Mathematics of extinction
The "Mathematics of extinction" explores the quantitative aspects of species extinction, delving into how mathematical models and statistical techniques can help understand and predict the fate of endangered species. Extinction occurs when the last individual of a species dies, and while some species persist for millions of years, human activities are believed to accelerate extinction rates. Factors influencing whether a species can avoid extinction include population size dynamics, geographical separation, and the ecological interactions between species.
Mathematical models like geometric growth, logistic growth, and predator-prey dynamics offer insights into population trends and the likelihood of survival or extinction. The concept of local extinction highlights that a species can disappear from one area while surviving in another, with potential for recolonization. Alarmingly, scientists estimate that we may be experiencing a sixth mass extinction, with thousands of species vanishing annually at rates far exceeding historical levels. Understanding these mathematical principles is crucial for conservation efforts and addressing the ongoing biodiversity crisis.
Mathematics of extinction
Summary: Causes and factors of extinction can be quantified and modeled using mathematical and statistical techniques.
Extinction occurs when the last member of a species dies. A species survives for much longer than any of its members. For example, a human can live up to about 120 years, whereas the human species (Homo sapiens) is thought to have existed for hundreds of thousands of years. It is not known how long our species will endure and indeed most species on Earth have already become extinct. There are many causes of extinction, some natural and others as a result of human activities. Many factors influence whether an endangered species can avoid extinction. These factors can be quantified and modeled using mathematical and statistical techniques. A species can disappear in some parts of its habitat but not in others. Not all species have existed on Earth for the same length of time some appear only briefly while others manage to persist for incredibly long periods of time. Human activities may be increasing the rate at which other species become extinct.
Rise of Extinction
A species is endangered when it consists of a small number of members. In such cases, individuals may have trouble finding each other because of geographical separation. For a species that is endangered, it is of interest to know whether the species is likely to become extinct. It is customary to let N(t) represent the size of a population at time t. The fact that the species is endangered implies that N(t) takes positive values close to zero. If N(t) is eventually measured to be zero, then the species has become extinct. However, if N(t) rebounds to larger positive values, then the species persists. In general, stochastic effects largely determine whether an endangered species will become extinct. Given population data N(t) at different times t, one may compute the mean (μ) of the population growth rate.
For example, if t =10 then
A positive (or negative) value of μ indicates that the population is growing (or declining) on average. Combining this information with the standard deviation (σ) of R(t) allows one to assess the risk for extinction, which is typically highest when μ is negative and σ is small. Complex models of population dynamics exist to predict whether a species will persist or become extinct. These include geometric growth models in which a population multiplies at a fixed rate, logistic growth models in which populations slowly attain steady-state sizes, and Lotka-Volterra predator-prey models for interactions between multiple species, named for Alfred Lotka and Vito Volterra.
Local Extinction
A species can become extinct in one area (such as an island) and still persist elsewhere (such as a continent). If the species is able to recolonize the former area, then this is known as a “rescue effect.” If local extinction events become synchronized as a result of global climate change, for example then the risk of a species becoming globally extinct is much higher.
Rate of Extinction
Scientists estimate that there may be 10 million species alive today and yet they account for fewer than 1 in 1000 species that have ever lived. The average time to extinction for a species, as measured from the time of its first appearance, is close to 10 million years. When the time to extinction for a species is much longer, such as more than 100 million years, then later members are said to be living fossils.
Mass Extinction
A mass extinction occurs when a large number of species become extinct in a short period of time. Although rare, the fossil record indicates that these events have occurred at least five times, the most famous being the mass extinction of non-flying dinosaurs 65 million years ago in what was probably a meteor impact. Many scientists believe that we are currently in the midst of a sixth mass extinction, with up to 40,000 species becoming extinct each year a rate that is roughly 100-1000 times higher than in prehistoric times.
Bibliography
Allen, Linda J. S. An Introduction to Mathematical Biology. Upper Saddle River, NJ: Prentice Hall, 2007.
Bright, Michael. Extinctions of Living Things (Timeline: Life on Earth). Portsmouth, NH: Heinemann, 2008.
Erickson, J., and A. E. Gates. Lost Creatures of the Earth. New York: Facts on File, 2001.
Hallam, T. Catastrophes and Lesser Calamities: The Causes of Mass Extinctions. New York: Oxford University Press, 2005.
Hecht, J. Vanishing Life: The Mystery of Mass Extinctions. New York: Atheneum, 2009.
Thieme, Horst R. Mathematics in Population Biology. Princeton, NJ: Princeton University Press, 2003.