Danish Ali Ahmed
daa11@le.ac.uk
Leicester University 2012
(Supervisor: Dr. Sergei Petrovskii)
CONTENTS

Section 1
Introduction to pest trapping
2. Modelling of movement and the Diffusion process
3. Finite Difference method – Explicit/Implicit scheme
4. Mean field approach
5. Individual based model
6. Circular field with a circular trap installed at the centre (MFA)
7. Square field with a circular trap installed at the centre (IBM)
8. Comparison between MFA and IBM under similar geometry
9. Stochastic Fluctuations
10. Conclusion
1.
CONTENTS
•
Section 2
1.
The 2-D Diffusion equation and the flux of the system
The cross shaped trap
Investigating the effect of the position of an initial point source release
on trap catches
Modelling Click Beetle (Agriotes obscurus, Females) trap catches
Other research work (Some listed problems)
2.
3.
4.
5.
Section 1:
1) Introduction to pest trapping
• An essential component of monitoring integrated pest management
•
•
•
•
programmes is the trapping of insects.
There are mainly two types of traps commonly used in the research field of
ecological studies and pest management. These are passive and baited
traps. Passive trapping relies on the movement of individual insects into a
capturing zone whereas baited traps lure the insects through the usage of
an attractant.
We will be interested in passive traps.
Scientists have regularly used passive traps to monitor populations. The
most common approaches include the ’mark-release-capture’ method and
the other uses the concept of counts as density measures.
The main aim is to construct a comprehensive understanding and
interpretation of trap counts, with a well defined and consistent
mathematical framework.
• Over the last two decades research has been directed towards the study of
movement, however mathematicians still lack a consistent framework for
the modelling of such a phenomenon. This is not due to neglect of such a
notion, however it is known that the study of movement is intrinsically
difficult since it consists of at least two scales - temporal and spatial.
• Once passive trapping is well understood we can investigate more
complicated systems which incorporate wind effects and also baited
trapping with an attractant (food/lighting).
• A natural extension would lead to the study of multiple traps.
• Common models include discrete individual based models (Lagrangian
viewpoint) and Diffusion processes (Eulerian viewpoint).
2) Modelling of movement and the Diffusion process
 Two main conceptual approaches assigned to the modelling of





movement.
The Lagrangian and the Eulerian viewpoint.
Lagrangian point of view concentrates on the moving
individual. Includes the characterization of velocity,
acceleration, turning angles . . .
Eulerian models are characterized by population densities and
fluxes at spatial points.
Diffusion models are examples of Eulerian models.
Individual based models (IBM) are considered as Lagrangian
models.
 Both Lagrangian and Eulerian models complement eachother.
 The Diffusion process is the process by which matter is
transported from one part of a system to another as a result of
random particle motion.
 The motion of particles is completely random, independent
and isotropic.
 The fundamental idea of Diffusion is based on the Fickian
hypothesis.
 Fickian hypothesis: Rate of transfer of diffusive substance
through a unit area of a section is proportional to the
concentration gradient measured normal to the section
u
F  D
x
F = rate of transfer per unit area of cross section.
u = Concentration of diffusing substance.
x = Spatial co-ordinate measured normal to the section.
D = Diffusion coefficient.
 Once derived (See crank 1975) we obtain the Diffusion
equation
u
 
u 

D

t
x 
x 
Or more generally



u
   D u
t

in n dimensions.
• We will be interested in the 1D and 2D Diffusion equations.
u
 u
1D :
D 2
t
x
2
  2u  2u 
u
2D :
 D 2  2 
t
y 
 x
where D = D(t) is possibly time dependent.
3) Finite difference method.
 An analytic solution cannot always be found for the Diffusion
equation provided an initial condition and suitable boundary
conditions.
 We resort to the discretization of the geometry and solve the
system numerically using finite difference methods.
 The 1D Diffusion equation with constant diffusion coefficient
u
 2u
D 2
t
x
can be formally discretized using the differencing technique.
The finite difference method (Explicit and Implicit)

