PREDATOR-PREY EQUATION

Predator-prey equation, known as the Lokta-Voltere Model describe the dynamics of biological systems in which two species interact, one as a predator and the other as prey.

ALFRED JAMES LOTKA

(MARCH 2, 1880 – DECEMBER 5, 1949) WAS A US MATHEMATICIAN, PHYSICAL CHEMIST, AND STATISTICIAN, FAMOUS FOR HIS WORK IN POPULATION DYNAMICS AND ENERGETICS.

Vito Volterra
(3 May 1860 – 11 October 1940) was an Italian mathematician and physicist, known for his contributions to mathematical biology and integral equations,being one of the founders of functional analysis.

The Lotka–Volterra model makes a number of assumptions about the environment and evolution of the predator and prey populations:

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• The prey population finds ample food at all times.
• The food supply of the predator population depends entirely on the size of the prey population.
• The rate of change of population is proportional to its size.
• During the process, the environment does not change in favour of one species and genetic adaptation is inconsequential.

PREY:
dx/dt = ax - βyx

PREDATOR
the equation is
dy/dx = δ xy - γx

First you need to know what each variation stands for:
y = is the number of any predator (example: a wolf)
x= is the number of any prey (example: a bunny)
(dy/dt) and (dx/dt) = represents the growth rate of each family in time
t (and y, I guess) = represents time
(α), (β), (γ) and (δ) = are positive parameters that represents the interaction between both species

First you need to use the Taylor Series to get a linear solution of the equation
f(x,y) = A0 - A1x - A1y
f(x,y) = B0 + B1x - B2y
With this coefficients you can study the models of competition, illness and mutualism (in biology) in a ecosystem

According to the prey equitation it assumes that the preys have a food system of limited food for defined times and it reproduces exponentially unless that any predator exists. This exponential growth is represented in the equation by the term of "ax."
The term of the equation "βxy" comes to represent the encounter of both species and their interaction.
The equation can represents the change of number of preys by their own growth unless of the encounters with predators

In this equation "δxy" represents the growth of predators meanwhile "γy" represents the natural death of the predators in exponential form; on how much predators is necessary to the number of victims or preys to grow to mantain population

GLOBAL CONTEXT

SOURCES

• https://en.m.wikipedia.org/wiki/Lotka–Volterra_equations
• http://www.scholarpedia.org/article/Predator-prey_model