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differential

Real life use of Differential Equations

Differential equations have a remarkable ability to predict the world around us.  They are used in a wide variety of disciplines, from biology, economics, physics, chemistry and engineering. They can describe exponential growth and decay, the population growth of species or the change in investment return over time.  A differential equation is one which is written in the form dy/dx = ……….  Some of these can be solved (to get y = …..) simply by integrating, others require much more complex mathematics.

Population Models

One of the most basic examples of differential equations is the Malthusian Law of population growth dp/dt = rp shows how the population (p) changes with respect to time.  The constant r will change depending on the species.  Malthus used this law to predict how a species would grow over time.

More complicated differential equations can be used to model the relationship between predators and prey.  For example, as predators increase then prey decrease as more get eaten. But then the predators will have less to eat and start to die out, which allows more prey to survive.  The interactions between the two populations are connected by differential equations.

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The picture above is taken from an online predator-prey simulator .  This allows you to change the parameters (such as predator birth rate, predator aggression and predator dependance on its prey).  You can then model what happens to the 2 species over time.  The graph above shows the predator population in blue and the prey population in red – and is generated when the predator is both very aggressive (it will attack the prey very often) and also is very dependent on the prey (it can’t get food from other sources).  As you can see this particular relationship generates a population boom and crash – the predator rapidly eats the prey population, growing rapidly – before it runs out of prey to eat and then it has no other food, thus dying off again.

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This graph above shows what happens when you reach an equilibrium point – in this simulation the predators are much less aggressive and it leads to both populations have stable populations.

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There are also more complex predator-prey models – like the one shown above for the interaction between moose and wolves.  This has more parameters to control.  The above graph shows almost-periodic behaviour in the moose population with a largely stable wolf population.

Some other uses of differential equations include:

1) In medicine for modelling cancer growth or the spread of disease
2) In engineering for describing the movement of electricity
3) In chemistry for modelling chemical reactions
4) In economics to find optimum investment strategies
5) In physics to describe the motion of waves, pendulums or chaotic systems.

With such ability to describe the real world, being able to solve differential equations is an important skill for mathematicians.  If you want to learn more, you can read about how to solve them here.

If you enjoyed this post, you might also like:

Langton’s Ant – Order out of Chaos How computer simulations can be used to model life.

Does it Pay to be Nice? Game Theory and Evolution. How understanding mathematics helps us understand human behaviour

Essential resources for IB students:

1) Revision Village

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Revision Village has been put together to help IB students with topic revision both for during the course and for the end of Year 12 school exams and Year 13 final exams.  I would strongly recommend students use this as a resource during the course (not just for final revision in Y13!) There are specific resources for HL and SL students for both Analysis and Applications.

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There is a comprehensive Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and then provides a large bank of graded questions.  What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial.  Really useful!

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The Practice Exams section takes you to a large number of ready made quizzes, exams and predicted papers.   These all have worked solutions and allow you to focus on specific topics or start general revision.  This also has some excellent challenging questions for those students aiming for 6s and 7s.

Each course also has a dedicated video tutorial section which provides 5-15 minute tutorial videos on every single syllabus part – handily sorted into topic categories.

2) Exploration Guides and Paper 3 Resources

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I’ve put together four comprehensive pdf guides to help students prepare for their exploration coursework and Paper 3 investigations. The exploration guides talk through the marking criteria, common student mistakes, excellent ideas for explorations, technology advice, modeling methods and a variety of statistical techniques with detailed explanations. I’ve also made 17 full investigation questions which are also excellent starting points for explorations.  The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.

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All content on this site has been written by Andrew Chambers (MSc. Mathematics, IB Mathematics Examiner).

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