Particle systems with locally dependent branching : long-time behaviour, genealogy and critical parameters

We consider the long-time behaviour of spatially extended random populations with locally dependent branching. We treat two classes of models: 1) Systems of continuous-time random walks on the d-dimensional grid with state dependent branching rate. While there are k particles at a given site, a branching event occurs there at rate s(k), and one of the particles is replaced by a random number of offspring (according to a fixed distribution with mean 1 and finite variance). 2) Discrete-time systems of branching random walks in random environment. Given a space-time i.i.d. field of random offspring distributions, all particles act independently, the offspring law of a given particle depending on its position and generation. The mean number of children per individual, averaged over the random environment, equals one The long-time behaviour is determined by the interplay of the motion and the branching mechanism: In the case of recurrent symmetrised individual motion, systems of the second type become locally extinct. We prove a comparison theorem for convex functionals of systems of type one which implies that these systems also become locally extinct in this case, provided that the branching rate function grows at least linearly. Furthermore, the analysis of a caricature model leads to the conjecture that local extinction prevails generically in this case. In the case of transient symmetrised individual motion the picture is more complex: Branching random walks with state dependent branching rate converge towards a non-trivial equilibrium, which preserves the initial intensity, whenever the branching rate function grows subquadratically. Systems of type 1) and systems of type 2) with quadratic branching rate function show very similar behaviour. They converge towards a non-trivial equilibrium if a conditional exponential moment of the collision time of two random walks of an order that reflects the variability in the branching mechanism is finite almost surely. The equilibrium population has finite variance of the local particle number if the corresponding unconditional exponential moment is finite. These results are proved by means of genealogical representations of the locally size-biased population. Furthermore, we compute the threshold values for existence of conditional exponential moments of the collision time of two random walks in terms of the entropy of the transition functions, using tools from large deviations theory. Our results prove in particular that - in contrast to the classical case of independent branching - there is a regime of equilibria with variance of the local number of particles.Wir betrachten das Langzeitverhalten von zufälligen, räumlich ausgebreiteten Populationen mit lokal abhängiger Verzweigung, speziell werden zwei Klassen von Modellen untersucht: 1) Zeitkontinuierliche Systeme von Irrfahrten auf dem d-dimensionalen Gitter mit zustandsabhängiger Verzweigungsrate. Wenn an einem Ort gerade k Teilchen sind, findet dort ein Verzweigungsereignis mit Rate s(k) statt, und eines der Teilchen wird durch eine zufällige Anzahl Nachkommen (gemäß einer vorgegebenen Verteilung mit Mittelwert 1 und endlicher Varianz) ersetzt. 2) Zeitdiskrete Systeme von verzweigenden Irrfahrten in zufälliger Umgebung. Wir betrachten ein Raum-Zeit-Feld von unabhängigen Kinderzahlverteilungen, gegeben dieses Feld verhalten sich alle Teilchen unabhängig, die Verteilung der Anzahl Nachkommen eines Teilchens hängt von seiner Position und seiner Generation ab. Die mittlere Anzahl Nachkommen pro Individuum, gemittelt über die zufällige Umgebung, ist exakt eins. Das Langzeitverhalten wird durch das Zusammenspiel von Bewegungs- und Verzweigungsmechanismus bestimmt: Bei rekurrenter symmetrisierter Individualbewegung sterben Systeme vom zweiten Typ stets aus. Wir beweisen ein Vergleichresultat für konvexe Funktionale von Systemen vom Typ 1, das impliziert, dass auch dort im rekurrenten Fall lokales Aussterben vorherrscht, sofern die Verzweigungsratenfunktion mindestens linear wächst. Darüberhinaus erhärten wir anhand eines Karikaturmodells die Vermutung, dass lokales Aussterben generisch vorliegt. Im Fall transienter symmetrisierter Individualbewegung bietet sich ein reichhaltigeres Bild: Verzweigende Irrfahrten mit zustandsabhängiger Verzweigung konvergieren gegen ein nicht-triviales Gleichgewicht, das die Anfangsintensität erhält, sofern die Verzweigungsratenfunktion subquadratisch wächst. Wir zeigen eine Parallele zwischen Systemen vom Typ 2 und Systemen vom Typ 1 mit quadratischer Verzweigungsratenfunktion. Wenn ein bedingtes exponentielles Moment der Kollisionszeit zweier unabhängiger Irrfahrten von einer Ordnung, die von der Variabilität im Verzweigungsmechanismus abhängt, fast sicher endlich bleibt, so konvergieren die Systeme gegen nichttriviale Gleichgewichte. Die zweiten Momente der lokalen Teilchenanzahlen im Gleichgewicht sind genau dann endlich, wenn auch das entsprechende unbedingte Moment endlich ist. Wir erzielen diese Resultate mittels genealogischer Darstellungen der lokal größenverzerrten Populationen. Darüberhinaus berechnen wir unter Verwendung von Hilfsmitteln aus der Theorie der großen Abweichungen die Schwellwerte für die Existenz der bedingten exponentiellen Momente der Kollisionszeit in Termen der Entropie der Übergangsmatrizen der Irrfahrt. Dies zeigt insbesondere, dass - im Gegensatz zum klassischen Fall unabhängiger Verzweigung - ein Regime von Gleichgewichten mit unendlicher Varianz der lokalen Anzahl existiert

