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- By Rose Teteki Abbey, K. C. Abraham, David Tuesday Adamo, LeRoy H. Aden, Efrain Agosto, Victor Aguilan, Gillian T. W. Ahlgren, Charanjit Kaur AjitSingh, Dorothy B E A Akoto, Giuseppe Alberigo, Daniel E. Albrecht, Ruth Albrecht, Daniel O. Aleshire, Urs Altermatt, Anand Amaladass, Michael Amaladoss, James N. Amanze, Lesley G. Anderson, Thomas C. Anderson, Victor Anderson, Hope S. Antone, María Pilar Aquino, Paula Arai, Victorio Araya Guillén, S. Wesley Ariarajah, Ellen T. Armour, Brett Gregory Armstrong, Atsuhiro Asano, Naim Stifan Ateek, Mahmoud Ayoub, John Alembillah Azumah, Mercedes L. García Bachmann, Irena Backus, J. Wayne Baker, Mieke Bal, Lewis V. Baldwin, William Barbieri, António Barbosa da Silva, David Basinger, Bolaji Olukemi Bateye, Oswald Bayer, Daniel H. Bays, Rosalie Beck, Nancy Elizabeth Bedford, Guy-Thomas Bedouelle, Chorbishop Seely Beggiani, Wolfgang Behringer, Christopher M. Bellitto, Byard Bennett, Harold V. Bennett, Teresa Berger, Miguel A. Bernad, Henley Bernard, Alan E. Bernstein, Jon L. Berquist, Johannes Beutler, Ana María Bidegain, Matthew P. Binkewicz, Jennifer Bird, Joseph Blenkinsopp, Dmytro Bondarenko, Paulo Bonfatti, Riet en Pim Bons-Storm, Jessica A. Boon, Marcus J. Borg, Mark Bosco, Peter C. Bouteneff, François Bovon, William D. Bowman, Paul S. Boyer, David Brakke, Richard E. Brantley, Marcus Braybrooke, Ian Breward, Ênio José da Costa Brito, Jewel Spears Brooker, Johannes Brosseder, Nicholas Canfield Read Brown, Robert F. Brown, Pamela K. Brubaker, Walter Brueggemann, Bishop Colin O. Buchanan, Stanley M. Burgess, Amy Nelson Burnett, J. Patout Burns, David B. Burrell, David Buttrick, James P. Byrd, Lavinia Byrne, Gerado Caetano, Marcos Caldas, Alkiviadis Calivas, William J. Callahan, Salvatore Calomino, Euan K. Cameron, William S. Campbell, Marcelo Ayres Camurça, Daniel F. Caner, Paul E. Capetz, Carlos F. Cardoza-Orlandi, Patrick W. Carey, Barbara Carvill, Hal Cauthron, Subhadra Mitra Channa, Mark D. Chapman, James H. Charlesworth, Kenneth R. Chase, Chen Zemin, Luciano Chianeque, Philip Chia Phin Yin, Francisca H. Chimhanda, Daniel Chiquete, John T. Chirban, Soobin Choi, Robert Choquette, Mita Choudhury, Gerald Christianson, John Chryssavgis, Sejong Chun, Esther Chung-Kim, Charles M. A. Clark, Elizabeth A. Clark, Sathianathan Clarke, Fred Cloud, John B. Cobb, W. Owen Cole, John A Coleman, John J. Collins, Sylvia Collins-Mayo, Paul K. Conkin, Beth A. Conklin, Sean Connolly, Demetrios J. Constantelos, Michael A. Conway, Paula M. Cooey, Austin Cooper, Michael L. Cooper-White, Pamela Cooper-White, L. William Countryman, Sérgio Coutinho, Pamela Couture, Shannon Craigo-Snell, James L. Crenshaw, David Crowner, Humberto Horacio Cucchetti, Lawrence S. Cunningham, Elizabeth Mason Currier, Emmanuel Cutrone, Mary L. Daniel, David D. Daniels, Robert Darden, Rolf Darge, Isaiah Dau, Jeffry C. Davis, Jane Dawson, Valentin Dedji, John W. de Gruchy, Paul DeHart, Wendy J. Deichmann Edwards, Miguel A. De La Torre, George E. Demacopoulos, Thomas de Mayo, Leah DeVun, Beatriz de Vasconcellos Dias, Dennis C. Dickerson, John M. Dillon, Luis Miguel Donatello, Igor Dorfmann-Lazarev, Susanna Drake, Jonathan A. Draper, N. Dreher Martin, Otto