where
The coefficients A(t, T) and B(t, T) in this case are determined by the following ordinary differential equations:
and:
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
Transformation of this equation leads to the logistic ODE:
(3.13)
where n is the population in units of carrying capacity
For systems, we can consider the predator-prey model in an environment consisting of foxes and rabbits:
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.