The use of modified Leslie matrix andgenetic algorithms to model jellyfish

population (Pelagia noctiluca, Scyphozoa)

Matevž Malej1, Alenka Malej2, Alenka Malej31 University of Primorska, Faculty of management2 Actual IT, Koper3 NIB, Marine biology station Piran

The 8th European Conference on Ecological modeling,ECEM ’07, Trieste, Italy, november 2007

MareInItaly.it 2004-08-24 Meduse nel salento Anche quest’anno la proliferazione di meduse nelle acque dei nostri mari sta rendendo difficile la balneazione ai numerosi turisti desiderosi di fare un tuffo in mare per mitigare i fastidiosi effetti della calura estiva. Ultimamente, nella penisola salentina il fenomeno si è dimostrando tanto grave da costringere la maggior parte dei bagnanti a rimanere sulla terraferma e a rinviare una tradizionale gara di nuoto. A spargere il terrore è sempre lei la “Pelagia noctiluca”.

language newspaper for the French - Italian Riviera and the Principality of Monaco

18.06.2007

Pelagia noctiluca on the Côte d'Azur (08.08.2006 Some facts about the jellyfish shoal from Italy still present on the beachesHundreds of Pelagia noctiluca jellyfish that swept in on a current from Italy at the weekend are still present on the Côte d’Azur, with many beaches still flying their orange flags to indicate the presence of the “mauve stinger”. Although their sting is not venomous to humans, they have in some rare cases led to severe allergic reactions so are best avoided by bathers. They generally leave a whip-lash-like scar on the body, with the sting being felt on impact. It later becomes very itchy and blisters. The word Pelagia in Greek means “of the sea” while nocti stands for night and luca means light. The name refers to the organism’s ability to glow in the dark when it’s disturbed. The Pelagia noctiluca is very common in warm or temperate waters, including the Mediterraneum, the Red Sea and Atlantic ocean, and while it lives mainly in open waters it is often carried by currents nearer the coast. Shoals of 45 kilometres in length, involving thousands, are not uncommon.

08.08.2006

Pelagia noctiluca in port of Cap d’Ail

Photograph by Claire Lathbury

Coriere della sera IT SCIENZE Nel palermitano un trentenne ha riportato ustioni su tutto il corpo

Spiagge italiane invase dalle meduse La più pericolosa è la «Pelagia noctiluca», con lunghi tentacoli e un «ombrello» di soli 10 centimentri. In Sicilia ègià allarme MILANO - Per i popoli orientali sono una vera prelibatezza. Essiccate, tagliate a fettine e fatte marinare nella salsa di soia, le meduse sono un contorno perfetto. Lo sanno bene le autorità della Florida che, nel tentativo di costruirci un business, giàda qualche tempo hanno stanziato 80mila dollari per insegnare a 80 pescatori come raccoglierle, essiccarle e conservarle. Tutto cominciò con la solita emergenza estiva, quandil Golfo fu invaso da queste strane «astronavi marine», come le chiamano gli yankee, che mettono in pericolo la tranquillità dei bagnanti.

Sunday, Aug. 13, 2006

The Sea Stings Back By LISA ABEND and GEOFF PINGREE

Stifling heat, sunburn — to the peculiar pleasures of Spain's beaches in August, add the sting of the jellyfish. In the last [an error occurred while processing this directive]couple of weeks, fleets of bloblike Pelagia noctiluca have reached beaches from Barcelona to Málaga. In Catalonia alone, the Red Cross has treated 14,044 bathers for the painful stings. Local governments in Benidorm and elsewhere have posted signs in three languages warning of the dangers. The Interior Ministry has publicized advice for those who are stung — wash the affected area with salt water, don't rub it, seek assistance. (In rare cases, an allergic reaction to the sting can prove deadly.) The Environment Ministry is sending out boats armed with large nets to snare the jellyfish before they reach shallow water. And Spain is not the only country on the defensive: red warning flags have been hoisted on beaches in France, Sicily and along the Italian Riviera. Josep-María Gili, a marine biologist at the Institute of Ocean Science in Barcelona, attributes the surge in the jellyfish population to overfishing of its predators. But he has another explanation for why so many jellyfish are reaching the beach: global warming. "Less rain and higher temperatures have made the coastal waters as salty and warm as those in the middle of the ocean," says Gili. "The invasion of the jellyfish is a message telling us that we have to take better care of the sea."