Dreydoppel, Angelyn Dries, A. J. Droge, Francis X. D'Sa, Marilyn Dunn, Nicole Wilkinson Duran, Rifaat Ebied, Mark J. Edwards, William H. Edwards, Leonard H. Ehrlich, Nancy L. Eiesland, Martin Elbel, J. Harold Ellens, Stephen Ellingson, Marvin M. Ellison, Robert Ellsberg, Jean Bethke Elshtain, Eldon Jay Epp, Peter C. Erb, Tassilo Erhardt, Maria Erling, Noel Leo Erskine, Gillian R. Evans, Virginia Fabella, Michael A. Fahey, Edward Farley, Margaret A. Farley, Wendy Farley, Robert Fastiggi, Seena Fazel, Duncan S. Ferguson, Helwar Figueroa, Paul Corby Finney, Kyriaki Karidoyanes FitzGerald, Thomas E. FitzGerald, John R. Fitzmier, Marie Therese Flanagan, Sabina Flanagan, Claude Flipo, Ronald B. Flowers, Carole Fontaine, David Ford, Mary Ford, Stephanie A. Ford, Jim Forest, William Franke, Robert M. Franklin, Ruth Franzén, Edward H. Friedman, Samuel Frouisou, Lorelei F. Fuchs, Jojo M. Fung, Inger Furseth, Richard R. Gaillardetz, Brandon Gallaher, China Galland, Mark Galli, Ismael García, Tharscisse Gatwa, Jean-Marie Gaudeul, Luis María Gavilanes del Castillo, Pavel L. Gavrilyuk, Volney P. Gay, Metropolitan Athanasios Geevargis, Kondothra M. George, Mary Gerhart, Simon Gikandi, Maurice Gilbert, Michael J. Gillgannon, Verónica Giménez Beliveau, Terryl Givens, Beth Glazier-McDonald, Philip Gleason, Menghun Goh, Brian Golding, Bishop Hilario M. Gomez, Michelle A. Gonzalez, Donald K. Gorrell, Roy Gottfried, Tamara Grdzelidze, Joel B. Green, Niels Henrik Gregersen, Cristina Grenholm, Herbert Griffiths, Eric W. Gritsch, Erich S. Gruen, Christoffer H. Grundmann, Paul H. Gundani, Jon P. Gunnemann, Petre Guran, Vidar L. Haanes, Jeremiah M. Hackett, Getatchew Haile, Douglas John Hall, Nicholas Hammond, Daphne Hampson, Jehu J. Hanciles, Barry Hankins, Jennifer Haraguchi, Stanley S. Harakas, Anthony John Harding, Conrad L. Harkins, J. William Harmless, Marjory Harper, Amir Harrak, Joel F. Harrington, Mark W. Harris, Susan Ashbrook Harvey, Van A. Harvey, R. Chris Hassel, Jione Havea, Daniel Hawk, Diana L. Hayes, Leslie Hayes, Priscilla Hayner, S. Mark Heim, Simo Heininen, Richard P. Heitzenrater, Eila Helander, David Hempton, Scott H. Hendrix, Jan-Olav Henriksen, Gina Hens-Piazza, Carter Heyward, Nicholas J. Higham, David Hilliard, Norman A. Hjelm, Peter C. Hodgson, Arthur Holder, M. Jan Holton, Dwight N. Hopkins, Ronnie Po-chia Hsia, Po-Ho Huang, James Hudnut-Beumler, Jennifer S. Hughes, Leonard M. Hummel, Mary E. Hunt, Laennec Hurbon, Mark Hutchinson, Susan E. Hylen, Mary Beth Ingham, H. Larry Ingle, Dale T. Irvin, Jon Isaak, Paul John Isaak, Ada María Isasi-Díaz, Hans Raun Iversen, Margaret C. Jacob, Arthur James, Maria Jansdotter-Samuelsson, David Jasper, Werner G. Jeanrond, Renée Jeffery, David Lyle Jeffrey, Theodore W. Jennings, David H. Jensen, Robin Margaret Jensen, David Jobling, Dale A. Johnson, Elizabeth A. Johnson, Maxwell E. Johnson, Sarah Johnson, Mark D. Johnston, F. Stanley Jones, James William Jones, John R. Jones, Alissa Jones Nelson, Inge Jonsson, Jan Joosten, Elizabeth Judd, Mulambya Peggy Kabonde, Robert Kaggwa, Sylvester Kahakwa, Isaac Kalimi, Ogbu U. Kalu, Eunice Kamaara, Wayne C. Kannaday, Musimbi Kanyoro, Veli-Matti Kärkkäinen, Frank Kaufmann, Léon Nguapitshi Kayongo, Richard Kearney, Alice