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= | number of rabbits at time t |
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= | number of foxes at time t |
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= | birth rate of rabbits |
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= | death rate of rabbits |
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= | unit birth rate of rabbits |
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= | death rate of foxes |
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= | birth rate of foxes |
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= | 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)).
3.3.4 Logistic Function
A logistic function (or logistic curve) is an S-shaped sigmoid curve defined by the equation:
(3.15)
where
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A special case is when
We can verify from this equation that the logistic function satisfies the non-linear initial value problem:
(3.16)
The logistic function models processes in a range of fields such as artificial neural networks (learning algorithms, where it is called an activation function), economics, probability and statistics, to name a few.
3.4 EXISTENCE THEOREMS FOR STOCHASTIC DIFFERENTIAL EQUATIONS (SDEs)
A random process is a family of random variables defined on some probability space and indexed by the parameter t where t belongs to some index set. A random process is a function of two variables:
where T is the index set and S is the sample space. For a fixed value of t, the random process becomes a random variable, while for a fixed sample point x in S the random process is a real-valued function of t called a sample function or a realisation of the process. It is also sometimes called a path.
The index set T is called the parameter set, and the values assumed by
The index set T can be discrete or continuous; if T is discrete, then the process is called a discrete-parameter or discrete-time process (also known as a random sequence). If T is continuous, then we say that the random process is called continuous-parameter or continuous-time. We can also consider the situation where the state is discrete or continuous. We then say that the random process is called discrete-state (chain) or continuous-state, respectively.
3.4.1 Stochastic Differential Equations (SDEs)
We give a short introduction to stochastic differential equations (SDEs) as they are closely related to ODEs. We discuss SDEs in more detail in Chapter 13.
We introduce the scalar random processes described by SDEs of the form:
