We can use mathematical modelling to predict the movements of many species in the wild πΎπΎ. But first…why is it important for us to be able to do this?! π€Well, knowing how and why animals move through their environment has huge ecological, behavioural and evolutionary ramifications such as those in social interactions, spread of disease, gene flow and migration.

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Now for the maths! π€ To make a model, we start by collecting some data through tracking the animal in question π». This technology may, for example, give us the position of the animal at each minute during an hour π°. We can then plot these movements in (x,y) coordinates. Since we only have the position of the animal every minute (and not every second or nano second etc), the path we obtain will be jagged where each βstepβ goes to the next in a straight line with a given distance and angleππ. Using this path, we can create graphsπ (histograms) showing the likelihoodβs of different step distances and angles.

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We use this information to

make our model by telling the computer each of the step lengths and angles that are possible, but also that some should be more likely to be chosen than others! π»βοΈ

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While this is the basis of our model, we can also include other information we know about the preferences of how a particular animal in question moves. For example, leopards π like to be on their own so may be more likely to move away from areas with other leopards, but elephants πππ are social and live in groups so are more likely to be looking for their friends. Jaguars like hiding in the cover of the trees so may be more likely to head deep into forests π³; however, meerkats would more likely prefer to sunbath in the open βοΈ. Many other factors could also be considered, including predation risk, weather, foraging ability etc!

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To make our model a little easier, we break up the area we are looking at into a grid π (to reduce the number of spots the animal can move to from its current position). We give each square of the grid a value for each of the characteristics above between 0-1 i.e. close to 0 = low foraging ability (not much food in the area) and near to 1 = very good foraging ability (lots of food here!)π₯

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Finally we can assign a probability that the animal will move to each of the squares by…multiplying the probability of its distance βοΈ by the probability of its angle βοΈ by the values for the other preferences (numbers between 0-1) and divide β this by the sum of the probabilities of each square in the grid. By repeating this method our model is able to make predictions about an animals route, providing essential knowledge for a wildlife conservationist πΎπΎπΎ

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#appliedmathematicsΒ #usingmathsΒ #mathbiologyΒ #ecologyΒ #conservationΒ #evolutionΒ #helpinganimalsΒ #stemΒ #keeplearningΒ #scicomm

## Happy New Year!π₯ππ The maths of fireworks and leap years π

Itβs 2020!! Fireworks are a staple at new year and these beautiful displays wouldnβt be possible without maths π! 2020 is also a leap year – but do you know the maths behind why we have leap years?? And did you know that they arenβt always every 4 years!? π€

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First for the maths of fireworksππ. Well fireworks work like anything you throw into the air – they follow a curve and eventually start falling back to the ground. It is very important to get the maths right when making fireworks as we wouldnβt want them going off when they have fallen back down! Using maths, you can calculate the height that the firework explodes given the force that propels it into the air (will need to be larger for bigger fireworks). The fuse π§¨ needs to be the correct length so that it burns for just the right amount of time before going off at the fireworks highest point. What we call the βstarsβ are the things that light up in each direction to produce amazing patterns in the sky. Since firework shells are usually spherical, the arrangement of the stars inside the shells are propelled out symmetrically to give these lovely displays. π

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Why do we have leap years?! Our calendar year is 365 days long; however, the Earth π actually takes approximately 365.25 days to orbit the Sun βοΈ. Therefore, to keep our calendar π
in line with how far along the Earth is in its orbit, we need to add an extra 0.25 days to each year. The easiest way to do this is to add one whole day every 4 years….and that is exactly what we do – in our leap years! In fact, the orbit of the Earth can be even more accurately measured as taking approximately 365.2422 days. Therefore, adding an extra quarter day every year is too much! To get us back in line, we do something that not a lot of people know…we skip having some leap years! More specifically, we skip having a leap year if the year:

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1. Is a multiple of 100;

2. Is not a multiple of 400.

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So the year 2100 for example will not be a leap year!π

