Original article Biomath 2 (2013), 1312061, 1–10

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 ISSN 1314-684X

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h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / Biomath Forum

Parameter Identification in Population Models for
Insects Using Trap Data

Claire Dufourd∗, Christopher Weldon†, Roumen Anguelov∗, Yves Dumont§
∗Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, South Africa,

{claire.dufourd, roumen.anguelov}@up.ac.za
†Department of Zoology and Entomology, University of Pretoria, Pretoria, South Africa,

cwweldon@zoology.up.ac.za
§CIRAD, Umr AMAP, Montpellier, France,

yves.dumont@cirad.fr

Received: 18 October 2013, accepted: 6 December 2013, published: 23 December 2013

Abstract—Traps are used commonly to establish the
presence and population density of pest insects. Deriving
estimates of population density from trap data typically
requires knowledge of the properties of the trap (e.g. active
area, strength of attraction) as well as some properties
of the population (e.g. diffusion rate). These parameters
are seldom exactly known, and also tend to vary in
time, (e.g. as a result of changing weather conditions,
insect physiological condition). We propose using a set
of traps in such a configuration that they trap insects
at different rates. The properties of the traps and the
characteristics of the population, including its density,
are simultaneously estimated from the insects captured
in these traps. The basic model is an advection-diffusion
equation where the traps are represented via a suitable
advection term defined by the active area of the traps.
The values of the unknown parameters of the model are
derived by solving an optimization problem. Numerical
simulations demonstrate the accuracy and the robustness
of this method of parameter identification.

Keywords-partial differential equation; advection-
diffusion equation; parameter identification; inverse
problem; trap interference; population density.

I. INTRODUCTION

This work is motivated by the need to develop a
reliable and efficient method for detecting the presence
and estimating population density of the invasive fruit fly,
Bactrocera invadens Drew, Tsuruta & White (Diptera:

Tephritidae) in South Africa. Bactrocera invadens is a
fruit fly species introduced from Asia to Africa where it
was first described and recorded in Kenya in 2003 [9],
[20]. In 2010, B. invadens was detected in the northern
part of the Limpopo province in South Africa [21]. Its
capacity for rapid population growth, high invasive po-
tential [16], and wide range of fruit hosts [26] represents
a major threat for all fruit industries in South Africa.

Fruit flies are a perennial problem in South Africa be-
cause in addition to B. invadens there are three endemic
species that already represent economic pests. Fruit flies
have historically been controlled in South Africa by the
application of insecticide cover sprays. Current practice,
however, involves the use of alternative control strategies
due to regulation- and consumer-driven requirements for
fruit to be free of insecticide residues. The primary
techniques used in fruit fly control are the application
of bait sprays [21], M3 bait stations [22], or the ‘male
annihilation technique’ [21]. All three techniques work
on the same principal: a food or sex attractant, which
is fed on by adult flies, is mixed with an insecticide
such as malathion or GF120. With regard to B. invadens,
male annihilation technique has been applied to control
incursions in South Africa [21]. Another control strategy
for this pest may include mass-trapping, which uses male
attractants to capture and kill males of a population,
leading to reduced female mating and possibly causing
local extinction of the population [12]. Alternatively, the

Citation: Claire Dufourd, Christopher Weldon, Roumen Anguelov, Yves Dumont, Parameter Identification in
Population Models for Insects Using Trap Data , Biomath 2 (2013), 1312061,
http://dx.doi.org/10.11145/j.biomath.2013.12.061

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C Dufourd et al., Parameter Identification in Population Models for Insects Using Trap Data

Sterile Insect Technique (SIT) may represent a useful
approach to control incursions of B. invadens. SIT in-
volves the release of large numbers of sterilized males
that compete with wild males for female fertilization,
which leads to no production of viable offspring [18]
and its success can be measured using the ratio of sterile:
wild insects captured in and array of surveillance traps
[17].

