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Modelling of Natural Systems: Individual assignment COM3001.
1 Question 1: Dynamical systems
Consider the following Lorentz dynamical system, defined as
dx(t)/dt = σ(y − x)
dy(t)/dt = x(ρ − z) − y
dz(t)/dt = xy − βz (1)
with σ = 10, ρ = 28 and β = 8/3.
• Exercise 1 [20 marks]
– Research and summarize the significance of the Lorenz system in mathemati cal modeling, discussing its historical background and real-world applications. The literature review should be approximately one page in length and should highlight the historical significance of the model. In addition to discussing its origins, aim to include observations that are directly relevant to the exercises you will undertake. To strengthen the review, reference more recent appli cations or developments of the model, demonstrating its continued relevance and evolution in research.
– Support in detail how to use numerical methods to simulate the system, intro ducing the necessary mathematical passages. Choose an appropriate initial condition and simulate the system using different numerical methods and time step sizes. Present your results using one or more figures, showing the differences between the accomplished simulations.
First, you should provide a detailed explanation of how to apply numerical methods to simulate multidimensional dynamical systems. Once the method ology is clearly outlined, proceed to generate one or more figures with multiple panels, each illustrating the results obtained using different numerical integration methods and discretisation steps. Your visualisations should include phase portraits to highlight the system’s qualitative behavior under each method. Optionally, you may also include time series plots ( x(t), y(t), z(t) as time varies) to show how the system evolves over time—but the main goal is to clearly demonstrate the differences between numerical methods. Ensurethat all figures are easy to read:
∗ Use appropriately sized fonts for axis labels and legends.
∗ Make sure that trajectories are clearly distinguishable.
In addition to visual comparisons, you are encouraged to quantify the differences between simulations and include these comparisons in a separate figure or subplot. This will support a more rigorous evaluation of the numerical methods used.
• Exercise 2 [30 marks]
– Identify the most suitable numerical method and time step size for accurately simulating the system. Using your chosen numerical method and discretization step, generate phase portraits showing the characteristic butterfly pattern for at least two different initial conditions.
Here, you should design a procedure to select an appropriate numerical methodand discretisation step for the chosen dynamical system, continuing from thecomparison of different methods and step sizes explored in the previous section. Your procedure should be informed by results. Justify your selectionby discussing the outcomes of your comparisons, highlighting which methodand time step best balance accuracy, stability, and computational efficiencyfor your specific system. Finally, include at least two phase portraits of thesystem, generated from different initial conditions, to illustrate the system’s behavior.
– Explain what it means for a system to be chaotic. Use simulations and supporting figures to demonstrate the chaotic nature of the Lorenz system.
Demonstrate that classic features of chaotic systems—such as sensitivity to initial conditions—can be observed in the system under consideration. I rec ommend exploring and illustrating the chaotic nature of the system using two different approaches. Be sure to include a figure (or figures) that clearly sup port your findings.
2 Question 2: Prey-Predator System
Consider the agent based modelling paradigm for the prey-predator model introduced
in the lectures and do the following exercises.
• Exercise 1 [20 marks]
– Simulate the prey-predator model in an agent-based model paradigm and report your result graphically. The results should reproduce the typical os cillatory behaviour of prey-predator interactions. Interpret why these results differ from those obtained using equation-based modelling, providing as many reasons as possible.
Report a clear a figure on the system behaviour and explain the results shown. Then, compare this approach to equation based modelling and provide multiple reasons on their differences.
– If you repeat the simulations multiple times, the results will be different. Explain why and estimate the variability between different runs, supporting your reasoning.
Provide an estimate of how much different simulations vary by quantifying differences in key variables. This can be done by computing metrics such as the mean and standard deviation across multiple runs. You may choose variables that represent:
∗ Global properties of a simulation, such as the maximum number of prey observed during the entire run, and/or
∗ Time-specific quantities, such as the number of prey at a particular time point.
A complete answer should include both quantitative analysis (e.g., summary statistics, plots) and a reflection on the observed variability
• Exercise 2 [30 marks]
– Examine how varying a specific parameter affects the behavior of the dynamical system. For example, you could analyze how the probability of species extinction changes as a parameter α changes from α0 to higher values (you should choose the secific values). Use statistical analysis computing the average and the standard deviation of the outcome across multiple simulations for each parameter value, reporting the results in a figure. Finally, Explain the observed trends and relationships between α and the system’s behavior.
After selecting appropriate variables of interest (for instance, the probability of species extinction in the example above), vary a chosen parameter and perform multiple simulations for each parameter value. For each case, estimate relevant statistics (e.g., mean, standard deviation) of the selected vari ables. Finally, summarize your findings in a clear and well-labeled figure that presents the results. The figure should allow for easy comparison across pa rameter values and highlight how the system’s behavior changes in response to those variations. In the accompanying text, be sure to interpret the figure, explaining the trends or patterns observed.
3 Instructions
Prepare a report (maximum of six pages without considering the supplementary material) mimicking research journal papers’ structure and writing style. For instance, you can use the IEEE templates https://www.ieee.org/conferences/publishing/ templates.html. It is not necessary to use the template, but please report your name at the top of the report and the citations at the end like in a publication. Refer to the citations in the main text. Please remember to report clear figures, with axis labelled and adequate font sizes.
Plagiarism and Collusion
This assignment must be completed individually - any submitted work suspected of being the product of collusion will be thoroughly investigated, and those involved will be penalised according to university regulations. Credit will not be given to copied material (unchanged or minimally modified) from published sources, including websites. Any references that you use should be cited. Please refer to the guidance on “Collaborative work, plagiarism and collusion” in the Undergraduate and MSc Handbook. Note that this assignment will be checked via TurnItIn.
Deadline
Please upload a digital copy onto Blackboard by this deadline: 28/04/2025 , 23 : 59pm.