@article{wright_fayad_selgrade_olufsen_2020, title={Mechanistic model of hormonal contraception}, volume={16}, ISSN={["1553-7358"]}, DOI={10.1371/journal.pcbi.1007848}, abstractNote={Contraceptive drugs intended for family planning are used by the majority of married or in-union women in almost all regions of the world. The two most prevalent types of hormones associated with contraception are synthetic estrogens and progestins. Hormonal based contraceptives contain a dose of a synthetic progesterone (progestin) or a combination of a progestin and a synthetic estrogen. In this study we use mathematical modeling to understand better how these contraceptive paradigms prevent ovulation, special focus is on understanding how changes in dose impact hormonal cycling. To explain this phenomenon, we added two autocrine mechanisms essential to achieve contraception within our previous menstrual cycle models. This new model predicts mean daily blood concentrations of key hormones during a contraceptive state achieved by administering progestins, synthetic estrogens, or a combined treatment. Model outputs are compared with data from two clinical trials: one for a progestin only treatment and one for a combined hormonal treatment. Results show that contraception can be achieved with synthetic estrogen, with progestin, and by combining the two hormones. An advantage of the combined treatment is that a contraceptive state can be obtained at a lower dose of each hormone. The model studied here is qualitative in nature, but can be coupled with a pharmacokinetic/pharamacodynamic (PKPD) model providing the ability to fit exogenous inputs to specific bioavailability and affinity. A model of this type may allow insight into a specific drug’s effects, which has potential to be useful in the pre-clinical trial stage identifying the lowest dose required to achieve contraception.}, number={6}, journal={PLOS COMPUTATIONAL BIOLOGY}, author={Wright, A. Armean and Fayad, Ghassan N. and Selgrade, James F. and Olufsen, Mette S.}, year={2020}, month={Jun} }
 @article{graham_selgrade_2017, title={A model of ovulatory regulation examining the effects of insulin-mediated testosterone production on ovulatory function}, volume={416}, ISSN={["1095-8541"]}, DOI={10.1016/j.jtbi.2017.01.007}, abstractNote={Polycystic ovary syndrome (PCOS), a common cause of infertility in women, is often accompanied by abnormal reproductive and metabolic hormone levels. Specifically, androgens such as testosterone are elevated in many PCOS women, and the syndrome itself is frequently associated with insulin resistance, which leads to hyperinsulinemia, i.e., elevated insulin. Although the precise role of insulin in ovulatory function is unclear, its role in ovulatory dysfunction is often linked to the effects of increased ovarian androgen production. We present a mathematical model of the menstrual cycle that incorporates regulation by the pituitary-ovarian axis and mechanisms of ovarian testosterone production. We determine a physiological role for testosterone in the normal ovulatory cycle and study the role of hyperinsulinemia in pathological regulation of the cycle. Model results indicate increased ovulatory disruption with elevated insulin-mediated testosterone production and suggest that variations in the response of ovarian follicles to essential signals can alter the degree to which hyperinsulinemia disrupts the ovulatory cycle. The model also provides insight into the various PCOS phenotypes and the severity of ovulatory dysfunction.}, journal={JOURNAL OF THEORETICAL BIOLOGY}, author={Graham, Erica J. and Selgrade, James F.}, year={2017}, month={Mar}, pages={149–160} }
 @article{panza_wright_selgrade_2016, title={A delay differential equation model of follicle waves in women}, volume={10}, ISSN={["1751-3766"]}, DOI={10.1080/17513758.2015.1115564}, abstractNote={ABSTRACT This article presents a mathematical model for hormonal regulation of the menstrual cycle which predicts the occurrence of follicle waves in normally cycling women. Several follicles of ovulatory size that develop sequentially during one menstrual cycle are referred to as follicle waves. The model consists of 13 nonlinear, delay differential equations with 51 parameters. Model simulations exhibit a unique stable periodic cycle and this menstrual cycle accurately approximates blood levels of ovarian and pituitary hormones found in the biological literature. Numerical experiments illustrate that the number of follicle waves corresponds to the number of rises in pituitary follicle stimulating hormone. Modifications of the model equations result in simulations which predict the possibility of two ovulations at different times during the same menstrual cycle and, hence, the occurrence of dizygotic twins via a phenomenon referred to as superfecundation. Sensitive parameters are identified and bifurcations in model behaviour with respect to parameter changes are discussed. Studying follicle waves may be helpful for improving female fertility and for understanding some aspects of female reproductive ageing.}, number={1}, journal={JOURNAL OF BIOLOGICAL DYNAMICS}, author={Panza, Nicole M. and Wright, Andrew A. and Selgrade, James F.}, year={2016}, pages={200–221} }