   2u
u u
 D

t

n 1
j
•
n
j

n 1
j

 1     u
x 2
2

n
j

, j  0,1,...., J , n  0,1,...., N


 is the parameter which distinguishes different schemes with
0    1.
• x  jx
• t  nt
(Spatial step)
(Time step)
•  2u j  u j  1x   2u jx   u j  1x 
Explicit scheme
•  0
u nj 1  u nj
t

 u
D
n
2
j
2
x 
• Explicit method is known to be numerically stable and
convergent provided the courant condition is satisfied.
D
• Numerical error:
t
1

x 2 2

u   t    x 
2

Implicit scheme
•  1
u
n 1
j
u
t
n
j

 u
D
2
n 1
j
2
x 
• Scheme is always numerically stable and convergent.
• More numerically intensive than the explicit method.
• Requires solving a system of numerical equations in each time
step.
• Numerical error:

u   t    x 
2

4) Mean field approach (MFA).
 Consider the 1D Diffusion equation with constant diffusion
coefficient
u
 2u
 D 2 ,0  x  L
t
x
 Initial condition: u ( x,0)  U 0 ,U 0  Initial population density
 Boundary condition:
u
u (0, t )  0
( L, t )  0
x
 Using the explicit scheme the governing equation becomes
u nj 1  u nj
t
 u nj1  2u nj  u nj1 
t
1

 D
D

2
2

 with



x
2



x


Discretization of (Initial/Boundary) conditions:
 Initial condition: u( x,0)  U 0  u j  0
0
 Boundary condition at x = 0,L:
u(0, t )  0  u Nj  0
u L, t 
 0  u Jn  u Jn 1
t
5) Individual based model (IBM).
 Alternative way to look at the dynamics of trap counts is to
simulate the movement of each individual in the field
(Lagrangian viewpoint).
 
 Particle moves under curvilinear track r  r (t )
 Observations allow recording at certain moments,
t0 , t1 ,...., tn 
correspondingly the curvilinear path is mapped into a series of
broken lines
 

r0 , r1 ,...., rn 
ti  ti 1  ti
• Assume ti  t  const  ti  it
• Time step
Increment in time is constant.

•
•
•
•
•

Individual at a position ri  xi , yi  at time ti will be at position




ri 1  ri  r i at time ti 1
The movement vector can be described by
r i  xi , yi   i ,i 
in rectangular / polar co-ordinates respectively.
 i is the step length and  i the turning angle.
xi , yi or i ,  i  are random variables, therefore the
movement is a stochastic process.
Assume no correlation between two subsequent steps
Statistical properties of movement are independent of time.
Process is essentially Markov, i.e A process with no memory.
Each subsequent step does not carry any information based on
the previous step.
 The properties of the movement are fully determined by the
probability distribution functions for x and y. We denote
these distributions as x  and y  respectively.
 Model the increments to be Gaussian (Normally distributed)
x  
 x 2 
, y  
exp  
2 
2
2x
 2 x 
1
 x 2 

exp  
2 
2
2x
 2 x 
1
where  x2 ,  y2 are the variances.
• Environment is isotropic, no advection and no drift.
x   y   0
 x2   y2   2
• The signature of the process is that the mean squared

displacement is proportional to time r 2   2Dt
i.e mean squared displacement grows linearly with time
 Recall that we stated that both Eulerian and Langrangian
viewpoints complement eachother.
 This agreement is formed using the signature of the process

 r 2  2 Dt   2
i.e a relation between Diffusion coefficient and  the
variance.
• Provided that D is fixed and t is fixed
 
  2 Dt
6) 2D: Circular field with a circular trap installed at the centre
(MFA)
 Solve the Diffusion equation in polar coordinates.
 The population of the species u(r,t) is to be radially symmetric
and axisymmetric.
 Consider the following problem:
BC’s:
 1 u  2u  ~
u
,   r  R
 D

