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					       Possible sources of Allee effect in ZM

       Details of the spawning process
       The spawning process of Zebra mussels has been described in a number of review papers,
e.g. (Mackie and Schloesser, 1996), (Nicholis, 1996), (Ram et al. 1996). Some details can be found
elsewhere. For our purposes the most important points of ZM spawning process are the following.
            Male and female adults release gametes into water. Note that this type of spawning
               is typical of marine invertebrates, but not of freshwater ones. The sperm cells find
               and fertilize eggs. Eggs can survive autonomously 2.54.75 hours, sperm cells up to
               22 hours (range is 222 hrs) (Sprung, 1987). It is not complete understood, what
               serves as a cue for the spawning event. Most probably it is a number of factors, such
               as food amount and water temperature. But synchronicity of spawning is most
               probably achieved by a chemical cue. It has been reported that serotonin in
               concentration of 10-4 and above may trigger spawning if other conditions are
               satisfied (Ram et al., 1996).
            Sperm cells move with the speed 40200 m/sec (curvilinear velocity). It has been
               reported that sperm cell are chemotactically targeted towards the eggs (Miller et al.,
               1994). This may result in successful fertilization even by strongly diluted sperm.
               Examples of very high fertilization success for a certain species of scallop with a
               very diluted sperm for the sperm concentration of 10–2 l-1, that is 10 sperm cells in
               a cubic centimeter were reported in (Styan and Butler, 2000). For a different scallop
               species there was practically no fertilization for the same sperm concentration. The
               difference may be accounted for by chemotaxis, and it can be concluded that it
               increases the probability of fertilization up to 1001000 times, especially at small
               sperm concentrations. This is in a good agreement with the following estimate: the
               diffusion coefficients of organic chemicals in water are of the order
                D ~ 105 1010 cm2/s. If we take the largest value (it is reasonable to assume that
               attractant should be a well-diffusive chemical), then during one hour or t~4000s the
               attractant may diffuse at the distance of the order Dt ~2mm. The diameter of the
               egg cell is about 0.1mm, therefore attractant creates around the egg a domain about
               1020 times greater diameter than the egg from which the fertilization can be done
               with a very high probability. This is equivalent to 100400 times increase of the egg
               cross-section, which may explain the order of the probability increase.
            One female can release 5662411670 (SE) eggs per induction (4 x per season)
               (Stoeckmann 2003) Other estimates give up to 1.5 million eggs/female/year, see
               review (Nicholis, 1996). One male can release 151008±97820 (SE) sperm per
               induction (4 x per season) (Stoeckmann 2003)
            After fertilizing eggs turn into planktonic larvae (veligers). They live about 533
               days, then attempt to settle (Sprung 1993). It appears that the settling stage has the
               highest mortality rate. European Lakes: survival rate of veligers in plankton is about
               80%, during settling about 1% (Mackie and Schloesser 1996). North America:
               9099% (up to 100%) mortality during settlement (Mackie and Schloesser 1996). In
               review (Nicholis, 1996) it is noted that different estimates of mortality during
               settlement vary between 20% and 100%.
              Very dense populations e.g. settlement of veligers in one area of lake Erie was
               50000 per m2 resulting in an adult density of approximately 15000 per m2 (Farleigh
               et al 1993). There is intense competition for space and probably food during the
               settlement process. Even bigger densities are mentioned in review (Ram et al., 1996)
                800000 per m2 (in North America) and 1700000 per m2 (in Europe).
              After settlement survival rate is rather high, and can be neglected for a while. Total
               lifespan is 1.52 years for North American ZM (Mackie and Schloesser, 1996)



       Possible ''weak points'' of the spawning process.
         From the described picture of spawning process it is possible to pick up several ''weak
points'' which in principle could be a reason for the Allee effect. We shall try to estimate the
minimum distance l which can separate a male and a female zebra mussels.