 @article{hendrix_selgrade_2014, title={Bifurcation analysis of a menstrual cycle model reveals multiple mechanisms linking testosterone and classical PCOS}, volume={361}, ISSN={["1095-8541"]}, DOI={10.1016/j.jtbi.2014.07.020}, abstractNote={A system of 16 differential equations is described which models hormonal regulation of the menstrual cycle focusing on the effects of the androgen testosterone (T) on follicular development and on the synthesis of luteinizing hormone (LH) in the pituitary. Model simulations indicate two stable menstrual cycles - one cycle fitting data in the literature for normal women and the other cycle being anovulatory because of no LH surge. Bifurcations with respect to sensitive model parameters illustrate various characteristics of polycystic ovarian syndrome (PCOS), a leading cause of female infertility. For example, varying one parameter retards the growth of preantral follicles and produces a "stockpiling" of these small follicles as observed in the literature for some PCOS women. Modifying another parameter increases the stimulatory effect of T on LH synthesis resulting in reduced follicular development and anovulation. In addition, the model illustrates how anovulatory and hyperandrogenic cycles which are characteristic of PCOS can be reproduced by perturbing both pituitary sensitivity to T and the follicular production of T. Thus, this model suggests that for some women androgenic activity at the levels of both the pituitary and the ovaries may contribute to the etiology of PCOS.}, journal={JOURNAL OF THEORETICAL BIOLOGY}, author={Hendrix, Angelean O. and Selgrade, James F.}, year={2014}, month={Nov}, pages={31–40} }
 @article{hendrix_hughes_selgrade_2014, title={Modeling Endocrine Control of the Pituitary-Ovarian Axis: Androgenic Influence and Chaotic Dynamics}, volume={76}, ISSN={["1522-9602"]}, DOI={10.1007/s11538-013-9913-7}, abstractNote={Mathematical models of the hypothalamus-pituitary-ovarian axis in women were first developed by Schlosser and Selgrade in 1999, with subsequent models of Harris-Clark et al. (Bull. Math. Biol. 65(1):157-173, 2003) and Pasteur and Selgrade (Understanding the dynamics of biological systems: lessons learned from integrative systems biology, Springer, London, pp. 38-58, 2011). These models produce periodic in-silico representation of luteinizing hormone (LH), follicle stimulating hormone (FSH), estradiol (E2), progesterone (P4), inhibin A (InhA), and inhibin B (InhB). Polycystic ovarian syndrome (PCOS), a leading cause of cycle irregularities, is seen as primarily a hyper-androgenic disorder. Therefore, including androgens into the model is necessary to produce simulations relevant to women with PCOS. Because testosterone (T) is the dominant female androgen, we focus our efforts on modeling pituitary feedback and inter-ovarian follicular growth properties as functions of circulating total T levels. Optimized parameters simultaneously simulate LH, FSH, E2, P4, InhA, and InhB levels of Welt et al. (J. Clin. Endocrinol. Metab. 84(1):105-111, 1999) and total T levels of Sinha-Hikim et al. (J. Clin. Endocrinol. Metab. 83(4):1312-1318, 1998). The resulting model is a system of 16 ordinary differential equations, with at least one stable periodic solution. Maciel et al. (J. Clin. Endocrinol. Metab. 89(11):5321-5327, 2004) hypothesized that retarded early follicle growth resulting in "stockpiling" of preantral follicles contributes to PCOS etiology. We present our investigations of this hypothesis and show that varying a follicular growth parameter produces preantral stockpiling and a period-doubling cascade resulting in apparent chaotic menstrual cycle behavior. The new model may allow investigators to study possible interventions returning acyclic patients to regular cycles and guide developments of individualized treatments for PCOS patients.}, number={1}, journal={BULLETIN OF MATHEMATICAL BIOLOGY}, author={Hendrix, Angelean O. and Hughes, Claude L. and Selgrade, James F.}, year={2014}, month={Jan}, pages={136–156} }
