Patchy agglomeration as a transition fro

The dynamics of the plankton population is shown to depend on the fraction of the phytoplankton population that aggregates to form colonies and on the number of the latter.. In fact, if

Patchy agglomeration as a transition from monospecies to recurrent

Joydev Chattopadhyaya, Samrat Chatterjeeb, Ezio Venturinob,

a

Agricultural and Ecological Research Unit, Indian Statistical Institute, 203 B.T Road, Kolkata 700108, India

b Dipartimento di Matematica, Universita’ di Torino, via Carlo Alberto 10, 10123 Torino, Italy

a r t i c l e i n f o

Article history:

Received 4 October 2007

Received in revised form

5 March 2008

Accepted 10 March 2008

Available online 14 March 2008

Keywords:

Phytoplankton–zooplankton

Toxic chemicals

Patch

Recurrent bloom

Red tides

Hopf-bifurcation

Coexistence

a b s t r a c t

We propose a model for explaining both red tides and recurring phytoplankton blooms Three assumptions are made, namely the presence of toxin producing phytoplankton, the satiation phenomenon in zooplankton’s feeding, modelled by a Holling type II response, and phytoplankton aggregation leading to formation of patches The dynamics of the plankton population is shown to depend on the fraction of the phytoplankton population that aggregates to form colonies and on the number of the latter

&2008 Elsevier Ltd All rights reserved

1 Introduction

Toxic or otherwise harmful algal blooms (HAB) are increasing

in frequency worldwide (Hallegraeff, 1993) and have negative

impact on aquaculture, coastal tourism and human health

(Anderson et al., 2000) The appearance of a bloom can have

devastating implications The complex and inconsistent

interac-tions between toxin producing phytoplankton (TPP) and their

grazers may be due to the level and solubility of toxicity However,

knowledge about interactions between TPP and their potential

grazers are only rudimentary (Edna and Turner, 2006) Also, we

know little about how phytoplankton blooms occur Their

formation mechanism is still not clear, in spite of the fact that

several theories have been formulated to explain it Among the

proposed explanations some researchers use a ‘top-down’

me-chanism (Pitchford and Brindley, 1999) whereas others are in

favour of a ‘bottom-up’ (Huppert et al., 2002; Robson and

Hamilton, 2004) approach It is worthy to remark that the

‘top-down’ view assumes that the phytoplankton bloom depends on

the grazing pressure while in ‘bottom-up’ mechanism the

availability of nutrient is the prime factor Apart from these

theories, also the release of toxic chemicals by TPP has been suggested to play pivotal role in the origin and control of bloom formation (Fehling et al., 2005)

In theory phytoplankton in the ocean are small relative to their predatory enemies and so they will not survive an encounter with

a grazer But, in reality phytoplanktons are not defenseless food-particles that are easily harvested by the consumers They use various anti-grazing strategies such as cell morphology (Hessen and Van Donk, 1993), presence of gelatinous substances, the formation of colonies (Lampert, 1987) and filamentous structures (Lynch, 1980), etc to counteract the grazing pressure by organ-isms of the higher trophic level The toxin liberated by the phytoplankton is also an anti-grazing strategy (Watanabe et al.,

1994), and is important for the existence of the phytoplankton and also for zooplankton species It is largely determined by the ways

in which the species of phytoplankton can resist mutual extinction due to competition or persistence despite grazing pressure from zooplankton (Mayeli et al., 2004)

It is now known that increased spine length and cells in a colony of members of a phytoplankton species (like genus Scenedesmus), when zooplankton grazing is intense, helps in reducing zooplankton filtering rates The effect of these defense mechanisms at the population level has been observed in a few studies (Mayeli et al., 2004) The study of the defense mechanism through the formation of colonies or patches becomes more important if such colonies or patches have the ability to release toxin chemicals, like in case of dinoflagellate (Smayda and Shimizu, 1993) Toxic chemicals released through chemical signals

Contents lists available atScienceDirect

journal homepage:www.elsevier.com/locate/yjtbi

Journal of Theoretical Biology

0022-5193/$ - see front matter & 2008 Elsevier Ltd All rights reserved.

