**A simple mathematical model
of plankton net avoidance and its validation using peculiarities of the biology
of the hyperiid amphipod Vibilia armata**

(english translation and comments on my paper Un modèle mathématique de l'évitement d'un filet à plancton, son application pratique, et sa vérification indirecte en recourant au parasitisme de l'amphipode hypéride Vibilia armata Bovallius (

**Prologue**

I wrote this paper 36 years ago. Now (2010) I am retired. In retrospect I
think that this article had several shortcomings (I was young and
inexperienced)

- the paper is too long; it contains the matter of at least 3 papers;

- its usefulness is not obvious at first;

- some results which may be of interest (the well-known Clarke-Bumpus plankton
net cannot correctly sample copepods!) are not quickly apparent.

The french language may be a problem -- I was young and most people said if you
submit a doctorate thesis in a french university you should publish in french.
An english translation exists which was published in __Fish. mar. Serv.
Transl. Ser.__, No. 3116 by the Canada Dept.of Environment, Marine Ecology
Laboratory, Dartmouth, Nova Scotia. I never saw it, so I will give my own
translation (I hope the english is correct).

Here is a shorter more informal version which may be easier to understand.

**1. Introduction**

*In 1963-64 I used plankton tows made with a 1-meter net almost every day
in the bay of Villefranche-sur-Mer to record the occurrence of pelagic amphipods.
The small boat of the station zoologique brought back alive soon the contents
of the net bucket to the nearby laboratory. Small salps were numerous at that
time with many of them with parasitic amphipods inside or grasped on their
tunic. The commonest species was Vibilia armata. Its association with salps
is known since the XIXth century.*

*Adult and subadult males of V. armata are readily recognizable by their
2nd pair of antennae which are different from those of the female. Adult males
of this species were scarce in the 1-m net samples, contrary to tows made
with a much larger net I also used, the Isaacs-Kidd midwater trawl.*

*Adult females of V. armata deposit their larvae on the tunic of a salp,
while males do not stay a long time on salps. They devour them as they do
with other gelatinous preys thus behaving like free-living predators.*

*I made the following conjecture: If there are in the ocean almost the same
number of males than females, and the females hold tight to their host when
they sense the approaching net contrary to the males, large nets should (in
average) catch more males.
I compiled my own data and other found in the literature. The ratio of Vibilia
adult males to adult females in the samples was in fact increasing with the
diameter of the net. *

*Then I tried to state it more mathematically. When it detects the net,
an individual flees with a certain angle. If it reaches the edge of the ring
before the net path intersects its track it is caught (Fig. 1). Of course
this presuppose many catches made at several periods of the year, at different
depths and times of the day. This assumes also an ideal net with a 100 percent
filtration coefficient (which is never the case), that the net is towed at
perfectly constant speed, etc. We must of course tolerate some degrees of
approximation.*

**Avoidance of plankton nets**

__PART I : Equations__

[*insert Fig.1 of the paper here*]

Fig. 1 - **A**,**B**: starting and ending position
of a *Vibilia* fleeing with an angle** ƒ**;
**U** and **u**: speeds of the net and the
*Vibilia*; **R**: radius of the net; **x0**: detection distance of the net by the
*Vibilia*.

Let define the avoidance **E** as (1 - **C**),
with **C** being the capture efficiency (= captured /
encountered).
**(1)**

(

The radius **r** inside which a *Vibilia* is always
caught is

**u** sin** ƒ **

**r** = **R** - **x0** ---------------
**(3)**

**U** - **u** cos** ƒ**

All the individuals caught are situated inside a circle of radius
**r**, while the net can potentially capture all the
*Vibilia* inside a circle of radius **R**.