P u b lis h e d o n T u e s d a y , A u g u s t 8 , 2 0 0 6 b y th e In d e p e n d e n t / U K

J e lly f is h P la g u e B la m e d o n C lim a te C h a n g e b y S te p h e n C a s t le

A plague of jellyfish along Europe's beaches has become the latest environmental hazard to be blamed on global warming.

Holidaymakers heading for Mediterranean beaches are being warned to prepare for an unprecedented invasion of the invertebrates whose sting can, in extreme cases, cause heart failure.

MEDITERRANEAN VISITOR: A jellyfish floated offMallorca, Spain, thissummer. An influx of jellyfish is related to global warming,ecologists say. DANI CARDONA/REUTERS

Western Mediterraneanover last two centuries massive outbreaks of Pelagiaon average at 12 years intervals (Goy et al. 1989)

Adriatic Sea – less predictablelast hundred years three periods of high Pelagia abundances1910-1914, 1976-1986, from 2004 on (Babič 1914, UNEP 1991)

Outbreaks accompanied by expansionoutside usual distribution range -advection into the northern Adriatic Sea

Pelagia research - three directions:• population dynamics in the NA• genetic analysis – Mediterranean Sea• dispersion pattern – model predictions

WM

CMEM

AD

PIRANNA

Model development

• 1. model: Malej A. & Malej M.: Population dynamics of the jellyfish Pelagianoctiluca (Forsskål, 1775), Marine eutrophication and population dynamics / 25th European Marine Biology Symposium 10th-15th September 1990, Lido degliEstensi, Ferrara, Italy. - Fredensborg : Olsen and Olsen, 1992. - (Internationalsymposisum series). - ISBN 87-85215-19-8. - (1992), str. 215-219.1 Leslie matrix, bio – parameters (growth, reproduction)

• 2. model: Malej A. & Malej A.: Invasion of the Jellyfish Pelagia noctiluca in the Northern Adriatic: a non-success story, Aquatic invasions in the Black, Caspian, and Mediterranean Seas / Edited by Henri Dumont, Tamara A. Shiganova and Ulrich Niermann. - Dordrecht ; Boston ; London : Kluwer AcademicPublishers, 2004. - (NATO Science Series ; vol. 35). - ISBN 140-2018-69-X. -str. 273-285. 3 seasonal Leslie matrices, bio – parameters, simulation method

• 3. model: Malej M. & Malej A. & Malej A.: The use of modified Leslie matrixand genetic algorithms to model jellifish population, ECEM ’074 seasoned Leslie matrices, bio – parameters opimised with geneticalgorhitms to fit field data; simulation method and stability analysis; integrates three fields of sience (math, biology, optimization)

Matrix model:stage-clasified life cycle of Pelagia noctiluca

ephyra immature medusa mature medusa

maturebell diam. 6.0 – 8.5 cm

maturebell diam. > 8.5 cm

conditionally maturebell diam. 3.5 – 6.0 cm

Death( )D1

Shrinkage( )S21

Growth ( )G12

Death( )D2

Shrinkage( )S32

Growth ( )G23

Death( )D3

Shrinkage( )S43

Growth ( )G34Reproduction( )R3

Death( )D4

Shrinkage( )S54

Growth ( )G45Reproduction( )R4

Death( )D5

Reproduction( )R5

ephyrae<1.0 cm

n1

Loop( )L1

immature1-3.5 cm

n2

Loop( )L2

Cond.mature

3.5-6.0 cmn3

Loop( )L3

mature6.0-8.5 cm

n4

Loop( )L4

mature>8.5 cm

n5

Loop( )L5

Model)1( +xn = ( )M x )(xn

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

+++++

)()()()()(

00000

0000

00

)1()1()1()1()1(

5

4

3

2

1

545

54434

43323

32212

543

5

4

3

2

1

xnxnxnxnxn

LGSLG

SLGSLG

RRR

xnxnxnxnxn

1 121D G= −

2 23 21 ( )D G L= − +)(1 1,1, −+ ++−= iiiiii SLGD 3, 4i =

)(1 5455 SLD −−=

)1( +xn population structure for month 1x +)(xn population structure for month x

( )M x matrix for month x

Genetic algorithm

Genetic algorithmSimulation of

populationdistribution

Simulated distribution

Evaluation function (Euclidian distance between measured distribution andsimulation result), selection (better solutions selected with greater probability),crossover, mutation (default)