2 
t
 r r r 
u ( r , t )  0 on r  ~
u ( r , t )
0
r
IC:
on r  R
u (r ,0)  U 0
 We calculate the flux of the system
t 2
K (t )  

0 0
t

u (~, t ' )
j ddt  2D 
dt '
r
0
 Use finite difference method (explicit scheme) to solve the
problem numerically.
 Use numerical integration to evaluate the flux curve K(t).
7) 2D: Square field with a circular trap installed at the centre
(IBM)
 Consider a square field with dimensions 0  x, y  L
L L
 Install a circular trap at the centre of the field  2 , 2  with radius ~
 Step lengths are normally distributed in both the x and y
2

directions with variance
 If an individual particle falls into the trap then it is removed
from the system
 The far field boundary acts as a reflective boundary
 To ensure that individuals which reach the far field do not
influence the trap counts we require that ~  L
~
L
 1,
~
R
 1
Additional comments for comparison between MFA and IBM
We require the following in order to compare the MFA and IBM:
• Typical lengths in the field are large in comparison to the
typical length of the trap
~
L
 1,
~
R
 1
• Recall: r 2   2 Dt   2   

2 Dt
• To reduce stochastic fluctuations we can increase the no. Of
individual particles introduced initially. For fixed field area this
refers to an increase in population density 
• We expect a smoother transition of influx as we increase 
8) Comparison between MFA and IBM under similar geometry.
Fix the following parameters:
• Constant Diffusion coefficient: D  1
• Length of square field: L  50
• Time increment for IBM: t  0.01
• Number of particles for IBM and MFA: N  250000,U 0  100
Calculated parameters:

• MSD: r 2   2 Dt   2    2 Dt    0.02
• Field area must be the same:
R
L2

 28.2
2
2

R

L
• Error in Field area:
100  0.07%
2
L
9) Stochastic fluctuations
 Let K t represent the flux of the particles in the IBM at time
step t
 Let K (t ) represent the flux of the particles in the mean field
approach over some time interval 0  t  t *
How do we reduce these stochastic fluctuations?
 Define the relative maximum of the stochastic fluctuations as
Sup K t  K (t )
   t0,t 



*
Tabulated values which illustrate the smooth transition of
influx as we increase 
Population density
Relative maximum of the stochastic
fluctuations
0.1
 ( )

16.86600
1
12.18340
5
7.09920
20
2.51662

10) Conclusion
 Successfully compared the individual based model and the mean
field approach, used for the trapping of insects. Calculated a
theoretical value for δ.
 Both processes essentially govern the dynamics of the pests
 Both Eulerian and Lagrangian methods complement each other
Some discussion points:
• Can we use these processes to describe or simulate pest trapping in
the field?
• If yes, can we find an analytical solutions for simple cases or simple
geometries?
• If no, can we enhance the models described to incorporate real
motion in the field?
• Super diffusive processes? Levy flights?
CONTENTS
•
Section 2
1.
The 2-D Diffusion equation and the flux of the system
The cross shaped trap
Investigating the effect of the position of an initial point source release
on trap catches
Modelling Click Beetle (Agriotes obscurus, Females) trap catches
1D Analytic solution via the Greens function for Super Diffusion
Introduction of Levy flights via the Cauchy distribution as opposed to
Brownian motion
Matching Super diffusive flux curves to flux curves produced by Cauchy
distributed step lengths
Further research
Other research work (Some listed problems)
2.
3.
4.
5.
6.
7.
8.
9.
1) The 2D Diffusion equation and the flux of the system
 The 2D Diffusion equation
  2u  2u 
u
2D :
 D 2  2 
t
y 
 x
 Flux of the system is defined in general as
t
N (t )  