The capture efficiency **C** (= captured / encountered) of the
net for *Vibilia* is thus

**C** = **[R** -**x0
u** sin** ƒ** / (**U** -
**u** cos** ƒ**)]² / **R**²

If we consider the diameter **D** of he net:

**C** = [**D** - 2 **x0
u** sin** ƒ** / (**U** -
**u** cos** ƒ**)]² / **D**²
**(4)**

The avoidance E can be defined as 1 - **C**, thus

**E** = 1 - **{[D** - 2**x0 u** sin** ƒ** / (**U**
- **u** cos** ƒ**)**]**² /
**D**²**}**
**(5)**

If we write **V** = **u** sin** ƒ** / (**U** - **u** cos** ƒ**), the equation
becomes
**(6)**

**E** = 1 - (**D** -
2**x0V**)² /
**D**²
**(7)**

With this equation it is possible to study each parameter.

**2. Influence of the diameter of the net**

To simplify a little, we consider -- for a given species, statistically, on
the long term -- the mean

value of the flee angle** ƒ**, the swimming speed
**u**, and the detection distance **x0** . This may be the case if an important number of
encounters at different times and places are averaged.

The function **E** = f(**D**) in the interval
[0,1] -- not [0, 0.1] as printed due to a typographical error -- is undefined
for **D** = 0, has the value **E** = 1 for
**D** = 2**x0V**, and is
asymptotic at **E** = 0 (Fig. 2).

[*insert Fig.2 here*]

*Fig. 2- E = f (D) for a given speed of
the net U. For example E is plotted here for
U = 100 cm/s, x0 =
50cm, u = 10 cm/s and ƒ = 45°. The dotted rising part has
no physical meaning (D -2x0V, the diameter of the avoidance circle), cannot be
negative.*

**Dmin** = 2**x0V**
is the diameter from which the net may begin to capture adult and subadult
male *Vibilia*, that is, the diameter for which the avoidance may be
greater than zero.

For a relatively low value of avoidance, say 0.1 (i,e only 10 percent of adult
and subadult males may avoid the net) the equation gives a diameter of 19.49
**Dmin** = 148cm. Beyond this point the
avoidance decreases very slowly while the diameter increases toward non-manageable
values.

**4. Speed of the net**

*[insert Fig.3 here]*

*Fig. 3- The avoidance E as a function of the speed of
the net U for 3 values of the diameter D and
as indicated for the other parameters x0, u and ƒ
.*

The function **E** = f (**U**) has a similar
graphic form than for the diameter (Fig. 3), the value **E** = 1
being attained for **U _{min}** =

2**x _{0} u **sin

This equation is shown in graphic form Fig. 3 for three different values of D and with some arbitrary, but likely, values for the other parameters.

**5. Determination of the avoidance E for a net of diameter
D**

In a water mass with a density of **N** individuals per m3 (of
the same species or of similar stage) a net of diameter
**D _{1}** will capture

**N1** / **N2** = **C1N** /
**C2N** = **C1** / **C2**

If we call **d** the ratio **N _{1}** /

(**D1** -2x0**V**)² / (D2 -2x0**V**)²

**d **= -------------- /
---------------
**(8)
D1² /
D2² **

We can solve this equation for 2x**0****V**:

1 - Square root
(**d**)

2x0**V** =
**D**min = **D1D2**
--------------------------
**(9)
D2 - D1**. Square root
(

This value substituted in equation (7) allows to compute **E
**for any net diameter corresponding to the same towing speed, and also
**D**0.1.

**6. Determination of E for a net of a given diameter D**

One cannot use the same reasoning for a net towed at two different speeds
**U**1 and **U**2 giving a capture ratio of **d**. This is
because **E** is not function of **U**, as it is for
**D**, but of **U** - **u** cos* ƒ*. The knowledge of

2x0 u sin

A graphic comparison allows nevertheless to see that, as long as u is less
than 1/5 of **U**1, it is possible to
confound (**U** - **u** cos* ƒ*) and

One can also demonstrate that to obtain at a speed **U1** the same capture efficiency that is obtained at
speed **U2** with a diameter
**D**, at **U1** one should
multiply **D** by

(**U2 **- **u** cos* ƒ*) / (

**6. All the parameters**

Theoretically (but practice is much more difficult) it is possible to
determine all the parameters. We will use two series of samples obtained in an
area with the same density of individuals per m3 (a condition unlikely to
occur) and two nets of different diameters **D1** and **D2**.

a) **D1** and **D2** are towed at speed **U1**. The ratio **d1** in individuals per m3 allows to calculate (equation
9) 2x0a) **D1** and **D2** are
towed at speed **U1**. The ratio
**d1** in individuals per m3 allows to
calculate (equation 9) 2**x0V1**.

b) The same nets, **D1** and **D2**, are towed at another speed **U2**. The ratio **d2** gives 2**x****0V2**.