A. Keefe, Ralph Keen, Catherine Keller, Anthony J. Kelly, Karen Kennelly, Kathi Lynn Kern, Fergus Kerr, Edward Kessler, George Kilcourse, Heup Young Kim, Kim Sung-Hae, Kim Yong-Bock, Kim Yung Suk, Richard King, Thomas M. King, Robert M. Kingdon, Ross Kinsler, Hans G. Kippenberg, Cheryl A. Kirk-Duggan, Clifton Kirkpatrick, Leonid Kishkovsky, Nadieszda Kizenko, Jeffrey Klaiber, Hans-Josef Klauck, Sidney Knight, Samuel Kobia, Robert Kolb, Karla Ann Koll, Heikki Kotila, Donald Kraybill, Philip D. W. Krey, Yves Krumenacker, Jeffrey Kah-Jin Kuan, Simanga R. Kumalo, Peter Kuzmic, Simon Shui-Man Kwan, Kwok Pui-lan, André LaCocque, Stephen E. Lahey, John Tsz Pang Lai, Emiel Lamberts, Armando Lampe, Craig Lampe, Beverly J. Lanzetta, Eve LaPlante, Lizette Larson-Miller, Ariel Bybee Laughton, Leonard Lawlor, Bentley Layton, Robin A. Leaver, Karen Lebacqz, Archie Chi Chung Lee, Marilyn J. Legge, Hervé LeGrand, D. L. LeMahieu, Raymond Lemieux, Bill J. Leonard, Ellen M. Leonard, Outi Leppä, Jean Lesaulnier, Nantawan Boonprasat Lewis, Henrietta Leyser, Alexei Lidov, Bernard Lightman, Paul Chang-Ha Lim, Carter Lindberg, Mark R. Lindsay, James R. Linville, James C. Livingston, Ann Loades, David Loades, Jean-Claude Loba-Mkole, Lo Lung Kwong, Wati Longchar, Eleazar López, David W. Lotz, Andrew Louth, Robin W. Lovin, William Luis, Frank D. Macchia, Diarmaid N. J. MacCulloch, Kirk R. MacGregor, Marjory A. MacLean, Donald MacLeod, Tomas S. Maddela, Inge Mager, Laurenti Magesa, David G. Maillu, Fortunato Mallimaci, Philip Mamalakis, Kä Mana, Ukachukwu Chris Manus, Herbert Robinson Marbury, Reuel Norman Marigza, Jacqueline Mariña, Antti Marjanen, Luiz C. L. Marques, Madipoane Masenya (ngwan'a Mphahlele), Caleb J. D. Maskell, Steve Mason, Thomas Massaro, Fernando Matamoros Ponce, András Máté-Tóth, Odair Pedroso Mateus, Dinis Matsolo, Fumitaka Matsuoka, John D'Arcy May, Yelena Mazour-Matusevich, Theodore Mbazumutima, John S. McClure, Christian McConnell, Lee Martin McDonald, Gary B. McGee, Thomas McGowan, Alister E. McGrath, Richard J. McGregor, John A. McGuckin, Maud Burnett McInerney, Elsie Anne McKee, Mary B. McKinley, James F. McMillan, Ernan McMullin, Kathleen E. McVey, M. Douglas Meeks, Monica Jyotsna Melanchthon, Ilie Melniciuc-Puica, Everett Mendoza, Raymond A. Mentzer, William W. Menzies, Ina Merdjanova, Franziska Metzger, Constant J. Mews, Marvin Meyer, Carol Meyers, Vasile Mihoc, Gunner Bjerg Mikkelsen, Maria Inêz de Castro Millen, Clyde Lee Miller, Bonnie J. Miller-McLemore, Alexander Mirkovic, Paul Misner, Nozomu Miyahira, R. W. L. Moberly, Gerald Moede, Aloo Osotsi Mojola, Sunanda Mongia, Rebeca Montemayor, James Moore, Roger E. Moore, Craig E. Morrison O.Carm, Jeffry H. Morrison, Keith Morrison, Wilson J. Moses, Tefetso Henry Mothibe, Mokgethi Motlhabi, Fulata Moyo, Henry Mugabe, Jesse Ndwiga Kanyua Mugambi, Peggy Mulambya-Kabonde, Robert Bruce Mullin, Pamela Mullins Reaves, Saskia Murk Jansen, Heleen L. Murre-Van den Berg, Augustine Musopole, Isaac M. T. Mwase, Philomena Mwaura, Cecilia Nahnfeldt, Anne Nasimiyu Wasike, Carmiña Navia Velasco, Thulani Ndlazi, Alexander Negrov, James B. Nelson, David G. Newcombe, Carol Newsom, Helen J. Nicholson, George W. E. Nickelsburg, Tatyana Nikolskaya, Damayanthi