Population dynamics is the branch of life sciences that studies the size and age composition of populations as dynamical systems, the biological and environmental processes driving them. Example scenarios are ageing population growth, or population decline. Population dynamics has traditionally been the dominant branch of mathematical biology, which has a history of more than 210 years, although more the scope of mathematical biology has expanded; the first principle of population dynamics is regarded as the exponential law of Malthus, as modeled by the Malthusian growth model. The early period was dominated by demographic studies such as the work of Benjamin Gompertz and Pierre François Verhulst in the early 19th century, who refined and adjusted the Malthusian demographic model. A more general model formulation was proposed by F. J. Richards in 1959, further expanded by Simon Hopkins, in which the models of Gompertz and Ludwig von Bertalanffy are covered as special cases of the general formulation.
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, tried to simulate some of these population dynamics. In the past 30 years, population dynamics has been complemented by evolutionary game theory, developed first by John Maynard Smith. Under these dynamics, evolutionary biology concepts may take a deterministic mathematical form. Population dynamics overlap with another active area of research in mathematical biology: mathematical epidemiology, the study of infectious disease affecting populations. Various models of viral spread have been proposed and analyzed, provide important results that may be applied to health policy decisions; the rate at which a population increases in size if there are no density-dependent forces regulating the population is known as the intrinsic rate of increase. It is d N d t 1 N = r where the derivative d N / d t is the rate of increase of the population, N is the population size, r is the intrinsic rate of increase.
Thus r is the maximum theoretical rate of increase of a population per individual – that is, the maximum population growth rate. The concept is used in insect population biology to determine how environmental factors affect the rate at which pest populations increase. See exponential population growth and logistic population growth. Exponential growth describes unregulated reproduction, it is unusual to see this in nature. In the last 100 years, human population growth has appeared to be exponential. In the long run, however, it is not likely. 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 at rate in which they exceed the ability at which humans can find food. In the future, humans would be unable to feed large populations; the biological assumptions of exponential growth is. Growth is not limited by resource predation. N t + 1 = λ N t. At λ = 1, we get 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; some species have continuous reproduction. D N d T = r N where d N d T is the rate of population growth per unit time, r is the maximum per capita growth rate, 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. “Logistics” comes from the French word logistique, which means “to compute”. Population regulation is a density-dependent process, meaning that population growth rates are regulated by the density of a population. Consider an analogy with a thermostat; when the temperature is too hot, the thermostat turns on the air conditioning 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. With density dependence, whether the population density is high or low, population dynamics returns the population density to homeostasis.
Homeostasis is the set point, or carrying capacity, defined as K. D N d T = r N where 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, population growth rate is highest at 1/2 K and the population growth rate is zero around K; the optimum harvesting rate is a close rate to 1/2 K. Above
Population ecology is a sub-field of ecology that deals with the dynamics of species populations and how these populations interact with the environment. It is the study of how the population sizes of species change over space; the term population ecology is used interchangeably with population biology or population dynamics. The development of population ecology owes much to demography and actuarial life tables. Population ecology is important in conservation biology in the development of population viability analysis which makes it possible to predict the long-term probability of a species persisting in a given habitat patch. Although population ecology is a subfield of biology, it provides interesting problems for mathematicians and statisticians who work in population dynamics; the most fundamental law of population ecology is Thomas Malthus' exponential law of population growth. A population will grow exponentially as long as the environment experienced by all individuals in the population remains constant.
This principle in population ecology provides the basis for formulating predictive theories and tests that follow: Simplified population models start with four key variables including death, birth and emigration. Mathematical models used to calculate changes in population demographics and evolution hold the assumption of no external influence. Models can be more mathematically complex where "...several competing hypotheses are confronted with the data." For example, in a closed system where immigration and emigration does not take place, the rate of change in the number of individuals in a population can be described as: d N d T = B − D = b N − d N = N = r N, where N is the total number of individuals in the population, B is the raw number of births, D is the raw number of deaths, b and d are the per capita rates of birth and death and r is the per capita average number of surviving offspring each individual has. This formula can be read as the rate of change in the population is equal to deaths.
Using these techniques, Malthus' population principle of growth was transformed into a mathematical model known as the logistic equation: d N d T = a N, where N is the biomass density, a is the maximum per-capita rate of change, K is the carrying capacity of the population. The formula can be read as follows: the rate of change in the population is equal to growth, limited by carrying capacity. From these basic mathematical principles the discipline of population ecology expands into a field of investigation that queries the demographics of real populations and tests these results against the statistical models; the field of population ecology uses data on life history and matrix algebra to develop projection matrices on fecundity and survivorship. This information is used for managing wildlife stocks and setting harvest quotas The population model below can be manipulated to mathematically infer certain properties of geometric populations. A population with a size that increases geometrically is a population where generations of reproduction do not overlap.