The ratio **¢** = 2**x****0V1** / 2**x****0V2** is the ratio of the projections of the two towing
speeds (**U2 **- **u** cos* ƒ*) / (

** u**
cos* ƒ* = (

With the value of **u** cos* ƒ* it
is possible to find 2x0

2x0 **u** sin*
ƒ* = 2

With 2x0 **u**
sin* ƒ* and

**7. Some practical notes on the parameters**

**- Diameter of the net **

As Barkley (1964) remarked, **D** should be the diameter of the
effective filtering section. This diameter is equal to the diameter of the
mouth opening multiplied by the square root of the filtering coefficient. This
correction should only be necessary when the filtering coefficient is lower
than about 0.90.

If 2**x****0V** is determined using two nets of different
diameters towed at the same speed, this correction is not needed if their
filtering coefficient are similar. If this is not the case it is necessary to
apply equation (9) using the corrected diameters. If 2x0 **u** sin* ƒ* is computed using the the catches of the same
net towed at two different speeds, the lesser speed should be greater than 60
cm/s because below this speed the filtering coefficient varies too much
(Tranter & Smith, 1968).

**- Towing speed of the net U**

**U** is the speed relative to the water at the position of the
organism. **U** should not be measured at the surface or in
relation to the bottom but recorded at the net mouth.

**- Fleeing speed of the organism u**

As stated by Barkley (1964), "we are not concerned with the details of the
movement of the organism but only with the initial and final conditions".
**u** is the straight line distance (or rather its projection on
the sagital plane) between the starting position and the position when the net
frame reaches the plane of the organism, divided by the time elapsed between
these positions. The sagital plane contains the net axis and the organism
position. If the organism does not swim in straight line or does no remain in
the sagital plane, **u** will be lower than the true fleeing speed
of the organism. Moreover, **u** is a mean resulting speed for all
trajectories encountered.

One can show that beyond a swimming speed **umax** = **DU** / (2x0 **u** sin* ƒ* +

**- Fleeing angle ƒ**

This angle is, on the sagital plane, the angle of the projection of the vector
**u** with the parallel of the net axis drawn at the position of
the organism. It goes from 0° to 180°, in the classical counterclockwise
turning.

For any organism with a given fleeing speed **u **there exists
an angle * ƒ** optimal for which the avoidance

__PART II : Indirect validation - The case of the hyperiid
amphipod Vibilia armata__

**1. Introduction**

*Vibilia armata *is a pelagic species (length about 8 mm) living as
parasitoid of salps (mainly *Salpa fusiformis* and *Thalia democratica*
in the bay of Villefranche, Mediterranan Sea). Adult and subadult males of this
species leave their host and are thus free-living in the plankton, while the
females firmly grasp the tunic of their host, even when slightly disturbed.
So males are able to avoid the plankton net while females have a high probability
to be caught.The size of the individuals of this species make unlikely any escapement
through the meshes.

If we hypothesize that the sex ratio at birth is 50:50 and males and females are (on average) uniformly dispersed on the water column, we have the opportunity to apply the equations seen in Part I: the number of adult males in the sample gives (in equation 1) the number of individuals captured and the number of adult females (by hypothesis equal to the number of males) gives an indication of the number encountered.

Let **S** be the ratio *[number of (adult) males / number of
(adult) females] *found in the samples. If the females, stuck on their host,
cannot avoid the net, the number of females caught (if they are in the sea distributed
like the males) is an estimate of the number of males encountered by the net.
If all the females are attached to salps, **S** *[number of
males captured / encountered] *corresponds to **C** in equation
(1)*.*

However, we do not know with certainty that all females are on salps. What
happens to the ratio **S** if a proportion **k** of
females are free and avoid the net like the males?