$

Work supported by MIUR Bando per borse a favore di giovani ricercatori

indiani (Samrat Chatterjee); MAE Indo-Italian program of cooperation in Science

and Technology, ‘Biomedical Sciences’ (Joydev Chattopadhyay).



Corresponding author Tel.: +39 011 670 2833.

E-mail addresses: joydev@isical.ac.in (J Chattopadhyay) ,

by aquatic organisms may cause indirect and avalanche effects on

the ecology of entire communities and ecosystems These signals

between microbial predators and prey may contribute to defense

and to food selection or avoidance, factors that probably affect the

trophic structure and algal blooms (Watanabe et al., 1994) For

example zooplankters like Copepods, being highly selective, often

can avoid eating the toxic phytoplankton and thus escape its

adverse effects (DeMott and Moxter, 1991) Thus the coupled

defense mechanism through patching and poison release results

will play an important role for the coexistence of the interacting

species But it is still not clear what are the different combinations

between these two anti-grazing strategies that may produce the

various planktonic dynamics, like coexistence, recurrent bloom,

monospecies bloom, etc., which are commonly observed in

nature

In the present paper a simple mathematical model of

TPP–zooplankton interactions with the predator response

func-tion taken as a monotonically increasing funcfunc-tion up to a certain

threshold density and then monotonically decreasing has been

proposed and analysed Here we assume the predation rate to be a

Holling type II functional response because in reality, the grazing

function is subject to saturation, so that some TPP escape from

predation by zooplankton and form a tide (Chattopadhyay et al.,

2002) This suppression of grazing is usually associated with

active hunting behaviour on the part of the predator, as opposed

to its passive waiting for food encounters For instance, the

raptorial behaviour of zooplanktons like Copepods is highly

complex, exhibiting a hunting behaviour (Uye, 1986) For these

cases a Holling type II functional form is an appropriate choice for

the predation rate

The organization of the paper is as follows We first discuss the

basic formulation of the model in Section 2 In Section 3 we

present some preliminary results that include the boundedness

and existence of different equilibrium points The stability and

Hopf-bifurcation analysis is given in Section 4 In Section 5 we

perform the numerical simulations with special emphasis on the

recurrent bloom phenomenon A final discussion concludes the

paper

2 The mathematical model

Let PðtÞ and ZðtÞ denote the TPP and zooplankton population

sizes, respectively The TPP population is assumed to follow the

law of logistic growth and the zooplankton consume

phytoplank-ton for their growth The prey is assumed to detect the presence of

the predator and to respond by grouping together and releasing

toxin chemicals, which diffuse in the surrounding water through

the surface of the patch

The phytoplankton density plays a central role in the above

defence mechanism, since both the number as well as the size of

patches may depend on it Here, we assume the patch size to be

proportional to the phytoplankton density, this being the real

novelty in this model In fact, if we make the alternate assumption

that the number of patches varies with the phytoplankton density,

while the size of each patch is constant, say R, with surface area

proportional to R2=3, and the number of patches increases with P,

the simplest situation being nðPÞ ¼ mP, we find the equation for Z

in (1) to have the last term linear in P But this equation results

then equivalent to other models already known in the scientific

literature, such as Chattopadhyay et al (2002) In a higher

dimensional system, the same model has been studied inSarkar

and Chattopadhyay (2003)

Assume then that the patch size is proportional to the

phytoplankton density This assumption is also quite reasonable

on biological grounds because many habitat fragmentation

experiments show that the patch size is proportional to the population density (Bowman et al., 2002) For example, inBender

et al (1998) the patch size is shown to depend on population density Root’s (1973) resource concentration hypothesis also shows that there is a positive relationship between patch size and population density