 @article{harris_selgrade_2014, title={Modeling endocrine regulation of the menstrual cycle using delay differential equations}, volume={257}, ISSN={["1879-3134"]}, DOI={10.1016/j.mbs.2014.08.011}, abstractNote={This article reviews an effective mathematical procedure for modeling hormonal regulation of the menstrual cycle of adult women. The procedure captures the effects of hormones secreted by several glands over multiple time scales. The specific model described here consists of 13 nonlinear, delay, differential equations with 44 parameters and correctly predicts blood levels of ovarian and pituitary hormones found in the biological literature for normally cycling women. In addition to this normal cycle, the model exhibits another stable cycle which may describe a biologically feasible "abnormal" condition such as polycystic ovarian syndrome. Model simulations illustrate how one cycle can be perturbed to the other cycle. Perturbations due to the exogenous administration of each ovarian hormone are examined. This model may be used to test the effects of hormone therapies on abnormally cycling women as well as the effects of exogenous compounds on normally cycling women. Sensitive parameters are identified and bifurcations in model behavior with respect to parameter changes are discussed. Modeling various aspects of menstrual cycle regulation should be helpful in predicting successful hormone therapies, in studying the phenomenon of cycle synchronization and in understanding many factors affecting the aging of the female reproductive endocrine system.}, journal={MATHEMATICAL BIOSCIENCES}, author={Harris, Leona A. and Selgrade, James F.}, year={2014}, month={Nov}, pages={11–22} }
 @article{margolskee_selgrade_2013, title={A lifelong model for the female reproductive cycle with an antimullerian hormone treatment to delay menopause}, volume={326}, ISSN={["1095-8541"]}, DOI={10.1016/j.jtbi.2013.02.007}, abstractNote={A system of 16 non-linear, delay differential equations with 66 parameters is developed to model hormonal regulation of the menstrual cycle of a woman from age 20 to 51. This mechanistic model predicts changes in follicle numbers and reproductive hormones that naturally occur over that time span. In particular, the model illustrates the decline in the pool of primordial follicles from age 20 to menopause as reported in the biological literature. Also, model simulations exhibit a decrease in antimüllerian hormone (AMH) and inhibin B and an increase in FSH with age corresponding to the experimental data. Model simulations using the administration of exogenous AMH show that the transfer of non-growing primordial follicles to the active state can be slowed enough to provide more follicles for development later in life and to cause a delay in the onset of menopause as measured by the number of primordial follicles remaining in the ovaries. Other effects of AMH agonists and antagonists are investigated in the setting of this model.}, journal={JOURNAL OF THEORETICAL BIOLOGY}, author={Margolskee, Alison and Selgrade, James F.}, year={2013}, month={Jun}, pages={21–35} }
 @article{margolskee_selgrade_2011, title={Dynamics and bifurcation of a model for hormonal control of the menstrual cycle with inhibin delay}, volume={234}, ISSN={["1879-3134"]}, DOI={10.1016/j.mbs.2011.09.001}, abstractNote={A system of 13 ordinary differential equations with 42 parameters is presented to model hormonal regulation of the menstrual cycle. For an excellent fit to clinical data, the model requires a 36 h time delay for the effect of inhibin on the synthesis of follicle stimulating hormone. Biological and mathematical reasons for this delay are discussed. Bifurcations with respect to changes in three important parameters are examined. One parameter represents the level of estradiol adequate for significant synthesis of luteinizing hormone. Bifurcation diagrams with respect to this parameter reveal an interval of parameter values for which a unique stable periodic solution exists and this solution represents a menstrual cycle during which ovulation occurs. The second parameter measures mass transfer between the first two stages of ovarian development and is indicative of healthy follicular growth. The third parameter is the time delay. Changes in the second parameter and the time delay affect the size of the uniqueness interval defined with respect to the first parameter. Saddle-node, transcritical and degenerate Hopf bifurcations are studied.}, number={2}, journal={MATHEMATICAL BIOSCIENCES}, author={Margolskee, Alison and Selgrade, James F.}, year={2011}, month={Dec}, pages={95–107} }