Regardless of the alternative control strategy used,
their successful application requires a good knowledge
of the distribution of the pest, and their dispersal ca-
pacity and density. The density of an insect population,
however, is a parameter that cannot easily be obtained
by direct field observations because traps usually sample
only a small proportion of all individuals. To overcome
this problem, it is often the case that captures of in-
sects in traps are compared to simulated data [14]. An
advection-diffusion equation is considered for modelling
the dynamics of fruit flies, where density is the initial
value of the model. Such a model requires knowledge
of the properties of the trap such as the active area
[5] and the strength of attraction, as well as some
properties of the population, like its diffusion rate [25].
These parameters are seldom exactly known, and also
tend to vary with changing weather [23] and landscape
heterogeneity [11],[10].

Determining the values of these parameters is actually
an inverse problem, that is, given the solution of the
model, or at least part of it, one or more of the model
parameters can be identified. The parameter identifica-
tion problem consists of finding a unique and robust
estimation for the parameter values. This problem leads
to solving a global optimization problem in order to
find the set of parameters that minimizes an objective
function. Mathematically, the existence and uniqueness
of this global minimum relies on the well-posedness
of the inverse problem, while its robustness relies on
its well-conditionedness. However, inverse problems are
typically ill-posed or conditioned, [19], [7], [24]. In
this paper, we show that by using different settings of
interfering traps we obtain a parameter identification
problem which can be solved numerically in a reliable
way. It is essential in this approach that interfering
traps generate different incoming streams of insects.
Thus, more information about the characteristics of the
insect population is provided. Indeed, as the relationship
between the setting and the traps is highly non linear and
not well understood, several settings of traps are consid-
ered and the robustness of the estimates are compared.
We demonstrate empirically that using this approach, the

problem of simultaneously identifying a set of unknown
parameters is well-posed and well-conditioned. The nu-
merical procedure falls under the well-known trial-and-
error method of regularization theory [28].

II. THE INSECT TRAPPING MODEL: THE DIRECT
PROBLEM

The model is formulated on a domain Ω ⊂ R2 which
is assumed to be isolated, i.e. there is no immigra-
tion and no emigration of insects. It is also assumed
that when there is no stimulus, the insects individually
follow a random walk. Because insects are often in
large abundance, we can apply a diffusion equation to
model the dispersal of insects at population level [29].
The traps set on Ω are attractive. Thus, the active area
of the trap [5] is the area where the concentration of
the attractant is above the threshold of concentration at
which the fruit flies can detect it. Therefore, in this area
the insects will be influenced to move in a preferred
direction towards the trap. This can be modelled using
an advection equation [3]. Finally, we assume that our
experiments take place over a short period of time, thus
we may omit reaction terms.

Using the above assumptions, the insect dynamics can
be modelled via an advection-diffusion equation.

∂u
∂t
−∇(s(x)∇u) + ∇(a(x)u) = 0,

∂u
∂n
|∂Ω = 0,

u|t=0 = u0.
(1)

u(t,x) denotes the population density at time t and at
the point x = (x1,x2) ∈ Ω.

The advection function a(x) is space-dependent and
determines the attractiveness of the trap with respect
to the distance to the center of the trap. The traps are
circular of radius Rtrap. Assume that the active area of a
trap is defined by a disk of radius Rmax from the center
of the trap. Then the insects that are beyond this disk
are not subjected to advection and we assume that their
dynamics are only governed by the diffusion term. As
the insect gets closer to the trap, the force of attraction
increases and reaches its maximum at a distance Rmin
from the center of the trap. If N is the number of traps
distributed on the domain, then:

a(x) =
N∑
T=1

aT (x),

aT (x) = amaxα(||xT −x||) xT−x||xT−x||,
(2)

where xT is the coordinate of trap T , and the function

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C Dufourd et al., Parameter Identification in Population Models for Insects Using Trap Data

α(d) is defined for d ∈ [0, +∞), as follows.

α(d) =

amax
d

sin
(

πd
2Rtrap

)
, if d < Rtrap

amax
d
, if Rtrap ≤ d < Rmin

amax
2d

(
cos
(
π d−Rmin
Rmax−Rmin

)
+1
)
, if Rmin ≤ d < Rmax

0 if Rmax ≤ d
(3)

The function α(d) is represented in Fig. 1. Note that
the value of the advection inside the trap does not
really matter, and we make α(d) decrease to 0 from the
distance Rtrap to ensure the continuity of a(x).