Based on the above discussion we assume that only a fraction k (where 0pkp1) of the phytoplankton aggregate to form N patches, so that the TPP population in each patch is ð1=NÞkP Next we consider the three-dimensional patch in the ocean to

be roughly spherical Its radius is then proportional to ½ð1=NÞkP1=3 The spherical shape of the patch is a sound biological assumption, because in the real world phytoplanktons are observed to form spherical colonies (Riebesell, 1993) For instance, this

phenomen-on occurs for phytoplanktphenomen-ons like Phaeocystis (Assmy et al., 2007) Thus, the surface of the patch results proportional to

½ð1=NÞkP2=3

¼rP2=3, with r  ðk=NÞ2=3 Finally, the predation rate

of the zooplankton population on the f  1  k ‘free’ phytoplank-ton population is assumed to follow what is called Michaelis– Menten or Holling type II functional response In these hypotheses the model reads

_

P ¼ rP  bP2 cfZP

a þ gPF1ðP; ZÞ, _

Z ¼ efZP

a þ gPmZ  erP

2=3

where all the parameters are non-negative The logistic growth of the TPP population is expressed via the parameters r and b; c represents the predation rate and e the conversion rate, cXe; m represents the natural mortality The measure of toxicity is represented by r

3 Some preliminary results 3.1 Positive invariance

By setting X ¼ ðP; ZÞT2R2 and FðXÞ ¼ ½F1ðXÞ; F2ðXÞT, with F: Cþ!R2and F 2 C1

ðR2Þ, Eq (1) becomes _

together with Xð0Þ ¼ X02R2

þ It is easy to check that whenever choosing Xð0Þ 2 R2þwith Xi¼0, for i ¼ 1; 2, then FiðxÞjXi¼0X0 Due

to the lemma of Nagumo (1942) any solution of Eq (2) with

X02R2þ, say XðtÞ ¼ Xðt; X0Þ, is such that XðtÞ 2 R2þfor all t40 3.2 Existence of equilibria

System (1) has only three equilibria Ei¼ ðPi;ZiÞ; i ¼ 0; 1; 2: the origin E0, the boundary equilibrium point E1¼ ðr=b; 0Þ Another feasible non-boundary equilibrium E2 Its positive coordinates are found in the P2Z phase plane by solving the nonlinear system eð1  kÞP=ða þ gPÞ  m  erP2=3¼0 and r  bP  cð1  kÞZ=

ða þ gPÞ ¼ 0 Solving these two equations we find Z2¼ ðr  bP2Þ

ða þ gP2Þ=cð1  kÞ, where P2 is the positive real root of the following equation:

fðPÞ  g3e3r3P5

þ3ag2e3r3P4

 fðeð1  kÞ  mgÞ33a2ge3r3gP3

þ f3maðeð1  kÞ  mgÞ2þa3e3r3gP2

3m2a2feð1  kÞ  mggP þ m3a3¼0 (3) From Descartes’ rule of sign, we observe that there exist either no positive root or more than one positive real root for Eq (3) depending on certain conditions on the parameters If these roots

are less than r=b, then there exist one or more positive equilibrium

point E2

For instance, let us consider the hypothetical set of parameter

values given inTable 1 The parameter values are chosen in such a

way that the number of patches becomes N ¼ 5 With this

parameter set, Eq (3) becomes

fðPÞ  0:164  107P5þ0:492  107P40:1904  105P3

þ0:4704  105P2

0:375  105P þ 106

Eq (4) has two positive roots 1.423 and 7.71 For P2¼1:423, we

have Z2¼0:412 and for P2¼7:17, we have Z2¼ 5:822 Thus the

value of parameters given in Table 1 gives a unique interior

equilibrium point E2 ð1:423; 0:412Þ

3.3 Boundedness of the solutions

Let us first recall the following lemma (Barbalat, 1959)