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We can use maths to check how accurate this method is:

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Every 400 years will have 100 (every 4 years) – 3 (not the 100th, 200th and 300th year) = 97 leap years. So we will have 400 – 97 = 303 βnormalβ years. Altogether, this will be (303 x 365 days) + (97 x 366 days) = 146,097 days. Every year will therefore have 146,097 / 400 = 365.2425 days. This is very close, but is still not quite right (still a tiny bit too highπ). However, is it so close that we wonβt notice the difference for many thousands of years. Scientists will then have to sit down and work out when else we should skip leap years…can you come up with a method that would give us 365.2422 days per year?? π€π€

#appliedmathematics#mathsinlife#happynewyear#leapyear#2020#fireworks#didyouknow#stem#keeplearning#scicomm

## Merry ChristMATHSππ

Before you find your seat at the dinner table today, why not do a bit of maths to ensure youβre a winner…the maths for winning the cracker pull every time π

There is a mathematical formula you can follow to win a Christmas cracker pull π. It has been found that the best way to pull a cracker is to pull down and at an angle βοΈ given by the following formula:

angle = (11 x cracker circumference)/cracker length + 5 x cracker quality,

where the cracker quality is a number between 1-3 (1 for low quality and 3 for high quality). This angle should be somewhere between 20-55 degrees.

Another fun ChristMATHS factππ if you were to add up all of the gifts that your true love gave to you in the 12 days of Christmas π΅ you would get one for each day of the year except for Christmas Day π€―π #christmas#maths#appliedmathematics#christmascrackers#12daysofchristmas#mathsinlife#winningstrategies#usingmaths#scicomm#merrychristmas

## Using maths to combat bacterial infections π¦

Let’s suppose you eat some undercooked meat π or drink some contaminated waterπ₯ by mistake. You might be fine, you might feel a bit sick for a few days but get better, but (if you are unlucky) you might have to go to the hospital and have a course of antibiotics π.

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Let’s talk about what happens inside your body when you get a bacterial infection. Your body recognises the bacteria as an ‘enemyβ and so produces chemicals that attract cells called white blood cells. Once white blood cells reach the bacteria, they engulf (eat) the bacteria until the infection is defeated (cleared). If the white blood cells can’t do this on their own, then antibiotics can help by either killing the bacteria or preventing them from replicating. Many types of bacteria, however, have some tricks up their sleeves to prevent such an easy defeatππΌ. For example, certain types of E. coli are able to hide from the immune system. The details of how the immune system interacts with bacteria is not always well understood…this is where maths comes to the rescue!π

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We can model an infection by writing mathematical equations and algorithms (rules) that describe what happens in the infection, based on what we already know about the biology π§¬. For example, we could say that white blood cells move around randomly until some of those chemicals show up, and then the white blood cells would move in the general direction of the bacteria. Once we have made our model, we then ‘press playβΆοΈ’ to watch a simulation of the infection. Incorporating the impact of different antibiotic strategies on the rate of infection clearance in these simulations, can help us to decide which strategies are best.

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These models can even be used to test the ability of made up drugs and strategies to clear infections (which is clearly much safer than testing these on people!). Using maths also saves on much of the money π΅ and time β° that would be spent in doing physical experiments π§«.

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Research on this topic was undertaken by James Preston during his PhD at the University of Nottinghamππ»

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#appliedmathematicsΒ #bacteriaΒ #mathmodelingΒ #mathsinlifeΒ #antibioticsΒ #scicomm

## What would happen if our planet started spinning the other way?! ππ Maths can tell us π

Like most planets in our solar system, if we were sat above the North Pole looking down on the Earth, we would see it rotating in an anti clockwise direction π. The forces involved in this spinning motion have shaped the environments and climates that we live in. But what would happen if it rotated the other way?π€π€