M. A. Niles, Bertil Nilsson, Nyambura Njoroge, Fidelis Nkomazana, Mary Beth Norton, Christian Nottmeier, Sonene Nyawo, Anthère Nzabatsinda, Edward T. Oakes, Gerald O'Collins, Daniel O'Connell, David W. Odell-Scott, Mercy Amba Oduyoye, Kathleen O'Grady, Oyeronke Olajubu, Thomas O'Loughlin, Dennis T. Olson, J. Steven O'Malley, Cephas N. Omenyo, Muriel Orevillo-Montenegro, César Augusto Ornellas Ramos, Agbonkhianmeghe E. Orobator, Kenan B. Osborne, Carolyn Osiek, Javier Otaola Montagne, Douglas F. Ottati, Anna May Say Pa, Irina Paert, Jerry G. Pankhurst, Aristotle Papanikolaou, Samuele F. Pardini, Stefano Parenti, Peter Paris, Sung Bae Park, Cristián G. Parker, Raquel Pastor, Joseph Pathrapankal, Daniel Patte, W. Brown Patterson, Clive Pearson, Keith F. Pecklers, Nancy Cardoso Pereira, David Horace Perkins, Pheme Perkins, Edward N. Peters, Rebecca Todd Peters, Bishop Yeznik Petrossian, Raymond Pfister, Peter C. Phan, Isabel Apawo Phiri, William S. F. Pickering, Derrick G. Pitard, William Elvis Plata, Zlatko Plese, John Plummer, James Newton Poling, Ronald Popivchak, Andrew Porter, Ute Possekel, James M. Powell, Enos Das Pradhan, Devadasan Premnath, Jaime Adrían Prieto Valladares, Anne Primavesi, Randall Prior, María Alicia Puente Lutteroth, Eduardo Guzmão Quadros, Albert Rabil, Laurent William Ramambason, Apolonio M. Ranche, Vololona Randriamanantena Andriamitandrina, Lawrence R. Rast, Paul L. Redditt, Adele Reinhartz, Rolf Rendtorff, Pål Repstad, James N. Rhodes, John K. Riches, Joerg Rieger, Sharon H. Ringe, Sandra Rios, Tyler Roberts, David M. Robinson, James M. Robinson, Joanne Maguire Robinson, Richard A. H. Robinson, Roy R. Robson, Jack B. Rogers, Maria Roginska, Sidney Rooy, Rev. Garnett Roper, Maria José Fontelas Rosado-Nunes, Andrew C. Ross, Stefan Rossbach, François Rossier, John D. Roth, John K. Roth, Phillip Rothwell, Richard E. Rubenstein, Rosemary Radford Ruether, Markku Ruotsila, John E. Rybolt, Risto Saarinen, John Saillant, Juan Sanchez, Wagner Lopes Sanchez, Hugo N. Santos, Gerhard Sauter, Gloria L. Schaab, Sandra M. Schneiders, Quentin J. Schultze, Fernando F. Segovia, Turid Karlsen Seim, Carsten Selch Jensen, Alan P. F. Sell, Frank C. Senn, Kent Davis Sensenig, Damían Setton, Bal Krishna Sharma, Carolyn J. Sharp, Thomas Sheehan, N. Gerald Shenk, Christian Sheppard, Charles Sherlock, Tabona Shoko, Walter B. Shurden, Marguerite Shuster, B. Mark Sietsema, Batara Sihombing, Neil Silberman, Clodomiro Siller, Samuel Silva-Gotay, Heikki Silvet, John K. Simmons, Hagith Sivan, James C. Skedros, Abraham Smith, Ashley A. Smith, Ted A. Smith, Daud Soesilo, Pia Søltoft, Choan-Seng (C. S.) Song, Kathryn Spink, Bryan Spinks, Eric O. Springsted, Nicolas Standaert, Brian Stanley, Glen H. Stassen, Karel Steenbrink, Stephen J. Stein, Andrea Sterk, Gregory E. Sterling, Columba Stewart, Jacques Stewart, Robert B. Stewart, Cynthia Stokes Brown, Ken Stone, Anne Stott, Elizabeth Stuart, Monya Stubbs, Marjorie Hewitt Suchocki, David Kwang-sun Suh, Scott W. Sunquist, Keith Suter, Douglas Sweeney, Charles H. Talbert, Shawqi N. Talia, Elsa Tamez, Joseph B. Tamney, Jonathan Y. Tan, Yak-Hwee Tan, Kathryn Tanner, Feiya Tao, Elizabeth S. Tapia, Aquiline Tarimo, Claire Taylor, Mark Lewis Taylor, Bishop Abba Samuel