UMEÅ UNIVERSITY September 1, 2016 Computational Science and Engineering Modeling and Simulation. Dynamical systems. Peter Olsson

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1 UMEÅ UNIVERSITY Septembe, 26 Computational Science and Engineeing Modeling and Simulation Dynamical systems Pete Olsson

2 Continuous population models fo a single species. Continuous gowth models The simplest possible: dn dt = biths deaths+migation. dn dt = bn dn N(t) = N e (b d)t, can t be tue foeve. Thee should be a self-limiting pocess when the population gets too lage. Intoduce a caying capacity, K: This is called logistic gowth in a population. dn ( dt = N N ). () K.. Steady states and lineaization Thee ae two steady states (equilibium states), N = and N = K. The fist one is unstable since lineaization aound N = gives dn dt = N and so N gows exponentially. The second steady state is stable. To see this, wite N = K + n, and lineaize: d(k +n) dt ( = (K +n) K +n ) K dn ( dt K n ) = n, K which gives an exponential decay, n(t) = n()e t. The caying capacity detemines the size of the steady state population wheeas / is the chaacteistic time scale of the esponse of the model to a change in the population.

3 ..2 Geneal fomulation In the geneal fomulation we have dn = f(n), (2) dt whee f(n) is a non-linea function of N. Then steady state solutions, N, ae solutions to f(n) =, they ae linealy stable if f (N ) < and linealy unstable if f (N ) >. dn/dt N To show this, conside the deviations fom N : n = N N and dn/dt = dn/dt: dn dt = f(n) f(n )+nf (N ) = nf (N ), with the solution n(t) = n()e f (N )t, which inceases if f (N ) >..2 Insect outbeak model: Spuce Budwom We ae now consideing the Spuce Budwom model which is a pactical model with two linealy stable steady state populations. The dynamics is given by dn dt = BN ( N ) p(n), KB whee p epesents pedation by bids. We now specialize to the following fom fo pedation: Pedation N p(n) = BN2 A 2 +N 2 which has a change fom low to high pedation at an appoximate theshold value N c = A. The dynamics now becomes dn dt = BN ( N KB ) BN2 A 2 +N 2. 2

4 .2. Nondimensionalisation This model has fou paametes, B, K B, B, and A and it is then essentially impossible to analyze how the behavio changes with the paamete values. To simplify things we intoduce the dimensionless quantities which leads to the equation u = N A, = A B B, q = K B A, τ = Bt A, ( du = u u ) u2 q +u = f(u;,q). 2 This now depends on only two paametes and q which ae pue numbes. Note that the time scale is also changed. Also note that thee ae othe possible ways to get a dimensionless population, the most obvious one being ũ = N/K B. Diffeent fomulations may give diffeent insights..2.2 Gaphical solution The steady states ae solutions of f(u;,q) = u ( u q ) = u2 +u 2, and since u = clealy is a solution this may be ewitten ( u q ) = u +u u +u 2 ( u q. 2 u ) The ight hand side of the equation is a cuve with a non-tivial shape which is independent of and q. The left hand side on the othe hand depends on the paametes but is simply the staight line (/q)u fom (,) to (q,). The solution to the equation is given by the intesections of these cuves as shown above. Fom diffeent and q one gets diffeent staight lines, and it is clea that the equation can have eithe one o thee solutions. Fo the case with thee solutions we denote them by u, u 2, and u 3. 3

5 To investigate which of these steady states solutions that ae stable, we conside du/ f(u;,q) and the sign of f/ u. Since f >, u u2 u = u 2 is linealy unstable. The othe two, u = u and u = u 3 ae stable as they have negative deivatives. f(u).5. =.4 q = 3 f(u) = u ( u q ) u 2 +u u.2.3 Hysteesis It is inteesting to examine the effect of a gadual change of the paamete as it tuns out that it leads to hysteesis. Stating with = the only solution is u = u =. When inceases the u - equilibium inceases monotonically until u themaximumofthecuve iseached +u 2 whee = h. When is inceased futhe this steady state disappeas and the equilibium value jumps to u 3. If is deceased again, the solution stays at u 3 even when the solutions u and u 2 eappea. It is only when = l that u 3 disappeas that the system jumps to the u -equilibium. h l u u 3 u l h.3 Delay Models One of the deficiencies of Eq. (2) is that the bith ate is assumed to act instantaneously wheeas thee is typically a time delay fom one geneation to the next because of the time to each matuity and the time fom conception to bith. One way to genealize Eq. () is to wite [ dn dt = N(t) N(t T) ], (3) K 4