During the tow the net encounters **N** males and **N**
females. It captures **CN** males,** CkN** free females,
and (**N** - **kN**) females attached to salps. **S**
in the samples corresponds to:

**
S** = **CN**
/ (**N** - **kN** + **CkN**) = **C**
/ (1 - **k** (1 - **C**))
**(13) **

When **k** = 0 (all females attached to salps), **S**
= **C**. If **k** is different from 0, the experimental
points will not follow a curve according to equation (4) but — if the
model is correct — an equation depending on **k**:

**[ **(**D**
- 2**x0V**)² /** D**²**
]** / **{**1 - **k** (1 - **[ **(**D**
- 2**x0V**)² /** D**² **]**
**}**
**(14) **

A graph of this function of the diameter is shown on Fig.4 for some values
of **k**.

[*insert Fig.4 here*]

It is apparent that, for low values of **k** (which is low by
hypothesis, say between 0 and 0.3) the function has the same shape than for
**k** = 0. It is then possible to verify if the experimental points
lie near the curve. This validation is permitted because the starting position
of the curve does not depend on **k** (because **S**
= 0 when **D** = 2**x0V** for
any **k** as can be seen from equation (14) and Fig. 4).

If k is low or nul then the parameter 2**x0V**
alone (which can be found experimentally: it is the diameter from which the
net begins to capture some males) is sufficient to draw the curve of the values
predicted by the model.

To find the most probable value of** k** I tried with a small
computer program to minimize the sum of squared deviations between the counts
in the samples and the values given by the equation.

**2. Material and methods**

*Vibilia armata *is a rare species in the plankton of Villefranche-sur-Mer.
A rough estimation gives 1 individual per 3,000 m3. The plankton net should
filter a very important amount of water in order to obtain **S **with
a good precision. Moreover a great assortment of samples should be counted to
temper a potential difference between males and females with depth, patchiness,
time of the day, time of the year, physiological state.

It is difficult to find data fulfilling all these requirements. I found in the literature and in my personal records usable data for 5 different net diameters (Table I). All the samples are from the Mediterranean. The greater part comes from long-term series made over more than a year with a frequency of 1 per week, or 1 every two week, or 1 per month depending on the project. For the smaller diameter a series of sample made at the Bouée-Laboratoire kindly supplied by J. Boucher and F. de Bovée. I also used the data (from the Mediterranean) of Stephensen (1918) during the Thor expedition.

[*insert Table I here - english translation should not be necessary*]

The samples of the Station Zoologique come mainly from the Bay of Villefranche and are also made along stations spaced along a transect Villefranche-Calvi (Corsica). The Bouée-Laboratoire (a long inhabited vertical cylinder with a platform and a winch for hauling a plankton net) was moored near the middle of this transect. The Thor sampled all the Mediterranean (Western and Eastern). The total number of samples used for this study amounts to 1259, plus the 311 tows from the Thor.

For the two nets with the largest diameter (Y.200 and IK) which filter a great
amount of water, the sampling respects the requirements: there are enough individuals
and a large range of situations. The data for the three other nets present more
problems. In the numerous samples (1028) of the RG net (RG and H in table I
correspond to custom nets made at the Station zoologique) there is a modest
number of adult *Vibilia*. This may be explained by the vertical distribution
of *V. armata *(which will be discussed later, in relation with the fact
that the sampling was mostly made by day). The two smaller nets, H and WP2,
have caught more individuals. This is due to the fact that this time the sampling
was made mostly during the night and because transects with a fair number
of captured adult *Vibilia* were selected . It is possible to show that
to catch with a 50 cm net the same number of male *Vibilia *than with
the Isaacs-Kidd (IK) 14,000 systematic plankton tows would have been necessary,
owing to the small volume filtered and to avoidance by the males. To select
samples with a large number of *Vibilia* obliges to hypothesize that
male and females were equally present in the sampled area.