In each generation there is an effective population size denoted as Ne which constitutes the number of individuals in the population that are able to reproduce and will reproduce in any reproductive generation in concern. In the population model below it is assumed. Assumption 01: Ne = N Nt+1 = Nt + Bt + It - Dt - Et Assumption 02: There is no migration to or from the population It = Et = 0 Nt+1 = Nt + Bt - Dt The raw birth and death rates are related to the per capita birth and death rates: Bt = bt × Nt Dt = dt × Nt bt = Bt / Nt dt = Dt / Nt Therefore: Nt+1 = Nt + - Assumption 03: bt and dt are constant. Nt+1 = Nt + - Take the term Nt out of the brackets. Nt+1 = Nt + Nt b - d = R Nt+1 = Nt + RNt Nt+1 = Take the term Nt out of the brackets again. Nt+1 = Nt 1 + R = λ Nt+1 = λNt Therefore: Nt+1 = λtNt The doubling time of a population is the time required for the population to grow to twice its size. We can calculate the doubling time of a geometric population using the equation: Nt+1 = λtNt by exploiting our knowledge of the fact that the population is twice its size after the doubling time.2Ntd = λtd × Nt λtd = 2Ntd / Nt λtd = 2 The doubling time can be found by taking logarithms.
For instance: td × log2 = log2 log2 = 1 td × log2 = 1 td = 1 / log2 Or: td × ln = ln td = ln / ln td = 0.693... / ln Therefore: td = 1 / log2 = 0.693... / ln The half-life of a population is the time taken for the population to decline to half its size. We can calculate the half-life of a geometric population using the equation: Nt+1 = λtNt by exploiting our knowledge of the fact that the population is half its size after a half-life.0.5Nt1/2 = λt1/2 × Nt λt1/2 = 0.5Nt1/2 / Nt λt1/2 = 0.5 The half-life can be calculated by taking logarithms. T1/2 = 1 / log0.5 = ln / ln R = b - d Nt+1 = Nt + RNt Nt+1 - Nt = RNt Nt+1 - Nt = ΔN ΔN = RNt ΔN/Nt = R 1 + R = λ Nt+1 = λNt λ = Nt+1 / Nt In geo
Mark and recapture
Mark and recapture is a method used in ecology to estimate an animal population's size. A portion of the population is captured and released. Another portion is captured and the number of marked individuals within the sample is counted. Since the number of marked individuals within the second sample should be proportional to the number of marked individuals in the whole population, an estimate of the total population size can be obtained by dividing the number of marked individuals by the proportion of marked individuals in the second sample; the method is most useful. Other names for this method, or related methods, include capture-recapture, capture-mark-recapture, mark-recapture, sight-resight, mark-release-recapture, multiple systems estimation, band recovery, the Petersen method, the Lincoln method. Another major application for these methods is in epidemiology, where they are used to estimate the completeness of ascertainment of disease registers. Typical applications include estimating the number of people needing particular services, or with particular conditions.
Allen. "Estimating the Number of People Who Inject Drugs in A Rural County in Appalachia". American Journal of Public Health. A researcher visits a study area and uses traps to capture a group of individuals alive; each of these individuals is marked with a unique identifier, is released unharmed back into the environment. A mark recapture method was first used for ecological study in 1896 by C. G. Johannes Petersen to estimate Pleuronectes platessa, populations. Sufficient time is allowed to pass for the marked individuals to redistribute themselves among the unmarked population. Next, the researcher captures another sample of individuals; some individuals in this second sample will have been marked during the initial visit and are now known as recaptures. Other animals captured during the second visit, will not have been captured during the first visit to the study area; these unmarked animals are given a tag or band during the second visit and are released. Population size can be estimated from as few as two visits to the study area.
More than two visits are made if estimates of survival or movement are desired. Regardless of the total number of visits, the researcher records the date of each capture of each individual; the "capture histories" generated are analyzed mathematically to estimate population size, survival, or movement. Let N = Number of animals in the population n = Number of animals marked on the first visit K = Number of animals captured on the second visit k = Number of recaptured animals that were markedA biologist wants to estimate the size of a population of turtles in a lake, she captures 10 turtles on her first visit to the lake, marks their backs with paint. A week she returns to the lake and captures 15 turtles. Five of these 15 turtles have paint on their backs; this example is =. The problem is to estimate N; the Lincoln–Petersen method can be used to estimate population size if only two visits are made to the study area. This method assumes that the study population is "closed". In other words, the two visits to the study area are close enough in time so that no individuals die, are born, or move into or out of the study area between visits.