6 whee the egulatoy effect depends on the population at an ealie time t T athe than the population at t. A moe easonable model would be to say that the effect should eally be an aveage ove past populations, e.g. dn [ dt = N(t) t ] w(t t )N(t )dt. K The weighting facto w will tend to zeo fo lage negative and positive t and will have a maximum at some time T. Fo w appoaching a delta function at time T, we will ecove Eq. (3). It tuns out that this model can show stable limit cycle peiodic solutions fo cetain values of the poduct T. This model has two steady states, N(t) = and N(t) = K. It is again convenient to use dimensionless units, u = N K, τ = t, T = T, which gives du = u(τ)[ u(τ T) ]..3. Linea stability analysis Using u = +n this becomes dn = [+n(τ)][ n(τ T)], and lineaizing, we get dn = n(τ T). Look fo solutions of the fom n(τ) = ce λτ λ = e λ T. The condition fo the solution to be unstable is Reλ >. That is clealy not possible if λ is eal-valued. We instead take λ = µ+iω and conside the eal and imaginay pats of λ = e λ T: µ = e µ T cosω T, ω = e µ T sinω T. (4) 5

7 We now equie µ >. Also note that if ω is a solution to Eq. (4) then so is ω, so we can take ω > without loss of geneality. Fom the fist equation wegetcosω T < which leadstoω T > π/2. Multiplying thesecond equation by T the condition ω T > π/2 implies Te µ T sinω T > π/2. Since sinω T and e µ T < when µ >, this leads to T T > π/2. When the constant u(t) = (i.e. n(t) = ) is unstable, the altenative becomes an oscillatoy behavio a stable limit cycle and that happens to be the case in this model..3.2 Voluntay execise Ty a simulation of Eq. (3) with T = 2, K = and =.5,., and.5. Use t =., stat with the initial condition N(t) = +.(t T/2), t T. Confim that the petubation fom N = K descibed above dies out when the bith ate is taken to be =.5 (i.e. T = ) but gows to stable oscillations fo the lage. Also detemine the peiod of the oscillations which is expected to be close to 4T fo T not too fa above π/2. These equations may also be studied with values close to π/2, e.g. T =.99π/2 and T =.π/2, but the simulations then have to be un substantially longe to see the espective t behavios. 6

8 2 Continuous Models fo Inteacting Populations Thegenealpictueisthattheeisawholewebofinteactingspecies. Wewill hee go fo the simplest possible situation and conside only two inteacting species. We ae mostly inteested in the pedato-pey situation whee one species feeds on the othe. 2. The Lotka-Voltea model To be concete we will conside abbits and foxes. The assumptions of the model ae: In the absence of foxes the abbits beed with dr/dt = ar, i.e. an exponential gowth, without bounds. The pesence of foxes means that the gowth ate is educed by a tem bfr. The elative gowth ate of the fox population is popotional to the population of abbits, df/dt = cfr. The foxes have a cetain death ate, df. Taken togethe this becomes dr dt df dt = ar bfr, = cfr df. This is the Lotka-Voltea system of equations. It was used by Voltea in 926 as a simple model to explain the oscillatoy levels of cetain fish catches. The same equations had aleady in 92 been deived by Lotka to descibe a hypotetical chemical eaction. 2.2 Rewite in dimensionless units To simplify the analysis we ewite the equations with f = bf a, = cr d, τ = at, α = d a, 7

