approximating the extinction threshold of spatial dynamics...

Approximating the extinction threshold of spatial dynamics ofmigratory birds

Xiang-Sheng Wang

Mprime Centre for Disease ModellingYork University, Toronto

(joint work with Jianhong Wu)

Approximating the extinction threshold of spatial dynamics of migratory birds First Previous Next Last 1

A beautiful pond in Keele campus of York University


Autumn, 2010


Winter, 2010

No migratory birds.


Spring, 2011


Three observations

• No significant migratory activities during summer or winter season.

• Almost all birds migrate during spring and autumn seasons.

• No significant birth activities during autumn and winter seasons.


Seasonal activities

• Spring migration : birds fly from the winter patch to the summer patch.

• Summer breeding : birds give births to newborns.

• Autumn migration : birds fly from the summer patch to the winter patch.

• Winter refuge : birds stay at the winter patch until spring comes.


Spring migration

• An: bird population at winter patch when spring migration starts.

• Bn: bird population at summer patch when spring migration ends.

• transition function for spring migration:

Bn = f1(An).


Summer breeding

• Bn: bird population at summer patch when spring migration ends.

• Cn: bird population at summer patch when autumn migration starts.

• summer breeding function:Cn = f2(Bn).


Autumn migration

• Cn: bird population at summer patch when autumn migration starts.

• Dn: bird population at winter patch when autumn migration ends.

• transition function for autumn migration:

Dn = f3(Cn).


Winter refuge

• Dn: bird population at winter patch when autumn migration ends.

• An+1: bird population at winter patch when spring migration starts in thecoming year.

• winter refuge function:An+1 = f4(Dn).


Discrete model

Bn: population

at the summer site

An(An+1): population

at the winter site

Cn: population

at the summer site

Dn: population

at the winter site

f1: spring migration

f2: summer breeding

f3: autumn migration

f4: winter refuge


Question 1

Do we have explicit formulas for the transition functions f1 and f3 of spring andautumn migrations in discrete model?


Continuous model

Bourouiba, L., Wu, J., Newman, S., Takekawa, J., Natdorj, T., Batbayar, N.,Bishop, C. M., Hawkes, L. A., Butler, P. J. & Wikelski, M. 2010 Spatial dynamicsof bar-headed geese migration in the context of H5N1. J. R. Soc. Interface 7,1627-1639.

Gourley, S., Liu, R. & Wu, J. 2010 Spatiotemporal distributions of migratory birds:patchy models with delay. SIAM J. Appl. Dyn. Syst. 9, 589-610.


Stopovers during spring migration

xs,i(t) = −(ms,i(t) + µs,i)xs,i(t) + αs,i−1ms,i−1(t− τs,i−1)xs,i−1(t− τs,i−1).

• xs,i(t): bird population at i-th (i = 1, 2, · · · , k) stopover

• ms,i(t): migration rate

• µs,i: natural death

• αs,i: flight mortality

• τs,i: time delay


Stopovers during autumn migration

xa,j(t) = −(ma,j(t) + µa,j)xa,j(t) + αa,j−1ma,j−1(t− τa,j−1)xa,j−1(t− τa,j−1).

• xa,j(t): bird population at j-th (j = 1, 2, · · · , l) stopover

• ma,j(t): migration rate

• µa,j: natural death

• αa,j: flight mortality

• τa,j: time delay


Winter refuge site

xs,0(t) = −(ms,0(t) + µs,0)xs,0(t) + αa,lma,l(t− τa,l)xa,l(t− τa,l).

• xs,0(t): bird population at winter refuge site


Summer breeding site

xa,0(t) = −(ma,0(t) + µa,0)xa,0(t) + αs,kms,k(t− τs,k)xs,k(t− τs,k)

+γ(t)xa,0(t)(1− xa,0(t)/K).

• xa,0(t): bird population at winter refuge site

• γ(t)xa,0(t)(1− xa,0(t)/K): logistic birth


Continuous model

ma,l−1

ma,1

ms,1

ms,k−1

ma,l

ma,0

ms,0

ms,k

xs,0

µs,0

xa,0

µa,0

birth

xa,l

µa,l

xa,1

µa,1

xs,1

µs,1

xs,k

µs,k


Periodic system of delay differential equations

xs,0(t) = −(ms,0(t) + µs,0)xs,0(t) + αa,lma,l(t− τa,l)xa,l(t− τa,l);

xs,i(t) = −(ms,i(t) + µs,i)xs,i(t) + αs,i−1ms,i−1(t− τs,i−1)xs,i−1(t− τs,i−1);

xa,0(t) = −(ma,0(t) + µa,0)xa,0(t) + αs,kms,k(t− τs,k)xs,k(t− τs,k)

+γ(t)xa,0(t)(1− xa,0(t)/K);

xa,j(t) = −(ma,j(t) + µa,j)xa,j(t) + αa,j−1ma,j−1(t− τa,j−1)xa,j−1(t− τa,j−1).


Threshold theorem

“Either there is a globally attracting periodic solution, or the bird populationbecomes extinct.”

Gourley, S., Liu, R. & Wu, J. 2010 Spatiotemporal distributions of migratory birds:patchy models with delay. SIAM J. Appl. Dyn. Syst. 9, 589-610.


Question 2

How to tie the threshold condition to model parameters explicitly (rather than theabstract spectral radius of a certain monodromy operator)?


