Computational modeling techniques

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1 Cmputatinal mdeling techniques Lecture 11: Mdeling with systems f ODEs In Petre Department f IT, Ab Akademi

2 Mdeling with differential equatins Mdeling strategy Fcus n the change in the values f all variables We describe the change as a functin f the current values f all variables If the prcess is deterministic and it takes place cntinuusly in time, the mdel may be frmulated in terms f differential equatins Octber 6, 2014 Cmputatinal mdeling techniques 2

3 Systems f differential equatins Very ften, mdels describe interactive situatins: several variables invlved, their change depends n all the ther variables An animal may serve as prey fr anther Tw species may depend n each ther fr mutual supprt Tw r mre species may cmpete fr the same resurce Mdel thrugh a system f differential equatins Each equatin gives the rate f change f ne variable as a functin f the current values f all variables Slve all equatins simultaneusly Bad news: the equatins are ften nn-linear and in general they cannt be slved analytically Gd news: many numerical techniques exist t analyze the behavir f such (systems f) equatins Octber 6, 2014 Cmputatinal mdeling techniques 3

4 Cntent Basic ntins Graphical slutins Examples cmpetitive hunter mdel predatr-prey mdel ecnmic aspects f an arms race Octber 6, 2014 Cmputatinal mdeling techniques 4

5 Autnmus systems f ODEs Definitin. A system f ODEs f the frm dx i /dt=f i (x 1,x 2, x n ) is called autnmus In ther wrds, n independent variable t n the right hand side the system is nt time-dependent; it nly depends n the variatin f its variables Octber 6, 2014 Cmputatinal mdeling techniques 5

6 Equilibria Equilibrium pints / steady states A system f ODEs dx i /dt=f i (x 1,x 2, x n ), i=1,2,,n a=(a 1,a 2,,a n ) is an equilibrium pint fr the system if f i (a)=0, fr all i Equivalently, if x i (0)=a i, fr all i, then the system has a slutin the cnstant functins x i (t)=a i Stable equilibrium a is stable if it is true that x(0) being clse t a implies that x(t) is clse t a, fr all t>0 a is called unstable therwise Asympttically stable a is asympttically stable if it is true that x(0) being clse t a implies that lim t x(t)=a Octber 6, 2014 Cmputatinal mdeling techniques 6

7 Lyapunv stability (ptinal slide) Cnsider an autnmus dynamical system dy/dx=f(y(x)), y(0)=y 0, where f:r n R n is cntinuus and has an equilibrium y e The equilibrium is (Lyapunv) stable if, fr every ε>0, there exists δ ε >0 such that, if y 0 -y e <δ ε, then y(x)-y e <ε, fr all x 0 The equilibrium is asympttically stable if there exists δ>0 such that, if y 0 -y e <δ, then lim x y(x)-y e =0 Octber 6, 2014 Cmputatinal mdeling techniques 7

8 Graphical slutins Cnsider autnmus systems f first-rder ODEs dx i /dt=f i (x 1,x 2, x n ) nt time dependent cnsider its slutin as describing a trajectry in the n-dimensinal plane, with crdinates (x 1 (t),x 2 (t), x n (t)) the n-dimensinal plane (x 1,x 2, x n ) is called a phase plane cnvenient t think abut it as the mvement f a particle autnmus system implies that the directin f mvement frm a given pint n the trajectry nly depends n that pint, nt n the time when the particle arrived in that pint Cnsequence: n trajectry can crss itself unless it is a clsed curve (peridic) Cnsequence: at mst ne trajectry ging thrugh any given pint Equivalently: tw different trajectries cannt intersect Octber 6, 2014 Cmputatinal mdeling techniques 8

9 Graphical slutins Cnsider autnmus systems f first-rder ODEs dx i /dt=f i (x 1,x 2, x n ) if (e 1,e 2, e n ) is an equilibrium pint, then the nly trajectry ging thrugh that pint is the cnstant ne Cnsequence: a trajectry that starts utside an equilibrium pint can nly reach the equilibrium asympttically, nt in a finite amunt f time A nn-cnstant mtin f a particle can have ne f the fllwing 3 behavirs: appraches an equilibrium pint mves alng r appraches asympttically a clsed path at least ne f the trajectry cmpnents becmes arbitrarily large as t tends t infinity Octber 6, 2014 Cmputatinal mdeling techniques 9

