Population process
Encyclopedia
In applied probability
, a population process is a Markov chain
in which the state of the chain is analogous to the number of individuals in a population (0, 1, 2, etc.), and changes to the state are analogous to the addition or removal of individuals from the population.
Although named by analogy to biological
populations, population processes find application in a much wider range of fields than just ecology
and other biological sciences. These other applications include telecommunication
s and queueing theory
, chemical kinetics
and financial mathematics, and hence the 'population' could be of packet
s in a computer network
, of molecule
s in a chemical reaction
, or even of units in a financial index
.
Population processes are typically characterized by processes of birth and immigration, and of death, emigration and catastrophe, which correspond to the basic demographic
processes and broad environment
al effects to which a population is subject. However, population processes are also often equivalent to other processes that may typically be characterised under other paradigm
s (in the literal sense of 'patterns'). Queues
, for example, are often characterised by an arrivals process, a service process, and the number of servers. In appropriate circumstances, however, arrivals at a queue are functionally equivalent to births or immigration and the service of waiting 'customers' is equivalent to death or emigration.
Typical population processes include birth-death process
es and birth, death and catastrophe processes.
Applied probability
Much research involving probability is done under the auspices of applied probability, the application of probability theory to other scientific and engineering domains...
, a population process is a Markov chain
Markov chain
A Markov chain, named after Andrey Markov, is a mathematical system that undergoes transitions from one state to another, between a finite or countable number of possible states. It is a random process characterized as memoryless: the next state depends only on the current state and not on the...
in which the state of the chain is analogous to the number of individuals in a population (0, 1, 2, etc.), and changes to the state are analogous to the addition or removal of individuals from the population.
Although named by analogy to biological
Biology
Biology is a natural science concerned with the study of life and living organisms, including their structure, function, growth, origin, evolution, distribution, and taxonomy. Biology is a vast subject containing many subdivisions, topics, and disciplines...
populations, population processes find application in a much wider range of fields than just ecology
Ecology
Ecology is the scientific study of the relations that living organisms have with respect to each other and their natural environment. Variables of interest to ecologists include the composition, distribution, amount , number, and changing states of organisms within and among ecosystems...
and other biological sciences. These other applications include telecommunication
Telecommunication
Telecommunication is the transmission of information over significant distances to communicate. In earlier times, telecommunications involved the use of visual signals, such as beacons, smoke signals, semaphore telegraphs, signal flags, and optical heliographs, or audio messages via coded...
s and queueing theory
Queueing theory
Queueing theory is the mathematical study of waiting lines, or queues. The theory enables mathematical analysis of several related processes, including arriving at the queue, waiting in the queue , and being served at the front of the queue...
, chemical kinetics
Chemical kinetics
Chemical kinetics, also known as reaction kinetics, is the study of rates of chemical processes. Chemical kinetics includes investigations of how different experimental conditions can influence the speed of a chemical reaction and yield information about the reaction's mechanism and transition...
and financial mathematics, and hence the 'population' could be of packet
Packet switching
Packet switching is a digital networking communications method that groups all transmitted data – regardless of content, type, or structure – into suitably sized blocks, called packets. Packet switching features delivery of variable-bit-rate data streams over a shared network...
s in a computer network
Computer network
A computer network, often simply referred to as a network, is a collection of hardware components and computers interconnected by communication channels that allow sharing of resources and information....
, of molecule
Molecule
A molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...
s in a chemical reaction
Chemical reaction
A chemical reaction is a process that leads to the transformation of one set of chemical substances to another. Chemical reactions can be either spontaneous, requiring no input of energy, or non-spontaneous, typically following the input of some type of energy, such as heat, light or electricity...
, or even of units in a financial index
Stock market index
A stock market index is a method of measuring a section of the stock market. Many indices are cited by news or financial services firms and are used as benchmarks, to measure the performance of portfolios such as mutual funds....
.
Population processes are typically characterized by processes of birth and immigration, and of death, emigration and catastrophe, which correspond to the basic demographic
Demography
Demography is the statistical study of human population. It can be a very general science that can be applied to any kind of dynamic human population, that is, one that changes over time or space...
processes and broad environment
Natural environment
The natural environment encompasses all living and non-living things occurring naturally on Earth or some region thereof. It is an environment that encompasses the interaction of all living species....
al effects to which a population is subject. However, population processes are also often equivalent to other processes that may typically be characterised under other paradigm
Paradigm
The word paradigm has been used in science to describe distinct concepts. It comes from Greek "παράδειγμα" , "pattern, example, sample" from the verb "παραδείκνυμι" , "exhibit, represent, expose" and that from "παρά" , "beside, beyond" + "δείκνυμι" , "to show, to point out".The original Greek...
s (in the literal sense of 'patterns'). Queues
Queueing theory
Queueing theory is the mathematical study of waiting lines, or queues. The theory enables mathematical analysis of several related processes, including arriving at the queue, waiting in the queue , and being served at the front of the queue...
, for example, are often characterised by an arrivals process, a service process, and the number of servers. In appropriate circumstances, however, arrivals at a queue are functionally equivalent to births or immigration and the service of waiting 'customers' is equivalent to death or emigration.
Typical population processes include birth-death process
Birth-death process
The birth–death process is a special case of continuous-time Markov process where the states represent the current size of a population and where the transitions are limited to births and deaths...
es and birth, death and catastrophe processes.