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Well mathematical equations can be used to model the changes in Earths atmosphere and oceans over time (this is how we can have weather forecasts! π¦). One of these models is called the Earth System Model (ESM) and this has been used to predict what our planet might be like (the temperatureπ₯, wind speedsπͺ, currentsπ, moistureπ§, concentrations of gasesπ«, land useπ etc…) in many years to come. There are a lot of measurements to consider over a whole lot of space (the entire Earth!π), so to make this a little easier the Earth is divided into a 3D gridπ (across its surface and in the vertical direction). Even so, however, these equations can only be solved using supercomputers! π¦ΈββοΈπ»

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So if we reverse the direction of the spin in these simulations, we can also use this model to test what would happen in this strange scenario π. As expected, it is found that countries would experience new climatic conditions due to changes in winds and currents. But it also turns out that if the Earth was rotating clockwise π, we would have fewer deserts and more forests. This would mean more plant life π± and more stores of carbon (which could reduce warming from climate change!)

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#appliedmathematicsΒ #theplanetsΒ #earthΒ #rotationΒ #backwardsΒ #mathsinspaceΒ #usingmathsΒ #climatechangeΒ #weatherΒ #simulationΒ #mathmodelingΒ #predictionsΒ #scicomm

## Weird &Wonderful random maths fact π Pokemon is packed full of maths!

Pokemon is a popular battle game played by so many – but do you realise how much maths it uses?? π€

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When one Pokemon defeats another in battle, they gain experience points. Once enough experience points have been gained, the Pokemonβs level increases, which in turn makes the Pokemon stronger, faster and harder to knock out ππΌ. Equations are used to decide how many experience points a Pokemon needs to advance to the next level. There are different equations for different Pokemon, making some easier to raise than others. Furthermore, there is a formula that decides how many experience points a Pokemon gains when it wins a fight. This equation depends on the level of the opponent, as well as the species of the opponent Pokemon.

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There are many other complicated equations involved in this game, including the equation that decides how much damage an attack does (based on the attack strength, opponent defence strength, move power, types of Pokemon and even a bit of luck π)

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Probability also appears in many aspects of the game, including the chance of an attack successfully hitting π―, the chance of encountering a specific Pokemon in the wild (stronger or more rare pokemon have a smaller chance of being encountered) and the chance of a wild Pokemon successfully being caught.

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So if you fancy a job designing video games for a company such as Nintendo, a strong understanding of algebra and probability is very important! π©πΌβπ»π¨π½βπ»

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#appliedmathematicsΒ #computergamingΒ #gameboyΒ #switchΒ #pokemonΒ #pokemonswordshieldΒ #pokemonrbyΒ #pikachuΒ #bulbapediaΒ #pokeballΒ #usingmathsΒ #scicomm

## Using maths to lower the cost of space travel π π

With current advances in technology, exploring our solar system is easier than it has ever been. But these missions are still quite rare since the amount of money needed for space travel remains very high! π΅π΅ A mathematical concept called the Arnold diffusion mechanism (Iβll explain this later), however, is being used to reduce the amount of fuel needed for these trips (one of the major costs of space travel π΅) by finding the most efficient routes (those that need the least fuel).

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But the most efficient route is in a straight line right?! π€ Well this is usually the case on earth but, with the gravitational forces present in space, a straight path isnβt always the best idea. Each planet and their moons π as well as stars βοΈ and asteroids βοΈ have gravitational forces surrounding them which pull the rockets without the need to use fuel. By using this, strange looping routes actually become more efficient than straight routes through space. So back to the Arnold diffusion concept – the idea is that if a small amount of force from the engine is applied at the right time and location, then it has a big impact on the movement of the rocket. Maths can be used, therefore, to find the precise times and positions, where a small fire of the engine will be enough to give large distances of space travel.π

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#space#usingmaths#appliedmathematics#science#stem#savingmoney#reducingfuels#rocket#solarsystem#keeplearning#mathsconcepts#scicomm

## Using the maths of fire ant colonies to help with engineering, construction and robotics πππππππ»

Colonies of fire ants are big on team work! During floods for example they have been found to form rafts by connecting together in a thick mass so that the water doesnβt get through! In a similar way, they can also build themselves into towers, balls or bridges when they need to help their fellow team mates.πͺπ½

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Within these masses each ant grabs onto their neighbours using sticky padded feet but they also constantly change neighbours so that they are always moving. When forming structures like a bridge therefore, any cracks are quickly healed by new ants filling these spaces. π π π –

We can use maths to figure out the rules of fire ant colony behaviours so that we can copy these in some pretty useful ways! This is done by modelling the changes in the positions of ants in a colony, including how they respond to stresses as well as their speed and movements when swapping neighbours.