The non-circular opening of some of the nets (Y.200 and IK) has been fitted to the nearest circle. The "Petersen's young-fish trawl" (Y.200) of the Thor Expedition has a circular opening, but 2 vertical 1.50 m poles make it rectangular (Schmidt, 1912, p. 8). By taking the mean of the vertical edges and the diagonal of the rectangle they make, it is possible to assimilate the opening to a circle of diameter about 1.80 m. The pentagonal opening of the10-foot Isaacs-Kidd has been considered as a 3.30 m circle.

For the horizontal tows the mean speed of the boat has been estimated from the distance traveled measured with the radar (average of several determinations). For the vertical tows the duration of the tow has been counted from the beginning of hauling up until the closure of the net.

The mean towing speed for each net is of course different due to the diversity
of the samples. It is possible to correct the speed of each net to set each
one to a common speed, say 2 knots, using the equations in pp. 62-63. However,
to apply these equations the knowledge of **u** cos * ƒ
*is necessary, if the fleeing speed of the organism is high in
comparison with the one of the net. This is why I first tried to experimentally
determine the swimming speed of an adult

**3. Estimation of the swimming speed of the male of Vibilia armata**

In a very faint lighting a live adult male of *V. armata *is placed
in a perspex tank 24 x 31 cm filled with 15°C sea water forming a thin layer
(2 cm) at the bottom. The amphipod moves either spontaneously or is teased mechanically
to obtain a fleeing reaction. A high-sensibility video camera allows to record
continuously its trajectory. With a dim lighting the animal swims close to the
bottom without breaking the surface (as was the case with a strong lighting).
The short sequences of rapid swimming were analyzed frame by frame, and as 1
frame is separated from the next by 1/50 s it is simple to deduce the swimming
speed of the *Vibilia*. One sequence is shown Fig. 5.

[*insert Fig. 5 here*]

*Fig. 5 - Example of a video-taped swimming sequence used to determine the
swimming speed of V. armata. Each cross marks the position of the front of the
animal every 1/50 s. The animal is already running when it arrives at position
1. In most of the sequences used, the trajectory of the amphipod is a straight
line, not a circular path as shown on this figure.*

Fifteen fleeing sequence (with a speed greater than usual) from two adult males were thus analyzed. The average of the 15 experiments was 11.7 cm/s (standard error 0.46) This swimming speed corresponds to a mean distance of 14 cm. This is not a speed sustainable only on a short distance. One male has shown a mean speed of 14.4 cm/s on a distance of 33 cm. The highest speed value was 22.2 cm/s during 0.1 s. This method was used with some adult females. Their mean speed was of the same magnitude than with the males.

We have seen p.54 that it is likely that **u** is lower than the
real fleeing speed, but it is difficult to know how much. Because the speed
measured in the laboratory could be less than the speed in the field, we will
keep the value of 11.7 cm/s which gives at least a crude estimate of **u**.

With these values one can see that the correction taking into account **u**
cos * ƒ *is unnecessary if the towing speed is above
60 cm/s. It may nevertheless be applied at all towing speeds, and we will take
arbitrarily 45° for

**4. Vertical distribution and nycthemeral migrations**

II cannot expose here a detailed study of the vertical migrations of *V.
armata*. However, I may resume the main conclusions drawn from a great number
of samples.

The depth distribution of *V. armata* is clearly different between day
and night. During the day the maximum of individuals lies around 600 m and the
amphipod are scarce in the first 200 m. In contrast during the night the majority
of Vibilia are concentrated in the first 100 m, and they become quickly infrequent
until 600m, depth at which they become rare. This situation is found all over
the year. Tows made with the Isaacs-Kidd midwater trawl (which captures as we
will see about the same number of males and females) does not show any substantial
difference in distribution between males and females. So we may conclude to
a true vertical migration and not a differential avoidance which would falsely
make believe to an absence in the upper layers during the day. Moreover when
the number of *Vibilia* per surface unit is integrated over all the water
column the figures are about the same during the day and during the night.

