Population dynamics is the branch of life sciences that studies the size and age composition of populations as dynamical systems , and the biological and environmentalprocesses driving them (such as birth and death rates , and by immigration and emigration ). Example scenarios are aging populations , population growth , or population decline .
History
Population dynamics has traditionally been dominant in the field of mathematical biology , which has a history of more than 210 years. The first principle of population dynamics is an exponential law of Malthus , as modeled by the Malthusian growth model . The early period was dominated by demographic studies such as Benjamin Gompertz and Pierre François Verhulst in the early 19th century, who refined the Malthusian demographic model.
A more general model formulation was proposed by FJ Richards in 1959, further expanded by Simon Hopkins , in which the models of Gompertz, Verhulst and Ludwig von Bertalanffy are covered as well. The LotkaVolterra predatorprey equations are another famous example, as well as the alternative ArditiGinzburg equations . The computer game SimCity and the MMORPG Ultima Online , among others, to simulate some of these population dynamics.
In the past 30 years, John Hasnard Smith has developed a new model of evolutionary game theory . Under these dynamics, evolutionary biology concepts may be taken to deterministic mathematical form. Population dynamics overlap with another active area of research in mathematical biology: the mathematical epidemiology , the study of infectious disease affecting populations. Various models of viral spread have been proposed and analyzed.
Intrinsic rate of increase
The rate at qui a population Increases in size if there are no densitydependent force regulating the population is Known As the intrinsic rate of Increase .
 {\ displaystyle {\ dfrac {dN} {dt}} {\ dfrac {1} {N}} = r}
Where {\ displaystyle dN / dt}is the rate of Increase of the population, N is the population size, and r is the intrinsic rate of Increase. This is the theoretical maximum rate of increase of a population per individual. The concept is commonly used in insect population to increase population. See also exponential population growth and logistic population growth. ^{[2]}
Common mathematical models
Exponential population growth
Exponential growth describes unregulated reproduction. It is very unusual to see this in nature. In the last 100 years, the population growth has appeared to be exponential. In the long run, however, it is not likely. Paul Ehrlich and Thomas Malthus believed that human population growth would lead to overpopulation and starvation due to scarcity of resources. They believed that human population would grow in the world. In the future, humans would be unable to feed large populations. The biological assumptions of exponential growth are consistent with growth rates. Growth is not limited by resource scarcity or predation. ^{[3]}
Simple discrete time exponential model
 {\ displaystyle N_ {t + 1} = \ lambda N_ {t}}
where λ is the discretetime per capita growth rate. At λ = 1, we get a linear line and a discretetime per capita growth rate of zero. At λ <1, we get a decrease in per capita growth rate. At λ > 1, we get an increase in per capita growth rate. At λ = 0, we get extinction of the species. ^{[3]}
Continuous time version of exponential growth.
Some species have continuous reproduction.
 {\ displaystyle {\ dfrac {dN} {dT}} = rN}
Where {\ displaystyle {\ dfrac {dN} {dT}}}is the rate of population growth per unit time, r is the maximum per capita growth rate, and N is the population size.
At r > 0, there is an increase in per capita growth rate. At r = 0, the per capita growth rate is zero. At r <0, there is a decrease in per capita growth rate.
Logistic population growth
Logistics comes from the French word logistics, which means to compute. Population regulation is a densitydependent process, meaning that population growth is regulated by the density of a population. Think of the analogy of a thermostat. When the temperature is too hot, the thermostat turns on the AC to decrease the temperature back to homeostasis . When the temperature is too cold, the thermostat turns on the heater to increase the temperature back to homeostasis. Likewise with density dependence, whether the population density is high or low, the population dynamics return to the population density to homeostasis. Homeostasis is the set point, or carrying capacity , defined as K. ^{[3]}
Continuoustime model of logistic growth
 {\ displaystyle {\ dfrac {dN} {dT}} = rN {\ Big (} 1 – {\ dfrac {N} {K}} {\ Big)}}
Where {\ displaystyle {\ Big (} 1 – {\ dfrac {N} {K}} {\ Big)}}is the density dependence, N is the number in the population, K is the set point for homeostasis and the carrying capacity. In this logistic model , growth population rate is Highest at 1/2 K and the population growth rate is around zero K . The optimum harvesting rate is a rate close to 1/2 K Where population will grow the fastest. Above K, population growth rate is negative. The logistic models also show density dependence, meaning the population growth rate. In the wild, you can not get these patterns to emerge without simplification. Negative density dependence Allows for a population That overshoots the carrying capacity to Decrease back to the carrying capacity, K . ^{[3]}
According to R / K selection theory, it may be specialized for rapid growth, or stability closer to carrying capacity.
Discrete time logistical model
 {\ displaystyle N_ {t + 1} = N_ {t} + rN_ {t} (1 {N_ {t} / K})}
This equation uses r INSTEAD of λ Because per capita growth rate is zero When r = 0. As r gets very high, there are oscillations and deterministic chaos . ^{[3]} Deterministic chaos is large changes in population dynamics when there is a very small change in r . This Makes it hard to make predictions at high r values Because a very small r error results in a massive error in population dynamics.
Population is always density dependent. Even a severe density independent event can not regulate populate, it can cause it to go extinct.
Not all population models are necessarily negative density dependent. The Allee effect allows for a positive correlation between population density and per capita growth rate in communities with very small populations. For example, a fish swimming on the sea is more likely to be the same fish swimming among a school of fish, because the pattern of movement is more likely to confuse and stun the predator. ^{[3]}
Individualbased models
Cellular automata are used to investigate mechanisms of population dynamics. Here are relatively simple models with one and two species.
Fisheries and wildlife management
In fisheries and wildlife management , population is affected by three dynamic rate functions.