The model assumes that no marks fall off animals between visits to the field site by the researcher, that the researcher records all marks. Given those conditions, estimated population size is: N ^ = K n k, It is assumed that all individuals have the same probability of being captured in the second sample, regardless of whether they were captured in the first sample; this implies that, in the second sample, the proportion of marked individuals that are caught should equal the proportion of the total population, marked. For example, if half of the marked individuals were recaptured, it would be assumed that half of the total population was included in the second sample. In symbols, k K = n N. A rearrangement of this gives N ^ = K n k, the formula used for the Lincoln–Petersen method. In the example = the Lincoln -- Petersen method estimates. N ^ = K n k = 10 × 15 5 = 30 The Lincoln–Peterson estimator is asymptotically unbiased as sample size approaches infinity, but is biased at small sample sizes.
An alternative less biased estimator of population size is given by the Chapman estimator: N ^ C =
In biology, a population is all the organisms of the same group or species, which live in a particular geographical area, have the capability of interbreeding. The area of a sexual population is the area where inter-breeding is possible between any pair within the area, where the probability of interbreeding is greater than the probability of cross-breeding with individuals from other areas. In sociology, population refers to a collection of humans. Demography is a social science. Population in simpler terms is the number of people in a city or town, country or world. In population genetics a sex population is a set of organisms in which any pair of members can breed together; this means that they can exchange gametes to produce normally-fertile offspring, such a breeding group is known therefore as a Gamo deme. This implies that all members belong to the same species. If the Gamo deme is large, all gene alleles are uniformly distributed by the gametes within it, the Gamo deme is said to be panmictic.
Under this state, allele frequencies can be converted to genotype frequencies by expanding an appropriate quadratic equation, as shown by Sir Ronald Fisher in his establishment of quantitative genetics. This occurs in Nature: localization of gamete exchange – through dispersal limitations, preferential mating, cataclysm, or other cause – may lead to small actual Gamo demes which exchange gametes reasonably uniformly within themselves but are separated from their neighboring Gamo demes. However, there may be low frequencies of exchange with these neighbors; this may be viewed as the breaking up of a large sexual population into smaller overlapping sexual populations. This failure of panmixia leads to two important changes in overall population structure: the component Gamo demos vary in their allele frequencies when compared with each other and with the theoretical panmictic original; the overall rise in homozygosity is quantified by the inbreeding coefficient. Note that all homozygotes are increased in frequency – both the deleterious and the desirable.
The mean phenotype of the Gamo demes collection is lower than that of the panmictic original –, known as inbreeding depression. It is most important to note, that some dispersion lines will be superior to the panmictic original, while some will be about the same, some will be inferior; the probabilities of each can be estimated from those binomial equations. In plant and animal breeding, procedures have been developed which deliberately utilize the effects of dispersion, it can be shown that dispersion-assisted selection leads to the greatest genetic advance, is much more powerful than selection acting without attendant dispersion. This is so for both autogamous Gamo demes. In ecology, the population of a certain species in a certain area can be estimated using the Lincoln Index. According to the United States Census Bureau the world's population was about 7.55 billion in 2019 and that the 7 billion number was surpassed on 12 March 2012. According to a separate estimate by the United Nations, Earth’s population exceeded seven billion in October 2011, a milestone that offers unprecedented challenges and opportunities to all of humanity, according to UNFPA, the United Nations Population Fund.
According to papers published by the United States Census Bureau, the world population hit 6.5 billion on 24 February 2006. The United Nations Population Fund designated 12 October 1999 as the approximate day on which world population reached 6 billion; this was about 12 years after world population reached 5 billion in 1987, 6 years after world population reached 5.5 billion in 1993. The population of countries such as Nigeria, is not known to the nearest million, so there is a considerable margin of error in such estimates. Researcher Carl Haub calculated that a total of over 100 billion people have been born in the last 2000 years. Population growth increased as the Industrial Revolution gathered pace from 1700 onwards; the last 50 years have seen a yet more rapid increase in the rate of population growth due to medical advances and substantial increases in agricultural productivity beginning in the 1960s, made by the Green Revolution. In 2017 the United Nations Population Division projected that the world's population will reach about 9.8 billion in 2050 and 11.2 billion in 2100.
