Prof. Alejandro Ucan-Puc
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?
$$\frac{dP}{dt}=kP-r.$$
But, in the real world the population that migrates is not constant in time, so the previous model is not so real.
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 $$\frac{dR}{dt}=sR.$$
Then the model would be $$\frac{dP}{dt}=kP-R,$$ $$\frac{dR}{dt}=sR.$$
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.
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 $$g_1(t,x,y)=c_1x+c_2y+f_1(t)$$ $$g_2(t,x,y)=k_1 x+k_2y+f_2(t).$$
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:
This model is known as the Lotka-Volterra for prey-predator.
Consider the following system of LODEs with $n-$equations and $n-$unknowns:
$$\begin{array}{ccc} a_{11} x_1 + a_{12} x_2 +\cdots + a_{1n} x_n +f_1(t)&=& x'_1 \\ a_{12} x_1 + b_{22} x_2 +\cdots + a_{2n} x_n +f_2(t) &=& x'_2 \\ \vdots & = & \vdots \\ a_{n1} x_1 + a_{n2} x_2 +\cdots + a_{nn} x_n +f_n(t)&=& x'_n \\ \end{array}$$
Last expressión can be expresed as:
$$\left(\begin{array}{c} x'_1 \\ x'_2 \\ \vdots \\ x'_n \end{array}\right) =\left(\begin{array}{c} a_{11} & a_{12} & \ldots & a_{1n}\\ a_{21} & a_{22} & \ldots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \ldots & a_{nn} \end{array}\right) \left(\begin{array}{c} x_1 \\ x_2 \\ \vdots \\ x_n \end{array}\right)+ \left(\begin{array}{c} f_1(t) \\ f_2(t) \\ \vdots \\ f_n(t) \end{array}\right)$$ $$ X' = A X+F$$
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.
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 $$\left(\begin{array}{c} 3e^{6t} \\ 5e^{6t} \end{array}\right).$$
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.
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 $c_1=c_2=\cdots=c_n=0.$
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.
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 $$X'=\left(\begin{array}{ccc} 1 & 0 & 1 \\ 1 & 1 & 0 \\ -2 & 0 & -1\end{array}\right) X$$
Coupled dynamic systems, multi-species predator-prey equations, state-space vector-matrix form X' = AX, and superposition principle.
Next Session: Eigenfunctions for Systems of LODEs