Lemma 1 Let g be a real valued differential function defined on

some half line ½a; þ1Þ, a 2 ð1; þ1Þ If (i) limt!þ1gðtÞ ¼ a;

jajo þ 1, (ii) g0ðtÞ is uniformly continuous for t4a, then

limt!þ1g0ðtÞ ¼ 0

We shall prove the following key lemma

Lemma 2 Assume at first that the initial condition of Eq (1) satisfies

Pðt0ÞXr=b Then either

(i) PðtÞXr=b for all tX0 and thus as t ! þ1,

ðPðtÞ; ZðtÞÞ ! E1¼ ðr=b; 0Þ, or

(ii) there exists t140 such that PðtÞor=b for all t4t1 If instead

Pðt0Þor=b, then PðtÞor=b for all tX0

Proof We consider first the case PðtÞXr=b for all tX0 From the

first equation of (1) we get

dP

dt¼rP  bP

2

cð1  kÞZP

Hence, for all tX0, we have that dPðtÞ=dtp0 Let

lim

If Z4r=b, then by theBarbalat (1959)lemma, we have

0 ¼ lim

t!1

dPðtÞ

dt ¼t!1lim PðtÞðr  bPðtÞÞ cð1  kÞZðtÞPðtÞ

a þ gPðtÞ

p lim

t!1½PðtÞðr  bPðtÞÞ ¼ ½Zðr  bZÞo0

This contradiction shows that Z ¼ r=b i.e.,

limPðtÞ ¼r

Of course, PðtÞ is differentiable and P0

ðtÞ is uniformly continuous for t 2 ð0; þ1Þ Thus, with Eq (7) all the assumptions of the Barbalat lemma are verified, so that

lim

t!1

dP

Combining then (7) with (1) we have lim

t!1

dPðtÞ

dt ¼ t!1lim

cð1  kÞZðtÞPðtÞ

Hence, Eqs (7)–(9) are in agreement if and only if limt!1ZðtÞ ¼ 0 This completes the case (i)

Suppose that assumption (i) is violated Then there exists t140

at which for the first time Pðt1Þ ¼r=b From Eq (1) we have dPðtÞ

dt





t¼t1

¼cð1  kÞZðt1ÞPðt1Þ

a þ gPðt1Þ o0

This implies that once a solution with P has entered into the interval ð0; r=bÞ then it remains bounded there for all t4t1, i.e., PðtÞor=b for all t4t1

Finally, if Pðt0Þor=b, then applying the previous argument it follows that PðtÞor=b for all t40, i.e (iii) holds true This completes the proof &

Lemma 3 Letting l ¼ ðr þ ZÞ2=4b there is Z 2 ð0; m such that for any positive solution ðPðtÞ; ZðtÞÞT of system (1) for all large t we have ZðtÞoM, with M ¼ l=Z

Proof Lemma 2 implies that for any ðPðt0Þ;Zðt0ÞÞ such that Pðt0ÞXr=b, then either a time t140 exists for which PðtÞor=b for all t4t1, or limt!1PðtÞ ¼ r=b Furthermore: if Pðt0Þor=b then PðtÞpr=b for all t40 Hence in any case a non-negative time, say t, exists such that PðtÞor=b þ , for some 40 and for all t4t Set W ¼ PðtÞ þ ZðtÞ Calculating the derivative of W along the solution of system (1), we find for t4t

dW

dt ¼rP  bP

2

cð1  kÞZP

a þ gP þ

eð1  kÞZP

a þ gP mZ  erP

2=3

Z prP  bP2

mZ, since cXe Taking Z40 we get dW

dt þZWpðr  bP þ ZÞP þ ðZ  mÞZ

Now if we choose Zpm, then dW

dt þZWpðr  bP þ ZÞPpðr þ ZÞ

2

4b l.

It is clear that the right-hand side of the above expression is bounded Thus, there exist a positive constant M, such that WðtÞoM for all large t The assertion of Lemma 2 now follows from the ultimate boundedness of P &

Let us define the following subset of R2

0;þ:

O ¼ ðP; ZÞ 2 R20;þ: Ppr

b;ZpM

Theorem 1 The set O is a global attractor in R20;þand, of course, is positively invariant

Proof Due to Lemmas 2 and 3 for all initial conditions in R2þ;0 such that ðPðt0Þ;Zðt0ÞÞdoes not belong to O, either there exists a positive time, say T, T ¼ maxft1;tg, such that the corresponding solution ðPðtÞ; ZðtÞÞ 2 int O for all t4T, or the corresponding solution is such that ðPðtÞ; ZðtÞÞ ! E1ðr=b; 0Þ as t ! þ1 But,

E12qO Hence the global attractivity of O in R20;þ has been proved &

Table 1

A hypothetical set of parameter values

The units of P and Z are g m 3 and time t is measured in days.

