Population modeling

Population modeling is an application of statistical models to the study of changes in populations.

Models allow us to better understand how complex interactions and processes work. Modeling of dynamic interactions in nature can provide a manageable way of understanding how numbers change overtime or in relation to each other. Ecological population modeling is concerned with the changes in population size and age distribution within a population as a consequence of interactions of organisms with the physical environment, with individuals of their own species, and with organisms of other species. . The world is full of interactions that range from simple to uncomfortably dynamic and many, if not all, of the earth’s processes affect human life. The earth’s processes are greatly stochastic and seem chaotic to the naked eye. However, a plethora of patterns can be noticed and are brought forth by using population modeling as a tool. . Population models are used to determine maximum harvest for agriculturist, to understand the dynamics of biological invasions, and have numerous environmental conservation implications. Population models are also used to understand the spread of parasites, viruses, and disease. The realization of our dependence on environmental health has created a need to understand the dynamic interactions of the earth’s flora and fauna. Methods in population modeling have greatly improved our understanding of ecology and the natural world.

History
Late 18th-century biologists began to develop techniques in population modeling in order to understand dynamics of growing and shrinking populations of living organisms. Thomas Malthus was one of the first to note that populations grew with a geometric pattern while contemplating the fate of humankind. One of the most basic and milestone models of population growth was the Logistic model of population growth formulated by Pierre François Verhulst in 1838. The logistic model takes the shape of a sigmoid curve and describes the growth of a population as exponential, followed by a decrease in growth, and bound by a carrying capacity due to environmental pressures. Population modeling became of particular interest to biologist in the 20th century as pressure on limited means of sustenance due to increasing human populations in parts of Europe were noticed by biologist like Raymond Pearl. In 1921 Pearl invited physicist A.J. Lotka to assist him in his lab. Lotka developed paired differential equations that showed the effect of a parasite on its prey. Mathematician Vito Volterra equated the relationship between two species independent from Lotka. Together, Lotka and Volterra formed the Lotka-Volterra model for competition that applies the logistic equation to two species illustrating competition, predation, and parasitism interactions between species. In 1939 contributions to population modeling were given by Patrick Leslie as he began work in biomathematics. Leslie emphasized the importance of constructing a life table in order to understand the effect that key life history strategies played in the dynamics of whole populations. Matrix algebra was used by Leslie in conjunction with life tables to extend the work of Lotka. Matrix models of populations calculate the growh of a population with life history variables. Later, Robert MacArthur and Edward Wilson characterized island biogeography. The Equilibrium Model of Island Biogeography describes the number of species on an island as equilibrium of immigration and extinction. The Logistic population model, the Lotka-Volterra model of community ecology, life table Matrix Modeling, the Equilibrium Model of Island Biogeography and variations there of are the basis for ecological population modeling today.

Equations
Logistic growth equation:


 * $$\frac{dN}{dt} = rN\left(1-\frac{N}{K}\right)\,$$

Lotka-Volterra equation:


 * $$\frac{dN_1}{dt} = r_1 N_1\frac{K_1-N_1 - \alpha N_2}{K_1}\,$$

Island biogeography:


 * $$S = \frac{IP}{I+E}$$

Species area:


 * $$\log(S) = \log(c)+z \log(A)\,$$