Wolde Tekestebirhan, Eugene TeSelle, M. Thomas Thangaraj, David R. Thomas, Andrew Thornley, Scott Thumma, Marcelo Timotheo da Costa, George E. “Tink” Tinker, Ola Tjørhom, Karen Jo Torjesen, Iain R. Torrance, Fernando Torres-Londoño, Archbishop Demetrios [Trakatellis], Marit Trelstad, Christine Trevett, Phyllis Trible, Johannes Tromp, Paul Turner, Robert G. Tuttle, Archbishop Desmond Tutu, Peter Tyler, Anders Tyrberg, Justin Ukpong, Javier Ulloa, Camillus Umoh, Kristi Upson-Saia, Martina Urban, Monica Uribe, Elochukwu Eugene Uzukwu, Richard Vaggione, Gabriel Vahanian, Paul Valliere, T. J. Van Bavel, Steven Vanderputten, Peter Van der Veer, Huub Van de Sandt, Louis Van Tongeren, Luke A. Veronis, Noel Villalba, Ramón Vinke, Tim Vivian, David Voas, Elena Volkova, Katharina von Kellenbach, Elina Vuola, Timothy Wadkins, Elaine M. Wainwright, Randi Jones Walker, Dewey D. Wallace, Jerry Walls, Michael J. Walsh, Philip Walters, Janet Walton, Jonathan L. Walton, Wang Xiaochao, Patricia A. Ward, David Harrington Watt, Herold D. Weiss, Laurence L. Welborn, Sharon D. Welch, Timothy Wengert, Traci C. West, Merold Westphal, David Wetherell, Barbara Wheeler, Carolinne White, Jean-Paul Wiest, Frans Wijsen, Terry L. Wilder, Felix Wilfred, Rebecca Wilkin, Daniel H. Williams, D. Newell Williams, Michael A. Williams, Vincent L. Wimbush, Gabriele Winkler, Anders Winroth, Lauri Emílio Wirth, James A. Wiseman, Ebba Witt-Brattström, Teofil Wojciechowski, John Wolffe, Kenman L. Wong, Wong Wai Ching, Linda Woodhead, Wendy M. Wright, Rose Wu, Keith E. Yandell, Gale A. Yee, Viktor Yelensky, Yeo Khiok-Khng, Gustav K. K. Yeung, Angela Yiu, Amos Yong, Yong Ting Jin, You Bin, Youhanna Nessim Youssef, Eliana Yunes, Robert Michael Zaller, Valarie H. Ziegler, Barbara Brown Zikmund, Joyce Ann Zimmerman, Aurora Zlotnik, Zhuo Xinping
- Edited by Daniel Patte, Vanderbilt University, Tennessee
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- The Cambridge Dictionary of Christianity
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- 05 August 2012
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- 20 September 2010, pp xi-xliv
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Convergence in Lotka-Volterra type diffusive delay systems without dominating instantaneous negative feedbacks
- Yang Kuang, Hal L. Smith
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- Journal of the Australian Mathematical Society. Series B. Applied Mathematics / Volume 34 / Issue 4 / April 1993
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- 17 February 2009, pp. 471-494
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This paper deals with the convergence aspect of diffusive delay Lotka-Volterra systems with infinite delays. It is well known that such a system has a globally asymptotically stable steady state if the negative feedbacks of the intraspecific competitions are dominant and instantaneous. It is shown here that such a globally asymptotically stable steady state continues to exist even if the instantaneous assumption is removed, provided that solutions of the system are eventually uniformly bounded and the delays involved in the intraspecific competitions are small. This work generalises several recent related ones.