M1 M2 M2 M3 M3 M3 M2 M4 M4 M3 M1M1

Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr

M1=[0 0 0.25 0.6 0.7 0.65 0.2 0.10 0 0 0 0.50 0.26 0.10 0 0 0 0.30 0.5 0.10 0 0 0 0.15 0.11];

M3=[0 0 0.3 0.5 0.7 0.65 0.2 0.10 0 0 0 0.8 0.35 0.10 0 0 0 0.30 0.5 0.10 0 0 0 0.15 0.10];

M2=[0 0 0.6 0.9 0.95 0.65 0.20 0.10 0 0 0 0.8 0.35 0.10 0 0 0 0.3 0.5 0.10 0 0 0 0.15 0.10];

M4=[0 0 0.10 0.1 0.10 0.35 0.1 0.10 0 0 0 0.5 0.25 0.10 0 0 0 0.3 0.5 0.10 0 0 0 0.15 0.6 ];

The dominant eigenvalues:λ1 = 0.88 λ2 = 1.09 λ3 = 0.98 λ4 = 0.74

Simulation of Pelagia abundance (N) over 100 months

months

N

Result of simulation analysis

• disappearance of Pelagia population from the NA in 8 yearsconforms to field data

• seasonal variations similar to field data, as indicated byeigenvalues:

λ1 = 0.88λ2 = 1.09λ3 = 0.98λ4 = 0.74

Sensitivity Analysis∆N/∆aij , ∆aij = 1%

parameter value sens. indicator parameter value . indicatorM1R3 0,25 1,2 M3R3 0,30 2,4M1R4 0,60 2,8 M3R4 0,50 2,5M1R5 0,70 0,7 M3R5 0,70 0,8M1G12 0,65 5,1 M3G12 0,65 5,3M1G23 0,50 4,4 M3G23 0,80 7,4M1G34 0,30 3,1 M3G34 0,30 4,7M1L2 0,20 1,5 M3L2 0,20 1,8M1L3 0,27 2,2 M3L3 0,35 4,7M1L4 0,50 5,0 M3L4 0,50 5,1M1L5 0,11 0,1 M3L5 0,10 0,1M1S54 0,10 0,2 M3S54 0,10 0,3M2R3 0,60 3,0 M4R3 0,10 0,4M2R4 0,90 2,9 M4R4 0,10 0,3M2R5 0,95 0,5 M4R5 0,10 0,1M2G12 0,65 4,7 M4G12 0,35 1,9M2G23 0,80 5,5 M4G23 0,50 3,4M2G34 0,30 3,0 M4G34 0,30 2,8M2L2 0,20 1,2 M4L2 0,10 0,6M2L3 0,35 3,1 M4L3 0,25 2,0M2L4 0,50 3,3 M4L4 0,50 3,7M2L5 0,10 0,1 M4L5 0,60 0,5M2S54 0,10 0,1 M4S54 0,10 0,2

Sensitivity AnalysisHow population growth rate over the entire cycle responds to

changes in vital rates at each phase within the cycle

Caswell and Trevisan, 1994:

Ah … periodic projection matrix from month hBh … projection matrix from month h to h + 1Ah = Bh-1 * Bh-2 * … * B1 * B12 … * Bh = Dh * Bh

Sensivity matrix of λ with respect to Ah :

Sensivity matrix of λ with respect to Bh :

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

= hijaλ

hAS

( ) hh AThB SDS ⋅=⎟

⎟⎠

⎞⎜⎜⎝

⎛

∂∂

= hijbλ

SB7 =0.0939 0.0591 0.0840 0.0458 0.00840.1404 0.0884 0.1257 0.0685 0.01260.1376 0.0866 0.1232 0.0671 0.01230.1513 0.0952 0.1354 0.0738 0.01360.0922 0.0580 0.0826 0.0450 0.0083