0 
t


j ddt    D u.n ddt
0 

where  is the trap boundary, n is a normal vector to the
trap boundary.
 The 2D Diffusion equation will be solved numerically using the
explicit finite difference technique. The flux will be evaluated
using numerical integration.
The initial condition.
• The most common design for studying dispersal is the
instantaneous point source release, in which marked
organisms are released instantaneously at one point in time
(usually t = 0) at a single point in space.
• Formally this is described using the dirac delta function
u( x, y, t  0)  N ( x  x0 , y  y0 )
 For the IBM we place N particles at a fixed position ( x0 , y0 )
 For the MFA it is not so straight forward. We model the point
source release using an initial Gaussian profile
 x  x0 2   y  y0 2 

u ( x, y, t  0) 
exp  
2
2

2
2



1
 We must ensure that  is not too small i.e on the order of the
mesh step sizex. Nor should it be too large as the wide
spread of the profile will not reflect the PSR.
2) The cross shaped trap
 A cross shaped trap is
installed at the centre of a
field of length L with
absorbing boundaries.
 The cross shaped trap has
wing length  and tip length 
with fixed perimeter
P  42   
 We introduce an initial point
source release at a distance r
from the centre of the cross
subtending an angle  from
the horizontal wing.
ε
r
ϒ
θ
L
L
 We model the spatio-temporal motion of the pests using the
Diffusion equation with the initial standard bi-variate Gaussian
distribution.
 We investigate how the orientation and radial distance of the
point source release effects the trap catches.
EVOLUTION: The evolution of the two dimensional Gaussian
distribution for fixed r = 6, θ = π/4 and times t = 0.5, 1.5, 2.5 and
2.9 days.
CLOSE UP: A ‘close up’ of the point source release for fixed r = 6, θ
= π/4 and time t = 2.9 days in the vicinity of the trap.
3) Investigating the effect of the position of an initial PSR on trap
catches
The effect on trap catches for varying radial distance and fixed
orientation θ = π/4 as time evolves.
The effect of orientation on trap catches for varying radial distance
at fixed time t = 1.5 days.
The effect of radial distance on trap catches for varying orientation
and fixed time t = 1.5 days
 Obtain a monotonously decreasing dependency of flux on radial distance,



