ml).29 IU/(lg ml).18 IU/(lg ml)

4 11 J.F. Selgrae / Mathematical Biosciences 225 (21) Auxiliary equations (A) E 2 ¼ e þ e 1 SeF þ e 2 PrF þ e 3 Lut 4 ; P 4 ¼ p þ p 1 Lut 3 þ p 2 Lut 4 ; INH ¼ h þ h 1 PrF þ h 2 Lut 3 þ h 3 Lut 4 : ða1þ ða2þ ða3þ The 41 parameters of (S) an (A) are liste in Table 1 an correspon to those which Pasteur [17] fit to the Welt ata except here Km ¼ 2 pg=ml; e ¼ 3 pg=ml;c 1 ¼ :8 L=IU ay 1 an c 2 ¼ :7 ðl=iuþ a ay 1 instea of 18, 4,.9 an.9, respectively, an the three time-elays are taken to be zero. With these new parameter values, simulations of the autonomous system (S) with (A) give a reasonable fit to the Welt ata (see Fig. 1 for E 2 an ). The cycle in Fig. 1 appears to be the only asymptotically stable perioic solution for the parameters in Table 1 in agreement with Pasteur s observations [17] for his best-fit parameter set in the system of elay ifferential equations. Although the Welt ata consist of aily hormone levels for 28 consecutive ays, the perio of the cycle in Fig. 1 is approximately 26 ays. The shorter perio reflects the fact that the surge occurs sooner in the cycle (from ay 1 to ay 15) because, without the time elay, synthesis respons more rapily to E 2 stimulation. The follicular phase extens from ay to ay 13, ovulation occurs aroun ay 14 an the luteal phase extens from ay 14 to ay 26. Hormone levels epicte in Fig. 1 represent those of a normally cycling woman an this cycle provies a goo starting point for bifurcation analysis. Since the solution in Fig. 1 is asymptotically stable, it can be reache after iterating for two cycles (52 ays) using the following initial conitions liste in the orer of the 13 state variables in (S), {4,12,2,11,5,1,1,1,1,1,1,1,1}. 3. Sensitive parameters an bifurcation iagrams The rise in E 2 uring the follicular phase primes the pituitary for secretion an the subsequent surge, which is necessary for ovulation. Sensitivity analysis [17,21] with respect to the E 2 follicular peak as system output inicate that the three most sensitive parameters are the half-saturation constant Km in the Hill function in (S1), the exponent a in (S5) an (S6) an the ovarian transfer parameter c 2 in (S5) an (S6). Both a an c 2 similarly affect the transfer of mass from the first follicular stage MsF to the secon follicular stage SeF an, if either is increase, a ecrease occurs in the follicular E 2 peak. Hence, we examine only how changes in Km an c 2 affect the ynamical behavior of (S) with (A). High levels of E 2 stimulate synthesis because of the presence of the Hill function in the numerator of the first term of (S1). Dur E 2 pg/ml time (ays) 12 1 IU/L time (ays) Fig. 1. Data (28 open circles per graph) from Welt et al. [24] an E 2 an moel simulations of system (S) with (A) giving a stable cycle of perio 26 ays.

