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 Lotka-Volterra predator-prey equations are another famous example, as well as the alternative Arditi-Ginzburg 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 density-dependent 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 discrete-time per capita growth rate. At λ = 1, we get a linear line and a discrete-time 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 density-dependent 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]

### Continuous-time 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]

### Individual-based models

Cellular automata are used to investigate mechanisms of population dynamics. Here are relatively simple models with one and two species.

Logical deterministic individual-based cellular automata model of single species population growth
Logical deterministic individual-based cellular automata model of interspecific competition for a single limited resource

## 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 non-human predation, disease and old age.

If 1 is the number of individuals at time 1 then

{\ displaystyle N_ {1} = N_ {0} + B-D + IE}

where 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 long-run 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 non-intuitive 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 daily-life contexts. Mostly used in multiple-input-multiple-output ( MIMO ) systems, they can be used for single-input-single-output ( 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 easy-to-implement solution for control-related problems. Multiple academic research has been carried out.

• Delayed density dependence
• Minimum viable population
• Maximum sustainable yield
• Nicholson-Bailey 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

1. 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): 3-20. doi : 10.1007 / s10750-012-1039-7 .
2. 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): 938-43. doi : 10.1603 / 0046-225X-34.4.938 .
3. ^ Jump up to:f Yang, Louie. Population Dynamics. Davis: UC Davis, 2014. page needed ]
4. Jump up^ Clark, Colin (1990). Mathematical bioeconomics: the optimal management of renewable resources . New York: Wiley. ISBN  0471508837 . [ page needed ]
5. Jump up^ Finley, C; Oreskes, N (2013). „Maximum sustained yield: A policy disguised as science“. ICES Journal of Marine Science . 70 (2): 245-50. doi : 10.1093 / icesjms / fss192 .
6. Jump up^ Holden, Matthew H; Ellner, Stephen P (2016). „Human judgment vs. Quantitative models for the management of ecological resources“. Ecological Applications . 26 (5): 1553-1565. doi : 10.1890 / 15-1295 . PMID  27755756 .
7. Jump up^ Standard, Pacific (2016-03-11). „Sometimes, Even Bad Models Make Better Decisions Than People“ . Pacific Standard . Retrieved 2017-01-28 .
8. Jump up^ Chakraborty, Kunal; Das, Kunal; Kar, T. K (2013). „An ecological perspective on marine reserves in prey-predator dynamics“ . Journal of Biological Physics . 39 (4): 749-76. doi : 10.1007 / s10867-013-9329-5 . PMC  3758828  . PMID  23949368 .
9. Jump up^ Lv, Yunfei; Yuan, Rong; Pei, Yongzhen (2013). „A prey-predator model with harvesting for fishery resource with reserve area“. Applied Mathematical Modeling . 37 (5): 3048-62. doi : 10.1016 / j.apm.2012.07.030 .

## References

• Macromodels of the World System Growth by Andrey Korotayev , Artemy Malkov, and Daria Khaltourina. ISBN  5-484-00414-4
• 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): 327-58.