Fig. 1. Graph of the function α(d)

The diffusion coefficient s(x) is also space-dependent.
It is assumed to be constant, s(x) = σ, outside the
active areas of the traps. Since the insects do not escape
from the traps there should be no diffusion across the
trap boundary. In order to ensure the existence and
uniqueness of the (weak) solution of (1) we assume that
inside a trap the function s(x) has a positive value ε
which is so small that the implied diffusion effect in the
time interval of observation can be neglected. In order
to further ensure continuity of s we take

s(x) = σ −
N∑
T=1

sT (||xT −x||),

sT (d) =

σ−ε, if d ≤ Rtrap
(σ−ε)

(
1− d−Rtrap

Rmin−Rtrap

)
, if Rtrap<d ≤ Rmin

0, if Rmin < d.
(4)

III. THE PARAMETER IDENTIFICATION PROBLEM

The trap parameters, Rmin, Rmax and amax, as well
as the diffusion parameter σ are seldom know. Therefore,
although our main goal is to estimate the initial popula-
tion density u0, these other parameters are also needed.
For simplicity, we assume that the initial population
density is a constant, that is, u0(x) = u0 ∈ R,∀x ∈ Ω.
Denote p = (u0,σ,Rmin,Rmax,amax) the vector of the

parameters to identify. Let P be a compact subset of R5
to which p belongs. When solving the direct problem (1),
we are given p and we find a function u satisfying the
differential equation in (1) and the respective boundary
and initial conditions. This way, we define a mapping
φ from the domain P of parameters to the space of
solutions:

u = φ(p) (5)

It is well-know that the solution operator φ is continuous
and injective. Therefore, using the compactness of P we
obtain that the operator φ−1 : φ(P) −→P is continuous
([28], p.29). Thus, the inverse problem to (1) is well-
posed. Then if a solution u of (1) is given, the value of
p = φ−1(u) can be determined by well-known methods,
e.g. minimizing a norm of u−φ(p). However, in practice
u is commonly not known, at least not on the whole
domain Ω × [0, +∞) [24]. What is usually available is
a function B(u) referred to as observation operator [4].

Therefore the parameter identification problem is
stated as follows: Given an observation ψ, find p such
that

(B ◦φ) (p) = ψ. (6)

In the setting of problem (1) the observation operator
consists of the insect count in traps at given times
t1, t2, ..., tK. More precisely, we have

B(u) =


 B1(u,t1) . . . B1(u,tK)... ...

BN (u,t1) . . . BN (u,tK)


 ,

where BT (u,tk) is the total number of insects captured
in trap T until time tk, T = 1, ...,N, k = 1, ...,K. Hence
the observation ψ in (6) is an N ×K real matrix

ψ =


 ψ1,t1 . . . ψ1,tK... ...

ψN,t1 . . . ψN,tK


 .

Note that B is an array of numbers representing averaged
values of u on particular areas of the domain at finite
number of points in time. Hence, the injectivity of B◦φ
is problematic. Furthermore, the observation ψ contains
both model and measurement error which further com-
plicates the well-posedness of equation (6). In particular,
ψ is not necessarily an element of (B◦φ)(P). As usual
for such situations, p is obtained as a solution of (6)
in a least square sense, that is, p is a solution of the
optimization problem,

Φ(p) = ‖(B ◦φ) (p) −ψ‖2F −→ min, (7)

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C Dufourd et al., Parameter Identification in Population Models for Insects Using Trap Data

where ‖A‖F =

(
m∑
i=1

n∑
j=1
|ai,j|2

)1/2
.

