A transition diagram is used to describe the behavior of systems. Birth-death equations were derived from transition diagram depicting the state of the birth-death processes. Queue models and characteristics of queue ...A transition diagram is used to describe the behavior of systems. Birth-death equations were derived from transition diagram depicting the state of the birth-death processes. Queue models and characteristics of queue models are also derivable from birth-death processes. These queue models consist of mathematical formulas and relationships that can be used to determine the operating characteristics （performance measures） for a waiting line. Schematic and transition diagrams of different single server queue models were shown. Relationships between birth-death processes, waiting lines （queues） and transition diagrams were given. While M/M/I/K queue model states was limited by K customers and had （K＋I） states, M/M/1/1 queue model had only two states. M/G/1/∝/∝ and M/M/1/∝/∝ shared similar characteristics. Many ideal queuing situations employ M/M/1 queueing model.展开更多
Cannibalism is widespread in size-structured populations. If cannibals and victims are in different life stages, dominant cohorts of cannibals can regulate recruitment. Arizona Tiger Salamanders, Ambystoma tigrinum ne...Cannibalism is widespread in size-structured populations. If cannibals and victims are in different life stages, dominant cohorts of cannibals can regulate recruitment. Arizona Tiger Salamanders, Ambystoma tigrinum nebulosum, exhibit facultative paedomorphosis in which salamander larvae either metamorphose into terrestrial adults or become sexually mature while still in their larval form. Although many salamanders exhibit cannibalism of larvae, the Arizona Tiger Salamander also exhibits cannibalism of young by the aquatic adults. We formulate a differential equations model of this system under the assumption that the terrestrial adults do not impact the system beyond their contribution to the birth of young larvae. We establish non-negativity, boundedness and persistence of the salamander population under certain assumptions. We consider the equilibrium states of the system in the presence or absence of a birth contribution from the terrestrial or metamorph adults. Constant per capita paedomorphosis leads to asymptotically stable equilibria. The per capita paedomorphosis rate of the larvae must be density dependent in order for periodic solutions to exist. Furthermore, the stage transition rate must be sufficiently decreasing in order to guarantee the existence of an unstable equilibrium. Periodic solutions are only possible in the presence of a unique non-trivial unstable equilibrium. Our results conform to previous theory on paedomorphosis which suggests general applicability of our results to similar systems.展开更多
文摘A transition diagram is used to describe the behavior of systems. Birth-death equations were derived from transition diagram depicting the state of the birth-death processes. Queue models and characteristics of queue models are also derivable from birth-death processes. These queue models consist of mathematical formulas and relationships that can be used to determine the operating characteristics （performance measures） for a waiting line. Schematic and transition diagrams of different single server queue models were shown. Relationships between birth-death processes, waiting lines （queues） and transition diagrams were given. While M/M/I/K queue model states was limited by K customers and had （K＋I） states, M/M/1/1 queue model had only two states. M/G/1/∝/∝ and M/M/1/∝/∝ shared similar characteristics. Many ideal queuing situations employ M/M/1 queueing model.
基金We acknowledge that part of this research has been made possible by support from the Murray State University Center for Institutional Studies and Research and the National Science Foundation （DEB 0109436 and DUE-1028125）.
文摘Cannibalism is widespread in size-structured populations. If cannibals and victims are in different life stages, dominant cohorts of cannibals can regulate recruitment. Arizona Tiger Salamanders, Ambystoma tigrinum nebulosum, exhibit facultative paedomorphosis in which salamander larvae either metamorphose into terrestrial adults or become sexually mature while still in their larval form. Although many salamanders exhibit cannibalism of larvae, the Arizona Tiger Salamander also exhibits cannibalism of young by the aquatic adults. We formulate a differential equations model of this system under the assumption that the terrestrial adults do not impact the system beyond their contribution to the birth of young larvae. We establish non-negativity, boundedness and persistence of the salamander population under certain assumptions. We consider the equilibrium states of the system in the presence or absence of a birth contribution from the terrestrial or metamorph adults. Constant per capita paedomorphosis leads to asymptotically stable equilibria. The per capita paedomorphosis rate of the larvae must be density dependent in order for periodic solutions to exist. Furthermore, the stage transition rate must be sufficiently decreasing in order to guarantee the existence of an unstable equilibrium. Periodic solutions are only possible in the presence of a unique non-trivial unstable equilibrium. Our results conform to previous theory on paedomorphosis which suggests general applicability of our results to similar systems.
