Question

3. Apparently, the consumption of toilet paper induces panic which induces a greater consumption of toilet paper. Let T be the amount of toilet paper available and P be the general level of panic of a population. I will describe a situation to you, and your goal is to write a pair of differential equations that represent this situation. reases • Toilet paper is produced at B rolls per day. • Panic increases as toilet paper decreases at rate a. • Toilet paper consumption increases at rate r as P increases • Panic dissipates at a rate d. CON reases ncreases. · Justify all the terms in your equations and how Write out a system of differential equations for they satisfy the above points. dt and dp dt · Just

Answer 1

Solotion ! 3 Given that pris the amount of toilet paper available. and p is the General level of panic. of the ni Popolation. Wow to find dI I We know that cu T2- i.e, the amount of toilet paper available will decrease as panic amount of peope increases

→ TX B i..., Toilet paper availability will increase. as the amount of toilet paper production increases. from these. T= okPB (Where to is the proportionality Constant) B = -484! [ id] - Woow to find the ne we know that . ( as the panic increases at a sale a with decrease in toilet papa) dy had et --wlad | Interne til den garalimaling

4 Model: Itinan The model becomes as de = PMD-d di PEMIDLOM dc (a) Compartmental Diagram for the model. Y BMI. y pri do PM - b) Dow to find the equilibrium for this model. we know that d? - BMI - JI - FCI,M) dt dom - PBMI -PM = 6 (9,) To find the equilibriosm let's consida.

=> BMI-d1=0 and fBMP-PM20 9 (3M-d) zo = 1:08, M: & >> M [PPT -?) -0 »M= 0 , I = 8 BI or SM .. (0) and learning) Qe the quiliboom Poids o To find the stability of each equilibrium. Igi thad first we do. Jacobian. (53= 1 of 2 - Brod B17 (6 g 7.5pm spp-p) Now consider the equilibsium points af (0,0) 1970 (33) .: The Characteristic equation will be (A+d) (1+) =0 - dard, ap

By assoming that both d and P as positive, the both eigen valves de negative; :') .:(0,0) is a stable node. → Decor of point o-edd for a gof . [Set Sopp 2 st of ? The chestacteristic Equalion win be 12-14 zo => jetpd Similarly if we assome p and d as positive the One cigen valve will be positive and the other Coill be negative. :( ) is unstable... For any doubts Please do Comment Please Like and Support Thainle You; mmen