Effect of directional movement on animal food search · 2017-04-06 · Gardiner, C. W. "Handbook of stochastic methods for physics, chemistry and the natural sciences." Applied Optics

Effect of directional movement on animal food search

Sarafa Iyaniwura

Institute of Applied Mathematics,University of British Columbia, Vancouver.

April 6, 2017

Sarafa Iyaniwura Math521: Numerical Analysis of PDEs April 6, 2017 1 / 13

Siniff, Donald Blair, and C. R. Jessen. ”A simulation model of animal movement

patterns.” Advances in ecological research 6 (1969): 185-219

Figure: A fox and a rabbit. Ref: Tom UlrichStone/Getty Images (left)

Question: What is the effect of directional motion on the time to get prey?


Literature review


Geometry of the region of animal movement

What do we want do to?

use first passage time theory to compare the search time to prey foran animal performing directional motion (biased random walk) andthat which performs a random walk (unbaised random walk)


First Passage Time

Let T1 be the mean first passage time for the Red Fox to find a prey andT2 is the second moment of the first passage time distribution. Then

D∆T1 + v · ∇T1 = −1, in Ωk⋃

i=1

Ωi ,

∇T1 · n = 0 on ∂Ω, and T1 = 0 on ∂Ωi , for i = 1, 2, . . . , k.

(1)

And the second moment satisfies

D∆T2 + v · ∇T2 = −T1, in Ωk⋃

i=1

Ωi ,

∇T2 · n = 0 on ∂Ω, and T2 = 0 on ∂Ωi , for i = 1, 2, . . . , k,

(2)

where D is the diffusion coefficient of the predator, and v is its velocity.Variance:

V = T2 − T 21 . (3)


Weak Formulation and Linear Finite Element

Let u ∈ V be a test function, then the weak formulation of theproblems are

D

∫W∇T1 · ∇u dx −

∫W

(v · ∇T1) u dx =

∫W

u dx

And the second moment,

D

∫W∇T2 · ∇u dx −

∫W

(v · ∇T2) u dx =

∫W

T1 u dx

where W = Ωk⋃

i=1Ωi .

this problem is solved using linear finite element method

D Kh−→T − V h−→T =−→f

where Kh and V h are the stiffness and advection matrices respectively.


Numerical results


Parameters: D = 0.41 km2/h, v = 0.085 km/h, R = 1 km, prey radius = 50cm)


Parameters:

D = 0.41 km2/h, v = 0.085 km/h, area of region = 4 km2, prey radius = 50cm)


Sources of error

Approximation error/interpolation error

triangulation of the domain

Quadrature Error

using numerical quadrature to evaluate integrals likeKi,j =

∫T∇φi · ∇φj dx that arise when constructing the stiffness

matrix


Conclusion and future work

Conclusion

depending on where the animal is starting from, directional motion mayor may not favour the animal

Future work

consider a more realistic scenario where the rabbits are also moving


Some references

Kurella, Venu, et al. ”Asymptotic analysis of first passage timeproblems inspired by ecology.” Bulletin of mathematical biology 77.1(2015): 83-125.

Siniff, Donald Blair, and C. R. Jessen. ”A simulation model of animalmovement patterns.” Advances in ecological research 6 (1969):185-219.

McKenzie, Hannah W., Mark A. Lewis, and Evelyn H. Merrill. ”Firstpassage time analysis of animal movement and insights into thefunctional response.” Bulletin of mathematical biology 71.1 (2009):107-129.

Gardiner, C. W. ”Handbook of stochastic methods for physics,chemistry and the natural sciences.” Applied Optics 25 (1986): 3145.



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Effect of directional movement on animal food search · 2017-04-06 · Gardiner, C. W. "Handbook of stochastic methods for physics, chemistry and the natural sciences." Applied Optics