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Knowing these things could massively help us in different areas of engineering including in developing robots that self assemble π€ and bridges that automatically fix cracks π.

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–#appliedmathematics#mathsinlife#usingmathinreallife#mathsinnature#ants#bridges#buildingbridges#selfhealing#robot#engineering#keeplearning#stem#scicomm

## Weird & Wonderful random maths fact π π³

There are 4 simple steps that anyone can take to check if a credit card number is real! π³

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Credit cards have 16 numbers. The first 15 are decided by the bank but the last number uses a mathematical algorithm based on the other numbers. This βcheck digitβ is used to check a credit card number is correct and that no errors have been made. β

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Follow the steps below with your credit card number to check this works! π€

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1. Beginning with the check digit (last digit), double every 2nd number (i.e. double the number to the left of the check digit and every other number after that, working from right to left)

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2. If any of the answers to the above gives a 2 digit number then add these 2 numbers together (i.e. 14 β‘οΈ 1+4=5)

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3. Add up all these values plus the digits of the credit card number that were not doubled to get one number.

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4. Adding the value of the check digit to this number should give a value that is divisible by 10! So if you didnβt already know your check number then you can find it by working out the 1 digit number you would need to add to your answer to give a multiple of 10! πππ» (i.e. if your answer is 46 then your check digit should be 4!)

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#mathsinlife#usingmaths#appliedmathematics#algorithm#creditcard#numbers#check#tryathome#stem#keeplearning#scicomm

## The Maths of Crowds π©βπ©βπ§βπ¦π¨βπ©βπ§βπ§π¨βπ¨βπ§βπ§π¨βπ¨βπ¦βπ¦π©βπ¦βπ¦π¨βπ©βπ§

You may think that, given a large number of people in a crowd, it is impossible to know how these people will move. Everyone has free will right?! and so we couldn’t possibly know what will happen. Well people are more predictable than you may know. By coming up with some basic behaviours that people are assumed to follow, mathematical equations have been made to predict the motion of people in crowds.ππ» So lets imagine a situation – how about you are simply shopping in a busy town centre like the one in the image. We are assumed to follow these general behaviours:

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1: People have a target place that they want to get to (your favourite shop might have a sale on!πππΌββοΈππ)

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2: There is some randomness in the way people move (would be pretty hard to walk in a completely straight line with lots of people around)

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3: People can’t walk through obstacles like walls (unfortunately π€)

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4: People that are with people they know will want to walk close to them (most people like to talk to the people that they go shopping with π«)

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5: People won’t want to get too close to strangers (can be a little awkward and uncomfortable π³)

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Using these behaviours, in what is called an Agent Based Model, can be important in ensuring safety in crowd situations. Overcrowding is exteremely dangerous and can lead to serious injury. These models are used in many situations where you might expect lots of people (such as at a football match or in the London underground) in order to ensure the design of the space is safe for that number of people.

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Similar models have also been made to explain the movement of a flock of birds π¦
π¦
π¦
or a shoal of fish π π π . Though I don’t think birds (or fish for that matter) are keen shoppers, rules they may follow are not too different. For example they want to fly close to each other to stay in the flock but not too close so they can avoid a crash.

#matharoundus#usingmaths#appliedmathematics#collectivebehavior#crowds#shoaloffish#flockofbirds#modelling#agentbasedmodeling#crowdsafety#stem#scicomm#keeplearning