Population dynamics

Population dynamicsFrom Wikipedia, the free encyclopedia∙1History∙2Intrinsic rate of increase∙3Common mathematical modelso 3.1Exponential population growtho 3.2Simple discrete time exponential modelo 3.3Continuous time version of exponential growth.o 3.4Logistic population growtho 3.5Continuous-time model of logistic growtho 3.6Discrete time logistical modelo 3.7Examples of individual-based models∙4Fisheries and wildlife management∙5See also∙6Notes∙7References∙8External linksThe 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.Where is the rate of increase of the population, N is the population size,and r is the intrinsic rate of increase. This is therefore the theoretical maximum rate of increase of a population per individual. The concept is commonly used in insectpopulation biology to determine how environmental factors affect the rate at whichpest populations increase. See also exponential population growth and logisticpopulation growth.[2]Exponential population growth[edit]Exponential growth describes unregulated reproduction. It is very 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. Paul Ehrlich and Thomas Malthus believed that human population growth would leadto 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 that the per capita growth rate is constant. Growth is not limited by resource scarcity or predation.[3] Simple discrete time exponential model[edit]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.[edit]Some species have continuous reproduction.where is the rate of population growth per unit time, r is the maximumper 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[edit]Logistics comes from the French word logistique, which means to compute.Population regulation is a density-dependent process, meaning thatpopulation growth rates are 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 thetemperature back to homeostasis. Likewise with density dependence.Whether the population density is high or low, population dynamics returnsContinuous-time model of logistic growthwhere is the density dependence, N is the number in thepopulation, 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 isa close rate to 1/2 K where population will grow the fastest. Above K, thepopulation growth rate is negative. The logistic models also showsdensity dependence, meaning the per capita population growth ratesdecline as the population density increases. In the wild, you can't getthese pattern to emerge without simplification. Negative densitydependence allows for a population that overshoots the carrying capacity to decrease back to the carrying capacity, K.[3]According to R/K_selection_theory organisms may be specialised forrapid growth, or stability closer to carrying capacity.Discrete time logistical model[edit]This equation uses r instead of λ because per capita growth rate iszero when r = 0. As r gets very high, thereare oscillations and deterministic chaos. [3] Deterministic chaos islarge changes in population dynamics when there is a very small r.This makes it hard to make predictions at high r values because avery small r error results in a massive error in population dynamics.Population is always density dependent. Even a severe densityindependent event cannot regulate populate, although it may cause itto go extinct.Not all population models are necessarily negative densitydependent. The Allee effect allows for a positive correlation betweenpopulation density and per capita growth rate in communities withvery small populations. For example, a fish swimming on its own ismore likely to be eaten than the same fish swimming among a schoolof fish, because the pattern of movement of the school of fish is morelikely to confuse and stun the predator.[3]Examples of individual-based models[edit]Logical deterministic individual-based cellular automata model of anecosystem with one species. The model demonstrates a mechanism ofS-shaped population growth.Logical deterministic individual-based cellular automata modelof interspecific competition for a single limited resource. A mechanism ofcompetitive exclusion of one species by another.∙Natality or birth rate, often recruitment, which means reaching a certain size or reproductive stage. Usually refers to the age a fish can be caught and counted in nets∙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 N1 is the number of individuals at time 1 thenwhere N0 is the number of individuals at time 0, B is the number of individuals born, D the number that died, I the number thatimmigrated, and E the number that emigrated between time 0and time 1.If we measure these rates over many time intervals, we candetermine how a population's density changes over time.Immigration and emigration are present, but are usually notmeasured.All of these are measured to determine the harvestable surplus, which is the number of individuals that can be harvested from a population without affecting long-term population stability oraverage population size. The harvest within the harvestablesurplus is termed "compensatory" mortality, where the harvestdeaths are substituted for the deaths that would have occurred naturally. Harvest above that level is termed "additive" 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 aim to reduce the size of a deer population through additive mortality. Bucks might be targeted to increase buck competition, or does might be targeted to reduce reproduction and thus overall population size.。