Let p∗ be the minimizer of Φ, that is

p∗ = arg min
p∈P

Φ(p) (8)

In the setting of the current model, both the observation
operator and the model (1) depend on the distribution
of the traps. Our aim is to find trap configurations
for which the minimizer of Φ is unique and can be
reliably determined by a numerical procedure. In order
to investigate the properties of (7), equivalently (8),
we solve the optimization problem iteratively using a
random multistart approach [30] over a set of initial
values of p ∈ P. For each randomly selected starting
value of p, a local minimum of Φ is found using the
Gauss-Newton line search algorithm. The solution of
(7) is identified as the estimated parameter values of
the minimum of the local minima. Thus, increasing
the number of starting values increases the chances of
finding the global minimum of the objective function.
Furthermore, for each trap setting, we consider how
well the global minimum can be discriminated from the
other local minima. This gives an important indication
on how well the parameter values can be identified in
the presence of noise and we refer to it as robustness.
Here we investigate the influence of the choice of the
settings of traps on the accuracy and robustness of the
method.

In general, one can expect that increasing N and K
leads to a more regular problem. However, as shown
in [2], an incoming stream can be quite accurately
identified by using relatively small number of obser-
vations. Furthermore, if the traps are far enough from
each other, the rows of B(u) are the same. Therefore,
in the considered setting, just increasing the size of
the matrix B(u) would not improve substantially the
regularity of the problem. Our approach is to have some
of the traps close enough so that due to interferences,
they produce different streams of trapped insects. This
will increase the rank of the matrix B which can be
reasonably expected to improve the regularity of the
problem. Empirical evidence supporting this conjecture
is provided in the next section, where the effect of
different configuration is examined.

IV. DESCRIPTION OF THE EXPERIMENTS

We consider four trap settings:
• (A) single trap,
• (B) nine traps in a square formation,

• (C) five traps in a Z-formation,
• (D) five traps in a kite formation.

These four setting are shown on Fig. 2. Note that in all
cases the traps are sufficiently far from the boundary of
the domain Ω so that in the considered period of obser-
vation the boundary condition in (1) remains reasonable.
Fig. 3 represents the population distribution after 15

Fig. 2. Distribution of the traps on Ω for each setting. The symbols
identify the traps that produce identical incoming streams within each
setting.

time units using setting (D). The interference between
the traps can be observed in Fig. 4, which is a zoom
in of Fig. 3. Note that in these experiments, because
of the symmetry of the trap distribution, identical trap
counts are observed in several traps. In Fig. 2, the
traps identified with the same symbols have identical
incoming streams. Therefore, by using the nine traps
of setting (B) for instance, we multiply the amount of
information by three compared to the one-trap setting
(A). In settings (C) and (D), five traps are used. Due to
their configuration, setting (C) produces three distinct
incoming streams, whereas setting (D) produces four
distinct incoming streams. The distinct incoming streams
obtained in each setting are represented in Fig. 5.

Considering the different settings of traps we run nu-
merical simulations in order to estimate, one, two, three,
and finally four parameters simultaneously using the
cummulative trap counts over 15 time units (K = 15).

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C Dufourd et al., Parameter Identification in Population Models for Insects Using Trap Data

Fig. 3. Insect distribution after 15 time units using setting (D).

Fig. 4. Interference between the traps using setting (D). Zoom in
of Fig. 3.

Since the parameter of main interest to identify here is
the initial population density u0, it is always among the
parameters to estimate. The diffusivity of the insects σ, is
the second parameter of main interest when dealing with
insect dispersal. The choice of the remaining parameters
to estimate is based on a sensitivity analysis. We selected
in priority parameters on which the model has the highest
sensitivity. Indeed, when solving the inverse problem, the
more sensitive the original problem on a parameter is,
the more accurate the parameter identification is. The
sensitivity of a parameter is a measure of the change
in the output of the model caused by a change in this
parameter value. However, since the parameters under
consideration have different order of magnitude, it is
more appropriate to measure their elasticity index [15],
i.e. the proportional change in the output of the model
caused by a change in the value of the parameter. The

Fig. 5. Cumulative number of captured insects in each trap, using
the trap settings (A), (B), (C) and (D) of Fig. 2.

