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====== Maths and Statistics ======

===== Integral and Differential Calculus =====

{{http://imgs.xkcd.com/comics/newton_and_leibniz.png?450}}((Don't understand? see [[http://www.explainxkcd.com/wiki/index.php?title=626:_Newton_and_Leibniz |here]].))

Calculus was created to describe in mathematical language how a quantity changes over time. It is an extremely useful and powerful tool for building **dynamics** models. That's why calculus has been used for over a century to understand the behavior of ecological systems.
 
The following are turorials to help you understand basic calculus concepts that we use in many mathematical models in ecology.



-->Growth rates, derivatives and exponential function#

Here you find that the exponential function is the limit of a discrete growth at a constant rate, when we make the time intervals very small. For this, we will go through the concept of derivatives and the notion of limit of a function.

  * [[en:ecovirt:roteiro:math:exponencial|Growth rates, derivatives and exponential function]]


 
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-->Antiderivatives and definite integral#

Know the integral, inverse derivative operation. Learn the difference between definite and indefinite integrals.

  * [[en:ecovirt:roteiro:math:integralr|Antiderivatives and definite integral]]

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-->Introduction to Differential Equations#

A differential equation is a relationship between the derivative of a function and some other mathematical function. Understand how these equations can be proposed and solved.

  * [[en:ecovirt:roteiro:math:eq_difr|Introduction to differential equations]]

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-->Numerical integration of differential equations#

Tutorials to solve differential equations with the help of computer programs. Computational numerical integration is the basic tool for mathematical modeling in biology.

  * [[en:ecovirt:roteiro:math:numeric_int_ipython|Numerical integration of differential equations]]

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-->Stability Analysis#

Does an ecological dynamic tend to a state of equilibrium? Does this equilibrium resist disturbances? Here's how to answer these questions with the help of calculus.

  * [[en:ecovirt:roteiro:math:stabilitysage|Stability analysis]]

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===== Introduction to stochastic processes =====

{{http://upload.wikimedia.org/wikipedia/commons/thumb/2/2b/Markovkate_01.svg/200px-Markovkate_01.svg.png }}

A stochastic dynamic happens when we have more than one possible state for a system, and we can //jump// to each one with a certain probability. Therefore, even systems that start out the same can differ over time. For example, populations under stochastic dynamics can have different sizes at any given time, each with a probability of happening. In this case, the population size is a [[http://en.wikipedia.org/wiki/Random_variable|random variable]].

Considering stochasticity is very important to understand ecological dynamics. With the stochastic models there were important theoretical advances, such as the [[en:ecovirt:roteiro:neutr:neutrarcmdr|neutral theory of biodiversity]]. Stochastic models also made the risk of extinction more evident in [[en:ecovirt:roteiro:den_ind:di_edr|small populations]] or under large [[en:ecovirt:roteiro:den_ind:di_edr| environmental variation]].


==== Random walks ====

{{:ecovirt:roteiro:neutr1.jpg?150 |}}

The [[http://en.wikipedia.org/wiki/Markov_chain|Markov Chains]] are used to describe ecological dynamics. They are models of stochastic processes in which time is discrete, and at each interval the system can change state, with a certain probability. The probabilities of changing from one state to another depend only on the present state ((Therefore, they can be expressed in transition matrices from time t to time t+1, as in [[en:ecovirt:roteiro:pop_str:pstr_mtexcel|]]

The following are simple case scripts for Markov Chains.


-->Random walk#
See why a walking drunk will do poorly, even if on average he walks in a straight line.

  * [[:ecovirt:roteiro:math:bebadoRcmdr|Random Walk Tutorial]]

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-->Zero-sum Game#
In a [[http://en.wikipedia.org/wiki/Zero-sum_game|zero-sum game]] you only win what others have lost. Discover the properties of this dynamic if gains and losses occur at random.

  * [[:ecovirt:roteiro:math:zerosumRcmdr|Dynamic Zero-Sum Tutorial]]

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