Co-Carriage Rates of Vancomycin-Resistant Enterococcus and Extended-Spectrum Beta-Lactamase-Producing Bacteria Among a Cohort of Intensive Care Unit Patients: Implications for an Active Surveillance Program
- Anthony D. Harris, Lucia Nemoy, Judith A. Johnson, Amy Martin-Carnahan, David L. Smith, Hal Standiford, Eli N. Perencevich
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- Infection Control & Hospital Epidemiology / Volume 25 / Issue 2 / February 2004
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- 02 January 2015, pp. 105-108
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- February 2004
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Objective:
To assess the co-colonization rates of extended-spectrum beta-lactamase (ESBL)-producing bacteria and vancomycin-resistant Enterococcus (VRE) obtained on active surveillance cultures.
Design:Prospective cohort study.
Setting:Medical and surgical intensive care units (ICUs) of a tertiary-care hospital.
Patients:Patients admitted between September 2001 and November 2002 to the medical and surgical ICUs at the University of Maryland Medical System had active surveillance perirectal cultures performed. Samples were concurrently processed for VRE and ESBL-producing bacteria.
Results:Of 1,362 patients who had active surveillance cultures on admission, 136 (10%) were colonized with VRE. Among these, 15 (positive predictive value, 11%) were co-colonized with ESBL. Among the 1,226 who were VRE negative, 1,209 were also ESBL negative (negative predictive value, 99%). Among the 1,362 who had active surveillance cultures on admission, 32 (2%) were colonized with ESBL. Among these, 15 (47%) were co-colonized with VRE. Of the 32 patients colonized with ESBL, 10 (31%) had positive clinical cultures for ESBL on the same hospital admission. For these 10 patients, the surveillance cultures were positive an average of 2.7 days earlier than the clinical cultures.
Conclusions:Patients who are colonized with VRE can also be co-colonized with other antibiotic-resistant bacteria such as ESBL-producing bacteria. Our study is the first to measure co-colonization rates of VRE and ESBL-producing bacteria. Isolating VRE-colonized patients would isolate 47% of the ESBL-colonized patients without the need for further testing. Hence, active surveillance for VRE should also theoretically diminish the amount of patient-to-patient transmission of ESBL-producing bacteria.
Frontmatter
- Hal L. Smith, Arizona State University, Paul Waltman, Emory University, Atlanta
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- The Theory of the Chemostat
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3 - Competition on Three Trophic Levels
- Hal L. Smith, Arizona State University, Paul Waltman, Emory University, Atlanta
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- The Theory of the Chemostat
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Summary
Introduction
The previous two chapters showed that competitive exclusion holds under a variety of conditions in the chemostat and its modifications. In this chapter it will be shown that if the competition is moved up one level – if the competition occurs among predators of an organism growing on the nutrient – then coexistence may occur. The fact that the competitors are at a higher trophic level allows for oscillations, and the coexistence that occurs is in the form of a stable limit cycle. Along the way it will be necessary to study a three-level food-chain problem which is of interest in its own right; it is the chemostat version of predator–prey equations. The presentation follows that of [BHW1].
Although most of the results can be established with mathematical rigor, there are some elusive problems. These center around the possibility of multiple limit cycles and the difficulty of determining the stability of such limit cycles. At this point one must simply make a hypothesis and resort to numerical evidence in any specific case. Determining the number of limit cycles is a deep mathematical problem, and even in very simple cases the solution is not known. Hilbert's famous sixteenth problem, concerning the number of limit cycles of a second-order system with polynomial right-hand sides, remains basically unresolved. In principle, the stability of a limit cycle can be determined from the Floquet exponents (see Section 4), but this is a notoriously difficult computation – indeed, generally an impossible one.
Appendices
- Hal L. Smith, Arizona State University, Paul Waltman, Emory University, Atlanta
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- The Theory of the Chemostat
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- 27 January 1995, pp 253-254
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9 - A Size-Structured Competition Model
- Hal L. Smith, Arizona State University, Paul Waltman, Emory University, Atlanta
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- The Theory of the Chemostat
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Summary
Introduction
The models considered in previous chapters of this monograph have ignored the fact that populations of microorganisms contain individuals with differing body size and that individuals of different size have different characteristics. Body size is clearly an important factor in determining an organism's energy requirements and its ability to uptake resources. Furthermore, if an organism can grow as well as reproduce, then it becomes important to determine how the organism allocates its energy resources between growth and reproduction.
Data from [Wi] – particularly figures 3, 6, 8, 10, 18, 19, and 21 – leave no doubt that, at least for certain populations of algae, individual cell volume varies significantly during the course of experiments in the chemostat. These data also suggest that steady-state size distributions are reached which have remarkably stable shapes with respect to changes in the control parameters for the chemostat (flow rates, temperature, CO2).