Assume now that ðPðt0Þ;Zðt0ÞÞ 2intðOÞ Then Lemma 2 implies that

PðtÞor=b for all t40 and also by Lemma 3 we know that ZðtÞoM for

all large t Finally note that if ðPðt0Þ;Zðt0ÞÞ 2qO, because Pðt0Þ ¼r=b

or Zðt0Þ ¼M or both, then still the corresponding solutions ðPðtÞ; ZðtÞÞ

must immediately enter intðOÞ or coincide with E1

We have proved that the trajectories of (1) are bounded Next we

shall study the stability property of different equilibrium points

4 Stability and Hopf-bifurcation analysis

The Jacobian matrix of system (1) has the form

Ji a11 a12

a21 a22

!

where

a11¼r  2bPicð1  kÞZi

ða þ gPiÞ2; a12¼cð1  kÞPi

a þ gPi

,

a21¼e ð1  kÞa

ða þ gPiÞ2

2

3rP

1=3 i

Zi,

a22¼eð1  kÞPi

a þ gPi

m erP2=3i

At the origin, the eigenvalues r, m are found showing its

instability At E1, we have the eigenvalues r,

e ð1  kÞr

ab þ rg

m

er

r b

 2=3

, thus giving conditional stability As a particular case, notice that

the second eigenvalue can become zero

Finally, at the interior equilibrium E2, the Jacobian becomes

J2

r  2bP2r  bP2

a þ gP2

cð1  kÞP2

a þ gP2

e ð1  kÞa

ða þ gP2Þ2

2

3rP

1=3 2

0

B

B

@

1 C C

Thus the eigenvalues in this case are obtained as roots of the

quadratic l2

trðJ2Þl þdetðJ2Þ ¼0, where

trðJ2Þ ¼r  2bP2r  bP2

a þ gP2

, and

detðJ2Þ ¼ecð1  kÞP2

a þ gP2

1  k

ða þ gP2Þ2

2

3rP

1=3 2

Now, trðJ2Þo0, iff

ro2bP2þr  bP2

a þ gP2

with P2or=b and we find that the Routh–Hurwitz criterion for

stability is satisfied if detðJ2Þ40, i.e if

1  k

a þ gP2

42

3rP

1=3

2

We can now summarize these findings in the following result

Theorem 2 In system (1), the trivial equilibrium point E0is always

unstable The axial equilibrium point E1is stable iff

ð1  kÞr

ab þ rgom

eþr

r

b

 2=3

The positive equilibrium point E2is locally asymptotically stable if the

following conditions hold:

robP2½2ða þ gP2Þ 1

P2

ða þ gP2Þ24 8r3

Next we shall perform the Hopf-bifurcation analysis near the interior equilibrium point E2 If the conditions required for such analysis are satisfied we can then conclude that the proposed system models the recurring bloom phenomenon

Theorem 3 Suppose the interior equilibrium point E2 exists and

f ¼ 1  k represents the number of phytoplanktons that are ‘free’ When f crosses a critical value, fc, given by

fc¼ðr  2bP  2Þða þ gP2Þ2

cZ2

(16) and if this critical value fcsatisfies the condition

fc42rða þ gP2Þ2

then system (1) experiences a Hopf-bifurcation around the positive steady state

Proof Notice that trðJ2Þjf ¼f

c¼0 and Q ¼ detðJ2Þjf ¼f

c40 if the condition (17) holds For f ¼ fc, the characteristic equation may be written as

