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Population dynamics is the type of mathematics used to model and study the size and age composition of populations as dynamical systems. Population dynamics is a branch of mathematical biology , and uses mathematical techniques such as differential equations to model behaviour.
Andrey Kolmogorov. In biomathematics, the Kolmogorov population model, also known as the Kolmogorov equations in population dynamics, is a mathematical framework developed by Soviet mathematician Andrei Kolmogorov in 1936 that generalizes predator-prey interactions and population dynamics.
The model can also be written in the form of a differential equation: = with initial condition: P(0)= P 0. This model is often referred to as the exponential law. [5] It is widely regarded in the field of population ecology as the first principle of population dynamics, [6] with Malthus as the founder.
An extension to these are the competitive Lotka–Volterra equations, which provide a simple model of the population dynamics of species competing for some common resource. In the 1930s Alexander Nicholson and Victor Bailey developed a model to describe the population dynamics of a coupled predator–prey system. The model assumes that ...
Population models are used to determine maximum harvest for agriculturists, to understand the dynamics of biological invasions, and for environmental conservation. Population models are also used to understand the spread of parasites, viruses, and disease. [2] Another way populations models are useful are when species become endangered.
The Lotka–Volterra system of equations is an example of a Kolmogorov population model (not to be confused with the better known Kolmogorov equations), [2] [3] [4] which is a more general framework that can model the dynamics of ecological systems with predator–prey interactions, competition, disease, and mutualism.
The rapid increase in the global population of the past century exemplifies Malthus's predicted population patterns; it also appears to describe socio-demographic dynamics of complex pre-industrial societies. These findings are the basis for neo-Malthusian modern mathematical models of long-term historical dynamics. [36]
In the study of age-structured population growth, probably one of the most important equations is the Euler–Lotka equation.Based on the age demographic of females in the population and female births (since in many cases it is the females that are more limited in the ability to reproduce), this equation allows for an estimation of how a population is growing.