A Troubled Transition: From President Morgan to President Waugh

Dickinson College\u27s twentieth-century journey has been marked primarily, though not entirely, by gains: increases in numbers of students and faculty, advances in the quality of the program offered, and a general broadening of opportunities for those enrolled in this program. Specific advances have been identified with particular presidential administrations, and have been gracefully limned by Charles Coleman Sellers\u27s general history of the college. For those interested in the academic policies of Dickinson College in this century, one administration stands out for the potential it embodied, but did not realize: the administration, in the early thirties, of Karl Tinsley Waugh. Waugh\u27s brief tenure at Dickinson offers a case study in the kinds of tensions and frustrations which can spring from any effort to orchestrate change, and it is presented here as a vignette of Dickinson history. Because of its brevity, Waugh\u27s administration was not a landmark in Dickinson history. But it might have been, and deserves on that account to be better known. To understand President Karl Tinsley Waugh and his travails, however, it is essential first to introduce his immediate predecessor and ultimately his nemesis, James Henry Morgan, and to place each of their presidencies, as well as their personalities, in the context of the other. [excerpt

The New York-New Jersey Boundary Controversy: John Marshall and the Nullification Crisis

In 1832 a long-standing boundary dispute between New York and New Jersey complicated the work of Chief Justice John Marshall and President Andrew Jackson. Long reviled by southern states\u27 rights advocates, including the president, Marshall in 1832 faced the prospect of having the Court\u27s decisions ignored by the state of Georgia. Federal authority was further challenged in the fall of 1832, when South Carolina nullified the tariff of 1828, thereby provoking a constitutional crisis. On December 10, 1832, to the amazement of many observers, Jackson issued a proclamation rejecting nullification and secession, and threatening military action if South Carolina did not change its course

Burnishing Buchanan\u27s Brand on his Birthday

James Buchanan’s brand needs refreshing. Outside his hometown, his name does not much register with Americans today. When it does, the reaction is usually negative. What a comedown from the high hopes associated with Old Buck’s election to the presidency in 1856. [excerpt

The Turbulent Sixties at Rutgers: An Interview with Richard P. McCormick

Richard P. McCormick’s professional life has been so intertwined with Rutgers University’s history that it is difficult to imagine anyone who knows more about Rutgers or who has put a greater imprint on the institution. Except for half a dozen years living in Philadelphia and Newark, Delaware, during the era of the Second World War, McCormick has been a significant presence at Rutgers for six decades. He arrived as a freshman at Rutgers College in 1934 and, after graduating in 1938, worked for the Department of History as a factotum while completing a master’s degree in history. Recruited to join the department as a junior faculty member in 1945 while still completing his doctorate at the University of Pennsylvania, McCormick has never left. He worked his way up the ranks of the faculty, served on virtually every major college committee during a thirty-seven-year teaching career at Rutgers, including a tumultuous term as department chair in the 1960s, and a three-year tenure as dean of Rutgers College in the 1970s. Although he formally retired from teaching at age sixty-five in 1982, McCormick has maintained an office on campus and since then has published a steady stream of articles, pamphlets, essays, and books, some on Rutgers University history. During his years teaching at Rutgers, McCormick was frequently invited to chair history departments at other institutions, assume deanships, and in one case, be seriously considered for a small college presidency. Always he declined, not because the offers were not flattering or were unattractive, but because, as he later told an interviewer, “I never wanted to leave Rutgers. . . . I was happy—and appreciated—here.” [excerpt

Governing New Jersey: Reflections on the Publication of a Revised and Expanded Edition of \u27The Governors of New Jersey\u27

New Jersey’s chief executive enjoys more authority than any but a handful of governors in the United States. Historically speaking, however, New Jersey’s governors exercised less influence than met the eye. In the colonial period few proprietary or royal governors were able to make policy in the face of combative assemblies. The Revolutionary generation’s hostility to executive power contributed to a weak governor system that carried over into the 19th and 20th centuries, until the Constitution was thoroughly revised in 1947. Before that date a handful of governors, by dint of their ideas and personalities, affected the polity in meaningful ways. Derived from a lecture delivered at Rutgers University’s Eagleton Institute on March 11, 2014, this essay focuses on the long history of the executive office, assessing individual governors and delineating the qualities that made them noteworthy, for good or ill

Paving the Way to Scandal: History Repeats Itself

Presidential candidate Marco Rubio of Florida enjoyed an assist this week managing the fallout from New York Times stories about his personal finances by an unlikely ally: Comedy Central host Jon Stewart, who dismissed the information as an example of “gotcha” politics, unworthy of current discussion. “How is this front page news?” Stewart said, calling the Times reports “inconsequential gossip.” [excerpt

Ike\u27s Leadership Lessons for New President

Just days into his presidency in the winter of 1953, Dwight Eisenhower met with his advisers and discussed a challenge from within the majority Republican caucus. If mishandled, it could have endangered his program for a stronger America. The issue, as he later related, was the demand of conservative Republican legislative leaders that Eisenhower balance the budget immediately and cut taxes no matter what the result. [excerpt

The Eisenhowers at Twilight: A Visit to the Eisenhower Farm, 1967

Dwight and Mamie Eisenhower relished life in Gettysburg. As he often remarked to friends, in retirement Ike sought to secure a piece of property that he could leave in better shape than he found it. The purchase in November 1950 of the 189-acre Redding Farm on the Millertown Road, only a short distance from Confederate A venue, was the outcome. Of course the Eisenhowers could have purchased a sizable farm in any number of locations. A Gettysburg address was predicated on their warm memories of a six-month sojourn in the borough in 1918 and recognition that Gettysburg was a convenient location for access to major cities. Lobbying by the Eisenhowers\u27 friends George and Mary Allen, who owned an 88-acre farm four miles south of the square in Gettysburg, along the Emmitsburg Road, also influenced the Eisenhowers\u27 pursuit of a Gettysburg property. [excerpt