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$$\begin{pmatrix} D(t) \\ P(t) \\ R(t) \\ 1 \end{pmatrix}= \begin{pmatrix} 1 & \frac{-1}{20} & 0 & 1 \\ \frac{1}{100} & 1 & \frac{-1}{10} & 0 \\ \frac{-1}{100} & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} D(t-1) \\ P(t-1) \\ R(t-1) \\ 1 \end{pmatrix}$$

when solving this system, the eigenvalues with magnitude part are[1.0162 1.0162 0.9677 1] the system of difference equations is given below. \begin{align*} D(t) &= D(t-1) + \left( 1 - \frac{P(t-1)}{20} \cdot 1 \right) \\ P(t) &= P(t-1) - \left( 1 - \frac{D(t-1)}{10} \cdot 1 \right) + \left( 1 - \frac{R(t-1)}{100} \cdot 1 \right) \\ R(t) &= R(t-1) + \left( 1 - \frac{D(t-1)}{100} \cdot 1 \right) \end{align*} when I try to make system positive, I cant do it even if I try 0.98379 in diagonal of above matrix to make eigenvalues equal to 1. can anybody explain me where I am wrong?

K B Dave
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Summyia
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  • can anybody help me in this? @eric towers? – Summyia Mar 27 '18 at 07:09
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    Why do you expect a generic linear system to be capable of a positivity condition (for all initial conditions)? – Eric Towers Mar 27 '18 at 17:38
  • can you please review this question https://math.stackexchange.com/questions/2710034/how-to-make-a-system-of-difference-positive-oscillatory.. I tried the method for different values – Summyia Mar 27 '18 at 17:43
  • can we shift the boundary in positive interval? because by changing the values of D' P' R', the value of x will also not be same as we used in previous method. I tried x y z instead of x and somehow numerically found the boundary of converging and diverging system but I want the positive solution space – Summyia Mar 27 '18 at 17:46
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    Again, why do you believe a linear system can impose a lower or upper bound (for all initial conditions)? Consider your first equation, so long as $P$ is less that $20$, $D$ increases, past any upper bound you may have in mind. There is nothing in this system that can upper or lower bound solutions. – Eric Towers Mar 27 '18 at 17:53
  • can you explain what do you mean by INITIAL CONDITIONS? – Summyia Mar 27 '18 at 17:56
  • @eric can you help me in this scenerio? https://math.stackexchange.com/questions/2710034/how-to-make-a-system-of-difference-positive-oscillatory – Summyia Mar 27 '18 at 18:02

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