Numerical Stability of the Escalator Boxcar Train under ... · Escalator Boxcar Train(EBT) One way to study the dynamics of a PSPMs numerically, is to divide the structure population

Numerical Stability of the Escalator BoxcarTrain under reducing System of Ordinary

Differential Equations

By

Tin Nwe Aye(Joint work with Linus Carlsson)

April 26, 2017

Tin Nwe Aye Numerical Stability of the Escalator Boxcar Train

Introduction

In mathematical biology, there is often a need to studyphysiological structured population models.

Physiologically structured population models(PSPMs)describes the dynamics of an arbitrary number ofbiological populations. Examples include models of fishesin lakes and trees in forests.

PSPMs investigate the study of change in populations inwhich individuals differ physiologically in one or moreways, e.g., in their size.


Introduction





Introduction





Introduction





For the dynamics of PSPMs, the density n(x , t) of individualsof state x at time t satisfies

∂

∂tn(x , t) +

∂

∂xg(x ,Et)n(x , t) = −µ(x ,Et)n(x , t)

g(xb,Et)n(xb, t) =

∫ ∞

xb

b(x ,Et)n(x , t)dx

n(x , 0) = n0(x)

Here, g is the growth rate, µ is the mortality rate, and b isthe fecundity rate.



∂

∂tn(x , t) +

∂


g(xb,Et)n(xb, t) =

∫ ∞

xb

b(x ,Et)n(x , t)dx

n(x , 0) = n0(x)




∂

∂tn(x , t) +

∂


g(xb,Et)n(xb, t) =

∫ ∞

xb

b(x ,Et)n(x , t)dx

n(x , 0) = n0(x)



Escalator Boxcar Train(EBT)

One way to study the dynamics of a PSPMs numerically,is to divide the structure population into distinct groupsof individuals that are more or less similar which are calledcohorts.

The cohorts clarify internal cohorts and boundary cohortsbecause of reproduction. The new born individuals areassumed to have the same physiological properties and areaccumulated in the boundary cohorts.

The number of individuals in the i th cohort is denoted byNi(t) and the mean individual state will be used anddenoted as Xi(t).












The dynamics of the internal cohorts and boundary cohortfor non reproduction are defined by

dNi

dt= −µ(Xi ,E )Ni (1)

dXi

dt= g(Xi ,E ) (2)

where E is the environmental feedback andi = 0, 1, 2, . . . ,M .If reproduction does occur, the dynamics of the boundarycohorts for the reproduction case follow

dN0

dt= −µ(X0,E )N0 +

M∑i=0

b(Xi ,E )Ni

dX0

dt= g(X0,E )


The dynamics of the internal cohorts and boundary cohortfor non reproduction are defined by

dNi

dt= −µ(Xi ,E )Ni (1)

dXi

dt= g(Xi ,E ) (2)

where E is the environmental feedback andi = 0, 1, 2, . . . ,M .If reproduction does occur, the dynamics of the boundarycohorts for the reproduction case follow

dN0

dt= −µ(X0,E )N0 +

M∑i=0

b(Xi ,E )Ni

dX0

dt= g(X0,E )


Process of internalizing the boundary cohort

In the course of time, both the number and size ofindividuals in the boundary cohort increase according tothe reproduction of individuals and the environment.

The boundary cohort must be internalized sufficientlyoften since an applicable large approximation error can beoccured because of above properties.

The number of internal cohorts will be increased due tointernalization


Process of merging internal cohorts

We merge the two cohorts together if the number ofindividuals in an internal cohort falls below a certainthreshold and that the size of the internal cohort closet tothis one, is close enough

The number of offsprings stays the same, compared to ifwe had not merged the cohorts


The Daphnia’s Life History Model

To illustrate the formulation of an EBT-model, theobjects for the PSPMs is the waterflea Daphnia pulex.

Daphnia behaviour is extremely influenced by the size ofan individual.

Larger individuals have higher food consumption, basalmetabolism and reproduction rate.

The mature Daphnia can shrink under particularconditions.


Error bounds when merging two cohorts

We consider two cohorts (Xa,Na) and (Xb,Nb).The difference, ∆x = Xb − Xa, between the sizes of these cohortsis assumed to be sufficiently small. The food available in thesystem will be assumed to be constant.From Equation (1), we have N ′(t) = −µN(t).According to the Equation (2) and the growth rate of Daphnia, thedynamics of cohorts for X (t):

X ′(t) = c1(1 − X (t)

K)


Error bounds when merging two cohorts

We consider two cohorts (Xa,Na) and (Xb,Nb).The difference, ∆x = Xb − Xa, between the sizes of these cohortsis assumed to be sufficiently small. The food available in thesystem will be assumed to be constant.From Equation (1), we have N ′(t) = −µN(t).According to the Equation (2) and the growth rate of Daphnia, thedynamics of cohorts for X (t):

X ′(t) = c1(1 − X (t)

K)


For non-merging cohort, the dynamics of cohort for populationfecundity bw (t) from the growth rate of Daphnia is

b′w (t) = c2Na(t)X 2a (t) + c2Nb(t)X 2

b (t).