5 J.F. Selgrae / Mathematical Biosciences 225 (21) ing the follicular phase, this function of E 2 provies a transition from a low synthesis rate given by V ; ¼ 5 IU=ay to a maximal rate of V ; þ V 1; ¼ 5 IU=ay. When E 2 reaches the value Km then the synthesis rate is V ; þ :5V 1; ¼ 275 IU=ay an the pituitary is synthesizing in substantial amounts. In aition, the positive sensitivity coefficient of Km with respect to the E 2 follicular peak [17,21] inicates that increasing Km increases follicular E 2. However, Selgrae et al. [21] suggest that follicular E 2 levels at the high en of the normal range may result in multiple stable cycles. To get a clear iea of how varying Km affects the number of stable cycles an the surge level, we raw the bifurcation iagram (Fig. 2) of the maximal value along a cycle against ifferent values of Km using XPPAUT [5]. Fig. 2 gives the bifurcation iagram with respect to the parameter Km. For values of Km between 5 an 77, the only stable solution is an equilibrium (the soli curve in the lower left of Fig. 2), which represents an anovulatory cycle because the level is too low. A sale-noe bifurcation (SN) of perioic solutions occurs at Km ¼ 77:1 resulting in a stable cycle (soli curve) an an unstable cycle (hollow curve) as Km increases. The stable cycle is ovulatory an continues to the normal cycle at Km ¼ 2 (enote by the in the figure). At Km ¼ 93:4, a Hopf bifurcation (HB at the left in Fig. 2) occurs which results in a stable perioic solution (soli curve) an an unstable equilibrium (the lighter curve that continues to the right from HB). This stable cycle exists only until Km ¼ 97:9 where it coalesces with the unstable cycle an both isappear via a sale-noe. When Km ¼ 211:7, another sale-noe bifurcation results in a small amplitue stable cycle (soli curve) an an unstable cycle (hollow curve). The stable cycle isappears at HB. The unstable cycle annihilates the large amplitue (normal) cycle as Km increases to (SN at the far right). The sigmoi shape curves on the left an the right in Fig. 2 which contain stable an unstable cycles are referre to as hysteresis curves or loops. For each Km value within a hysteresis curve, there is a stable normal cycle (large amplitue ) an a stable anovulatory cycle (small amplitue ). For Km in the interval 97:9 < Km < 211:7 between the lower SN s in Fig. 2 an, hence, between the hysteresis curves, there is only one stable cycle an it is ovulatory. We refer to this Km interval as the cycle uniqueness interval. For c 2 ¼ :7, the iameter of this interval is about 114. To guarantee that a woman has only a normal cycle, Table 2 Cycle uniqueness interval for ecreasing values of c 2. c 2 Interval iameter Km Bouns < Km < < Km < < Km < < Km < < Km < < Km < < Km < 363 her Km parameter must be within the cycle uniqueness interval. Decreasing Km from its value in Table 1, 2 pg/ml, keeps it within the interval an increases the surge. However, increasing Km only slightly to 212 moves Km to a region of multiple cycles an possible anovulation. Here we investigate how variations in the ovarian transfer parameter c 2 will change the size of the cycle uniqueness interval. The sensitivity coefficients for c 2 with respect to both the E 2 follicular peak an the peak are negative an significant [17,21]. Increasing c 2 causes a more rapi transfer of mass from the first follicular stage MsF to the secon stage SeF which iminishes the evelopment of not only MsF but of all subsequent ovarian stages. Hence, increasing c 2 reuces hormone prouction. Also, the cycle uniqueness interval is ecrease (see Table 2) an, for larger values of Km in the interval, the surge level may not be sufficient to cause ovulation. On the other han, ecreasing c 2 wiens the cycle uniqueness interval slightly (see Table 2) an moves it to the right along the Km -axis because more E 2 is secrete an sufficient is synthesize for larger half-saturation constants. As c 2 continues to ecrease, the cycle uniqueness interval begins to shrink because the hysteresis curves enlarge (see Fig. 3 for c 2 ¼ :5). When c 2 ¼ :5, the large amplitue stable cycle exist for a large interval of Km values but for a major portion of that interval there also exists a stable equilibrium or small amplitue stable cycle. Hence, the cycle uniqueness interval between the lower sale-noes (SN) has iameter of only 81. Suenly, as c 2 ecreases below.3, the cycle uniqueness interval expans greatly, e.g., if c 2 ¼ :25 (see Fig. 4) then the iameter is 263 (see Table 2). For c 2 ¼ :25, the hysteresis curve which was on the left in bifurcation iagrams for greater c 2 values has isappeare an this results in a much larger cycle uniqueness interval Km_ Fig. 2. Bifurcation iagram with respect to Km when c 2 ¼ :7. HB an SN enote Hopf an sale-noe bifurcations. The inicates the position of the cycle for the parameters of Table 1 an this cycle is the only stable solution at Km ¼ 2 pg=ml. The cycle uniqueness interval is 97:9 < Km < 211:7, i.e., the interval between the lower sale-noes Km_ Fig. 3. Bifurcation iagram with respect to Km when c 2 ¼ :5. The cycle uniqueness interval between the lower SN s is 153 < Km < 234.