The purpose of this chapter is to present a model of competition in the chemostat for a single resource that takes these factors into account, and to determine the extent to which these factors influence the outcome of competition. The model presented here is a special case of a more general class of models formulated in [DMKH] and treated in [MD, sec. 1.3]. This simplified model is one of several considered in an elegant paper by Cushing [Cu2]. Most of the results of this chapter are taken from [Cu2].
2 - The General Chemostat
- Hal L. Smith, Arizona State University, Paul Waltman, Emory University, Atlanta
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- The Theory of the Chemostat
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Summary
Introduction
In the previous chapter it was shown that the simple chemostat produces competitive exclusion. It could be argued that the result was due to the two-dimensional nature of the limiting problem (and the applicability of the Poincaré–Bendixson theorem) or that this was a result of the particular type of dynamics produced by the Michaelis–Menten hypothesis on the functional response. This last point was the focus of some controversy at one time, inducing the proposal of alternative responses. In this chapter it will be shown that neither additional populations nor the replacement of the Michaelis–Menten hypothesis by a monotone (or even nonmonotone) uptake function is sufficient to produce coexistence of the competitors in a chemostat. This illustrates the robustness of the results of Chapter 1. It will also be shown that the introduction of differing “death rates” (replacing the parameter D by Di in the equations) does not change the competitive exclusion result.
A brief mathematical digression on Liapunov stability theory will set the stage for the results of this chapter. Those familiar with the LaSalle corollary to Liapunov stability theory, also called the “invariance principle” by some authors, can skip immediately to Section 3.
Liapunov Theory
A more sophisticated mathematical approach is required for the results of this chapter because the equations considered cannot be reduced to planar systems. The theorem described in this section and used throughout this chapter is one of great power.
E - Some Techniques in Nonlinear Analysis
- Hal L. Smith, Arizona State University, Paul Waltman, Emory University, Atlanta
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- The Theory of the Chemostat
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- 27 January 1995, pp 282-293
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Subject index
- Hal L. Smith, Arizona State University, Paul Waltman, Emory University, Atlanta
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- The Theory of the Chemostat
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- 27 January 1995, pp 311-313
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Author index
- Hal L. Smith, Arizona State University, Paul Waltman, Emory University, Atlanta
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- The Theory of the Chemostat
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- 27 January 1995, pp 309-310
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The Theory of the Chemostat
- Dynamics of Microbial Competition
- Hal L. Smith, Paul Waltman
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- 27 January 1995
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The chemostat is a basic piece of laboratory apparatus, yet it has occupied an increasingly central role in ecological studies. The ecological environment created by a chemostat is one of the few completely controlled experimental systems for testing microbial growth and competition. As a tool in biotechnology, the chemostat plays an important role in bioprocessing. This book presents the theory of the chemostat as a model for larger ecological problems such as food chains, competition along a gradient, competition in the presence of an inhibitor, and the effects of time varying inputs. Models which take account of size structure, variable yields, and diffusion are also considered. The basic phenomena are modelled and analysed using the dynamical systems approach. Directions for research and open problems are discussed. Six appendices provide an elementary description of the necessary mathematical tools. Teachers, researchers, and students in applied mathematics, chemical engineering and ecology will find this book a welcome resource.
Preface
- Hal L. Smith, Arizona State University, Paul Waltman, Emory University, Atlanta
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- The Theory of the Chemostat
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The chemostat is a basic piece of laboratory apparatus, yet it occupies a central place in mathematical ecology. Its importance stems from the many roles it plays. It is a model of a simple lake, the ideal place to study competition in its most primitive form – exploitative competition. It is also used as a model of the wastewater treatment process. In its commercial form the chemostat plays a central role in certain fermentation processes, particularly in the commercial production of products by genetically altered organisms (e.g., in the production of insulin). The theoretical literature is scattered; papers appear in mathematical, biological, and chemical engineering journals. In addition to being known as a chemostat, other common names are “continuous culture” and, in the engineering literature, CSTR (continuously stirred tank reactor). This monograph is devoted to the theoretical description of ecological models based on the chemostat and to problems that can be described in that context, drawing from literature in all fields and presented within a common framework and with consistent notation.
In order to understand the mathematical importance of the chemostat, one must look at the broader picture of the subject of nonlinear differential equations. Linear differential equations have been studied for more than two hundred years; their solutions have a rich structure that has been well worked out and exploited in physics, chemistry, and biology. A vast and challenging new world opens up when one turns to nonlinear differential equations.