 @article{selgrade_2010, title={Bifurcation analysis of a model for hormonal regulation of the menstrual cycle}, volume={225}, ISSN={["1879-3134"]}, DOI={10.1016/j.mbs.2010.02.004}, abstractNote={A model for hormonal control of the menstrual cycle with 13 ordinary differential equations and 41 parameters is presented. Important changes in model behavior result from variations in two of the most sensitive parameters. One parameter represents the level of estradiol sufficient for significant synthesis of luteinizing hormone, which causes ovulation. By studying bifurcation diagrams in this parameter, an interval of parameter values is observed for which a unique stable periodic solution exists and it represent an ovulatory cycle. The other parameter measures mass transfer between the first two stages of ovarian development and is indicative of healthy follicular growth. Changes in this parameter affect the uniqueness interval defined with respect to the first parameter. Hopf, saddle-node and transcritical bifurcations are examined. To attain a normal ovulatory menstrual cycle in this model, a balance must be maintained between healthy development of the follicles and flexibility in estradiol levels needed to produce the surge in luteinizing hormone.}, number={2}, journal={MATHEMATICAL BIOSCIENCES}, author={Selgrade, James F.}, year={2010}, month={Jun}, pages={108–114} }
 @article{selgrade_harris_pasteur_2009, title={A model for hormonal control of the menstrual cycle: Structural consistency but sensitivity with regard to data}, volume={260}, ISSN={["1095-8541"]}, DOI={10.1016/j.jtbi.2009.06.017}, abstractNote={This study presents a 13-dimensional system of delayed differential equations which predicts serum concentrations of five hormones important for regulation of the menstrual cycle. Parameters for the system are fit to two different data sets for normally cycling women. For these best fit parameter sets, model simulations agree well with the two different data sets but one model also has an abnormal stable periodic solution, which may represent polycystic ovarian syndrome. This abnormal cycle occurs for the model in which the normal cycle has estradiol levels at the high end of the normal range. Differences in model behavior are explained by studying hysteresis curves in bifurcation diagrams with respect to sensitive model parameters. For instance, one sensitive parameter is indicative of the estradiol concentration that promotes pituitary synthesis of a large amount of luteinizing hormone, which is required for ovulation. Also, it is observed that models with greater early follicular growth rates may have a greater risk of cycling abnormally.}, number={4}, journal={JOURNAL OF THEORETICAL BIOLOGY}, author={Selgrade, J. F. and Harris, L. A. and Pasteur, R. D.}, year={2009}, month={Oct}, pages={572–580} }
 @article{selgrade_bostic_roberds_2009, title={Dynamical behaviour of a discrete selection-migration model with arbitrary dominance}, volume={15}, ISSN={["1563-5120"]}, DOI={10.1080/10236190802400741}, abstractNote={To study the effects of immigration of genes (possibly transgenic) into a natural population, a one-island selection-migration model with density-dependent regulation is used to track allele frequency and population size. The existence and uniqueness of a polymorphic genetic equilibrium is proved under a general assumption about dominance in fitnesses. Also, conditions are found which guarantee the existence of and determine the location of the global attractor for this model. The rate at which solutions approach the attractor is approximated. A measure of allelic diversity is introduced.}, number={4}, journal={JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS}, author={Selgrade, James F. and Bostic, Jordan West and Roberds, James H.}, year={2009}, pages={371–385} }
 @article{selgrade_roberds_2007, title={Global attractors for a discrete selection model with periodic immigration}, volume={13}, ISSN={["1023-6198"]}, DOI={10.1080/10236190601079100}, abstractNote={A one-island selection-migration model is used to study the periodic immigration of a population of fixed allele frequency into a natural population. Density-dependent selection and immigration are the primary factors affecting the demographic and genetic change in the island population. With the assumptions of complete dominance (CD) or no dominance (ND) and homozygote superiority in fitness, the existence and location of global attractors are established. Analysis of this model provides rudimentary information about the migration of transgenes into a natural population.}, number={4}, journal={JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS}, author={Selgrade, James F. and Roberds, James H.}, year={2007}, month={Apr}, pages={275–287} }