6 112 J.F. Selgrae / Mathematical Biosciences 225 (21) Km_ Km_ Fig. 4. Bifurcation iagram with respect to Km when c 2 ¼ :25. The cycle uniqueness interval is 5:9 < Km < 314, which is much larger than for greater c 2 values. Fig. 6. Bifurcation iagram with respect to Km when c 2 ¼ :35. A transcritical bifurcation occurs at Km ¼ 56:89. s enotes a stable cycle. The bifurcations occurring near c 2 ¼ :3 which permit the expansion of the cycle uniqueness interval inclue two new Hopf bifurcations an a transcritical bifurcation of perioic solutions. For c 2 :38, a small set of stable perioic solutions appears along the curve of equilibria to the left of the left hysteresis curve, see Fig. 5. The stable equilibrium at the far left in Fig. 5 unergoes a Hopf bifurcation at Km ¼ 51:2 resulting in a stable perioic solution which quickly isappears in another Hopf bifurcation at Km ¼ 6:95. We refer to this phenomenon as a Hopf bump. As c 2 ecreases, the Hopf bump enlarges an touches the hysteresis curve when c 2 ¼ :35, see Fig. 6. The intersection of the Hopf bump an the hysteresis curve prouces a transcritical bifurcation of perioic solutions in the parameter Km. Then as c 2 ecreases farther, the Hopf bump appears on the other sie of the hysteresis curve (Fig. 7) an quickly shrinks an isappears. This sequence of bifurcations results in a much larger cycle uniqueness interval as in Fig. 4 where c 2 ¼ :25. When c 2 ¼ :35, a transcritical bifurcation of perioic solutions occurs in the parameter Km at Km ¼ 56:89. There is an exchange of stability along the curves of cycles as they pass through the point Km ¼ 56:89, in the sense that for Km < 56:89 the upper curve consists of unstable cycles until Km ¼ 56:89 where the cycles change to stable but the lower curve consists of stable cycles for Km < 56:89 an changes to unstable as Km passes through A blow-up of this transcritical bifurcation is given by Fig. 8. Asc 2 ecreases, a stanar unfoling of the transcritical bifurcation [6] is epicte in Figs. 8 an 9, which are blow-ups of Figs. 5 7 near the bifurcation point. In Fig. 9(a), where c 2 ¼ :38, the Hopf bump lies below an to the left of the unstable branch of the hysteresis curve. At c 2 ¼ :35, the Hopf bump an the hysteresis curve touch at the point of transcritical bifurcation Km ¼ 56:89, see Fig. 8. Then as c 2 ecreases to.32, these curves separate into a narrow hysteresis curve with a sale-noe bifurcation, where cycle stability changes, an a Hopf bump to the right with another sale-noe, see Fig. 9(b). This unfoling of the transcritical bifurcation may be symbolize roughly by the family of hyperbolas (see [6, p. 25, #2]): 2 ðkm Þ 2 ¼ c 2 :35: Km_ Km_ Fig. 5. Bifurcation iagram with respect to Km when c 2 ¼ :38. A Hopf bump is present along the far left portion of the curve of equilibrium. s enotes a stable cycle. Fig. 7. Bifurcation iagram with respect to Km when c 2 ¼ :32. The Hopf bump is to the right of the left hysteresis curve, which is very narrow. s enotes a stable cycle an u, an unstable cycle.