elasticity indices of each parameter with respect to each
setting of traps is given in Table II. Note that the output
of the model is the most sensitive to a change in the
parameter for which the elasticity index is the greatest.
Therefore, in order the output of the model is more
sensitive to changes of values of amax than of Rmax for
which the model is more sensitive to changes of Rmin.
Therefore, we proceeded to the parameter identification
of p1 = (u0), p2 = (u0,σ), p3 = (u0,σ,amax) and
p4 = (u0,σ,amax,Rmax). Our aim is to find p∗κ that
satisfies (8), where κ denotes the number of parameters
to be estimated (κ = {1; 2; 3; 4}). We limited our
numerical simulations to the simultaneous estimation
of 4 parameters due to limitations in computational
capacity. Indeed, when solving the inverse problem, each
evaluation of the objective function requires solving
problem (1).

Since we do not have real field data, we simulated the
data over a period of 15 time units with the values of
parameters given in Table I. Assuming that the period

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C Dufourd et al., Parameter Identification in Population Models for Insects Using Trap Data

TABLE I
VALUES OF THE PARAMETERS USED TO SIMULATE THE FIELD

DATA

parameter unit value
u0 insects/unit area 27.5
σ m2/time unit 5.23
amax m/time unit 5.75
Rmax m 8
Rmin m 2

TABLE II
ELASTICITY OF THE PARAMETERS WITH RESPECT TO THE

INCOMING STREAMS OF INSECTS. LET ν DENOTE A PARAMETER,
AND z(ν) THE INCOMING STREAM OF INSECTS WITH RESPECT TO

ν, THEN, THE ELASTICITY INDEX OF PARAMETER ν IS
Eν = ν‖z(ν) −z(ν + ∆ν)‖2/ (‖z(ν)‖2∆ν)

Setting Trap u0 σ amax Rmax Rmin
(A) 1.00 0.92 1.42 1.26 0.60

(B)
Center 1.00 1.02 1.26 0.83 0.54
Corner 1.00 0.96 1.35 1.06 0.57
Median 1.00 0.98 1.31 0.95 0.55

(C)
Center 1.00 0.98 1.33 1.01 0.56
Side 1.00 0.95 1.38 1.15 0.59
Top/Bottom 1.00 0.95 1.36 1.08 0.58

(D)

Center 1.00 0.98 1.29 0.90 0.55
Side 1.00 0.95 1.36 1.10 0.58
Top 1.00 0.97 1.33 1.01 0.56
Bottom 1.00 0.94 1.39 1.17 0.59

between two consecutive collects of data is equal to
one time unit, we extract from the simulated data only
those that correspond the collect time. A noise of 5%
is added to the simulated data, and this gives ψ =
(ψi,t1 . . .ψi,tK )i=1..N , where N is the number of traps.
The use of simulated data also allows us to compare the
estimation with the real values of the parameters.

The numerical solution of the initial value boundary
problem (1) is obtained by using the Crank-Nicolson
scheme [8], [27]. The optimisation problem (8) is solved
by using the matlab function fminunc which performs
the Gauss-Newton line search algorithm [13]. Since this
function finds only a local minimum, we run it with
many randomly selected initial points in order to have a
reasonable certainty that the global minimum is among
the local minima found. Here we choose u(0)0 ∈ [5; 45],
σ(0) ∈ [4; 6], a(0)max ∈ [2; 8], R

(0)
max ∈ [3; 10] as ranges of

the initial parameter values.

V. RESULTS AND DISCUSSION

The results of the experiments described above are
presented in Table III. The values of the local minima of
Φ, and the respective estimated values of the parameter
vectors p̃κ, with κ = {1; 2; 3; 4}, are given according

to the trap setting. According to our method, when
several local minima of Φ are found, the identified set of
parameter values corresponds to the global minimizer of
Φ, that is, the minimizer of the smallest local minimum.
The ∗ on the right of the table indicates the values of
the parameters identified as global minimizers of Φ. In
these experiments, we investigate the accuracy and the
robustness of the identified parameter values. The accu-
racy of p∗κ is measured by calculating the relative error to
the exact solution p̄κ (Table I) i.e. Erel=‖p̄κ−p∗κ‖/‖p̄κ‖.
In this setting the concept of robustness of the method
represents how well the global minimum value of Φ is
discriminated against the values of Φ of other minima.

