Saturday, March 31, 2012

Study of Two species Interactions using Lotka Volterra Model

An ecological network is a presentation of the biotic interactions in an ecosystem, in which species(nodes) are connected by pairwise interactions(links). Species compete, evolve and disperse simply for the purpose of seeking resources to sustain their struggle for their very existence. Depending on their specific settings of applications, they can take the forms of resource-consumer, plant-herbivore, parasite-host interactions etc. When seemingly competitive interactions are carefully examined, they are often in fact some forms of predator-prey interaction in disguise. Thus these predator-prey interaction models give the building blocks of large and complex ecological networks.

Generic Predator-Prey Model:
Consider two populations whose sizes at a reference time t are denoted by x(t), y(t) respectively. The functions x and y might denote population numbers or concentrations (number per area) or some other scaled measure of the population sizes, but are taken to be continuous functions. Changes in population size with time are described by the time derivatives dx/dt and dy/dt respectively, and a general model of interacting populations is written in terms of two autonomous differential equations:

dx/dt = xf(x, y)
dy/dt = yg(x,y )

The functions f and g denote the respective per capita growth rates of the two species. It is assumed that df/dy < 0 and dg/dx > 0. This general model is often called Kolmogorov's predator-prey model.

Fig: An Example of Periodic activity generated by Predator-Prey Model.

Lotka Volterra Model:
In 1926, the famous Italian mathematician Vito Volterra proposed a differential equation model to explain the observed increase in predator fist (and corresponding decrease in prey fish) in the Adriatic Sea during World War I. At the same time in the United States, the equations studied by Volterra were derived independently by Alfred Lotka (1925) to describe a hypothetical chemical reaction in which the chemical concentrations oscillate. The Lotka-Volterra model is the simplest model of predator-prey interactions It is based on linear per capita growth rates, which are written as

f = b - py
g = rx - d

  • The parameter b is the growth rate of x (the prey) in the absence of interaction with species y (the predators). Prey numbers are diminished by these interactions: The per capita growth rate decreases (here linearly) with increasingly y, possibly becoming negative.
  • The parameter p measures the impact of predation on (dx/dt)/x.
  • The parameter d is the death (or emigration) rate of species y in the absence of interaction with species x.
  • The term rx denotes the net rate of growth (or immigration) of the predator population in response to the size of the prey population.
The Prey-Predator model with linear per capita growth rates is

dx/dt = (b - py)x
dy/dt = (rx - d)y

This system is referred to as Lotka Volterra Model and represents one of the earliest models in mathematical ecology.
Solving these equations analytically require complicated mathematical skills. On the top of that, some of the differential equations have no analytical solution at all. A better approach would be to use numerical methods like Euler's method.

Fig: Example showing cycles formed by population dynamics of two species generated from the model

This model is not very realistic. It does not consider any competition among prey or predators. As a result, prey population may grow infinitely without any resource limits. Predators have no saturation: their consumption rate is unlimited. The rate of prey consumption is proportional to prey density. Thus, it is not surprising that model behavior is unnatural showing no asymptotic stability. However numerous modifications of this model exist which make it more realistic.

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