7 J.F. Selgrae / Mathematical Biosciences 225 (21) IU/L Km_ Fig. 8. Blow up of transcritical bifurcation at Km ¼ 56:89 when c 2 ¼ :35. There is an exchange of stability as the curves of cycles cross. Darker curves are stable solutions an lighter or hollow curves, unstable solutions. s enotes a stable cycle time (ays) Fig. 1. ata (28 open circles) from Welt et al. [24] an moel simulation when c 2 ¼ :25 an Km ¼ 3 pg=ml giving a stable cycle of perio 27.5 ays. Initial conitions liste in the orer of the state variables of (S) are {4,12,2,11,5,1,1,1,1,1,1,1,1}. Hence, this rapi sequence of bifurcations as c 2 ecreases opens a large cycle uniqueness interval of Km values where only one stable cycle exists. When c 2 ¼ :25, this interval has a iameter of 263 along the Km -axis as compare to 114 when c 2 ¼ :7. Although hormone levels for cycles when c 2 ¼ :25 are somewhat abnormal, surges are sufficient for ovulation (see Fig. 1 where Km ¼ 3 pg=ml). Notice that the surge in Fig. 1 is broaer than normal an luteal levels are higher than follicular levels. Also, for this cycle, E 2 is twice the normal level which may inicate the presence of several ominant follicles. 4. Biological implications an conclusion Here we stuy bifurcation behavior of the Welt moel (elays equal zero) with respect to variations in the most sensitive parameters, Km an c 2. Km is the half-saturation parameter in the Hill function for synthesis (S1) an this value represents the level of E 2 for significant synthesis in the moel. An surge is necessary for a normal menstrual cycle an for ovulation. We introuce the notion of a cycle uniqueness interval which is an interval of Km values where there is a unique stable perioic solution an it represents an ovulatory cycle. If Km lies outsie the interval then either no surge occurs or there are two stable cycles one is ovulatory an the other is anovulatory. The best-fit parameter set for the McLachlan moel [8] has its Km value outsie the cycle uniqueness interval an, consequently, has two stable cycles. A large cycle uniqueness interval will mean that a wie range of follicular E 2 levels will promote the surge an will guarantee an ovulatory cycle. We observe that the bifurcation iagram of maximum graphe against the Km value has hysteresis curves at high an low Km an the cycle uniqueness interval lies between sale-noe bifurcations at the lower ens of these hysteresis curves (Figs. 2 an 3). The parameter c 2 measures mass transfer between the first two stages of ovarian evelopment an is inicative of healthy follicular growth. Too large a c 2 value results in mass transfer which is too rapi for the normal evelopment of the first an subsequent stages of the ovaries. Also, varying c 2 changes the cycle uniqueness interval (Table 2). Decreasing c 2 from.7 (its value in Table 1) wiens the cycle uniqueness interval slightly an moves it towar larger Km values because more E 2 is secrete an sufficient is Km_ Km_ Fig. 9. (a) Blow up for c 2 ¼ :38. (b) Blow up for c 2 ¼ :32. s inicates a stable cycle an u, an unstable cycle. SN enotes two sale-noes.

8 114 J.F. Selgrae / Mathematical Biosciences 225 (21) synthesize for larger half-saturation constants. As c 2 continues to ecrease, the cycle uniqueness interval begins to shrink because the hysteresis curves enlarge (Fig. 3) until a suen quarupling of the interval iameter occurs near c 2 ¼ :3. The mathematical explanation for this quarupling is the formation of a Hopf bump of perioic solutions (Fig. 5) followe by a transcritical bifurcation (Fig. 6). Having c 2 quite small allows for early an excessive follicular growth an follicular E 2 concentration twice that of the Welt ata [24]. So, although there is an surge for c 2 less than.3 (Fig. 4), E 2 levels may be high enough to be consiere harmful if these levels persist over many cycles. To attain a normal ovulatory menstrual cycle a balance must be maintaine between healthy evelopment of the follicles an flexibility in E 2 levels neee to prouce the surge. References [1] D.T. 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