9 and they become d df It is easy to see that this gives two steady states: 2.3 Phase space plot = ( f), (5) = αf( ). (6) (,f) = (,), (,f) = (,). The figue to the ight shows one possible solution fo the Lotka-Voltea equations with the paamete α =. This is fo the initial values () = 2.5 and f() =.. This shows oscillatoy behavio as a consequence of the coupling between the two species. A diffeent kind of figue a phase spaceplot isshown totheight. That is obtained by plotting the two vaiables hee the non-dimensional densities of foxes and abbits against one anothe. That kind of plot gives mostly the same infomation (though the time scale is eliminated) but as we will see it is vey convenient fo analyzing these kinds of systems. f f. τ Even though the Lotka-Voltea equations give the desied oscillations it has a seious dawbacks the behavio is vey sensitive to petubations it theefoe mainly woks as a stating point fo developing moe ealistic models. 8

10 The sensitivity to petubations is clealy seen in egions whee the cuves ae close togethe. E.g. aound (,f) = (,.2) in this figue, a small decease in f takes us to a diffeent cuve with athe diffeent popeties. We now tun to two diffeent methods that help us detemine popeties of systems of coupled diffeential equations f without finding the exact solution. These methods ae () phase space analyses and (2) stability analyses of the steady states. The latte becomes somewhat moe involved fo the case of two coupled vaiables than what we saw in the pevious chapte. 2.4 Gaphical phase space analysis Fom the Lotka-Voltea equations we see that the ate of change of and f is only a function of these same vaiables and this suggests plotting ( d, df ) as a vecto field. (We have hee suppessed the infomation of the magnitude of these vectos, only keeping the diection of the field at each point.) The figues below show these vecto fields fo both the odinay Lotka- Voltea equations to the left and some modified equations to the ight. By following the vectos it is possible to constuct the possible tajectoies. 2 2 f f Besides the vectos the ight figue also shows the nullclines which ae the cuves in the (,f)-plane whee d/ = o df/ =. The points whee two nullclines meet ae the steady states, and the analysis of thei popeties is impotant fo detemining the behavio of the system. 9

11 2.5 Linea analysis of the steady state Fo the case of only one vaiable we lineaize by witing f(x) f(x )+(x x )f (x ), andthis is especially useful when f(x ) = when the behavio is govened by the deivative f (x ). Fo two vaiables we wite f = (f,f 2 ) and x = (x,x 2 ) and the coesponding expession become f(x) f(x )+(x x ) f, (7) x x whee the ightmost symbol is a way to wite the Jacobian fo f = x x ( f x f 2 x 2 f 2 f x x 2 ) x = A. (8) To apply this to ou pesent situation we use the compact notation d df dx = f(x), = f (,f), = f 2 (,f). (Hee the symbol f unfotunately has two diffeent meanings, as it is both the function and the density of foxes. The meaning should howeve hopefully be clea fom the context.) Let x denote a steady state, i.e. a state with f(x ) =. It then follows that the dynamics may be witten dx f(x )+A(x x ) = A x, whee x is the deviation fom the steady state x.

12 2.6 Stable o unstable In the one-dimensional case we found that the stability of the steady state was govened by f (x ). In the pesent case it is instead the eigenvalues of the matix that detemine the behavio. Fo the following, we change the notation and let x denote the deviation fom the fixed point. Thee ae then eigenvectos x ν and eigenvalues λ ν such that Ax ν = λ ν x ν. An abitay initial state may now be witten in tems of the eigenvectos as 2 x() = c ν ()x ν, ν= and the time dependence then becomes dx = Ax = ν c ν Ax ν = ν c ν λ ν x ν. The solution becomes x(τ) = ν c ν (τ)x ν, whee c ν (τ) = c ν ()e λντ. 2.7 Eigenvalues The eigenvalues ae detemined fom the equation With this becomes which gives det(a λi) =. A = ( a a 2 a 2 a 22 (a λ)(a 22 λ) a 2 a 2 =, a a 22 a 2 a 2 λ(a +a 22 )+λ 2 =, and, with TA = a +a 22, we find λ ± = ( ) TA± (TA) 2 4 deta. (9) 2 ),