 Natality or birth rate , often recruitment, which means reaching a certain size or reproductive stage. A reference to the age of fish can be taken and counted in net.
 Population growth rate , which measures the growth of individuals in size and length. More important in fisheries, where population is often measured in biomass.
 Mortality , which includes harvest mortality and natural mortality. Natural mortality includes nonhuman predation, disease and old age.
If N _{1} is the number of individuals at time 1 then



 {\ displaystyle N_ {1} = N_ {0} + BD + IE}


where N _{0} is the number of individuals at time 0, B is the number of individuals born, D the number that died, I the number that immigrated, and E the number that emigrated between time 0 and time 1.
If we measure these rates over many time intervals, we can determine how much population density changes over time. Immigration and emigration are present, but are usually not measured.
All of these are measured in the surplus population, which is the number of individuals that can be harvested. The harvest within the harvestable surplus is termed „compensatory“ mortality, where the harvest deaths are naturally substituted. It is more addictive than mortality, because it adds to the number of deaths that would have occurred naturally. These terms are not necessarily judged as „good“ and „bad,“ respectively, in population management. For example, a fish & game agency might be able to reduce the size of a population through additive mortality. Bucks might be targeted to increase buck competition,
For the management of many fish and wildlife populations, the goal is often to achieve the largest possible longrun sustainable harvest, also known as maximum sustainable yield (or MSY). Given a population dynamic model, such as any of the above, it is possible to calculate the size of the population that produces surplus harvestable surplus at equilibrium. ^{[4]} While the use of population dynamics is a controversial issue among scientists, ^{[5]} it has been shown to be more effective than incorrect models and natural resource management. ^{[6]}^{[7]} To give an example of a nonintuitive result, fisheries Produce more fish When there is a nearby refuge from human predation in the form of a nature reserve , resulting and in Higher catches than if the whole area Was open to fishing. ^{[8] }^{[9]}
For control applications
Population dynamics have been widely used in several control theory applications. With the use of evolutionary game theory , the games are widely used for different industrial and dailylife contexts. Mostly used in multipleinputmultipleoutput ( MIMO ) systems, they can be used for singleinputsingleoutput ( SISO ) systems. Some examples of applications are military campaigns, resource allocation for water distribution , dispatch of distributed generators, lab experiments, transport problems, communication problems, among others. Furthermore, with the appropriate contextualization of industrial problems, population dynamics can be efficient and easytoimplement solution for controlrelated problems. Multiple academic research has been carried out.
See also
 Delayed density dependence
 Minimum viable population
 Maximum sustainable yield
 NicholsonBailey model
 Overshoot (population)
 Pest insect population dynamics
 Cycle population
 Population dynamics of fisheries
 Population ecology
 Population genetics
 Population modeling
 Ricker model
 r / K selection theory
 Sigmoid curve
 Societal collapse
 System dynamics
Notes
 Jump up^ Brotz, Lucas; Cheung, William W. L .; Kleisner, Kristin; Pakhomov, Evgeny; Pauly, Daniel (2012). „Increasing jellyfish populations: Trends in Large Marine Ecosystems“. Hydrobiologia . 690 (1): 320. doi : 10.1007 / s1075001210397 .
 Jump up^ Jahn, Gary C; Almazan, Liberty P; Pacia, Jocelyn B (2005). „Effect of Nitrogen Fertilizer on the Intrinsic Rate of Increase of Hysteroneura setariae (Thomas) (Homoptera: Aphididae) on Rice (Oryza sativaL.)“. Environmental Entomology . 34(4): 93843. doi : 10.1603 / 0046225X34.4.938 .
 ^ Jump up to:^{a }^{b }^{c }^{d }^{e }^{f} Yang, Louie. Population Dynamics. Davis: UC Davis, 2014. ^{[ page needed ]}
 Jump up^ Clark, Colin (1990). Mathematical bioeconomics: the optimal management of renewable resources . New York: Wiley. ISBN 0471508837 . ^{[}^{ page needed}^{ ]}
 Jump up^ Finley, C; Oreskes, N (2013). „Maximum sustained yield: A policy disguised as science“. ICES Journal of Marine Science . 70 (2): 24550. doi : 10.1093 / icesjms / fss192 .
 Jump up^ Holden, Matthew H; Ellner, Stephen P (2016). „Human judgment vs. Quantitative models for the management of ecological resources“. Ecological Applications . 26 (5): 15531565. doi : 10.1890 / 151295 . PMID 27755756 .
 Jump up^ Standard, Pacific (20160311). „Sometimes, Even Bad Models Make Better Decisions Than People“ . Pacific Standard . Retrieved 20170128 .
 Jump up^ Chakraborty, Kunal; Das, Kunal; Kar, T. K (2013). „An ecological perspective on marine reserves in preypredator dynamics“ . Journal of Biological Physics . 39 (4): 74976. doi : 10.1007 / s1086701393295 . PMC 3758828 . PMID 23949368 .
 Jump up^ Lv, Yunfei; Yuan, Rong; Pei, Yongzhen (2013). „A preypredator model with harvesting for fishery resource with reserve area“. Applied Mathematical Modeling . 37 (5): 304862. doi : 10.1016 / j.apm.2012.07.030 .
References
 Macromodels of the World System Growth by Andrey Korotayev , Artemy Malkov, and Daria Khaltourina. ISBN 5484004144
 Turchin, P. 2003. Complex Population Dynamics: a Theoretical / Empirical Synthesis. Princeton, NJ: Princeton University Press.
 Weiss, Volkmar (2007). „The Population Cycle Drives Human History – From a Eugenic Phase into a Dysgenic Phase and Eventual Collapse“ . The Journal of Social, Political and Economic Studies . 32 (3): 32758.