Theoretical ecology is the scientific discipline devoted to the study of ecological systems using theoretical methods such as simple conceptual models, mathematical models, computational simulations, and advanced data analysis. Effective models improve understanding of the natural world by revealing how the dynamics of species populations are often based on fundamental biological conditions and processes. Further, the field aims to unify a diverse range of empirical observations by assuming that common, mechanistic processes generate observable phenomena across species and ecological environments. Based on biologically realistic assumptions, theoretical ecologists are able to uncover novel, non-intuitive insights about natural processes. Theoretical results are often verified by empirical and observational studies, revealing the power of theoretical methods in both predicting and understanding the noisy, diverse biological world.
Escherichia coli, also known as E. coli, is a Gram-negative, facultative anaerobic, rod-shaped, coliform bacterium of the genus Escherichia that is commonly found in the lower intestine of warm-blooded organisms. Most E. coli strains are harmless, but some serotypes (EPEC, ETEC etc.) can cause serious food poisoning in their hosts, and are occasionally responsible for food contamination incidents that prompt product recalls. Most strains do not cause disease in humans and are part of the normal microbiota of the gut; such strains are harmless or even beneficial to humans (although these strains tend to be less studied than the pathogenic ones). For example, some strains of E. coli benefit their hosts by producing vitamin K2 or by preventing the colonization of the intestine by pathogenic bacteria. These mutually beneficial relationships between E. coli and humans are a type of mutualistic biological relationship — where both the humans and the E. coli are benefitting each other. E. coli is expelled into the environment within fecal matter. The bacterium grows massively in fresh faecal matter under aerobic conditions for three days, but its numbers decline slowly afterwards.
Bacterial growth is proliferation of bacterium into two daughter cells, in a process called binary fission. Providing no event occurs, the resulting daughter cells are genetically identical to the original cell. Hence, bacterial growth occurs. Both daughter cells from the division do not necessarily survive. However, if the surviving number exceeds unity on average, the bacterial population undergoes exponential growth. The measurement of an exponential bacterial growth curve in batch culture was traditionally a part of the training of all microbiologists; the basic means requires bacterial enumeration by direct and individual, direct and bulk (biomass), indirect and individual, or indirect and bulk methods. Models reconcile theory with the measurements.
The carrying capacity of an environment is the maximum population size of a biological species that can be sustained by that specific environment, given the food, habitat, water, and other resources available. The carrying capacity is defined as the environment's maximal load, which in population ecology corresponds to the population equilibrium, when the number of deaths in a population equals the number of births. The effect of carrying capacity on population dynamics is modelled with a logistic function. Carrying capacity is applied to the maximum population an environment can support in ecology, agriculture and fisheries. The term carrying capacity has been applied to a few different processes in the past before finally being applied to population limits in the 1950s. The notion of carrying capacity for humans is covered by the notion of sustainable population.
A logistic function or logistic curve is a common S-shaped curve with equation
Exponential growth is a process that increases quantity over time. It occurs when the instantaneous rate of change of a quantity with respect to time is proportional to the quantity itself. Described as a function, a quantity undergoing exponential growth is an exponential function of time, that is, the variable representing time is the exponent.
In population ecology and economics, maximum sustainable yield (MSY) is theoretically, the largest yield that can be taken from a species' stock over an indefinite period. Fundamental to the notion of sustainable harvest, the concept of MSY aims to maintain the population size at the point of maximum growth rate by harvesting the individuals that would normally be added to the population, allowing the population to continue to be productive indefinitely. Under the assumption of logistic growth, resource limitation does not constrain individuals' reproductive rates when populations are small, but because there are few individuals, the overall yield is small. At intermediate population densities, also represented by half the carrying capacity, individuals are able to breed to their maximum rate. At this point, called the maximum sustainable yield, there is a surplus of individuals that can be harvested because growth of the population is at its maximum point due to the large number of reproducing individuals. Above this point, density dependent factors increasingly limit breeding until the population reaches carrying capacity. At this point, there are no surplus individuals to be harvested and yield drops to zero. The maximum sustainable yield is usually higher than the optimum sustainable yield and maximum economic yield.
The Lotka–Volterra equations, also known as the predator–prey equations, are a pair of first-order nonlinear differential equations, frequently used to describe the dynamics of biological systems in which two species interact, one as a predator and the other as prey. The populations change through time according to the pair of equations:
Pierre François Verhulst was a Belgian mathematician and a doctor in number theory from the University of Ghent in 1825. He is best known for the logistic growth model.
Population dynamics is the type of mathematics used to model and study the size and age composition of populations as dynamical systems.
Population ecology is a sub-field of ecology that deals with the dynamics of species populations and how these populations interact with the environment, such as birth and death rates, and by immigration and emigration.
A metapopulation consists of a group of spatially separated populations of the same species which interact at some level. The term metapopulation was coined by Richard Levins in 1969 to describe a model of population dynamics of insect pests in agricultural fields, but the idea has been most broadly applied to species in naturally or artificially fragmented habitats. In Levins' own words, it consists of "a population of populations".
The Allee effect is a phenomenon in biology characterized by a correlation between population size or density and the mean individual fitness of a population or species.
The Luria–Delbrück experiment (1943) demonstrated that in bacteria, genetic mutations arise in the absence of selective pressure rather than being a response to it. Thus, it concluded Darwin's theory of natural selection acting on random mutations applies to bacteria as well as to more complex organisms. Max Delbrück and Salvador Luria won the 1969 Nobel Prize in Physiology or Medicine in part for this work.
When a quantity grows towards a singularity under a finite variation it is said to undergo hyperbolic growth. More precisely, the reciprocal function has a hyperbola as a graph, and has a singularity at 0, meaning that the limit as is infinite: any similar graph is said to exhibit hyperbolic growth.
Intraspecific competition is an interaction in population ecology, whereby members of the same species compete for limited resources. This leads to a reduction in fitness for both individuals, but the more fit individual survives and is able to reproduce. By contrast, interspecific competition occurs when members of different species compete for a shared resource. Members of the same species have rather similar requirements for resources, whereas different species have a smaller contested resource overlap, resulting in intraspecific competition generally being a stronger force than interspecific competition.
The Gompertz curve or Gompertz function is a type of mathematical model for a time series, named after Benjamin Gompertz (1779–1865). It is a sigmoid function which describes growth as being slowest at the start and end of a given time period. The right-side or future value asymptote of the function is approached much more gradually by the curve than the left-side or lower valued asymptote. This is in contrast to the simple logistic function in which both asymptotes are approached by the curve symmetrically. It is a special case of the generalised logistic function. The function was originally designed to describe human mortality, but since has been modified to be applied in biology, with regard to detailing populations.
A Malthusian growth model, sometimes called a simple exponential growth model, is essentially exponential growth based on the idea of the function being proportional to the speed to which the function grows. The model is named after Thomas Robert Malthus, who wrote An Essay on the Principle of Population (1798), one of the earliest and most influential books on population.
A chemostat is a bioreactor to which fresh medium is continuously added, while culture liquid containing left over nutrients, metabolic end products and microorganisms is continuously removed at the same rate to keep the culture volume constant. By changing the rate with which medium is added to the bioreactor the specific growth rate of the microorganism can be easily controlled within limits.
A population model is a type of mathematical model that is applied to the study of population dynamics.