In the future, the world's population is expected to peak, after which it will decline due to economic reasons, health concerns, land exhaustion and environmental hazards. According to one report, it is likely that the world's population will stop growing before the end of the 21st century. Further, there is some likelihood that population will decline before 2100. Population has declined in the last decade or two in Eastern Europe, the Baltics and in the Commonwealth of Independent States; the population pattern of less-developed regions of the world in recent years has been marked by increasing birth rates. These followed an earlier sharp reduction in death rates; this transition from high birth and death rates to low birth
Population dynamics of fisheries
A fishery is an area with an associated fish or aquatic population, harvested for its commercial or recreational value. Fisheries can be farmed. Population dynamics describes the ways in which a given population grows and shrinks over time, as controlled by birth and migration, it is the basis for understanding changing fishery patterns and issues such as habitat destruction and optimal harvesting rates. The population dynamics of fisheries is used by fisheries scientists to determine sustainable yields; the basic accounting relation for population dynamics is the BIDE model, shown as: N1 = N0 + B − D + I − Ewhere N1 is the number of individuals at time 1, N0 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, E the number that emigrated between time 0 and time 1. While immigration and emigration can be present in wild fisheries, they are not measured. A fishery population is affected by three dynamic rate functions: Birth recruitment.
Recruitment means reaching reproductive stage. With fisheries, recruitment refers to the age a fish can be caught and counted in nets. Growth rate; this measures the growth of individuals in length. This is important in fisheries where the population is measured in terms of biomass. Mortality; this includes natural mortality. Natural mortality includes non-human predation and old age. If these rates are measured over different time intervals, the harvestable surplus of a fishery can be determined; the harvestable surplus is the number of individuals that can be harvested from the population without affecting long term stability. The harvest within the harvestable surplus is called compensatory mortality, where the harvest deaths are substituting for the deaths that would otherwise occur naturally. Harvest beyond, additive mortality, harvest in addition to all the animals that would have died naturally. Care is needed. Over-simplistic modelling of fisheries has resulted in the collapse of key stocks.
The first principle of population dynamics is regarded as the exponential law of Malthus, as modelled by the Malthusian growth model. The early period was dominated by demographic studies such as the work of Benjamin Gompertz and Pierre François Verhulst in the early 19th century, who refined and adjusted the Malthusian demographic model. A more general model formulation was proposed by F. J. Richards in 1959, by which the models of Gompertz and Ludwig von Bertalanffy are covered as special cases of the general formulation; the population size is the number of individual organisms in a population. The effective population size was defined by Sewall Wright, he defined it as "the number of breeding individuals in an idealized population that would show the same amount of dispersion of allele frequencies under random genetic drift or the same amount of inbreeding as the population under consideration". It is a basic parameter in many models in population genetics. Ne is less than N. Small population size results in increased genetic drift.
Population bottlenecks are. Overpopulation may indicate any case in which the population of any species of animal may exceed the carrying capacity of its ecological niche. Virtual population analysis is a cohort modeling technique used in fisheries science for reconstructing historical fish numbers at age using information on death of individuals each year; this death is partitioned into catch by fisheries and natural mortality. VPA is virtual in the sense that the population size is not observed or measured directly but is inferred or back-calculated to have been a certain size in the past in order to support the observed fish catches and an assumed death rate owing to non-fishery related causes; the minimum viable population is a lower bound on the population of a species, such that it can survive in the wild. More MVP is the smallest possible size at which a biological population can exist without facing extinction from natural disasters or demographic, environmental, or genetic stochasticity.
The term "population" refers to the population of a species in the wild. As a reference standard, MVP is given with a population survival probability of somewhere between ninety and ninety-five percent and calculated for between one hundred and one thousand years into the future; the MVP can be calculated using computer simulations known as population viability analyses, where populations are modelled and future population dynamics are projected. In population ecology and economics, the maximum sustainable yield or MSY is, the largest catch that can be taken from a fishery stock over an indefinite period. Under the assumption of logistic growth, the MSY will be at half the carrying capacity of a species, as this is the stage at when population growth is highest; the maximum sustainable yield is higher than the optimum sustainable yield. This logistic model of growth is produced by a population introduced to a new habitat or with poor numbers going through a lag phase of slow growth at first.
Once it reaches a foothold population it will go through a rapid growth rate that will start to level off once the species approaches carrying capacity. The idea of maximum sustained yield is to decrease population density to the point of highest growth rate possible; this changes the number of the population, but the new number c