 @article{clark_kulenovic_selgrade_2005, title={On a system of rational difference equations}, volume={11}, ISSN={["1563-5120"]}, DOI={10.1080/10236190412331334464}, abstractNote={We investigate the global character of solutions of the system of difference equations with positive parameters and non-negative initial conditions.}, number={7}, journal={JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS}, author={Clark, CA and Kulenovic, MRS and Selgrade, JF}, year={2005}, month={Jun}, pages={565–580} }
 @article{selgrade_roberds_2005, title={Results on asymptotic behaviour for discrete, two-patch metapopulations with density-dependent selection}, volume={11}, ISSN={["1563-5120"]}, DOI={10.1080/10236190412331335508}, abstractNote={A 4-dimensional system of nonlinear difference equations tracking allele frequencies and population sizes for a two-patch metapopulation model is studied. This system describes intergenerational changes brought about by density-dependent selection within patches and moderated by the effects of migration between patches. To determine conditions which result in similar behaviour at the level of local populations, we introduce the concept of symmetric equilibrium and relate it to properties of allelic and genotypic fitness. We present examples of metapopulation stability, instability and bistability, as well as an example showing that differentially greater migration into a stable patch results in metapopulation stability. Finally, we illustrate a Naimark-Sacker bifurcation giving a globally asymptotically stable invariant curve for the 4-dimensional model.}, number={4-5}, journal={JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS}, author={Selgrade, JF and Roberds, JH}, year={2005}, month={Apr}, pages={459–476} }
 @article{franke_selgrade_2003, title={Attractors for discrete periodic dynamical systems}, volume={286}, ISSN={["0022-247X"]}, DOI={10.1016/S0022-247X(03)00417-7}, abstractNote={A mathematical framework is introduced to study attractors of discrete, nonautonomous dynamical systems which depend periodically on time. A structure theorem for such attractors is established which says that the attractor of a time-periodic dynamical system is the union of attractors of appropriate autonomous maps. If the nonautonomous system is a perturbation of an autonomous map, properties that the nonautonomous attractor inherits from the autonomous attractor are discussed. Examples from population biology are presented.}, number={1}, journal={JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS}, author={Franke, JE and Selgrade, JF}, year={2003}, month={Oct}, pages={64–79} }
 @article{clark_kulenovic_selgrade_2003, title={Global asymptotic behavior of a two-dimensional difference equation modelling competition}, volume={52}, ISSN={["0362-546X"]}, DOI={10.1016/S0362-546X(02)00294-8}, abstractNote={We investigate the global asymptotic behavior of solutions of the system of difference equationsxn+1=xna+cyn,yn+1=ynb+dxn,n=0,1,…,where the parameters a and b are in (0,1), c and d are arbitrary positive numbers and the initial conditions x0 and y0 are arbitrary nonnegative numbers. We show that the stable manifold of this system separates the positive quadrant into basins of attraction of two types of asymptotic behavior. In the case where a=b we find an explicit equation for the stable manifold.}, number={7}, journal={NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS}, author={Clark, D and Kulenovic, MRS and Selgrade, JF}, year={2003}, month={Mar}, pages={1765–1776} }
 @article{clark_schlosser_selgrade_2003, title={Multiple stable periodic solutions in a model for hormonal control of the menstrual cycle}, volume={65}, ISSN={["0092-8240"]}, DOI={10.1006/bulum.2002.0326}, number={1}, journal={BULLETIN OF MATHEMATICAL BIOLOGY}, author={Clark, LH and Schlosser, PM and Selgrade, JF}, year={2003}, month={Jan}, pages={157–173} }
 @article{selgrade_roberds_2001, title={On the structure of attractors for discrete, periodically forced systems with applications to population models}, volume={158}, ISSN={["0167-2789"]}, DOI={10.1016/s0167-2789(01)00324-4}, abstractNote={This work discusses the effects of periodic forcing on attracting cycles and more complicated attractors for autonomous systems of nonlinear difference equations. Results indicate that an attractor for a periodically forced dynamical system may inherit structure from an attractor of the autonomous (unforced) system and also from the periodicity of the forcing. In particular, a method is presented which shows that if the amplitude of the k-periodic forcing is small enough then the attractor for the forced system is the union of k homeomorphic subsets. Examples from population biology and genetics indicate that each subset is also homeomorphic to the attractor of the original autonomous dynamical system.}, number={1-4}, journal={PHYSICA D}, author={Selgrade, JF and Roberds, JH}, year={2001}, month={Oct}, pages={69–82} }
