Lecture 2: Differential-Delay equations.

Transcription

1 Lecture : Differential-Delay equations. D. Gurarie A differential equation, or syste:, ; of the syste:, y f y t y t y, predicts a (near) future state 0 0 y t dt y f y t dt, fro its current state, and thus carries no past eory. This approach is suitable for any echanical systes, based on Newton s law: the current state (position + elocity) deterines its future dynaics. But in any biological systes reaction does not coe as iediate response to stiulation. It often appears with certain tie-lags, so such systes are capable to accuulate past eory. equation: Matheatically we can include tie-lag, e.g. T in the odel, ia differential-delay y t f y t, y t T, t,... () Such equation typically exhibit ore coplicated dynaic patters, copared to their DE counterparts. Linear differential-delay equations The siplest st order linear differential-delay equation (DDE) yt ry t T by t... () can be soled by the ethod of characteristic polynoial, sae way as all linear DEs (-st or higher order) with constant coefficients. Indeed, any linear DE, gien by differential operator n n L D a D... a n, has exponential solutions e t n n L y y a y... an y 0, whose exponent soles characteristic equation: n n p a... a n 0 So an n-th order DE has at ost n real (or coplex) exponential solutions, and its general solution is a linear cobination of those: t Typical exaples are y t C e.

2 (i) linear growth/decay: y ay 0, with at (ii) (ii) linear oscillator: x bx x 0 y t Ce, or, with quadratic i b b t t, or cos sin p, and coplex roots ; / - daping rate; / / 4 - frequency y t e e C t C t - daped oscillation. y t Applying this ethod to DDE (), we get solution t e, where obeys a transcendental equation T re b 0 (3) Unlie -st order DE such equation will typically hae (infinitely) any real and coplex solutions. In that sense a DDE (een first order) is an infinite diensional syste, unlie n-th order DE (n-diensional syste). Solutions of (3) are gien by a generalized log- function, called Product-Log, or Plog, z inerse function of the coplex ap: w ze z Plog w, so z z ae 0 z Plog(- a) (4) This function is built in Matheatica (see noteboo) and other standard pacages, Analysis of Plog: DDE: y ay t bt Plog b Plog rte a T (5) T T 0 Real roots Equation (4) has at ost one (or two) real roots, proided at / e e -0.4 p x xe Fig. : Function x

3 Moreoer, positie a hae single positie root, while negatie range / e at 0, has two negatie roots. Based on real roots, one would conclude that equilibriu y 0 of (3) is unstable for positie growth rate a, and stable for negatie (lie in DEs). The situation, howeer is ore coplicated, as both cases hae infinitely any coplex roots, that can deterine their stability. Coplex roots Coplex z u i has real and iaginary parts satisfying: Hence, we get equation for I z;re z ui e u i a u cot Haing soled st equation (6) for at a fixed leel a a ia nd equation: u a u a sin cot f e a...(shown in Fig.). Stability analysis Sequences u, hae asyptotic forulae n n a ; u ln /... we copute real part As a increases they undergo a sequence of bifurcations at critical leels / unstable coplex pairs z0 ;...; z,re z j 0 The entire bifurcation pattern of DDE: y ay t Range a u Rez yt 0 (6) a with Stability xt xt 0,/ e 0 - real ce c e... Stable+ daped osc. e uj zj /, / zt j Re 0 ce daped osc. j [ /,5 / ] u 0 u u j c e i t zt j jc je st (Hopf) bifurcation [5 /,9 / ] u u 0 u ui t zt j j c e c e nd bifurcation j j

4 Fig.: Function f sin cot e (blue), whose leels deterine I z u Re z cot (purple)., and Analysis of tie lag: DDE: y ry t T 0 P rt i, and we get equations for the iaginary and real parts of T Here log T sin T T Tcot f T e rt cot T As aboe syste undergoes a sequence of bifurcations at critical lags T 5 9 ; ; ;... r r r with critical frequencies I, and aplitudes T 0. (7) Nonlinear DDE We apply the aboe (linear) analysis to seeral exaples of nonlinear DDE.. Logistic DDE: y t T y t r y t N. It still can sere as a population odel, but tie-lag in the relatie growth rate, has to do with delayed deelopental stages. Soe insects (and other organiss) undergo seeral stages, e.g. lara, pupae etc. So if yt represents adult population at tie t, its growth rate (at t ) depends on the state of the syste (i.e. adult feales) at tie t T, the tie it taes to deelop fro egg to adult.