The comparison of the day and night catches (Table I) discloses a remarkable
peculiarity. In the night samples the ratio of males increases, as anticipated,
with the diameter of the net, a near equality of sexes being obtained with the
IK which is no longer avoided by the adult males. There are not enough day data
for the nets H and WP2. However, it may be seen that for the RG and the Y.200,
the proportion of males captured is higher during the day. Digging further in
Table I, one can see that, for the 50 upper meters the sexes are caught in near-equality:
for the RG net 45 females and 39 males (in 43 positive tows / 498); for the
Y.200, 48 females and 49 males (in 19 positives tows / 51). It is clear that
most of the rare females caught by day in the upper layers were free, not clinging
on a salp. More deeper, where individuals of this species are in greater number
during the day, some females must be resting on a salp because their number
compared to the males increases in the samples. In the other hand during the
night most *Vibilia* are caught near the surface and a large proportion
(which can be evaluated by computing k) lays on a salp. These results are in
agreement with Franqueville (1971) who finds *Salpa fusiformis* between
300 and 800 m during the day, and near the surface at night.

To study the variations of **S **with the diameter it is clear
that only the night tows can be used. The few data from the RG net must be left
out.

**5. Results**

*[insert Fig. 6 here]*

*Fig. 6 - Values observed for *V. armata* of the ratio S
in all night tows, for different net diameters. These diameters are corrected
to correspond to a unique towing speed of 2 knots. The small vertical bars are
the confidence intervals at the 5 % level. The curve corresponding to the equation
(14), fitted using the least squares method of p. 69) is plotted on this figure.*

The values of the ratio **S** (for the night tows only) are plotted
Fig. 6 with their confidence intervals at the 5 % level. **S**
represents according to our hypothesis the ratio (males captured / encountered).
The proportion of captured adult males is clearly function of the net diameter.
Furthermore the data points agrees closely with the curve from equation (14).

The method established p. 69 permits to compute the combination of 2**x0V**
and **k** which gives the best fit with the curve (Fig. 6). These
parameters are 2**x0V ** (the value below
which all adult males avoid the net) =** **14 cm and **k**
(proportion of free-swimming females) = 0.30.

With all approximations done (the speed of the Y.200 net is estimated and not
directly measured) the value of k should only be considered as an indication.
For example if **S** is made equal to 0.92 for the IK (*i.e.*
considering all night and day tows which is permitted because the avoidance
is almost nil for this net) then the best fit is obtained with 2**x0V
**=** **12.5 cm and **k** = 0.

With a value of 2**x0V **equal to 14 it
is possible to use the equation (7) to compute the avoidance **E**
for each net at its corresponding towing speed. These numbers are shown on the
last line of Table I. It can be seen that only the IK arrives at a correct
sampling of *V. armata* males, a large proportion of which avoid the
nets of about 50 cm in diameter.

If we take, arbitrarily, a mean value of 45° for * ƒ*,
and for

**6. Discussion**

A. The implied hypotheses

*a) The adult males, at difference with the females, are scared by the approaching
net.* The adult male of *V. armata* is a free-swimming predator.
In the laboratory I have seen it completely devouring a blastozooid of *Thalia
democratica* in a few minutes. Siphonophores and even copepods are also
eaten. During rearing experiments one may observe that adult males leave a salp
when it is slightly disturbed, while on the contrary females tightly hold their
host.

it is thus plausible to think that the adult males freely wander in the environment
while adult and subadult females stay to the salp zooid on which they achieve
their development. This is corroborated by the low value of **k**.

*b) The proportion of the sexes in the ocean is close to 1:1*. This
is almost the case with the Isaacs-Kidd catches. The low proportion of males
caught by the smaller nets is obviously an artifact of the fishing engine.

*c) Each net has encountered the same average number of males and females.
*This is likely considering the high number of samples, made in various
conditions. Swarms composed mostly of females could bias the results but this
is not tenable because in the samples made with the larger nets (Y.200 and IK)
which caught a large number of *Vibilia *no strong overbalance of sexes
is apparent.

B. The results

With the approximate hypotheses made the data points lie close to the theoretical curve. I have used existing data for my demonstration, not a carefully designed experiment where identical nets (excepted for the diameter) are towed at a monitored constant speed.

I think that with the corrections proposed above it is possible to use these heterogeneous data.

