Stochastic Modelling Flashcards
What is a Stochastic Model? (How is it different from the models considered so far)
So far we have dealt with two types of population models:
- Continuous Models (Ordinary Differential Equations, Delayed Differential Equations)
- Discrete Models (Difference Equations, Delayed Difference Equations.)
Both of these types of models are determinisitic models, meaning the population evolves with certainty in time.
Stochastity is the process of introducing probability, or randomness, into the model to account for different behaviours.
What are the two types of behaviour that stochastic models account for?
- Demographic Stochasity
- Environmental Stochasity
What is Demographic Stochasity?
Randomness from the inherently discrete nature of individuals, which has the largest impact on small populations as the behaviour of one individual is more impactful.
What is Environmental Stochasity?
Randomness resulting from changes that impact the entire population, such as changes in the environment, this does not have a diminished effect on large populations.
How is the population defined in terms of a stochastic variable?
What is the setup of the stochastic birth model?
- Considering the population where growth is only due to the birth of individuals.
- Considering the population at time t + delta t, such that only one individual can be born in delta t.
What does Beta represent?
Beta represents the birth probability per individual per unit time.
What is the expression for one birth occuring when the population size is 1?
What is the expression for one birth in a population of n individuals?
- The first term represents the probability of a birth from one individual.
- The second term represents the probability of having no births from the remaining members of the population.
What is the simplified expression for one birth in a population of n individuals?
What is the probability differential equation for the birth model?
What are typical initial conditions for the stochastic birth model?
What is the exact solution to the stochastic birth model ODE?
Why is averaged behaviour important?
Averaged behaviour is important because it is often not possible to exactly solve the differential equation that includes the stochastic process.
So we can look at averages quantities (mean, variance, skewness etc) to investigate the behaviour of the model without needing to exactly solve it.
What is the general equation for the ‘moments’ of a probability distribution?
What is the general equation for the mean of a probability distribution?
The mean refers to the k=1 (first order) moment, and in general is given by:
What equation for the rate of change of the mean is found for the Stochastic birth model?
The following differential equation for the mean is obtained:
What is the equation for the mean of the Stochastic birth model?
The differential equation for the mean can be solved to obtain an exponential growth equation with the birth rate being the growth rate:
What is the definition of variance?
The variance is given by the following equation of probability distribution moments:
What is the equation for the rate of change of the variance found for the stochastic birth model?
What is the equation for the variance of the stochastic birth model?
This can be found by solving the ODE using the integrating factor method and applying the delta function inital conditions:
What are the governing equations of the stochastic birth and death model?
What is the equation for norm conservation?
What is the definition of a probability generating function?
The probability generating function contains complete information about the probability distribution.
It is defined by the following power series in z:
What is the first important property of the probability generating function?
What is the second important property of the probability generating function?
What is the third important property of the probability generating function?
What is the equation for the mean in terms of the probability generating function?
What is the equation for the variance in terms of the probability generating function?
What partilal differential equation is found for the stochastic birth & death process?
What is the general solution to the stochastic birth & death model PDE?
What is the particular solution to the stochastic birth & death model PDE?
What is the equation for the extinction probabilites associated with the stochastic birth & death model?
What is the mean of the stochastic birth & death model?
What is the variance of the stochastic birth & death model?