13 This shows that the behavio is detemined by TA and deta. Thee ae then seveal diffeent possible behavios. A complete list is given in Appendix A of Muay, Mathematical Biology. We hee just focus on the cases whee the eigenvalues ae complex, which is what we get if 4 deta > (TA) 2. (The equation in Fig. A.2 appeas to be wong.) In the phase space plot the solutions ae ciculating aound the fixed point in one of the following ways: A stable spial towads to fixed point, if TA <. An ellipse centeed at the fixed point, if TA =. An unstable spial out fom the fixed point, if TA >. 2.8 Stability analysis of the Lotka-Voltea equations We now wanty to apply this stability analysis to the Lotka-Voltea equations. Ou equations ae d = f (,f) = f, df = f 2 (,f) = α(f f), and the Jacobian becomes ( ) ( ) f / f A = / f f =. f 2 / f 2 / f αf α( ) This should now be evaluated at the espective fixed points The tivial fixed point (,f) = (,) At (,f) = (,) we get A = ( α TA = α and deta = α. The eigenvalues ae then λ ± = ( ) α± ( α) 2 +4α = 2 2 ( α±(+α)), which gives ) λ + =, λ = α, and since one of these eigenvalues has the eal component >, x(τ) gows exponentially. 2,

14 2.8.2 The non-tivial fixed point (,f) = (,) At (,f) = (,), which is the othe fixed point, we get A = ( α and TA = and deta = α. The eigenvalues ae and the solution is λ ± = ±i α, x(τ) = c x e i ατ +c 2 x 2 e i ατ, which shows that the solution close to the neighbohood of (,) is peiodic with peiod 2π/ α. Since Reλ = fo both eigenvalues the steady state is neutally stable. 2.9 Realistic Pedato-Pey Models One of the unealistic assumptions in the Lotka-Voltea model is that gowth of abbits is unbounded in the absence of pedation. The othe is that thee is no limit to the pey consumption. The task in the compute lab is to modify the equations in diffeent ways to see how the behavio changes. ), 3

15 2. Realistic Pedato-Pey Models One of the unealistic assumptions in the Lotka-Voltea model is that gowth of abbits is unbounded in the absence of pedation. The othe is that thee is no limit to the pey consumption. Define modified equations: d df = f (,f) = ( g) f +s, = f 2 (,f) = α f +s αf. To detemine the non-tivial fixed point which we denote by (,f ) we use df/ = in the second equation and get /( + s ) = which gives = /( s). In the fist equation d/ = gives f = ( g )(+s ). To detemine the eigenvalues we fist get the following esult d d f +s = f +s s f (+s) = 2 f (+s) 2, which we use in Eq. (8) to get A = 2g f (+s) 2 α f (+s) 2 +s α +s α, () 2.. How does the behaviou depend on g and s? The next step to analyze this model is to get out the eigenvalues and detemine fo what paamete values diffeent kinds of behavious should be expected. To detemine the eigenvalues we would need to put in the expessions fo and f in Eq. () and then use the diffeent elements in A in Eq. (9). This quickly becomes athe messy and we theefoe tun to a gaphical method to detemine the egion in paamete space hee the new paametes g and s whee a cetain behavio would be expected. The method consists of calculating the popeties fo many diffeent paametesatthesametime. Inmatlab(osome othe plotting pogam) one defines two vectos with values of g and s between cetainlimits. Onethencalculates, f, g = /Rgass det A > (ta) 2 T A > s = /R sat

16 TA, and deta fo each of these paamete paisandfinally plotsthepoints(g,s) which fulfill cetain conditions. The figue shows two such egions. The solid dots ae the points fo which 4 deta > (TA) 2, and thus give complex eigenvalues. The geen squaes ae the points fo which TA > and whee we expect the fixed point to be unstable. In the egion whee both these conditions ae fulfilled thee is a possibility of a stable limit cycle. The following figues ae the esults obtained with g =.25 and s =.5. One of the cuves stats athe close to the(unstable) fixed point and spials outwad until it stabilizes as it appoaches the stable limit cycle. The othe cuve stats outside the limit cycle and spials inwads. The figues below show the time dependence fo both and f, stating fom the diffeent stating points, but eventually eaching the same behavio f τ f f τ 5

