Re: Re: Delay Differential Equations

*To*: mathgroup at smc.vnet.net*Subject*: [mg105272] Re: [mg105232] Re: [mg105198] Delay Differential Equations*From*: DrMajorBob <btreat1 at austin.rr.com>*Date*: Wed, 25 Nov 2009 23:01:28 -0500 (EST)*References*: <200911241048.FAA00661@smc.vnet.net>*Reply-to*: drmajorbob at yahoo.com

Using Daniel Lichtblau's methods, we can find out when the type A population reaches zero: Needs["DifferentialEquations`InterpolatingFunctionAnatomy`"]; Clear@p p = p /. First@ NDSolve[{p'[t] == -0.0367*p[t - 37]*UnitStep[t - 37] - 0.0123*p[t], p[t /; t <= 0] == 30000}, p, {t, 0, 120}, Method -> {"EventLocator", "Event" -> p[t]}]; end = InterpolatingFunctionDomain[p][[1, -1]] Plot[p@t, {t, 0, end}, AxesOrigin -> {0, 0}] 54.2857 But this allows dead cells to mutate (and reduce the number of type A cells), so it's probably not what you really want. Regardless of cell type, this gives the number ALIVE at time t: Clear[alive, p] p = p /. First@NDSolve[{ p'[t] == -0.0123*p@t, p[0] == 30000}, p, {t, 0, 120}]; alive[t_?NonNegative] := p[t]; alive[t_?Negative] = 0; Plot[alive@t, {t, -37, 120}, AxesOrigin -> {0, 0}] and this purports to determine the proportion of living (or dead) cells that are type A, at time T: Clear[typeA] typeA = typeA /. First@NDSolve[ {typeA'[t] == -0.0367*typeA[t - 37], typeA[t /; t <= 0] == UnitStep@t}, typeA, {t, 0, 120}]; FindRoot[typeA@t, {t, 0, 60}] Plot[typeA@t, {t, 0, 120}, AxesOrigin -> {0, 0}] {t -> 64.248} But that function goes negative as well, and not at the same time as before! So... Daniel Huber's simpler take is probably the right one: Clear[p] p = p /. NDSolve[{p'[t] == -0.0123 p[t] - 0.0367 Boole[t > 37] p[t], p[0] == 30000}, p, {t, 0, 120}][[1]]; Plot[p@t, {t, 0, 120}] We can get a symbolic solution in this case, too: Clear[p] p = p /. First@ DSolve[{p'[t] == -0.0123 p[t] - 0.0367 Boole[t > 37] p[t], p[0] == 30000}, p, t]; p[t] // InputForm Piecewise[{{30000/E^(0.0123*t), t <= 37}}, 116640.59642871171/E^(0.049*t)] (Accurate for t>=0.) Bobby On Wed, 25 Nov 2009 01:31:53 -0600, Daniel Lichtblau <danl at wolfram.com> wrote: > Yun Zhao wrote: >> Hi Daniel, >> >> Thanks for your reply. But my question is I was trying to solve an >> equation, and the solution just doesnt make much sense, and I looked in >> the help files, but they didn't really help me much. The problem is >> this: >> >> I have some cells of type A, and at some point in time, cells of type A >> are going to slowly change into cells of type B. So, population size of >> cell type A will decrease over time due to two factors: (1) becoming >> cells of type B and (2) dying over time. So for a differential >> equation, i have two rates, one is the transformation rate (0.0367), one >> is the death rate (0.0123). But I want to delay the transformation by >> 37 hours. So from time = 0 to time = 37, all that is changing in cells >> of type A is that they are dying at the death rate, then after 37 hours, >> they are also disappearing at the transformation rate. >> >> So my differential equation is like this: >> >> *solrandom1=NDSolve[{p'[t]* >> >> *==0.0367*p[t-37]-0.0123*p[t],p[t/;t<0]=30000},p,{t,0,120}]* >> >> What I expect when I plot p(t) over t is a gradual decrease from t=0 to >> t=37. Then more decrease from t=37 to t=120. p(t) should never be less >> than zero. But when I ran that code in Mathematica, and plotted it >> using this command >> >> *Plot[Evaluate[p[t]/.solrandom1],{t,0,120},PlotRange**Ã¯â??Â®Automatic]* >> >> p(t) was actually less than zero. I am not able to figure out why this >> is. >> >> Please tell me what I did wrong. Thanks. > > > First is that the delay term you have is not negative. So I'd expect an > increase. When I do the following I indeed see a rising population. > > solrandom1 = > NDSolve[{p'[t] == 0.0367*p[t - 37] - 0.0123*p[t], > p[t /; t <= 0] == 30000}, p, {t, 0, 120}] > > Plot[Evaluate[p[t] /. solrandom1], {t, 0, 120}, > PlotRange -> Automatic] > > Second is that your claim of no transformations for 37 hours is wrong. > You are modeling that by 0.0367*p[t - 37], and p[t] is 30000 for t<=0. > So this contribution is not going to be zero (whether you use a positive > or negative thereof). You can repair this by using the system below. > > solrandom1 = > NDSolve[{p'[t] == -0.0367*p[t - 37]*UnitStep[t - 37] - 0.0123*p[t], > p[t /; t <= 0] == 30000}, p, {t, 0, 120}] > > Next is that there is nothing to stop this solution from going negative. > For that you'd need an EventHandler. Could do as below. > > solrandom1 = > NDSolve[{p'[t] == -0.0367*p[t - 37]*UnitStep[t - 37] - 0.0123*p[t], > p[t /; t <= 0] == 30000}, p, {t, 0, 120}, > Method -> {"EventLocator", "Event" -> p[t]}] > > Last is that these methods are things in the documentation I had noted > in my last reply. > > Daniel Lichtblau > Wolfram Research > >> On Tue, Nov 24, 2009 at 1:28 PM, Daniel Lichtblau <danl at wolfram.com >> <mailto:danl at wolfram.com>> wrote: >> >> Yun Zhao wrote: >> >> Hi, >> >> Does anyone have experience working with delay differential >> equations in >> Mathematica 7? I found some help files on wolfram and >> Mathematica help, but >> the information available there were very limited. If anyone can >> refer me to >> other online or textual sources, I would really appreciate it. >> Thank you >> very much. >> >> Mike >> >> >> Not sure what you saw in the Documentation Center. These are perhaps >> the most useful, worth checking if you have not encountered them >> already. >> >> howto/SolveDelayDifferentialEquations >> tutorial/NDSolveDelayDifferentialEquations >> >> Daniel Lichtblau >> Wolfram Research >> >> > > -- DrMajorBob at yahoo.com

**References**:**Delay Differential Equations***From:*Yun Zhao <yun.m.zhao@gmail.com>

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