 @article{schlosser_selgrade_2000, title={A model of gonadotropin regulation during the menstrual cycle in women: Qualitative features}, volume={108}, ISSN={["0091-6765"]}, DOI={10.2307/3454321}, journal={ENVIRONMENTAL HEALTH PERSPECTIVES}, author={Schlosser, PM and Selgrade, JF}, year={2000}, month={Oct}, pages={873–881} }
 @article{roberds_selgrade_2000, title={Dynamical analysis of density-dependent selection in a discrete one-island migration model}, volume={164}, ISSN={["0025-5564"]}, DOI={10.1016/s0025-5564(00)00002-x}, abstractNote={A system of non-linear difference equations is used to model the effects of density-dependent selection and migration in a population characterized by two alleles at a single gene locus. Results for the existence and stability of polymorphic equilibria are established. Properties for a genetically important class of equilibria associated with complete dominance in fitness are described. The birth of an unusual chaotic attractor is also illustrated. This attractor is produced when migration causes chaotic dynamics on a boundary of phase space to bifurcate into the interior of phase space, resulting in bistable genetic polymorphic behavior.}, number={1}, journal={MATHEMATICAL BIOSCIENCES}, author={Roberds, JH and Selgrade, JF}, year={2000}, month={Mar}, pages={1–15} }
 @article{selgrade_1998, title={Using stocking or harvesting to reverse period-doubling bifurcations in discrete population models}, volume={4}, ISSN={["1023-6198"]}, DOI={10.1080/10236199808808135}, abstractNote={This study considers a general class of 2-dimensional, discrete population models where each per capita transition function (fitness) depends on a linear combination of the densities of the interacting populations. The fitness functions are either monotone decreasing functions (pioneer fitnesses) or one-humped functions (climax fitnesses). Four sets of necessary inequality conditions are derived which guarantee generically that an equilibrium loses stability through a period-doubling bifurcation with respect to the pioneer self-crowding parameter. A stocking or harvesting term which is proportional to the pioneer density is introduced into the system. Conditions are determined under which this stocking or harvesting will reverse the bifurcation and restabilize the equilibrium. A numerical example illustrates how pioneer stocking can be used to reverse a period-doubling cascade and to maintain the system at any attracting cycle along the cascade.}, number={2}, journal={JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS}, author={Selgrade, JF}, year={1998}, pages={163–183} }
 @article{selgrade_roberds_1997, title={Period-doubling bifurcations for systems of difference equations and applications to models in population biology}, volume={29}, ISSN={["1873-5215"]}, DOI={10.1016/S0362-546X(96)00041-7}, number={2}, journal={NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS}, author={Selgrade, JF and Roberds, JH}, year={1997}, month={Jul}, pages={185–199} }
 @article{selgrade_namkoong_1992, title={Dynamical behavior for population genetics models of differential and difference equations with nonmonotone fitnesses}, volume={30}, number={8}, journal={Journal of Mathematical Biology}, author={Selgrade, J. F. and Namkoong, G.}, year={1992}, pages={815} }
 @article{selgrade_ziehe_1987, title={CONVERGENCE TO EQUILIBRIUM IN A GENETIC MODEL WITH DIFFERENTIAL VIABILITY BETWEEN THE SEXES}, volume={25}, ISSN={["1432-1416"]}, DOI={10.1007/BF00276194}, abstractNote={A single locus, diallelic selection model with female and male viability differences is studied. If the variables are ratios of allele frequencies in each sex, a 2-dimensional difference equation describes the model. Because of the strong monotonicity of the resulting map, every initial genotypic structure converges to an equilibrium structure assuming that no equilibrium has eigenvalues on the unit circle.}, number={5}, journal={JOURNAL OF MATHEMATICAL BIOLOGY}, author={SELGRADE, JF and ZIEHE, M}, year={1987}, pages={477–490} }
 @article{selgrade_namkoong_1984, title={Dynamical behavior of differential equation models of frequency and density dependent populations}, volume={19}, number={1}, journal={Journal of Mathematical Biology}, author={Selgrade, J. F. and Namkoong, G.}, year={1984}, pages={133} }