       Synchronicity of spawning
        Since gametes can survive only during several hours, to ensure high probability of
successful fertilization, the spawning events must trigger each other. It is reasonable to assume that
during spawning a certain amount of serotonin is released that can trigger the subsequent spawning.
The formula weight of serotonin (C12H14N2O2) is 218, therefore concentration of 1M corresponds
to 218g per litre of solution. The mean maximum dry weight for a zebra mussel is about
20100mg. So if we assume that the amount of released serotonin is about 1/4 of the mussel
weight, this gives at most 25mg=0.025g. For the limiting concentration of 10–4M which still has
biological effect this gives the volume 0.025/(21810–4)~1 litre. Therefore the subsequent spawning
is possible if both mussels are contained in the same litre of water. If we assume most favorable
case, that this litre is about 1cm high (order of the size of the mussel), then it covers the area about
0.1m2, which gives a minimum limiting density of synchronous spawning for ZM as about 10 per
m2 and the distance to nearest neighbor about 18cm. If we assume the shape of the volume as a half
of the sphere, this gives the distance to nearest neighbor about 8cm.

       Fertilization and settlement
         Since we consider very rare population of ZM which produces very little settling veligers,
we can neglect competition and related mortality during settlement. This gives very high survival
rate (up to 80%). For the subsequent analysis we take it a moderate value about 10% survival.
         The next question is how big is larvae spread. If the larvae has an ''attracting mechanism''
allowing them to settle near to each other, then about 10 survived larvae can settle a new colony
which will successfully spawn next year (attraction scenario). In this case about 100 eggs is enough
to settle.
         If there is no larvae attraction, then much larger numbers of settlers are necessary (random
spread scenario). Let us estimate this number.
         Spreading and settling. After fertilization ZM stays for 5–33 days at larvae stage, flowing
as a zooplankton. During this stage larvae can spread. The main mechanism in lakes must be
turbulent diffusion due to thermal convection and wind waves. The characteristic scale for these
processes is about 100102 meters. The diffusion coefficient for these processes depends on the
scale, and, according to the plot in (Okubo, 1980, p. 14) is of the order 101103cm2/sec. Taking for
characteristic time t~106sec (close to 12 days) and D~102cm2/sec, we obtain characteristic spread
distance of L ~ Dt ~ 10 4 cm=100m. Note that we do not take into account directed transport of the
whole larvae swarm, because the position of its centre is not important, important is the diameter of
the swarm at the moment of settlement.
        The settlement usually takes place at a moderate depth not too far from the shore. Let us
take the width of the part of the bottom suitable for settling as w~10m, then larvae from the swarm
can start settling at the area close to 2Lw~200m2. If any part of the bottom is suitable for settling,
then the larvae can settle at the area of ~3104m2. Let us take for the settling area 104m2, according
to the previous section, successful spawning requires density at least 10 per m2. Let us split the
settling area into ''settling sites'' with the critical area of 0.1m2, their number is N~104/0.1=105, and
consider the following problem: n larvae randomly and independently settle at N>>n sites. How big
n must be to ensure that at least at one site there are two settlers or more with the probability
PS99%?
        Let us estimate the probability P1 that at each site there is maximum one settler, then PS=1–
P1. n larvae can be distributed without repetitions over N sites N!/(N–n)! ways, total number of
distributions with repetitions is Nn, so
                   N!
         P
          1
              N  n ! N n
Let us assume that N>>1, N–n>>1, and use the Stirling formula N! 2N N N e  N , then
                                                         1
                                                 N n
                 N       N N n             N        2                      1        n  
       P                           e n              e  n  exp   N  n   ln 1 
                                                                                              n .
               N  n N  n   N n
                                            N n                              2  N n 
        1
                                                                                                
