Predator–Prey models
Predator-prey models are mathematical frameworks used to describe the dynamic interactions between two species—predators and their prey. These models typically consist of a set of differential equations that represent the changes in population sizes over time, with one equation reflecting the growth of the prey population and the other representing the growth of the predator population. The Lotka-Volterra model, developed independently by Alfred Lotka and Vito Volterra in the early 20th century, is one of the most well-known examples. It demonstrates how predator populations tend to oscillate in response to changes in prey populations, with predators lagging behind.
While these models provide valuable insights into ecological dynamics, they also have limitations. Critics point out that the basic equations do not account for factors such as food availability, spatial dynamics, and seasonal variations, which can significantly influence population changes. In response to these shortcomings, researchers have since developed more sophisticated models that incorporate additional variables and are capable of simulating complex interactions in natural ecosystems. Modern advancements include the use of computer simulations to create highly detailed models that better reflect real-world dynamics. Overall, predator-prey models serve as essential tools in ecology for understanding species interactions and population management.
Predator–Prey models
Summary: The interaction between the population sizes of a predator species and a prey species can be modeled using systems of equations.
Predator–prey models are systems of mathematical equations that are used to predict the populations of interacting species, one of which—the prey—is the primary food source for the other—the predator. One famous example that has been extensively studied is the relationship between the wolves and moose on Isle Royale in Lake Superior.
The Isle Royale populations are well suited for modeling the predator–prey relationship because there is little food for the wolves other than the moose and there are no other predators for the moose. In addition, the geographic isolation limits other factors that would complicate the mathematics in the equations, such as hunting or migration. This predator–prey interaction has been carefully studied since the 1950s and continues to be investigated into the twenty-first century.
Modeling Predator–Prey Populations
Most predator–prey models are composed of two equations, the first representing the change in the prey population, and the second the change in the predator population. Each equation has the following form: birth function minus death function.
If X(t) represents the quantity of prey at time t, and Y(t) represents the quantity of predators at time t, then the instantaneous rate of change in prey is
and the instantaneous rate of change in predators is
where f1 is the mathematical term that describes the births in the prey population, f2 describes the deaths in the prey population, f3 describes the births in the predator population, and f4 describes the deaths in the predator population.
There have been many predator–prey models proposed since the beginning of the twentieth century. The most famous and the earliest known is the Lotka–Volterra system, named for the two scientists who developed the same mathematical model independently, American Alfred Lotka (1880–1949) publishing the equations in 1925 and Italian Vito Volterra (1860–1940) publishing them in 1926. Lotka had degrees in physics and chemistry, and he believed that one could apply physical principles to biological systems. His work on predator–prey interactions is just part of extensive work he published in 1925 in the text titled Elements of Physical Biology. Lotka used a chemical reaction analogy to justify the terms in the model.
In the absence of predators, the prey should increase at a rate proportional to the current quantity of prey, X. In other words, more moose around to mate without being hunted means more calves would be born. Likewise, in the absence of prey, the predators should die off at a rate proportional to the current predator population, Y. In other words, with many wolves and no moose for food, more wolves would starve.
Lotka used a chemical reaction analogy to explain prey deaths and predator births: when a reaction occurs by mixing chemicals, the rate of the reaction is proportional to the product of the quantities of the reactants. Lotka argued that prey should decrease and predators should increase at rates proportional to the product of the quantity of prey and predators, XY. In other words, the moose deaths should be closely related to the rate of interaction of wolves and moose, and the wolf births should be as well because wolves need the moose for food to be healthy and have pups. The equations can be written as
for non-negative proportionality constants a, b, c, and d.
Volterra arrived at the same model using different reasoning. Volterra was a physicist whose daughter and son-in-law were biologists. While looking for a mathematical explanation for a problem his son-in-law was working on, Volterra became very interested in interactions of species and spent the rest of his professional life looking for a mathematical theory of evolution.
The Lotka–Volterra predator–prey model can be solved without a computer and yields a graph that makes sense. The population of the predator oscillates as does that of the prey, with the predator population trailing slightly behind. Too many prey results in more predators, who swamp the prey causing a decrease in prey. As the prey become scarce, the predators also start to die out, and the cycle begins again (see Figure 1).
While this result has reasonable qualitative behavior, many scientists have objected to the equations in this form. Some of the concerns about the model have included the following:
- • If there are no predators, the prey population would grow arbitrarily large
- • A reduction in the number of prey should cause more predator deaths rather than fewer predator births
- • For a fixed number of predators, the number of prey eaten is proportional to the number of prey present, implying that predators are always hungry and eat the same proportion of the prey no matter how large the number of prey gets
- • The food for the prey plays a role in the births and deaths of the prey, and should be included in the model
- • No spatial considerations are incorporated in the model, so factors such as migration or seeking safety in herds are ignored
- • These equations do not take into account gestation periods and seasonal changes in birth rates
- • The constants a, b, c, and d are difficult to estimate for a given situation without a large amount of data collected from field observations
Much work has been done since the 1930s to modify the equations to address these concerns and to apply the equations to data from specific situations, such as the moose and wolves of Isle Royale. In the twenty-first century, scientists use sophisticated computer models to model predator–prey interactions using increasingly intricate equations to incorporate more realistic assumptions in the mathematics.
Bibliography
Kingsland, Sharon E. Modeling Nature, Episodes in the History of Population Ecology. Chicago: University of Chicago Press, 1995.
Lotka, Alfred J. Elements of Physical Biology. Baltimore, MD: Williams and Wilkins Publishers, 1925.
Volterra, Vito. “Variations and Fluctuations of the Number of Individuals in Animal Species Living Together.” In Animal Ecology. Edited by R. Chapman. New York: McGraw-Hill, 1926.
Vucetich, John A. “The Wolves and Moose of Isle Royale.” http://www.isleroyalewolf.org/wolfhome/home.html.