Systems of LODEs

class: center, middle, inverse, title-slide

.title[
# Systems of LODEs
]
.subtitle[
## Session 06
]
.author[
### Alejandro Ucan
]
.date[
### 2023-10-07
]

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# Goals

* Know different problems involving systems of ODEs. <br/><br/>
* Know the matrix form of a system of ODEs. <br/><br/>
* Study the solution (vector of functions) of a system.

---
# Recall

> To model the population growth we use a function `$P(t)$` that represents the population at time `$t.$` The main hypothesis is that the speed of growth is proportional to the population, `$$\frac{dP}{dt}=kP.$$` What if we consider a population that migrates to another country? <br/><br/>
 `$$\frac{dP}{dt}=kP-r.$$` <br/><br/>
But, in the real world the population that migrates is not constant in time, so the previous model is not so real.

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# A more realistic model

> Assume that `$R(t)$` defines the migrate population at time `$t,$` so my previous ODE changes to `$$\frac{dP}{dt}=kP-R,$$` and if we have information about the growth of the migrate population, for example `$$\frac{dR}{dt}=sR.$$` <br/><br/> 
Then the model would be `$$\frac{dP}{dt}=kP-R,$$` `$$\frac{dR}{dt}=sR.$$`

---
# System of linear ODEs

> A system of ODEs is a pair of ODEs that involves two functions, in general the form is `$$\frac{dx}{dt}=g_1(t,x,y)$$` `$$\frac{dy}{dt}=g_2(t,x,y)$$`  where `$g_1$` and `$g_2$` could be any kind of functions. <br/><br/><br/>
Similar to the single ODEs, the systems can be classified in the type of the function `$g_1$` and `$g_2.$` We say that the system is __linear__ if the functions `$g_1$` and `$g_2$` are linear functions `$$g_1(t,x,y)=c_1x+c_2y+f_1(t)$$` `$$g_2(t,x,y)=k_1 x+k_2y+f_2(t).$$`

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# Prey-Predator System

> Assume that we have two species that interact in an ecosystem, such that `$x(t)$` and `$y(t)$` measures the population at time `$t.$` Where `$x(t)$` is a vegetarian species and `$y(t)$` is a carnivore (consume `$x(t)$`) then `$$\frac{dy}{dt}=-ay+bxy$$` `$$\frac{dx}{dt}=cx-dxy$$`  where: <br/><br/>
* `$a$` is the rate of natural death for the carnivore. <br/><br/>
* `$b$` is the rate of growth of the carnivore population per vegetarian. <br/><br/>
* `$c$` is the rate of natural death for the vegetarian. <br/><br/>
* `$d$` is the rate of growth of the vegetarian population per carnivore. <br/><br/>

This model is known as the __Lotka-Volterra for prey-predator.__

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# How does it look a solution for that system?

<div class="figure">
<img src="https://strimas.com/post/2017-10-13-lotka-volterra/index_files/figure-html/time-plot-1.png" alt="Solutions for the Lotka-Volterra system." width="60%" />
<p class="caption">Solutions for the Lotka-Volterra system.</p>
</div>
---
# Systems of LODEs

> Consider the following system of LODEs with `$n-$`equations and `$n-$`unknowns:
 `$$\begin{array}{ccc} a_{11} x_1 + a_{12} x_2 +\cdots + a_{1n} x_n +f_1(t)&=& x'_1 \\ a_{12} x_1 + b_{22} x_2 +\cdots + a_{2n} x_n +f_2(t) &=& x'_2 \\ \vdots & = & \vdots \\ a_{n1} x_1 + a_{n2} x_2 +\cdots + a_{nn} x_n +f_n(t)&=& x'_n \\ \end{array}$$`

---
## Matrix form

> Last expressión can be expresed as:

`$$\left(\begin{array}{c} x'_1 \\ x'_2 \\ \vdots \\ x'_n \end{array}\right) =\left(\begin{array}{c} a_{11} & a_{12} & \ldots & a_{1n}\\ a_{21} & a_{22} & \ldots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots  \\ a_{n1} & a_{n2} & \ldots & a_{nn} \end{array}\right) \left(\begin{array}{c} x_1 \\ x_2 \\ \vdots \\ x_n \end{array}\right)+ \left(\begin{array}{c} f_1(t) \\ f_2(t) \\ \vdots \\ f_n(t) \end{array}\right)$$`
$$ X' = A X+F$$

---
# How is the solution?

> The solution of a system is a vector of functions `$$X=\left(\begin{array}{c} x_1(t) \\ x_2(t) \\ \vdots \\ x_n(t) \end{array}\right)$$` that satisfies the previous equations.
<br/><br/>

---
#### Example

> Lets verify that `$$\left(\begin{array}{c} e^{-2t} \\ -e^{-2t}  \end{array}\right)$$` is a solution for `$$X'=\left(\begin{array} 1 & 3 \\ 5 & 3 \end{array}\right) X$$` and also `$$\left(\begin{array}{c} 3e^{6t} \\ 5e^{6t}  \end{array}\right).$$`

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# Superposition and Linear Independence

> __Superposition:__ if `$X_1$` and `$X_2$` are solutions of the system, then `$$c_1 X_1+c_2 X_2$$` is also a solution. <br/><br/>
__Definition:__ we say that the vector solution `$X_1, \, X_2,\, \cdots,\, X_n$` are __linealmente indepentiendes__ if the linear combination `$$c_1 X_1+c_2 X_2+\cdots + c_n X_n=O$$` where `$O$` is the zero function, we have that `$c_1=c_2=\cdots=c_n=0.$`

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#### Example:

> Prove that `$$X_1=\left(\begin{array}{c} e^{-2t} \\ -e^{-2t}  \end{array}\right)\quad \mbox{y} \quad \left(\begin{array}{c} 3e^{6t} \\ 5e^{6t}  \end{array}\right)$$` are linearly independent.

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#### Example:

> Prove that `$$X_1=\left(\begin{array}{c} \cos(t) \\ \frac{-\cos(t)}{2}+\frac{\sin(t)}{2}\\ -\cos(t)-\sin(t) \end{array}\right) \mbox{ y } \left(\begin{array}{c} 0 \\ e^t \\ 0  \end{array}\right)$$` is a linearly independent solution for `$$X'=\left(\begin{array}{ccc} 1 & 0 & 1 \\ 1 & 1 & 0 \\ -2 & 0 & -1\end{array}\right) X$$`