Since N–n>>n, we can use approximation ln(1+x)x–x2/2, that is
                         1  n          1 n                   nn  1
                                                   2
                                                                              n2
       ln P1   N  n                         n                  
                        2  N  n 2  N  n  
                                                               2N  n  4 N  n 2
Neglecting terms containing N-2, and assuming n large enough such that n(n–1)n2, we obtain
                  n2
       ln P1        ln 1  PS  ,
                 2N
and hence the minimum suitable n is
                          1
       nmin  2 N ln            .
                       1  PS
If we take PS=1–e–50.993, N=105, then
         nmin=103,
therefore about 1000 surviving larvae can start a new colony with the probability >99%. If we take
into account the fact that the larvae prefer solid surfaces for settlement, the limiting n can be even
less, because of the smaller number of available sites.
         Taking survival rate at 20% (too small according to the literature), we need successful
fertilization of about 5000 eggs, or only about 10% of the mean egg release of a single female, and
this can be done by about 3% of sperm cells released by a single male.
       Fertilization kinetics model and estimates
        Models of fertilization kinetics has been considered in a number of papers. The basic model
has been proposed in (Vogel et al. 1982), in (Styan, 1998) an attempt has been made to add
accounting for polyspermy, which decreases the fertilization efficiency. From our point of view,
some assumptions made in (Styan, 1998) may be not very accurate, and we wanted to develop a
different model of this process.
        There are two types of statistics involved: 1) how many encounters of spermatozoa and eggs
has happened and 2) how many spermatozoa are attached to a certain egg. In the cited papers it has
been assumed that both kinds of events can be approximately described by a Poisson distribution.
While for the first kind of events this assumption seems quite reasonable, for the second one it
needs justification. Therefore the questions of how many eggs have been hit and how many of
those have been hit only once may prove more complicated. And the deviations from Poisson
distribution may be most important in cases which are typical of the Allee effect: very rare sperm
or very rare eggs.
        The most detailed description could be given by a stochastic model, but in most realistic
cases it proves too complicated. So we used a chemical kinetics-type model for averages.

       Kinetics of averages
        The conceptual model used for fertilization is the following. There are sperm cells S and
unfertilized eggs E. If they meet, the sperm sticks to the egg and with some probability fertilizes it.
If the sperm has penetrated the egg surface, fertilization begins. After some time a blocking
mechanism switches on that prevents secondary penetrations, and the egg remains fertilized. But
for some time the cell remains vulnerable to a secondary penetration, and in such case a
polyspermy may take place, when the egg eventually dies. Like in (Vogel et al. 1982) we assume
that unfertilized eggs can survive only during some time T, and we neglect differences in survival
times. This gives the following system of ''reactions'', where EV, EF, and ED denote ''vulnerable''
(capable of being damaged by a second fertilization), fertilized, and non-surviving eggs:
                 
         S  E k1 E ,
                 
         S  E k2 EV ,
                
        S  EV k1 EV ,
                
        S  EV k2 E D ,
            
        EV k3 E F ,
                 k1 
                    
        S  E F k2  E F ,
                     k1 
                        
        S  ED k2  ED .
Adding initial conditions S(0)=S0, E(0)=E0, EV(0)=ED(0)=0 and the corresponding system of
differential equations
              k1  k 2 S E  EV  ED   k1  k 2 SE0 ,
        dS
         dt
        dE
              k 2 SE ,
         dt
        dEV
               k 2 SE  k3 EV  k 2 SEV ,
          dt
       dEF
             k 3 EV ,
        dt
       dED
             k 2 SEV .
        dt
The main characteristic of the fertilization is fertilization efficiency
           E T 
       F         .
              E0
      Let us nondimensionalize model:
       S  S 0 s, E  E0 x, EV  E0 y, EF  E0 z ,
          k1  k 2 E0 ,                   a  k2 S0 ,
       ds
            s,
       dt
       dx
             asx,
        dt
       dy
             asx  k3 y  asy,
        dt
       dz
             k3 y,
        dt
       s0   x0   1, y 0   z 0   0.
This immediately gives s=e–t. The second equation can be written as
       dx a                        as  1 
             x, x  exp           .     
       ds                                  
       Similarly

       y s  
                  a
                  k3
                                        a      
                        s k3 /   s exp s  1,
                                                  
                                                

        y t  
                     a
                     k3
                                            a
                                             
                                                         
                                                          
                          e  k3t  e t exp e t  1 ,
                                                                               
                                                                                       ,

        z t   k3  y  d 
                         t       k3ae  a /  1 k3 /  1
                                    k3 
                                                              as 
                                              et s  1 exp  ds,
                                                                                  
