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DEPARTMENT OF COMPUTER SCIENCE Spring Semester 2017-2018 MODELLING AND SIMULATION OF NATURAL SYSTEMS 2 hours
Answer TWO questions.
All questions carry equal weight. Figures in square brackets indicate the per- centage of available marks allocated to each part of a question.
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1. This question covers general concepts for modelling and simulating natural systems and numerical methods for solving differential equations.
a) General concepts for modelling natural systems.
(i) What is an agent based model ? What are the key considerations for designing an agent based model ? [10%]
(ii) Ants are social insects that follow a number of well-defined behaviours to allow a colony to survive and may be modelled using an agent based modelling approach. Give five examples of types of individual behaviour that might be important for developing an agent based model of an ant colony. [5%]
(iii) Give five examples (other than an ant colony) of natural systems that agent based modelling could be used to model. [5%]
(iv) You are presented with two models which have been used to fit a set of data. One of these models has 25 free parameters and error 10−5 %. The other has 3 free parameters and 2% error. Give a definition of what a free parameter is. What are the disadvantages of a model with too many free parameters? Briefly discuss which of the two models you would use. On which principle would you base your decision?
b) Numerical methods for solving differential equations (Euler’s method).
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(i)
(ii)
Explain Euler’s method. As part of your answer, provide the corresponding algorithm. How should one choose the discretisation time-step ∆t? Justify your answer. [20%]
A first order linear differential equation can be solved with Euler’s method,
or analytically (via integration). How do the two solutions differ in terms of accuracy? [5%]
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[30%]
(iii)
The RK methods consist of numerical methods that solve initial value problems of the form dx = f (t, x(t)). They can be described by a set of equations:
dt
xn+1 = xn + φ∆t,
wherexn isaknownvalueoffunctionxattimetn,∆t=tn+1−tn,andφis
the increment function, defined as:
φ=α1k1 +α2k2 +…+αiki,
where i is an (integer) index, α1,α2, … are the parameters and k1, k2,… are
defined by the following equations:
k1 =f(tn,xn),
k2 =f(tn +p1∆t,xn +q11k1∆t),
…
with parameters p1 and q11. Which values must αi, pi, and qi have so that the general equations correspond to Euler’s method and the Midpoint method? Justify your answer. As part of the justification you will need to provide the mathematical formulas for Euler’s method and the Midpoint method. [HINT: The definitions of k3, k4 etc. are not required for answering this question.]
[25%]
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2. This question covers general concepts for solving and constructing ordinary differential equa- tions and their applications in natural systems. It also covers basic Computational Neuro- science topics.
a) The question requires you to construct ordinary differential equations to model a number of natural systems.
(i) Assume that the population of mosquitos m on a lake grows at a rate propor- tional to their current population. Write down a differential equation for the rate of change of m. State the conditions that any constants you include must satisfy. [5%]
(ii) The average global birth rate in 2010 was b and the death rate was m . Write down a differential equation for the growth rate of the global population p. [5%]
(iii) A circular patch of oil on the surface of some water has a radius r metres at time t minutes. Suppose that the rate of increase of r is proportional to 1/r. Express this statement as a differential equation. State the conditions that any constants you include must satisfy. [5%]
(iv) The length of a leaf during the period of its growth is proportional to the amount of water it contains. Suppose the leaf absorbs water from its parent plant at a rate proportional to its length l and that it loses water by evaporation
at a rate proportional to the square of its length. Form a differential equation
to describe the rate of change of leaf length over time. State the conditions that any constants you include must satisfy. [5%]
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b)
Assume that the population of aphids p in a greenhouse can be modelled using a
logistic growth equation, i.e.
dp = rp(1 − kp) dt
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c)
where t denotes the time in weeks and r, k are positive constants.
(i) Show how to derive the following solution to the differential equation:
p = c exp (rt) 1+ckexp(rt)
where c is a constant of integration to be determined from initial conditions. [20%]
(ii) Given k = 0.001 and that the initial size of the population p is 100, determine the value of the constant of integration c. [5%]
(iii) Find the equilibrium points of the equation. Are they stable or unstable? [10%] (iv) What is the maximum size of the population of aphids in the greenhouse? [5%]
(i) Explain what a spiking neuron model is. As part of your answer describe the integrate-and-fire model, providing the equations for the case of a neuron that receives input from other neurons and the case where a constant input
is assumed. In the latter case of a constant input, perform the steady state analysis of the corresponding differential equation. Describe what happens if the firing threshold of the neuron is set at a value higher than the constant input. Also describe what happens if the firing threshold of the neuron is set
at a value lower than the constant input. [25%]
(ii) Briefly describe the problem of neuronal coding. Present three codes according
to which the information may be transmitted in brain systems. [15%]
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3. a)
In the Hudson Bay area of North America, there is an ecosystem consisting of two interacting species, the snowshoe hare (prey) and the Canada lynx (predator). Write down and fully define two equations describing how you would expect the numbers of these species to change over time, based on the following simple assumptions:
• The hare have an unlimited food supply.
• Their only predator is the lynx.
• The lynx are totally dependent on the hare as their only food supply.
• The lynx have unlimited appetites.
• Without predation the population of hares increases exponentially.
By which name is this system is known? Find the equilibrium points of the system. [40%]
Another approach to modelling this system would be using an Agent Based Model (ABM). Explain two advantages and two disadvantages of using an ABM approach as compared with the equation based approach above. [15%]
A formal description of the ABM considers agents to be finite state machines (poten- tially with memory), and changes in state can be thought to be state transitions. Ants are examples of social insects that interact and follow particular foraging behaviours in order to allow the colony to survive. For the ant colony example, write down:
(i) Two types of agent that you could feasibly use in an ABM simulation of an ant colony.
[5%]
(ii) For each of these agent types, two possible states that the agent could exist in
[10%]
Another example of a system that can be modelled using an ABM technique is the behaviour of biological cells in laboratory-based tissue culture. In this case, agents would represent individual cells and rules would represent behaviour such as migration, cell division or cell death. For this system, give an example of three parameters that you might use in your model.
[5%]
In general terms (i.e. not limited to the biological cells) state how we can obtain model parameters. Describe one way in which you could investigate how the value of
a particular parameter affects the predicted emergent behaviour in an ABM and how you could improve the model in light of these results. [15%]
Briefly discuss a mathematical method by which you could evaluate the effect of a specific parameter to the system’s behaviour in a differential equation-based model, which is conceptually similar to the technique you use in the ABM approach.
b)
c)
d)
e)
f)
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