TABLE III
LOCAL MIMIMA OF THE OBJECTIVE FUNCTION Φ, WITH THE

FOUR SETTINGS OF TRAPS WITH RESPECT TO THE NUMBER OF
PARAMETERS ESTIMATED SIMULTANEOUSLY. TS STANDS FOR THE

TRAP SETTING USED (FIG. 2). THE * INDICATES THE SET OF
PARAMETERS IDENTIFIED AS THE GLOBAL MINIMIZER OF Φ.

p TS Φ ũ0 σ̃ ãmax R̃max Erel

p
1

=
(u

0
) (A) 240004 27.63 – – – 0.5% *

(B) 259813 27.43 – – – 0.3% *
(C) 572150 27.45 – – – 0.2% *
(D) 620115 27.31 – – – 0.7% *

p
2

=
(u

0
,
σ

) (A) 219149 21.49 3.31 – – 22.5% *
222543 25.43 4.73 – –

(B) 256632 27.01 5.15 – – 1.8% *
(C) 572066 27.52 5.24 – – 0.1% *

826874 19.69 2.45 – –
(D) 619615 27.46 5.26 – – 0.2% *

p
3

=
(u

0
,
σ
,
a
m
a
x
) (A) 204279 17.79 4.30 7.89 – 34.9% *

214067 21.14 6.44 7.90 –
(B) 256473 26.70 5.16 5.81 – 2.8% *

297564 18.37 4.48 7.65 –
(C) 572055 27.61 5.23 5.73 – 0.4% *

636640 17.35 4.18 8.07 –
(D) 598689 25.17 5.48 6.31 – 8.4% *

647548 17.43 4.45 8.14 –

p
4

=
(u

0
,
σ
,
a
m
a
x
,
R
m
a
x
)

(A) 210462 33.31 3.25 5.07 5.83 22.1% *
210466 34.38 3.08 4.90 5.71
210471 32.75 3.35 5.17 5.89
210503 32.02 3.47 5.30 5.98
210533 31.53 3.56 5.39 6.03
210679 30.22 3.81 5.65 6.20

(B) 255885 26.93 5.09 5.64 8.15 2.1% *
277462 18.24 4.00 6.80 8.56

(C) 571732 27.51 5.28 5.85 7.88 0.6% *
609828 17.47 3.84 7.05 8.61

(D) 597514 25.40 5.37 6.07 8.18 7.2% *
615604 17.66 4.06 7.08 8.61

As a preliminary experiment, Table IV presents the
identified values and relative errors of p when a single
parameter value is unknown. p = (u0), p = (σ), p =
(amax) and p = (Rmax) are successively identified using
each setting of traps. These results are in agreement with

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C Dufourd et al., Parameter Identification in Population Models for Insects Using Trap Data

the sensitivity analysis provided in Table II, i.e. the more
sensitive the output of the model to a certain parameter,
the more accurate its estimation.

TABLE IV
IDENTIFIED VALUES OF THE SINGLE VALUE PARAMETER p AND

ITS RELATIVE ERROR WITH RESPECT TO THE TRAP SETTING, TS.
THE VALUES OF THE ESTIMATES ARE ROUNDED TO TWO DIGITS

AFTER THE DECIMAL POINT.

TS p ũ0 σ̃ ãmax R̃max R̃min Erel

(A)

p = (u0) 27.63 – – – – 0.48%
p = (σ) – 5.20 – – – 0.60%
p = (amax) – – 5.77 – – 0.37%
p = (Rmax) – – – 8.03 – 0.41%
p = (Rmin) – – – – 2.02 0.81%

(B)

p = (u0) 27.43 – – – – 0.24%
p = (σ) – 5.24 – – – 0.21%
p = (amax) – – 5.74 – – 0.17%
p = (Rmax) – – – 7.98 – 0.24%
p = (Rmin) – – – – 1.99 0.64%