17 3 Discete Population Models fo a Single Species 3. Intoduction: simple models Many species have no ovelap between successive geneations, population gowth take place though discete steps: N t+ = N t F(N t ) = f(n t ). The simplest case: suppose that F(N t ) = > : N t+ = N t N t = t N. () This is usually not vey ealistic, though it could wok fo the ealy stages of gowth of cetain bacteia. 3.2 Moe geneal One geneally expects the function f(n t ) to have a maximum at N m and decease fo N t > N m. One such model is ( N t+ = N t N ) t, K but has the dawback that N t+ < if N t > K, which is of couse not ealistic. Anothe bette model is: [ ( N t+ = N t exp N t K )], >, K >, which is a modification of Eq. () with a motality facto 3.3 Gaphical solution exp( N t /K). 6

18 Giventhefunctionf(N t )andthestating value N the sequence N, N 2,..., may be detemined gaphically: The steady state solutions N ae intesections of the cuve N t+ = f(n t ) and the staight line N t+ = N t. If N is a stable o unstable solution is detemined by the eigenvalue of the system f (N t ), since that detemine the effect of a small petubation fom the steady state: f(n +δ) = N +δf (N ). A small petubation will lead to oscillations if f (N ) <. A small petubation will die out if f (N ) <. 3.4 Discete Logistic Model and Chaos Rescale with u t = N t /K andexamine the behavio of u t+ = u t ( u t ), >. We ae inteested in solutions with u t >, only. The steady states and the coesponding eigenvalues λ = f (u ) ae u =, λ =, u 2 =, λ 2 = 2. ut The cuves hee ae fo =,.5, The solid dots show the espective steady state solutions, u Stable steady state The detemining facto fo the stability is the eigenvalue, λ = f (u 2) = 2 ; the system is stable fo λ <. The figues hee show the behavio fo u t 7

19 = 2.8 and the initial value u =.72. The successive iteations take u t towads the stable solution u = ( )/.643. ut+ ut u t 2 t Unstable steady state The value = 3.2 on the othe hand gives λ =.2 and we expect the solution u to be unstable. Stating at u =.72, which is close to u 2 = ( )/.688, we see, just as expected, that successive iteations take us away fom the steady state. As the figues below show this gives a new behavio with an oscillation with peiod=2. ut+ ut u t 2 t Peiod doubling To examine this behavio in moe detail we conside the map fom u t to u t+2 defined by C u t+ = u t ( u t ), u t+2 = u t+ ( u t+ ). ut+2, ut+ A B The figue to the ight is fo = 3.2 and we see that thee ae then thee solutions to u t+2 = u t, which we denote by u A, u t 8

20 u B, and u C. Of these u B is an unstable solution since it has f (u B ) >. (Note that u B = u 2 ). On the othe hand, u A and u C ae stable. Note what this means: u t = u A will give u t+2 = u A, wheeas u t = u C similaly gives u t+2 = u C. Togethe with the ealie figue we may conclude that that system oscillates, u A, u C, u A, u C,... This phenomenon which is hee seen as a change fom a simple steady state to an oscillation between two diffeent values is called a bifucation. The discussion above may now be taken one moe step in the same diection as shown in this figue, which is C fo = 3.5: when inceases the solutions u A and u C move somewhat, and the B A steady states eventually become unstable as f (u C ) < and f (u A ) <. When that happens we again get a peiod doubling and the system epeats itself afte u t 4 units of time. It is then possible to epeat the discussion in this section fo u t+4. We would then find fou solutions to u t+4 = u t which ae stable fo = 3.5 but tun unstable at a slightly lage. This instability gives an oscillation of peiod eight, and it now seems that this peiod doubling may be continued without limit. It tuns out that the distance in between successive bifucations deceases apidly, and as we appoach c the oscillations have peiod 2 k with k. Fo > c the behavio is apeiodic and we thee ente the egion of chaos Chaos It tuns out that the behavio changes in unexpected ways as a function of : Fo < 3 thee is a unique solution ( )/. Fo 3 < + 6( 3.45) the system has peiodic fluctuates between two values. Fo + 6 < < 3.54 (appoximately) the system has peiodic oscillations between fou values. 9 ut+2, ut+

21 Fo 3.54 < < 3.57 the system oscillates between 8, 6, 32, values, etc. At 3.57 is the onset of chaos. We can no longe see any oscillations of finite peiod and slight vaiations in the initial value yields damatically diffeent esults ove time. ut