Eq (18) has two roots Z1¼ þi ffiffiffiffi

Q p and Z2¼ i ffiffiffiffi

Q p For all f, the general roots are of the form

Z1ðf Þ ¼ b1ðf Þ þ ib2ðf Þ; Z2ðf Þ ¼ b1ðf Þ  ib2ðf Þ

Now we shall verify the transversality condition d

dfðReðZjðf ÞÞÞf ¼f

ca0; j ¼ 1; 2

Substituting Zjðf Þ ¼ b1ðf Þ þ ib2ðf Þ into Eq (18) and calculating the derivatives, we have

2b1ðf Þb0

1ðf Þ  2b2ðf Þb0

2ðf Þ þ Q0ðf Þ ¼ 0, 2b2ðf Þb0

1ðf Þ þ b1ðf Þb0

Solving (19), we get d

dfðReðZjðf ÞÞÞf ¼f

c¼ A 2ðb2þb2Þa0, where A ¼ bðf ÞQ0

ðf Þ Hence, the transversality condition holds This implies that a Hopf-bifurcation occurs at f ¼ fc and the theorem follows &

Thus, we observe that when the fraction of TPP population which does not form the patch, i.e., f ¼ 1  k, crosses a certain critical value there is a chance for the occurrence of the periodic solution, i.e our model can show the recurrent bloom phenom-enon To verify these results, we now perform numerical experiments

5 Numerical simulation Theorem 2 ascertains that system (1) is locally asymptotically stable around the interior equilibrium point under certain parameter conditions We perform our numerical simulations with the set of parameter values given inTable 1, because for these values system (1) is locally asymptotically stable around the interior equilibrium point E2 ð1:423; 0:412Þ, see Fig 1 First we take k ¼ 0:65, retaining the other parameter values fixed, and observe periodic solutions, seeFig 2 This supports our analytical claim that this model can show the recurrent bloom phenomenon

Our aim now is to observe the role played by toxic chemicals in

the system We begin with the parameter f ¼ 1  k We have

already observed from Theorem 3 that a Hopf-bifurcation occurs

when the parameter f crosses a certain critical value To verify this

result numerically we compute the bifurcation diagram for both

species, seeFig 3 To do it, we solve (1) for 10 000 time units and

examine only the last 3000 time units to eliminate the transient

behaviour We then plot the successive maxima and minima of

every species taking f ¼ 1  k as the bifurcation parameter, while

the other parameters are kept fixed at the level given inTable 1 At

the same time we vary r together with f in a way such that the

number of patches formed remains the same, i.e., N ¼ 5 For lower

values of the parameter f we observe that the system is stable

around the positive equilibrium point, but with an increase in the

value of f, a Hopf-bifurcation occurs and the system shows

periodic oscillations This supports Theorem 3 On the other hand

for values of f even larger, a double period cyclic phenomenon

is shown, as in some cases two maxima appear in the plots, see

Fig 3 Analysing this bifurcation diagram we observe also that for

lower values of f, there is a huge increase in the TPP population

and the zooplankton population is washed away from the system This phenomenon represents a monospecies bloom

In the above simulation we have considered a fixed number of colonies, N ¼ 5 It is interesting to see also what happens to the dynamical nature of the system, when the number of colonies or patches N changes To observe the role of N, we keep k ¼ 0:75 fixed and vary r so that N always remains an integer, again retaining the same other parameter values as given inTable 1 If the TPP population forms a single patch it is very difficult for the zooplankton population to survive, Fig 4 This shows the occurrence of the red tides, i.e the TPP monospecies blooms The system is instead found stable around the interior coexistence equilibrium point, seeFig 5, only for very low values

of N, namely 2pNp4, while for larger ones, persistent oscillations occur,Fig 6

Thus, we may conclude that the fraction of phytoplankton that aggregate to form patches plays an important role in the occurrence of recurrent blooms, in the coexistence of all the species and in the occurrence of monospecies blooms More specifically, for stability of the system around the interior equilibrium point, the fraction k ¼ 1  f of the TPP population that aggregates to form patches must be between certain lower and upper threshold values If the fraction k is less than the lower threshold value, then it may cause recurrent blooms with possibly double periods and if it is higher than the upper threshold value then there is a bloom of TPP population which causes also the extinction of the zooplankton population and represents a red tide bloom