The value found for **Dmin**, about 14
cm for a towing speed of 2 knots, leads for **D0.1**
(diameter for which the avoidance is only 10%) a value of 273 cm for this
towing speed. If we use for **u** 17.7 cm/s and for * ƒ*
45° the correction exposed pp. 62-63 shows that to obtain the same avoidance
at 3 knots a net diameter of 177 cm would be enough. The 6-foot IKMT, the aperture
of which may be considered equivalent to a 2 m circle, towed at 3 knots should
be convenient to sample

The reaction distance **x0 **was estimated
to about 80 cm, which corresponds more or less to the attachment point of the
bridles to the towing cable (this does not apply to the IK whose aperture is
more uncluttered). The distance **x0 **is
perhaps less for this net; the asymptotic position of the corresponding point
on Fig. 6 does not indicate if at a lower speed this net would not be more efficient.

Before closing this discussion I would warn to not transpose blindly these
findings to other hyperiid species or to draw carelessly some conclusions about
their biology because the numbers of males and females are unbalanced. In the
case of *Hyperia* (= now *Lestrigonus*) *schizogeneios*,
which has a somewhat comparable biology -- adult males males less attached to
the host than the females -- (Laval, 1972), there are much more males than females
in the samples (Stephensen, 1924). This is probably due to male swarms on the
net path (e.g. 120 males only found in a Thor sample). In some gammarid species,
as observed by Fage, 1928 and Macquart-Moulin, 1968, and postulated by Pirlot,
1939 for numerous species, the adult males leave their host at night and gather
near the surface. In other cases things are more complex [I intended to develop
this point but I never did].

__PART III : Application of the equations on published data__

__Note.__ *After 36 years I find the next section of the paper rather
indigest... This is because I examine several candidate data sets to conclude
that most do not respect the necessary conditions to apply the equation. I
will not continue to translate. It is simpler to summarize the main conclusions.
The most important condition is that each net should be towed through an identical
environment of organisms of the chosen species. If multiple tows are considered
the counts of the organisms can be averaged. Ideally each net should be towed
at constant speed (a requirement very difficult to satisfy (see Aron &
Collard (1969)). In the rare cases where several successive plankton tows
are made (in a tentative to ascertain the "measure") it is surprising
to constate how much they differ from each other. After elimination of samples
with very low catches and tows made at varying speeds (their values of 2 x0V
are not linearly correlated) it is possible to retain the following data sets:*

**- Fleminger & Clutter (1965)**

This experiment was made in somewhat artificial conditions. The plankton was first pumped in situ and placed in a pool and recaptured with 3 nets of diameter 22, 32, and 45 cm. The distance traveled by the nets was short (of the order of 5 meters) and the towing speed (30 cm/s) was low. It is nevertheless possible to arrive at some interesting values of avoidance:

The smaller net (22 cm) is almost completely avoided by the copepod *Acartia
clausi*, while 86 % of them avoid the 32 cm-net and 69 % escape the 45-cm
net, this with a very low towing speed.

The copepod *Corycaeus anglicus* also avoids almost completely the smaller
net, 90 % avoid the middle net and 74 % the largest one.

In another species of copepod, *Paracalanus parvus*, 95 % of the individuals
avoid the 32-cm net, and 80 % the 45-cm net.

In darkness 70 % of mysids avoid the smaller net, 53 % escape the middle one,
and 40 % are retained by the larger net.

**- Winsor & Clarke (1940)**

These data are taken in the ocean with 3 nets attached to a frame tied to the towing cable: two very small nets (12.7 cm) one 1 m above and the other one 1 m below a net of diameter 75 cm. For my computations the two small nets are averaged. Each tow was replicated 10 times. Despite what I said against using successive tows I will try to exploit these data. For the computations the captures of the large net will be scaled to the volume filtered by the small net. Between successive tows the avoidances (which depend on the towing speed) should be linearly correlated. We will only retain the tows significantly correlated (if not, other factors than avoidance must be involved).