For merging cohorts, we naturally add the number of individuals inboth cohorts, i.e.,

Nm0 = Na0 + Nb0

In the case when we merge the two cohorts, we get the dynamicsof the fecundity as

b′m(t) = c2Nm(t)X 2m(t) (3)

In view of Equation (3), we initialize the merged cohort size to

Xm0 =

√Na0X

2a0 + Nb0X

2b0

Na0 + Nb0


For non-merging cohort, the dynamics of cohort for populationfecundity bw (t) from the growth rate of Daphnia is

b′w (t) = c2Na(t)X 2a (t) + c2Nb(t)X 2

b (t).

For merging cohorts, we naturally add the number of individuals inboth cohorts, i.e.,

Nm0 = Na0 + Nb0

In the case when we merge the two cohorts, we get the dynamicsof the fecundity as

b′m(t) = c2Nm(t)X 2m(t) (3)

In view of Equation (3), we initialize the merged cohort size to

Xm0 =

√Na0X

2a0 + Nb0X

2b0

Na0 + Nb0


Theorem

Under the above assumptions we get

bm = bw + O(∆x(∆t)2)

Proof: For merging cohort, we can also get newborn individualsby using Taylor’s expansion and above equations.

bm = c2(Na0Xa02 + Nb0Xb0

2)∆t − c2µ(Na0Xa02 + Nb0Xb0

2)∆t2

2

+ c1c2Na0

√Na0Xa0

2 + Nb0Xb02

Na0 + Nb0

∆t2

+ c1c2Nb0

√Na0Xa0

2 + Nb0Xb02

Na0 + Nb0

∆t2

− 1

K

(Na0Xa0

2 + Nb0Xb02)

∆t2 + O(∆t3)


We can also caluclate newborn individuals by using Taylor’sexpansion

bw = c2(Na0Xa0

2 + Nb0Xb02)

∆t − c2µ(Na0Xa02 + Nb0Xb0

2)

+ c1c2(Na0Xa0 + Nb0Xb0)∆t2 − c1c2

(Na0Xa0

2

K+

Nb0Xb02

K

)∆t2

+ O(∆t3)

Using the continuity of the square root function, we get

Xa0 =

√Na0Xa0

2 + Nb0Xb02

Na0 + Nb0

+ O(∆x),

Xb0 =

√Na0Xa0

2 + Nb0Xb02

Na0 + Nb0

+ O(∆x)

Thus, the number of new born individuals for merging cohortsconverges to the number of new born individuals for nonmergingcohort when ∆x and ∆t goes to zero.


We can also caluclate newborn individuals by using Taylor’sexpansion

bw = c2(Na0Xa0

2 + Nb0Xb02)

∆t − c2µ(Na0Xa02 + Nb0Xb0

2)

+ c1c2(Na0Xa0 + Nb0Xb0)∆t2 − c1c2

(Na0Xa0

2

K+

Nb0Xb02

K

)∆t2

+ O(∆t3)

Using the continuity of the square root function, we get

Xa0 =

√Na0Xa0

2 + Nb0Xb02

Na0 + Nb0

+ O(∆x),

Xb0 =

√Na0Xa0

2 + Nb0Xb02

Na0 + Nb0

+ O(∆x)

Thus, the number of new born individuals for merging cohortsconverges to the number of new born individuals for nonmergingcohort when ∆x and ∆t goes to zero.


Simulation of the EBT and Daphnia model

MATLAB is selected to be used for this project and thefunction ode45 is chosen to solve the ODEs because of itsaccuracy and speed.

We used a least square method to fit the best monomialfor the simulation time depending on the number of timesteps.

We found that the relationship between the simulationtime and the number of time steps was linear.


Result of the simulation

TimeSpan(days)

Merging ElapsedTime(seconds)

InternalCo-horts

j(mgC/L) m(mgC/L) v(mgC/L)

2 Yes 15 28 0.3005 0.0330 0.3335

1 Yes 26 47 0.2636 0.0326 0.2962

1/2 Yes 47 59 0.3025 0.0382 0.3407

1/4 Yes 96 98 0.2832 0.0391 0.3234

2 No 21 748 0.3087 0.0330 0.3418

1 No 65 1366 0.3062 0.0335 0.3398

1/2 No 142 1680 0.3137 0.0382 0.3519

1/4 No 463 3164 0.3132 0.0392 0.3523


Biomass of juvenile and mature for merging cohorts


Conclusion and Future Work

In this paper, we merge cohorts that are sufficiently closetogether.

The reduced system of ODE’s will not give rise to largechanges in the general solution.

The running time of the simulation, when merging, isproportional to the time step.

In future work, we aim to build EBT-solver which includesthe automatic feature of merging and splitting for moregeneral models.


References

A. M. de Roos and L. Persson. From individual life history topopulation dynamics using physiologically structured models.https:// staff fnwi.uva.nl/a.m. deroos/downloads/EBT/EBTsyllabus pdf, 2004, Accessed:2014-19-09.

A. Brannstrom, L. Carlsson, and D. Simpson. On theconvergence of the escalator boxcar train. SIAM Journal onNumerical Analysis,, 51(6):3213-3231, 2013.

A. M. de Roos, A gentle introduction to models ofphysiologically structured populations, inStructured-Population Models in Marine, Terrestrial, andFreshwater Systems, S .Tuljapurkar and H.Caswell, eds.,Chapman & Hall, NewYork, 1997, pp. 119-204.


Thank you for your attention.


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Numerical Stability of the Escalator Boxcar Train under ... · Escalator Boxcar Train(EBT) One way to study the dynamics of a PSPMs numerically, is to divide the structure population