(C)

p = (u0) 27.45 – – – – 0.19%
p = (σ) – 5.24 – – – 0.19%
p = (amax) – – 5.74 – – 0.13%
p = (Rmax) – – – 7.99 – 0.17%
p = (Rmax) – – – – 1.99 0.05%

(D)

p = (u0) 27.31 – – – – 0.69%
p = (σ) – 5.27 – – – 0.71%
p = (amax) – – 5.72 – – 0.49%
p = (Rmax) – – – 7.95 – 0.60%
p = (Rmin) – – – – 1.96 1.76%

A. Do interfering trap-settings provide better results
than non-interfering trap settings?

From Table III we can see that, using setting (A)
i.e. without trap interference, the number local minima
found increases, as the number of parameters to identify
simultaneously increases. Conversely, using the setting
with trap interferences, a maximum of two minima were
found.

Moreover, the minima found using one trap are in a
maximum range of 5% of the optimal value of φ and
within this range, the norms of the different minima can
differ by 15%. More precisely, for p = p2 (resp. p = p3),
the minima are found withing a range of 2% (resp. 15%)
of the optimal value of φ. In particular, when p = p4, 6
minima are found within a range of 0.1% of the optimal
value of φ where the norm of the minima can differ
by 10.6%. This shows that, without interferences, as the
number of parameters to identify increases, the reliability
of the estimates decreases.

Furthermore, we can see from Table III, that the
one-trap setting provides a poor accuracy of the es-
timates when several parameter values are identified
simultaneously, compared to the other setting. In fact,

using one trap, the relative error of the estimates is
above 22% when two or more parameter values are
estimated, whereas, using the other settings, the relative
error is always below 8.4%. Therefore, when two or
more parameters are to be identified, we may conclude
that interfering trap settings provide estimates with better
accuracy than non-interfering trap settings. However,
from Table III, note that if a single parameter needs to be
identified, setting (A), using only one trap, would provide
a sufficiently accurate and reliable estimate. In this case,
adding more traps is not really helpful. This suggests that
one must choose an appropiate setting depending on the
parameters that need to be identified.

By investigating further on the results, when p = p3
using one trap, we simulated the incoming streams using
the global minimizer of Φ (p3 = p∗3), (Table III). The
curve obtained with the latter simulation as well as the
curve representing the incoming stream simulated using
the real value of p , given in Table I (p3 = p̄3), are
represented in Fig. 6. As we can see, the two curves are
hardly distinguishable, meaning that two different sets of
parameters can lead to very similar incoming streams.
This illustrates the ill-conditionedness of the problem
where two very similar streams of trapped insects are
produced using two very different sets of parameters
(Erel = 34.9%, Table III).

Fig. 6. Simulated cumulative number of captures when estimating
three parameters with setting (A), using the real values, p̄3, and the
identified values, p∗3.

B. Interfering trap-setting strategies

We have shown above that using a setting of traps that
are interfering provide better results for parameter identi-

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C Dufourd et al., Parameter Identification in Population Models for Insects Using Trap Data

fication in terms of robustness and accuracy when several
parameters are identified simultaneously. However, little
is known on the actual role of the interferences between
the traps and their effect on the regularity of the problem.
In order to understand the effect of these interactions,
several interfering trap settings are compared.

We study the results obtained using the nine-trap
setting (B) and the setting (D) of five traps. One could
expect better accuracy and robustness of the estimates
using setting (D) since in this setting four distinct
incoming streams are produced whereas only three are
produced using setting (B). However, apart from the case
p = p1, the relative error of the estimate using setting
(D) is always lower than using setting (B). Considering
the fact that the noisy data are averaged over the traps
producing the same streams, in setting (B), the corner
and median noisy data are averaged over three traps,
whereas only the noisy data of the two side traps are
averaged using setting (D). Therefore, the data used to
identify the parameters using setting (B) are smoother
than those used with setting (D). This reduces the effect
of the noise and may explain why more accurate and
robust estimations are obtained using nine traps.

