Ordinary Differential Equations

Transcription

1 Ordinary Differential Equations Michael H. F. Wilkinson Institute for Mathematics and Computing Science University of Groningen The Netherlands December 2005

2 Overview What are Ordinary Differential Equations (ODEs)? Equilibria: existence and properties. What techniques exist to solve ODEs? How do we draw up ODEs: A case study on Predator-Prey systems. A variant: Delay Differential Equations (DDEs)

3 Ordinary Differential Equations ODEs can be used to model the behaviour of a system through time The systems studied range from spring-mass systems to ecology and economics. ODEs have the general form: d x dt = f( x, t). (1) Quite frequently, we deal with time independent ODEs, which have the form The simplest form they can take are linear ODEs: d x dt = f( x). (2) d x dt = A x. (3) with A a matrix.

4 Equilibria Equilibria occur where Three types of equilibria occur: d x dt = f( x, t) = 0. (4) Stable: the system returns to the equilibrium after small perturbation. Unstable: the system diverges from the equilibrium state after small perturbation. Neutrally Stable: the system neither diverges from, nor converges towards the equilibrium after small perturbation. In the case of time independent ODEs we can determine the stability by local stability analysis quite easily. In the first step, we linearize the ODE around the equilibrium.

5 Equilibria Let the equilibrium point be x 0. The ODE given by d x dt = f( x) (5) can be approximated by Taylor expansion. We first introduce x = x x 0. We then approximate (5) by in which J is the Jacobian matrix: J = f 1 d x dt = J x (6) f 1 x n x f n x 1 f n x n x= x0 (7)

6 Equilibria The solution to such a set of linear differential equations is just x (t) = a i Ei e λ it, (8) with E i the eigenvectors of J, λ i the corresponding eigenvalues, and a i the amplitude of that particular eigenmode of the system. Clearly, if all the eigenvalues have a negative real part, the system returns to the equilibrium position and the system is stable. If one or more eigenvalues have positive real parts then it is unstable. If none have positive and one or more zero real parts it is neutrally stable.

7 Equilibria To determine eigenvalues we need to solve the equation f 1 f x 1 λ 1 f x 2 1 x n f 2 f 2 f det x 1 x 2 λ 2 x n = 0 (9) f n f x 2 n x n λ x= x0 f n x 1 This yields an n th order polynomial c 0 + c 1 λ + c 2 λ c n λ n = 0 (10) the roots of which are the eigenvalues. Note that if all c i are real and have the same sign, the system is stable.

8 Solving ODEs Usually, we are concerned with systems which are too large for analytical treatment, but can be handled numerically. Most often we are concerned with initial-value problems: given the state of the system at time t 0, compute the state at a number of points in time t n > t 0. The simplest method is the Euler method x(t n+1 ) = x(t n ) + h f( x(t n ), t n ) + O(h 2 ), (11) in which h is the time step and O(h 2 ) indicates an error which is proportional to h 2. An improvement is the midpoint method: k1 = h f( x(t n ), t) (12) x(t n+1 ) = x(t n ) + h f( x(t n ) + k 1 /2, t n + h/2) + O(h 3 ) (13)

9 Solving ODEs Better still is the 4 th order Runge-Kutta method: k1 = h f( x(t n ), t) (14) k2 = h f( x(t n ) + k 1 /2, t n + h/2) (15) k3 = h f( x(t n ) + k 2 /2, t n + h/2) (16) k4 = h f( x(t n ) + k 3, t n + h) (17) x(t n+1 ) = x(t n ) + k k k k O(h5 ) (18) Various strategies for computing an optimal time step h exist. Problems occur in so-called stiff ODEs. In many cases (chemistry, biology) we need to check for zero crossings!

10 Predator-Prey Systems One of the best studied systems is the so-called predator-prey (or host-parasite) system. It is fundamental to understanding food webs in ecology. Its general form is in which dx 1 dt = F (x 1) G(x 1, x 2 ) (19) dx 2 dt = ηg(x 1, x 2 ) H(x 2 ) (20) F (x 1 ) is the growth rate of x 1, G(x 1, x 2 ) is the rate of predation of x 1 by x 2, η is the efficiency with which prey biomass is turned predator biomass, and H(x 2 ) is the starvation rate of x 2.

11 Predator-Prey Systems In its simplest form, we have the Lotka-Volterra system in which ( dx 1 dt = r 1 1 x ) 1 x 1 fx 1 x 2 (21) K 1 dx 2 dt = ηfx 1x 2 d 2 x 2 (22) f determines the rate at which predator and prey encounter eachother, r 1 is the maximum relative growth rate of x 1, K 1 is the carrying capacity of the ecosystem for x 1, and d 2 is the starvation rate of x 2. This system can be stable, neutrally stable, or unstable, depending on the parameters.

12 Predator-Prey Systems Other models for F include the Monod-model for bacteria S F (x 1, S) = µ max K S + S x 1 (23) in which S is the substrate (food) concentration, µ max the maximum growth rate, and K S a saturation constant. Other models for G include: Holling Type II Holling Type III Jost f f x 1 K 2 +x 1 x 2 x 2 1 x K 2 +x x 2 1 x K 2 +k 2 x 1 +x Ivlev f(1 e kx 1)x 2 f same as Monod for vertebrates similar to above All these models try to limit maximum growth in some way.

