Numerical Methods in Computational Finance. Daniel J. Duffy

have the form:

      where and are given functions of t.

      The coefficients A(t, T) and B(t, T) in this case are determined by the following ordinary differential equations:

(3.11)

      and:

(3.12)

The first Equation (3.11) for B(t, T) is the Riccati equation and the second one (3.12) is solved easily from the first one by integration.

      3.3.3 Predator-Prey Models

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.