Numerical phenomenon

“Repeated deadly outbreaks of H5N1 on stopovers during the autumn migrationof bar-headed geese could lead to a larger reduction in the size of the equilibriumbird population compared with that obtained after repeated outbreaks during thespring migration.”

Bourouiba, L., Wu, J., Newman, S., Takekawa, J., Natdorj, T., Batbayar, N.,Bishop, C. M., Hawkes, L. A., Butler, P. J. & Wikelski, M. 2010 Spatial dynamicsof bar-headed geese migration in the context of H5N1. J. R. Soc. Interface 7,1627-1639.


Numerical phenomenon

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10 000

15 000

20 000

25 000

30 000

35 000

40 000

45 000

0 1000 2000 3000 4000 5000

tota

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time (in days)

cs of bar-headed geese migration L. Bourouiba et al. 9


Question 3

Can we provide any mathematical justification for the numerical phenomenon thatautumn disease outbreak is comparably more serious than spring?


Three questions

• Do we have explicit formulas for the transition functions f1 and f3 of springand autumn migrations in discrete system?

• How to tie the threshold condition to model parameters explicitly (rather thanthe abstract spectral radius of a certain monodromy operator)?

• Can we provide any mathematical justification for the numerical phenomenonthat autumn disease outbreak is comparably more serious than spring?


Answers

• We approximate the continuous system by a discrete system which providesexplicit formulas for transition functions f1 and f3.

• The threshold of the continuous system can be approximated by that ofdiscrete system.

• Sensitivity analysis demonstrates that autumn disease outbreak is more seriousthan spring.


Three basic assumptions

• The migratory activity is insignificant during summer breeding season andwinter refuge time;

• The population left in the winter refuge site (resp. summer breeding site) afterspring (resp. autumn) migration is comparably negligible;

• The breeding activity does not occur during autumn and winter seasons.


Notations

According to seasonal activities of migratory birds, we divide a year by fourseasons:

t0spring(T1)−→ t1

summer(T2)−→ t2autumn(T3)−→ t3

winter(T4)−→ t0 + T.

t0 t1 t2 t3

spring

migration

begins

spring

migration

ends

autumnmigration

begins

autumnmigration

ends


Assumption 1: migration rates are piecewise constants

For i = 0, 1, · · · , k,

ms,i(t) =

Ms,i, t−i−1∑p=0

τs,p ∈ (t0, t1);

0, otherwise,

and for j = 0, 1, · · · , l,

ma,j(t) =

Ma,j, t−j−1∑q=0

τs,q ∈ (t2, t3);

0, otherwise.


Assumption 2: migration rates are sufficiently large

The quantity

ε :=

k∑i=0

e−Ms,iT1 +

l∑j=0

e−Ma,jT3

is sufficiently small.


Assumption 3: birth rate is also a piecewise constant function

γ(t) =

{γ0, t ∈ (t0 + τs, t2);

0, t ∈ [t0, t0 + τs] ∪ [t2, t0 + T ].


Three basic assumptions

t0t0 +

i−1∑p=0

τs,pt0 + τs t1

t1 +i−1∑p=0

τs,pt2t2 +

j−1∑q=0

τa,qt3t3 +

j−1∑q=0

τa,qt3 + τa t0 + T

γ0 > 0

Ms,i > 0 Ma,j > 0

The quantity

ε :=

k∑i=0

e−Ms,iT1 +

l∑j=0

e−Ma,jT3

is sufficiently small.


Approximate discrete dynamic system

Bn: population

at the summer site

An(An+1): population

at the winter site

Cn: population

at the summer site

Dn: population

at the winter site

f1: spring migration

f2: summer breeding

f3: autumn migration

f4: winter refuge


Approximate discrete dynamic system

Bn = f1(An) = Ane(γ0−µa,0)T1

k∏i=0

αs,iMs,i

Ms,i + µs,i − µa,0 + γ0+ · · · ;

Cn = f2(Bn) =(1− µa,0

γ0)Ke(γ0−µa,0)(T2−τs)

(1− µa,0γ0

) KBn − 1 + e(γ0−µa,0)(T2−τs);

Dn = f3(Cn) = Cne−µs,0T3

l∏j=0

αa,jMa,j

Ma,j + µa,j − µs,0;

An+1 = f4(Dn) = Dne−µs,0(T4−τa).


Asymptotic formula of extinction threshold

R0 ∼ rs · ps · tr, as ε→ 0.

• reproductive rate: rs = eγ0(T1+T2−τs).

• survival probability:

ps = (

k∏i=0

αs,i)× e−µa,0(T1+T2−τs) × (

l∏j=0

αa,j)× e−µs,0(T3+T4−τa).

• transition rate:

tr = (

k∏i=0

Ms,i

Ms,i + µs,i − µa,0 + γ0)× (

l∏j=0

Ma,j

Ma,j + µa,j − µs,0).


Sensitivity analysis

• Suppose Ms,i =Ma,j and µs,i = µa,j for some i and j. We have from theinequality

| dR0

dµs,i| < | dR0

dµa,j|

that R0 is more sensitive to the change of µa,j than that of µs,i.

• Therefore, the bird population is more vulnerable to a serious disease occurringat a stopover during the autumn migration.

• This provides a mathematical explanation for the phenomenon observed fromnumerical simulation by Bourouiba et al. (2010).


Conclusion

• Mathematical model approximates biological activities.

• Sometimes, it is difficult to analyze a complicated model.

• If there are some small (or large) quantities in the complicated model, we couldapproximate it by a simpler model via asymptotic analysis.


Thank you!


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