•
i.e the further the point source release from the centre of the cross shaped
trap, the lower the trap counts.
The optimal trap counts are obtained at θ = π/8,3π/8 from the values of θ
investigated. A clear indication that individuals that approach the trap from
these angles have a greater chance of being trapped.
Individuals that approach the trap from a horizontal (θ = 0) or vertical (θ =
π/2) direction have a lower chance of being trapped.
Due to the symmetry of the problem we analyse the range
0 ≤ θ ≤ π/4, we conclude that the optimal angle of approach must lie in the
range 0 < θ < π/4.
Simulations for the flux of the system illustrate that for a point source
release situated far from the centre of the trap are of linear form (r >> 8).
A transition is shown between linear to non-linear flux dependency on
time as we shift the release point closer to the trap centre.
4) Modelling Click Beetle (Agriotes obscurus, Females) trap
catches.
 The Centre for Agricultural and Rural Sustainability at the
University of Plymouth and Agriculture and Agri-Food Canada
have carried out mark release recapture studies with the click
beetle (A. obscurus).
 In this section we aim to model the experimental trap catches
using the Diffusion model and we pay special importance to
the parameter D.
 For modelling purposes it is still unclear whether a constant
Diffusion coefficient D does suffice, alternatively there is a
possibility that D could be modelled as time dependent (Super
Diffusion), spatially dependent or even incorporating a
combination of temporal-spatial dependencies.
The experiment
• The experimental study was carried out
on a Fallow field of dimensions 72m ×
72m with an open boundary.
• Three types of pitfall traps were installed
in the field. Twenty-four linear traps 0.5m
long were installed at regular spatial
intervals 1m parallel to each side. Within
the 70m × 70m area remaining in the field
60 circular pitfall traps were systematically
arrayed and 25 cross shaped traps
positioned so that no trap aligned with
any other using a randomised latin square
design.
• A point source release of N = 100
A. obscurus (females) were released at the
centre (36,36) of this field.
• Trap catches were monitored at unequal
time intervals, i.e traps were inspected
and emptied at particular times, after
inspection they were instantly reinstalled.
• Monitoring of these traps formed a
cumulative trap count.
The results
Time (hrs), t
Rescaled time, t
Trap catches N(t)
1
1/24
3
3
1/8
4
25
25/24
13
48
2
22
96
4
29
The model
 We model this complex situation by concentrating on the cross shaped trap closest to the
point source release.
•
•
•
•
•
•
•
Investigation of the cumulative trap counts formed by the trap in the vicinity of the
point source release is justified from the assumption that the effects from other traps in the
(far) field
are negligible i.e all other (cross-shaped) traps were greater than 10m away from the release.
The position of this (local) cross shaped trap is (40,32), a distance of 5√2m ≈ 5.66m from the
centre of the
field at an angle of 45o from the horizontal.
We model the situation using the geometry set up earlier (cross shaped trap) where we set
the field length L = 10m, radial distance r =(5√2)/9m and θ = π/4.
The idea here is to map the field experiment geometry into an easily manageable geometry
prepared for simulations, without affecting the results.
The population density of the pests u evolves according to the 2-D Diffusion equation in
cartesian co-ordinates. We impose u = 0 on the field boundary and the no flux conditions on
the trap boundary ∂Ω.
The point source release is modelled using the Gaussian profile with standard deviation σ =
1/2.
We discretize the Diffusion equation using the method of finite differences under these set
parameter values.
Modelling experimental trap catches using the Diffusion
equation with constant diffusion coefficient.
Modelling trap catches using the super Diffusion model.
Model 1: D(t) = at + b, varying ‘a’ with fixed ‘b’.
Model 2: D(t) = a√t + b, varying ‘a’ with fixed ‘b’.
Model 3: D(t) = at2 + b, varying ‘a’ with fixed ‘b’.
Further study
 We compared the simulation results with empirical data
obtained from field experiments on beetles. There is a
disagreement between data and curves predicted by the
diffusion model (with constant diffusion coefficient).
 This indicates that the dispersal pattern of pests does not
follow the standard Brownian diffusion. This suggests that the
beetles possibly could exhibit a more complicated pattern of
movement using “fat-tail” distributions or Lévy flights.
 We need to understand the effects of Super Diffusion and the
Levy flight process using individual based and mean field
approaches. This would provide a better understanding of the
spreading of insects.
 In particular this will explain whether the spreading is due to
behavioural aspects or simply due to there being a larger
chance that insects make bigger steps.
5) Other research work (Some listed problems)
 Study of trap counts for different trap shapes
Detailed study of how the shape of the trap affects the trap
counts. In particular we looked at square shaped traps, slot
shaped (rectangular) traps, circular traps and cross shaped traps.
• The effect of the corners in a trap: Analytic study of the wedge
domain in the vicinity of the corner
Solved the 2D Diffusion equation using separation of variables in
a wedge domain.
  2u 1 u
u
1  2u
 D 2 
 2
t
r r r  2
 r



Under the conditions:
~
u(r ,0, t )  0, u(r, , t )  0, u(a, , t )  0, u(r , ,0)  U 0 , u(0, , t )
is finite.
 Schematic approach for the
calculation of flux at sharp
corners
From a numerical and
computational perspective,
how we calculate the flux
along straight edges is clear. It
is at discontinuities (such as
sharp corners) problems arise.
This has been an interesting
mode of study in the field of
computational fluid dynamics.
With equal interest in the
application of pest trapping.
We have a concise study of
how corners affect trap counts
using different schematic
approaches.
I would like this opportunity to thank everyone for listening to
the presentation, with a special thanks to the organisers of the
MESS seminar.
Thankyou
Danish Ali Ahmed
daa11@le.ac.uk