*C. finmarchicus
C. finmarchicus Metridia Centropages*

cop. V adult
females *lucens*
*typicus*

**E** small net
50 %
57 %
37 %
21 %

**E** large net
9.7 %
11 %
6.9 %
3.7 %

*Table II - Percentages of avoidance E for the small net
(17.7 cm) and the large net (75 cm) at the theoretical towing speed of 2 knots
as indicated by Winsor & Clarke (1940)*

At the difference of Winsor and Clarke (1940) it can be concluded that the
12.7 cm-net (which will become the well-known Clarke-Bumpus net) at the towing
speed of about 2 knots, is unsuitable for sampling copepods. We may remark that
the adult female of *Calanus finmarchicus* avoids better the net than
the copepodite V, which is another assurance of the consistence of the results.
The relative ordering of the species is also consistent with the results of
Singarajah (1969) who tested the avoidance capabilities of planktonic species
with an aspiring pipet. This author showed that *Calanus helgolandicus*,
a species similar to *C. finmarchicus*, rates far from *Centropages
typicus* for its avoidance capabilities.

**Conditions necessary to apply the equations**

a) Do not use successive (replicated) tows made with the same net or with different nets.

b) Tows made with two different nets attached side by side to the same cable may be adequate provided that

1) the adjusted catches of the small net are not greater than the ones of the larger net.

2) only the tows made at constant speed (or better with a speed
relative to the water) are retained. This may be verified *a posteriori*
by computing the correlation between* *the* *2**x0V***.*
The speed of the net must be known with the best possible precision.

3) the two nets work in similar hydrodynamic conditions. To simplify the computations it is helpful that the two nets have the same filtering coefficient.

The condition that the two nets encounter the same repartiton of the organism
is the most stringent. It may be met in the long term by adding together numerous
tows made in the largest range of depths, time of the day, time of the year.
As it is necessary to choose two nets of very different diameters to increase
the precision, the smaller net inevitably filters a much smaller volume than
the greater one. If at small scale the organism is overdispersed then the
number of individuals captured by the large net may be insufficient for the
small net: the encounters are not statistically distributed with the same
manner on the mouth surface. Cassie (1959) have shown that some not random
variations of the zooplankton abundance may be observed over a distance of
10 cm. The comparison of the two nets can only be made with a large enough
density of organisms relatively to the smallest filtered volume. To increase
the latter it is possible to use two small nets and to use their average for
each tow. An important point should not be neglected: if we consider several
tows made with a couple small-large net, it is not correct to compute the
average of the catches of the small net and the average of those of the large
net and then to calculate the ratio **d** of these numbers. If
we average different tows we may mix tows with a sufficient density of organisms
with tows with too small numbers, which mean nothing and bias the computation.
Moreover there is the risk to mix tows made at constant speed with tows made
at variable speed. This is why I was unable to use the data of Noble (1970,
table 2, p.1036).

**General conclusion**

The model presented here, which was conceived independently of the one of Barkley
(1972), the existence of which has only been brought to my attention when
my own work was finished, may be considered as a generalization of his equation
for all possible values of the angle * ƒ *and
not only for

The principal interest of a model is to suggest new questions.

Some of them only extend the discussion on the parameters.

1) The method exposed above to arrive to an estimate of **E**
by comparison of the captures of two nets requires that **x0**,
**u**, and * ƒ* are the same for
the two nets. But is the reaction distance

2) Is the estimate of the averaged flee angle * ƒ*
different from the optimal angle

Another interest of a mathematical model is the possibility to better assess
the relative importance of the parameters. One of the main points resulting
from the above examination of the data of Winsor & Clarke (1940) is to
bring into focus the role of the towing speed **U**. Until now
this factor has been underestimated by the planktonologists. If for a fast
organism a net of diameter between **Dmin**
and **D0.1** is used, then it is mandatory
to keep the towing speed constant. Otherwise the quantitative results will
be biased, and the bias impossible to evaluate.

Literature cited

[*see the original paper*]

__Note:__ Throughout this text I used the symbol * ƒ*
instead of the symbol for theta only because the latter was not available
in my editing software.

Last revision: 2011-02-21