SB8 =0.1035 0.0841 0.0844 0.0509 0.00800.1533 0.1246 0.1250 0.0753 0.01190.1479 0.1202 0.1206 0.0727 0.01140.1596 0.1298 0.1302 0.0784 0.01240.0946 0.0769 0.0772 0.0465 0.0073

SB9 =0.0541 0.0889 0.0979 0.0495 0.00810.0867 0.1424 0.1568 0.0793 0.01300.0827 0.1357 0.1495 0.0756 0.01240.0890 0.1461 0.1610 0.0814 0.01330.0473 0.0777 0.0856 0.0433 0.0071

SB10 =0.0512 0.0537 0.0944 0.0470 0.00700.0920 0.0965 0.1697 0.0845 0.01270.0967 0.1015 0.1785 0.0889 0.01330.1097 0.1151 0.2024 0.1008 0.01510.0548 0.0575 0.1010 0.0503 0.0075

SB11 =0.0422 0.0397 0.0600 0.0391 0.00580.0873 0.0822 0.1241 0.0808 0.01190.1062 0.1000 0.1510 0.0983 0.01450.1203 0.1133 0.1710 0.1113 0.01640.0861 0.0811 0.1225 0.0797 0.0118

SB12 =0.1218 0.0655 0.0901 0.0606 0.01020.2190 0.1177 0.1620 0.1089 0.01840.2733 0.1469 0.2022 0.1359 0.02290.3012 0.1619 0.2228 0.1498 0.02530.1892 0.1017 0.1399 0.0941 0.0159

SB1 =0.0225 0.0822 0.0868 0.0824 0.02150.0454 0.1662 0.1755 0.1667 0.04350.0481 0.1760 0.1858 0.1765 0.04600.0562 0.2055 0.2170 0.2061 0.05380.0279 0.1020 0.1077 0.1023 0.0267

SB2 =0.0214 0.0280 0.0804 0.0785 0.02860.0291 0.0381 0.1094 0.1068 0.03890.0405 0.0531 0.1525 0.1489 0.05420.0565 0.0741 0.2129 0.2078 0.07560.0283 0.0371 0.1067 0.1041 0.0379

SB3 =0.1031 0.0340 0.0722 0.0819 0.01810.1283 0.0423 0.0898 0.1019 0.02250.1756 0.0580 0.1230 0.1395 0.03080.2431 0.0803 0.1703 0.1932 0.04270.1314 0.0434 0.0920 0.1044 0.0231

SB4 =0.1029 0.1044 0.0571 0.0830 0.01830.1229 0.1247 0.0683 0.0992 0.02190.1495 0.1517 0.0830 0.1206 0.02660.1944 0.1973 0.1080 0.1568 0.03460.0962 0.0976 0.0534 0.0776 0.0171

SB5 =0.0670 0.0814 0.0659 0.0526 0.01250.1184 0.1438 0.1164 0.0931 0.02220.1364 0.1658 0.1342 0.1073 0.02560.1568 0.1906 0.1542 0.1233 0.02940.0875 0.1063 0.0860 0.0688 0.0164

SB6 =0.0580 0.0677 0.0647 0.0483 0.00940.0874 0.1021 0.0976 0.0728 0.01420.1042 0.1217 0.1162 0.0868 0.01690.1193 0.1394 0.1331 0.0994 0.01940.0722 0.0843 0.0805 0.0601 0.0117

In summary

Modified Leslie matrix (Lefkovitch) model was succesfully appliedto describe long-term population data of the jellyfish Pelagianoctiluca in the NA.

We used Pelagia field data from 1984 – 1986 to set boudaries ofparameters for matrices which were further optimized using geneticalgorithm. This procedure also grouped matrices in fouraproximatley seasonal clusters.

Applying this model to Pelagia abundance indicated decline anddissaperance of population in about 8 years; the pattern thathappened in nature.

In future this model will be used to assess population changes in relation to changes of evironmental parameters (temprature, foodavailability, …).

1870

Thank you foryour attention

Acknowledgements

Reseasrch was funded by EU FP6 project SESAME and Slovenian ARRS.