                     0
                                                              
          z T .
        Therefore fertilization kinetics depends on three important parameter combinations
                                     
                                                     ,   k1  k 2 TE0 ,
            k         k3                    k2 S0
        p 3                 ,  
              k1  k 2 E0          k1  k 2 E0

          , p,   p 
                                     1        s   p 1
                                                               
                                                    1   s 1
                                                      e          ds 
                                     e   
                                                1 p

         p 
               1 e        1  z    p 1
                                                1    e      z
                                                                    dz
               0                    1 p
       Simplifications. To obtain more tractable expressions let us assume that  is large, so e–0,
and consider two limiting case, big k3, p>>1, and small k3, p0.
       a) practically no polyspermy, p>>1, then (1–z)p–1 is negligible compared to 1, so
                          1                                          k2 S0 
                     e z dz  1  e   1  exp     k  k E  ,               (**)
                          0
                                                                  1      2 0 

which gives expression from (Vogel et al. 1982), where polyspermy has not been taken into
account;
       b) polyspermy is very essential, p0,

        p
                                               
          1  z p 1  1  p 1  1  z1 p  p z ,
              1 p                1  p 1  z 1 p       1  z 1 p
         p
               1

               0
                         z
                    1  z 1 p
                                 e z dz   
                                               1 d

                                               0 dz
                                                                   
                                                        1  z  ze z dz 
                                                                 p




           1  z  ze z    1  z 
                                                   d
                                                               
                              1     1
                                                      ze z dz 
                      p                        p
                              0    0               dz
        
              1

              0
                 d
                  
                 dz
                          
                      ze z dz  e 
that is the expression used in (Styan, 1998).
         c) If p=2 and  big
                                           1  e   e 
           ,2,    2  ze z dz  2
                              1
                                                            .
                             0                     
Similarly, for all integer p the integral can be evaluated, though the number of terms is p.
         d) If both  and  are big (long interval and big amount of sperm), then only z<~–1<<1
contribute essentially in the integral,
                                            1  e   e  p
           , p,   p  ze dz  p
                                1
                                  z                             k
                                                                3 .
                                0                              k2 S0


       Order of magnitude estimates
        Available data do not allow for very accurate estimates. However, some order of magnitude
estimates for fertilization success can be done.
        The importance of polyspermy. The only available are the data on sea urchin (Styan and
Butler, 2000), and it follows that in their case it became essential when the ratio of sperm and eggs
concentrations was S0/E0~103. In our case this ratio can appear only when a few eggs enter the
cloud of sperm. Occurrence of such situations is a matter of pure chance, and we shall not consider
them. For smaller concentration ratios the polyspermy may be essential only if the stage of
vulnerability for zebra mussels is much longer than that for sea urchin, which seems unlikely.
Therefore we shall assume that the polyspermy is inessential for our estimates, and so the relation
(**) (Vogel et al, 1982) can be applied.
        Fertilizing efficiency estimates. In the absence of polyspermy and assuming that the most
important parameters are
                           ,   k1  k 2 TE 0
                 k2 S0
        