13 A Bacterial Predator-Prey System Suppose we have a bacterium X 1 using resource X 0, and which is preyed upon by predator Y. We assume that these species are present in a chemostat-like environment, in which all substances are well mixed, and which has a dilution rate D. Food (X 0 ) enters at a concentration S. Uptake of food by X 1 is modelled using the Monod-form. Uptake of X 1 by Y is modelled using the Holling-type-II form. Starvation of Y takes the usual form, with d y the starvation rate.

14 A Bacterial Predator-Prey System The differential equations for this model are dx 0 dt = D(S X 0 ) V 1 X 0 K 1 + X 0 X 1 (24) dx 1 X 0 X 1 = µ 1 X 1 V y Y DX 1 dt K 1 + X 0 K X + X 1 (25) dy dt = µ X 1 y Y (D + d y )Y K X + X 1 (26) with V x and µ x maximum uptake and growth rates respectively, and K x saturation constants.

15 A Bacterial Predator-Prey System This system can be in four phases Phase 0: Insufficient food is available to allow survival of X 1 ; stable equilibrium: X 0 = S, X 1 = 0, and Y = 0. Phase I: Sufficient food for survival of X 1, but insufficient number survive to allow survival of Y. Phase II: Both X 1 and Y coexist stably. Phase III: unstable coexistence of X 1 and Y. Important question: which parameters determine outcome?

16 A Bacterial Predator-Prey System The equilibrium with all species present is given by X 0 = 1 2 ( S K 1 V 1 K X µ y D d y ± ( S K1 V 1 K X µ y D d y ) 2 + 4K 1 S) (27) X 1 = Y = D + d y µ y D d y D D + d y µ y V y ( µ1 V 1 (S X 0 ) X 1 ) (28) (29) Only if all three solutions are positive do we have either a phase II or phase III system. The boundary between these phases II and III is determined by local stability analysis.

17 A Bacterial Predator-Prey System The dynamical behaviour of the predator prey system after introduction of predator Prey Predator Prey Predator Biomass (mg/l) Biomass (mg/l) time (h) time (h)

18 A Bacterial Predator-Prey System The boundaries of the phases I 0.18 I D (h 1 ) II III D (h 1 ) II III S (mg/l) S (mg/l)

19 Decoys in Predator-Prey Systems In a simple complication, we can add a third species X 2, which does not compete with X 1 but can collide with the predator. Assume that the predator can be in three states: free, bound to X 1 and bound to X 2. We can then draw up the following set of differential equations: d[x 1 Y ] dt d[x 2 Y ] dt dy free dt = k 1 [X 1 Y ] + rx 1 Y free (30) = k 2 [X 2 Y ] + rx 2 Y free (31) = (y x + 1)k 1 [X 1 Y ] + k 2 [X 2 Y ] r(x 1 + X 2 )Y free (32)

20 Decoys in Predator-Prey Systems We will assume that these reactions take place at a much faster rate than the predator prey dynamics We can then in quasi-steady state analysis say that k 1 [X 1 Y ] = rx 1 Y free and k 2 [X 2 Y ] = rx 2 Y free (33) Adding all versions of Y together yields dy dt = y x k 1 X 1 Y k 1 /r + X 1 + k 1 X 2 /k 2 = µ y X 1 Y K X + X 1 + K inh X 2 (34) Thus the presence of X 2 leads to an increase in the apparent saturation constant K X = K X + K inh X 2.

21 Decoys in Predator-Prey Systems The dynamical behaviour of the predator prey system after introduction of predator with decoys: K X = 2K X Prey Predator Prey Predator Biomass (mg/l) time (h) Biomass (mg/l) time (h)

22 Decoys in Predator-Prey Systems The boundaries of the phases with decoys: K X = 2K X I 0.18 I D (h 1 ) II D (h 1 ) II 0.06 III 0.06 III S (mg/l) S (mg/l)

23 Delay Differential Equations (DDEs) In ODEs the time evolution depends only on the current state of the system In DDEs the time evolution of the system depends on the current state, plus the state of the system at one or more points some time in the past. This type of equation is useful to model systems in which there are time delays between cause and effect. Examples are: incubation times of infections in models of epidemics, delay times between penetration of host by parasites and emergence of the new parasite generation delay between mating and birth, etc.

24 Delay Differential Equations (DDEs) DDEs have the general form: d x dt = f( x(t), x(t τ 1 ), x(t τ 2 ),..., x(t τ N )). (35) with τ 1, τ 2,..., τ N the set of delays in the system. The stability condition becomes f( x(t), x(t), x(t),..., x(t)) = 0. (36) Finding equilibria is easy, analysis of their properties generally done numerically. Numerical treatment requires different techniques than for ODEs. Initial value problems require information about the past for initialization!

25 Example DDE we use the bacterial predator-prey model as before, but use a delay equation to model the predator behaviour. The prey is converted to infected prey [X 1 Y ] as in the decoy model After a time delay τ the new generation of predators is released. The differential equations for this model are dx 0 dt = D(S X 0 ) V 1 X 0 K 1 + X 0 X 1 (37) dx 1 dt = µ 1 X 0 K 1 + X 0 X 1 rx 1 Y DX 1 (38) d[x 1 Y ] = e Dτ rx dt 1Y + rx 1 Y D[X 1 Y ] (39) dy dt = e Dτ (y x + 1)rX 1Y (D + d y )Y (40) in which X 1 and Y denote the concentrations of X 1 and Y at t τ.

26 Example DDE The delay effect can sometimes be approaximated by an average rate of formation of new prey, leading to a Holling type II version as in the original bacterial predator-prey model density (cm 3 ) density (cm 3 ) time (h) (a) time (h) (b) Part (a) shows the DDE appraoch, part (b) the ODE approximation.