COM**4 Bio-inspired Computing
Lecture 7a Introduction to Simulation
Objectives of Lecture 7a-c
1. Tointroducetheconceptofapredictive, mechanistic model
2. Tointroducetwoalternativeapproachesto modelling in the context of understanding population dynamics:
Equation-based Modelling
Individual (agent-based) Modelling
3. Topresenttheadvantages/disadvantagesofeach of these approaches.
Bi-directional process
SCIENCE
ENGINEERING
Bio-inspired computing
Pallavi Deshpande
Computational exploration of real-world systems
Biological Physical ?Social Financial ?Artificial Intelligence
Engineered hardware Algorithms
Software protection systems
What is a model?
Means different things to different people
Method for structuring and validating
knowledge
“ All models are wrong, but some models are useful”
Box, G.E.P., Robustness in the strategy of scientific model building, In Robustness in Statistics, R.L. Launer and G.N. Wilkinson, Editors.
Data driven models
Aim to derive knowledge from large datasets
Use statistics or machine learning methods to find a
relationship between inputs and outputs
Allows us to ask questions like “given a data set A,
what is the probability of outcome B?”
Does not attempt to consider details of mechanisms
“Top down” approach
Mechanistic Models
Starts from known or assumed mechanism and attempts to predict results (simulation)
Many approaches (ODEs, finite element, agent-based....)
Allows us to explore “what if?” scenarios
“Bottom up” approach
What do you notice about the populations of hares and lynxes?
Can you explain why this occurs?
Can a mathematical/ computational model simulate this ?
https://www.youtube.com/watch?v=swiSMSWgbKE
Population Models
A Simple Population Model
Considerapopulationofself-replicatingaggressiveorganisms that live in a space of fixed size.
Weareinterestedinknowinghowthesizeofthepopulation changes from day to day.
Let the population number on day t be N(t). This is our model variable.
Considertwoprocessesthatcontributetochangesinthe population size: birth and death.
A Simple Population Model
General form of the model is
N(t+1) = N(t) + number of births – number of deaths
Assumptions:
- population is large enough that we can approximate N(t), which is an integer, by a real number n(t).
- number of births per day is proportional to population size
(self-replicating organisms). where B is the birth rate (a model parameter).
- number of births = Bn(t),
A Simple Population Model
Assume there are two processes contributing to the death rate:
- organisms die naturally, at a rate proportional to the population size n(t)
- when organisms meet, they fight to the the death, and there is always a clear victor (one death per encounter).
Assume chance of meeting is proportional to square of the population density, which is proportional to n(t)2 (since they live in a space of fixed area).
Hence
death rate = D0 n(t) + D n(t)2,
where
D0 is the natural death rate ;
D depends on the area of the space in which the organisms live D0 , D are parameters (constants).
A Simple Population Model
Putting all this together, we obtain the model:
n(t+1) = n(t) + Bn(t) – D0n(t) – Dn(t)2
Or, more generally:
n(t+1) = f(n(t); B, D0, D)
model VARIABLE
model PARAMETERS
A Simple Population Model
This illustrates some important points:
1. We have made a number of simplifying assumptions in order to formulate the model.
2. We have identified what we believe to be the relevant processes that underlie the change in the population.
3. We have used parameters to encode the details of those processes. Even if we don’t know those details, we can study the outcome of the model for different values of those parameters.
What can we do with this model ?
n(t+1) = n(t) + Bn(t) – D0n(t) – Dn(t)2
We might want to use this to find out what the population n(t) will be on each day of the next week, given that we measure the population to be 500 on day 1.
We can specify values for the parameters and use the above formula iteratively to evaluate the population each day. This is simulation (synthesis).
Alternatively, we can use the mathematical expression above to deduce some general properties of the system. This is called analysis.
NEXT: Lecture 7b) Analysis and Synthesis for an equation-based model
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