             k1  k 2 E0
(see Vogel et al, 1982). For sea urchin the estimates in the cited paper were k1+k2410–4mm3/sec
and the ratio k2/(k1+k2)0.01. The sum of k can also be approximately estimated as the product of
the egg cross-section =r2 (r is the egg radius) times the sperm velocity v. If we take into account
only the “physical” egg radius r510–2mm, and v0.1mm/sec, then k1+k2810–4mm3/sec, which is
close to the estimate for the sea urchin. If we take into account the chemotactic shell around the
egg, and take its radius 10 times larger than the cell one, we obtain k1+k20.08mm3/sec. The actual
value for zebra mussel can be obtained only from experiments.
         For the duration of the fertilization process we can take the eggs lifetime T~104sec, because
for ZM sperm lives longer than eggs (in contrast with sea urchin, where sperm became practically
inactive after 1 hour). The values of S0 and E0 are very hard to estimate, because we need their
values in the intersection of the eggs and sperm clouds produced by spawning mussels.
Remembering the estimates of the serotonin activity, we can assume that the maximum volume for
both clouds can be about 1l. If we assume that eggs and sperm are uniformly mixed within their
clouds, and the mean cloud intersection is about 50%, then we can assume a mean fertilization
situation in a rare population as 50% of eggs (25000) and 50% of sperm (75000) diluted in 500ml
of water. This gives S00.15mm–3, E00.05mm–3, then for k1+k2810–4mm3/sec we obtain 0.5,
and for k1+k20.08mm3/sec 50, in both cases 0.03, which gives the fertilization success
1  e     0.03 , that is only about 750 fertilized eggs. Small  value may still diminish this
value. This number may be too small for dense enough settlement of veligers.
         We can conclude that under the assumptions made, the successful spawning of one male
and one female zebra mussels require them being close enough to mix up sperm and eggs in a small
volume, less than 500ml, and the distance between them probably has to be only a few centimeters.
         However, the fertilization constants k1 and k2 may be different, which may slightly loosen
the constraints on the mussels proximity. The estimate also may be loosened by a) taking into
account that there are several spawnings a year; b) nonrandom character of the settlement if the
veligers are attracted by certain areas of bottom. The opposite effect may be caused by accounting
for mussels death because of predation or parasitism.


       Conclusions
       Consideration of the details of zebra mussel spawning process allows us to suggest that it
may be successful only if they live in dense enough colonies. The critical distance between male
and female mussel is about 10 cm or less.


       References

        Fraleigh, P. C., Klerks, P. L., Gubanich, G., Matisoff, G. and Stevenson, R. C. 1993
Abundance and settling of zebra mussel (Dreissena polymorpha) veligers in western and central
lake Erie. In T.F. Nalepa and D. W. Schloesser (eds.) Zebra mussels: Biology, impacts, and control,
pp.129-141. Lewis Press. Boca Raton, Florida.
        Haag, W.R. and Garton, D.W. 1992. Synchronous Spawning in a Recently Established
Population of the Zebra Mussel, Dreissena-Polymorpha, in Western Lake Erie, USA.
Hydrobiologia 234: 103-110.
        Mackie, G.L. and Schloesser, D.W. 1996. Comparative Biology of Zebra Mussels in Europe
and North America: an Overview. American Zoologist 36: 244-258.
        Miller R.L., Mojares J.J., and Ram J.L. 1994. Species-specific sperm attraction in the zebra
mussel, Dreissena polymorpha, and the quagga mussel, Dreissena bugensis. Canadian J. Zoology,
72, 1764 1770.
        Sprung, M. 1993. The other life: an account of the present knowledge of the larval phase of
Dreisena polymorpha. In T.F. Nalepa and D. W. Schloesser (eds.) Zebra mussels: Biology, impacts,
and control, pp.39-53. Lewis Press. Boca Raton, Florida.
        Stoeckmann, A. 2003. Physiological Energetics of Lake Erie Dreissenid Mussels: a Basis
for the Displacement of Dreissena Polymorpha by Dreissena Bugensis. Canadian Journal of
Fisheries and Aquatic Sciences 60: 126-134.
        Nicholis S.J. 1996. Variations in the reproductive cycle of Dreissena Polymorpha in Europe,
Russia, and North America. Amer. Zool. 36: 311-325.
        Ram J.L., Fong P.P., Garton D.W. 1996. Physiological aspects of Zebra Mussel
reproduction: maturation, spawning and fertilization. Amer. Zool. 36: 326-338.
        Styan C.A., 1998. Polyspermy, egg size, and the fertilization kinetics of free-spawning
marine invertebrates. Amer. Nat. 152, 290–297.
        Styan C.A., Butler A.J. 2000. Fitting fertilization kinetics models for free-spawning marine
invertebrates. Marine biology 137, 943–951.
        Vogel H., Czihak G., Chang P., Wolf W, 1982. Fertilization kinetics of sea urchine eggs.
Math. Biosci. 58, 189–216.

				
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