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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