6 - The General Gradostat
- Hal L. Smith, Arizona State University, Paul Waltman, Emory University, Atlanta
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- The Theory of the Chemostat
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- 27 January 1995, pp 129-158
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Summary
Introduction
In the previous chapter the gradostat was introduced as a model of competition along a nutrient gradient. The case of two competitors and two vessels with Michaelis–Menten uptake functions was explored in considerable detail. In this chapter the restriction to two vessels and to Michaelis–Menten uptake will be removed, and a much more general version of the gradostat will be introduced. The results in the previous chapter were obtained by a mixture of dynamical systems techniques and specific computations that established the uniqueness and stability of the coexistence rest point. When the number of vessels is increased and the restriction to Michaelis–Menten uptake functions is relaxed, these computations are inconclusive. It turns out that unstable positive rest points are possible and that non-uniqueness of the coexistence rest point cannot be excluded. The main result of this chapter is that coexistence of two microbial populations in a gradostat is possible in the sense that the concentration of each population in each vessel approaches a positive equilibrium value. The main difference with the previous chapter is that we cannot exclude the possibility of more than one coexistence rest point.
Throughout this chapter we rely extensively on the results contained in Appendices A, B, and C. The presentation here follows closely that in [STW] (see also the review [SW2]).
The most straightforward generalization of the work of Chapter 5 would be simply to extend the number of vessels from two to an arbitrary number, say n.
F - A Convergence Theorem
- Hal L. Smith, Arizona State University, Paul Waltman, Emory University, Atlanta
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B - Differential Inequalities
- Hal L. Smith, Arizona State University, Paul Waltman, Emory University, Atlanta
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A - Matrices and Their Eigenvalues
- Hal L. Smith, Arizona State University, Paul Waltman, Emory University, Atlanta
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References
- Hal L. Smith, Arizona State University, Paul Waltman, Emory University, Atlanta
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5 - The Simple Gradostat
- Hal L. Smith, Arizona State University, Paul Waltman, Emory University, Atlanta
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Summary
Introduction
In the preceding chapter we saw how the chemostat could be modified to account for a new phenomenon – the presence of an inhibitor. In this chapter we extend the idea behind the simple chemostat to a new apparatus in order to model a property of ecological systems that it is not possible to model in the simple chemostat. The idea is to capture the essentials of the new phenomenon without destroying the tractability of the chemostat either as a mathematical model or as an experimental one. A very simple situation will be described here; a more complicated one – with a less explicit (in the sense of less computable) analysis – will be discussed in the next chapter. Just as the chemostat is a basic model for competition in the simplest situation, the apparatus here shows promise of being the model for competition along a nutrient gradient.
The “well mixed” hypothesis for the chemostat does not allow a nutrient gradient to be generated. A basic tenet is that the nutrient concentration is the same everywhere; hence any advantage in nutrient consumption is present everywhere. The model that incorporates a true gradient would be one involving partial differential equations; a new variable, space, must be accommodated. Systems of nonlinear partial differential equations are difficult mathematical objects to understand and analyze. Even numerical solutions pose added and significant difficulties. Moreover, even if an experimental gradient is devised, measurements that do not disturb the local environment take on new difficulties.
4 - The Chemostat with an Inhibitor
- Hal L. Smith, Arizona State University, Paul Waltman, Emory University, Atlanta
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Summary
Introduction
In the first two chapters the general theory of the chemostat was developed, and it was shown that competitive exclusion is the expected outcome. In Chapter 3, coexistence was shown to occur when the competition was at a higher trophic level; the mechanism was simply the oscillation of the object of the competition – the prey in the case being considered. In this chapter, we return to the basic chemostat model but add another factor, the presence of an inhibitor. The inhibitor affects the nutrient uptake rate of one of the competitors but is taken up by the other without ill effect. The use of Nalidixic acid in the experiments of Hansen and Hubbell [HH], discussed in Chapter 1, is an example. Its effect on one strain of E. coli was essentially nil while the growth rate of the other was severely diminished.
The interest in this subject goes far beyond laboratory examples. It is common for one strain of bacteria to be affected by an antibiotic while another is resistant. The antibiotic acts as the inhibitor in our discussion, and we want to know whether the resistant strain will eliminate the nonresistant strain. If it can, the antibiotic will not be effective in treating the disease. In commercial applications, genetically altered organisms are used to generate “products”.