5 Many interesting exaples of delayed odels appear in Physiology, so called periodic (or dynaic) diseases. We shall consider 3 such exaples:. Cheyne-Stoes respiration. Here ariable Ct represents CO -leel (concentration) in the blood. High leel of C turns in a negatie feedbac echanis, that is breathing that reoes excess CO. The feedbac response howeer, does not appear iediately, but has a tie lag (it taes the excess signal to reach breathing effector channels). The siplest ways to describe such feedbac regulation is ia DDE dc t dt with sigoid (switch) function: V y ; ( ) p BV C t T C t (8) a y y, called Hill function. Paraeter p represents CO production rate of (due to etabolis), and BV is C - dependent reoal rate. 3. Haeatopoiesis refers to generation of specialized blood cells, which is (partly) regulated by the blood cell population, called yt. Once again delays coe naturally here, and a suitable odel (Macey-Glass 977) cab be written as DDE where new cells are produced at a (delayed) rate y t y t T y t (9) y, and reoed (die out) at a rate y y a. Macey and Glass proposed function in the for: Function y for seeral alues y Fig.3: Production function for Haeatopoiesis * So production rate grows linearly for sall y, peas at y a / zero., then drops to

6 4. Feer control of alaria parasite. Malaria parasite undergoes seeral deelopental stages alternating between huan and osquito hosts. After a 7-day incubation period in a huan host, following infected bite, it releases large nuber (0,000-30,000) of shortlied extracellular blood stages, called erozoites. The erozoites rapidly inade red blood cells (RBC). Inside the RBC each parasite undergoes asexual replication oer a 48-7 hr period (depending on species). Then it bursts the cell, and releases a dozen of new erozoites to continue the cycle. Such exponential parasitic growth, and the associated destruction of RBC, is ostly responsible for the clinical syptos of alaria. As the parasite density (parasiteia) reaches a critical (pyrogenic) leel, 4 0 parasites/ 3 of blood, the patient will suffer periodic bouts of the acute feer. Feer is one of the natural (non-specific) body responses to inading parasites that attepts to aintain (control) their leel at or below a critical alue. It seres a first line of iuno-defense. To odel this process we introduce parasite density xt. In the feer-free condition it grows exponentially at a rate r.5ln.4/ day. When the feer switches on the growth turns into decay (with a delay of seeral hours). We assue decay rate proportional to r, r, ( ). So our odel becoes a DDE x t T x t r V x t N (0) with the Hill function V x, that switched on and off about pyrogenic leel N. x In all 3 cases we can find equilibria (sae as the corresponding DE w/o delay), and study their linearized stability.. Logistic delay (rescaled to N ) gies a linearized DDE of the for ut ru t T, where stable (carrying capacity) equilibriu undergoes bifurcation into liit cycle (Hopf) at rt / (see Matheatica noteboo). Cheyne-Stoes after rescaling u t V ut T with coefficients the equilibriu alue gies a linear DDE u Au t Bu t T * * * ;, where * indicates alues of A V B u V V u or deriatie V u at * u. The detailed analysis is gien in Matheatica noteboo. It shows

7 bifurcation of stable equilibriu into liit cycle behaior, associated with breathing abnoralities. 3. Haeatopoiesis gies linear DDS ut Bu t T Au t with A * ; B. Here we ary Hill paraeter 5 0, and find ore coplicated bifurcations fro equilibria to liit cycle, and chaos (details in Matheatica noteboo). Proble: Study stability and bifurcation in the rescaled feer odel, in ters of Hill paraeters, and diensionless tie rt. Find stability range, estiate period of oscillation at the bifurcation alue.