Читать книгу Numerical Methods in Computational Finance - Daniel J. Duffy - Страница 56

ODEs can be used as simple models of population growth, for example, by assuming that the rate of reproduction of a population of size P is proportional to the existing population and to the amount of available resources. The ODE is:

where r is the growth rate and K is the carrying capacity. The initial population is . It is easy to check the following identities:

Transformation of this equation leads to the logistic ODE:

(3.13)

where n is the population in units of carrying capacity and measures time in units of 1/r.

For systems, we can consider the predator-prey model in an environment consisting of foxes and rabbits:

(3.14)

where:

	=	number of rabbits at time t
	=	number of foxes at time t
	=	birth rate of rabbits
	=	death rate of rabbits
	=	unit birth rate of rabbits
	=	death rate of foxes
	=	birth rate of foxes
	=	unit birth rate of foxes.

The ODE system (3.14) is a model of a closed ecological environment in which foxes and rabbits are the only kinds of animals. Rabbits eat grass (of which there is a constant supply), procreate and are eaten by foxes. All foxes eat rabbits, procreate and die of geriatric diseases.

System (3.14) is sometimes called the Lotka–Volterra equations, which are an example of a more general Kolmogorov model to model the dynamics of ecological systems with predator-prey interactions, competition, disease and mutualism (Lotka (1956)).