10 Example dx/dt=-x+y, dy/dt=-x-y Analytic slutin: x(t)=e -t sin(t), y(t)=e -t cs(t) Equilibrium pint: dx/dt=dy/dt=0 leads t x=y=0 Nte: x 2 (t)+y 2 (t)=e -2t, i.e., the trajectry is a circular spiral with decreasing radius appraching 0 Girdan et al. A first curse in mathematical mdeling. (3 rd editin), Page 415 Octber 6, 2014 Cmputatinal mdeling techniques 10

11 Example: a cmpetitive hunter mdel Assume we have a small pnd that we desire t stck with game fish, say trut and bass. The prblem we want t slve is whether it is pssible fr the tw species t cexist Hypthesis Unlimited amunt f fd available The space is a limitatin fr the c-existance f the tw species The mre there is f the ther species, the smaller the grwth rate; Assume the causes t be in the cmpetitin fr space mdel it as prprtinal in the number f pssible interactins between the tw species Mdel frmulatin The change in the level f trut X(t): dx/dt = ax(t) bx(t)y(t) Similar reasning fr the level f bass Y(t): dy/dt=my(t) nx(t)y(t) Octber 6, 2014 Cmputatinal mdeling techniques 11

12 Example: a cmpetitive hunter mdel Mdel: dx/dt = ax(t) bx(t)y(t), dy/dt=my(t) nx(t)y(t) Questin: can the tw ppulatins reach an equilibrium where bth are nn-zer Answer: ax-bxy=0, my-nxy=0 Slutin: either x=y=0, r x=m/n, y=a/b Difficulty: impssible t start with exactly the equilibrium values (they might nt even be integers) s, we cannt expect t start in an equilibrium pint study the prperty f the equilibrium, hping it is a stable ne Octber 6, 2014 Cmputatinal mdeling techniques 12

13 Equilibrium pints: (0,0), (m/n, a/b) Additinal questin: what is the behavir if we start clse t the equilibrium pint? Example: a cmpetitive hunter mdel (cntinued) dx/dt = ax(t) bx(t)y(t), dy/dt=my(t) nx(t)y(t) Slutin: we study the tendency f X(t), Y(t) t increase/decrease arund the equilibrium pint. Fr this, we study the sign f the derivatives f X(t), Y(t) dx/dt 0 ax-bxy 0 a/b Y dy/dt 0 my-nxy 0 m/n X Girdan et al. A first curse in mathematical mdeling. (3 rd editin), Page 421 Octber 6, 2014 Cmputatinal mdeling techniques 13

14 Bass Y a/b m/n X Trut Bass Y a/b m/n X Trut Octber 6, 2014 Cmputatinal mdeling techniques 14

15 Graphical analysis f the trajectry directins Y Bass a/b m/n Trut X Octber 6, 2014 Cmputatinal mdeling techniques 15

16 Graphical analysis f the trajectry directins arund the equilibria Y Bass a/b m/n Trut X Octber 6, 2014 Cmputatinal mdeling techniques 16

17 Y Graphical analysis f the trajectry directins Bass a/b Bass win Trut win m/n X Cnclusin: the c-existance f the tw species is highly imprbable Octber 6, 2014 Cmputatinal mdeling techniques 17

18 Limits f graphical analysis Nt always pssible t determine the nature f the mtin near an equilibrium based n graphical analysis Example: the behavir in Fig 11.9 thrugh graphical analysis is satisfied by all 3 trajectries in Fig Example: The trajectry in Fig 11.10c culd be either grwing unbundedly r apprach a clsed curve Girdan et al. A first curse in mathematical mdeling. (3 rd editin), Page Octber 6, 2014 Cmputatinal mdeling techniques 18

19 A predatr-prey mdel Als knwn as the Ltka-Vlterra mdel, ( ) A mdel where we have tw species, ne being the primary fd surce fr the ther: predatr-prey mdel Histrical references: used in chemistry, eclgy, bimathematics, ecnmy Hypthesis The prey ppulatin finds unlimited amunt f fd at all times The fd supply f the predatr ppulatin depends entirely n the size f the prey ppulatin Grwth rate f each ppulatin taken in islatin is prprtinal t its size Predatrs have limitless appetite Octber 6, 2014 Cmputatinal mdeling techniques 19