In setting (C), five traps are used producing three
distinct incoming streams as for the nine-trap setting (B).
The estimates obtained using setting (C) are the most
accurate compared to all the other experiments that were
carried out (Erel = 6% for p = p4). The robustness,
however, is not as good as for the nine trap setting. For
instance, when p = p3, two local minima of φ are found
within 10% (resp. 14%) of its optimal value for setting
(C) (resp. setting (B)). Despite this, the global minimum
of φ is clearly identified. This counter intuitive result
shows that the interference phenomenon is not trivial.
In particular, we can see in Fig. 5 that the 3 incoming
streams produced by the nine trap setting are more
distinct than those obtained using the five-trap setting. A
possible explanation may relate to the sensitivity of the
parameters. Indeed, the parameters amax and Rmax are
more sensitive using setting (C) than when using setting
(B) (Table II). Such a result suggest that there must be
an optimal setting which would allow to obtain robust
and accurate parameter estimation using as few traps as
possible.

These results show that trap interference can be used
to make the problem more regular, however, the re-
lationship between the regularity of the problem and
the interfence between the traps is highly nonlinear and
therefore difficult to analyse.

VI. CONCLUSION

Parameter Identification is challenging, in particular
when the direct problem is defined by PDEs, since it
often leads to solving inverse problems that are ill-
posed or ill-conditioned. We demonstrate numerically
that the interferences between traps can be used to make
the problem well-posed and well-conditioned using a
trial-and-error approach. This method enables the iden-
tification of parameter values that describe population
characteristics, i.e the population density, its diffusion
rate, attractiveness of the traps and their maximum radius
of attraction. By choosing biologically realistic ranges
of parameter values to estimate, the parameters are
identified as the global minimizer of function Φ using
starting values over these ranges. We show numerically
that trap interferences can be used not only to increase
the accuracy of the estimates, but also to find a global
minimum which can be well descriminated from the
other local minima. The trial-and-error method over the
choice of the trap setting, followed with the multistart
approach to solve the optimization method increases the
chances of succeeding in solving the parameter identifi-
cation problem. For each trap setting trial, we found nu-
merically the global minimum of the objective function
and we measured how well it is discriminated from the
other minima. The setting where the global minimum can
be the best discriminated from the other minima provides
an inverse problem that is well-conditioned and thus
provides the most robust solution to the parameter iden-
tification problem. Note that, once a “good” setting of
trap is identified, other methods, such as random search
methods, could be interesting alternatives to finding the
global minimum of the objective function [30], [6], [1].

Furthermore this work investigates how experiments
using traps should be conducted in the field so that
sufficient information is recorded, using as few traps as
possible. Indeed, the optimal setting of traps depends on
the parameters that need to be identified. For instance,
we showed that a setting using only one trap would
be sufficient to identify a single parameter. However
when several parameters are unknown, interfering traps
provide more accurate and more reliable estimates. In
particular, setting (C) stands out from the other settings.
Due to a highly nonlinear relationship between the trap
interferences and the regularity of the problem, it is
challenging to find an optimal setting of traps providing
highly accurate and robust estimates. A global sensitivity
of the model to the parameters will be considered in a
future work to give more insight into this relationship.

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C Dufourd et al., Parameter Identification in Population Models for Insects Using Trap Data

This is a promising method, not only from the numerical
and theoretical perspective for parameter identification,
but may also prove to be of practical importance for the
determination of insect population density in the field
with the use of traps.

ACKNOWLEGMENT

The support of the National Research Fundation of
South Africa and Citrus Research International Ltd.
(Project: 1075) is acknowledged. Y. Dumont would also
like to thank the support of the “SIT feasibility program”
in Reunion, jointly funded by the French Ministry of
Health and the European Regional Development Fund
(ERDF).

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http://dx.doi.org/10.11145/j.biomath.2013.12.061

	Introduction
	The Insect Trapping Model: The Direct Problem
	The Parameter Identification Problem
	Description of the experiments
	Results and Discussion
	Do interfering trap-settings provide better results than non-interfering trap settings?
	Interfering trap-setting strategies

	Conclusion
	References