20 A predatr-prey mdel: whales vs. krill Assumptins and mdel frmulatin (the Ltka-Vlterra mdel, ) The krill ppulatin x Assume that the cean supprts an indefinite grwth f the krill (we d nt mdel explicitly the planktn) The ppulatin declines prprtinally t the number f interactins between krill and whales dx/dt=ax-bxy The whale ppulatin y Whale die f natural causes; cnstant mrtality rate Whales reprduce at a rate prprtinal t the number f interactins between krill and whales dy/dt=-my+nxy Equilibrium pints dx/dt=dy/dt=0 (x,y)=(m/n,a/b) r (x,y)=(0,0) Octber 6, 2014 Cmputatinal mdeling techniques 20

21 A predatr-prey mdel: graphical analysis Girdan et al. A first curse in mathematical mdeling. (3 rd editin), Page 430 Cnclusin: fr the analysis f the behavir arund the nn-zer pint we need further analysis: is it peridic, (asympttically) stable, r unstable? Octber 6, 2014 Cmputatinal mdeling techniques 21

22 Predatr-prey mdel: analytic slutin Nte: analytic slutin pssible in this case it reveals the peridic behavir f the mdel skip it here; nly lk at the results f the numerical integratin Octber 6, 2014 Cmputatinal mdeling techniques 22

23 A predatr-prey mdel: numerical integratin Octber 6, 2014 Cmputatinal mdeling techniques 23

24 A predatr-prey mdel: phase prtrait Octber 6, 2014 Cmputatinal mdeling techniques 24

25 A predatr-prey mdel: numerical integratin Girdan et al. A first curse in mathematical mdeling. (3 rd editin), Page Octber 6, 2014 Cmputatinal mdeling techniques 25

26 Example: Ecnmic aspects f an arms race Prblem. Tw cuntries engaged in an arms race. Interested in whether the arms race will lead t uncntrlled spending and eventual win by the cuntry with better ecnmic assets Assumptins x is the annual defense expenditures f cuntry 1, y f cuntry 2 Driving factrs fr rate f increase f the defense budget x the bigger it is, the less it shuld grw: negative influence assumed t be prprtinal t its current budget x (ecnmic cnstraint) the bigger the budget f the adversary, the mre it shuld increase: psitive influence assumed t be prprtinal the current budget f the adversary y (military cnstraint) sme cnstant level f grwth is needed anyway (precautinary cnstraint) Mdel dx/dt=-ax+by+c dy/dt=mx-ny+p Octber 6, 2014 Cmputatinal mdeling techniques 26

27 Example: Ecnmic aspects f an arms race (cntinued) Mdel dx/dt=-ax+by+c dy/dt=mx-ny+p Equilibrium pint: -ax+by+c=0; mx-ny+p=0 x=(bp+cn)/(an-bm), y=(ap+cm)/(anbm) Just t make a chice in drawing the plts, assume that an-bm>0 Plt the tw lines and cnsider the intersectin f the tw half planes Girdan et al. A first curse in mathematical mdeling. (3 rd editin), Page 447 Regin A: dx/dt>0, dy/dt<0 Regin B: dx/dt<0, dy/dt<0 Regin C: dx/dt>0, dy/dt>0 Regin D: dx/dt<0, dy/dt>0 Octber 6, 2014 Cmputatinal mdeling techniques 27

28 Example: Ecnmic aspects f an arms race (cntinued) dx/dt=-ax+by+c dy/dt=mx-ny+p Regin A: dx/dt>0, dy/dt<0 Regin B: dx/dt<0, dy/dt<0 Regin C: dx/dt>0, dy/dt>0 Regin D: dx/dt<0, dy/dt>0 Cnclusin: asympttically stable pint Girdan et al. A first curse in mathematical mdeling. (3 rd editin), Page 447 Octber 6, 2014 Cmputatinal mdeling techniques 28

29 Learning bjectives Be able t frmulate the evlutin f change in terms f an ODE-based mdel Be able t identify the equilibrium pints f simple ODE-base mdels Understand the cncepts f stable, asympttically stable equilibrium pints Octber 6, 2014 Cmputatinal mdeling techniques 29