To further substantiate our theoretical analysis with the numerical approach, we provide phase plane diagrams corre-sponding to Figs 1, 2 and 6 The phase plane diagram corresponding toFig 1shows that the equilibrium points E0and

E1 are saddle points while the interior equilibrium point E2 is a spiral sink, seeFig 7 The phase plane diagram ofFig 2shows that the equilibria E0 and E1 are saddle points while the interior equilibrium point E2 is a spiral source, the solution moving cyclically around it, seeFig 8 The phase plane diagram related to

Fig 4shows that E0is a saddle while E1is a nodal sink, seeFig 9 Notice that in this situation the interior equilibrium point does not exist

0

0.5

1

1.5

2

2.5

Time

TPP

Zooplanton

Fig 1 The figure depicts the local stability of system (1) around the interior

equilibrium point E 2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

time

TPP

Zooplankton

Fig 2 The figure depicts coexistence of all the species through periodic

0 1 2 3

0 0.2 0.4 0.6 0.8

(1−k)

(1−k)

Fig 3 The figure depicting the bifurcation diagram with f ¼ 1  k as the bifurcation parameter.

6 Conclusion

In aquatic systems the recurrent bloom phenomenon is commonly observed in nature (Assmy et al., 2007; Hallegraeff, 1993; Riebesell, 1993; Uye, 1986), studied in laboratory experi-ments (Fehling et al., 2005) and investigated via mathematical models (Pitchford and Brindley, 1999; Robson and Hamilton,

0

0.5

1

1.5

2

2.5

3

time

TPP

Fig 4 For N ¼ 1 we show here the monospecies TPP bloom.

0

1

2

3

N

0

0.2

0.4

0.6

0.8

N

Fig 5 The bifurcation diagram for low values of N as the bifurcation parameter.

0

1

2

3

N

0

0.2

0.4

0.6

0.8

N

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

TPP

E 1

E 2

E 0

Fig 7 Phase plane diagram corresponding to Fig 1.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

TPP

1

Fig 9 Phase plane diagram corresponding to Fig 4.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

TPP

E 1

E 2

E 0

Fig 8 Phase plane diagram corresponding to Fig 2.

2004) Toxin producing phytoplankton is now also recognized as a

major actor in the formation of such blooms (Edna and Turner,

2006; Huppert et al., 2002; Smayda and Shimizu, 1993) and the

impact of its negative economic consequences is beginning to be

evaluated (Anderson et al., 2000) As already remarked above and

in the Introduction, quite a good number of papers have appeared

to explain the bloom phenomenon (Pitchford and Brindley, 1999;

Robson and Hamilton, 2004) Experimental and field evidence

show that the recurrent bloom phenomena is also associated with

characteristics of the patch such as its size and shape (Bender

et al., 1998; Bowman et al., 2002; Hessen and Van Donk, 1993;

Lynch, 1980; Mayeli et al., 2004) In the frame of our knowledge,

however, so far no effort has been made to formulate this

situation in terms of mathematical modelling

Several research papers have already appeared, see

Chatto-padhyay et al (2002)andSarkar and Chattopadhyay (2003)and

the literature cited therein, to establish the role of TPP in the

context of the plankton bloom mechanism However, the effect of

phytoplankton aggregation, observed in nature (Hessen and Van

Donk, 1993) has always been neglected in the modelling efforts

One of the proposed models of Chattopadhyay et al (2002),

accounting for recurrent blooms in particular, is somewhat related

with some aspects of this paper In fact, case 5 ofChattopadhyay

et al (2002) assumes the Holling type II response in feeding,

together with a linear function of P modelling the poisoning effect

The latter corresponds to taking a constant size for each patch

and assuming the number of patches to be proportional to the

phytoplankton density This assumption represents indeed the

main difference with the present model The mechanism we

propose relies in fact on the three basic assumptions consisting in

the presence of TPP, its patching agglomeration for self-defense

and the consequent nonlinear functional response P2=3accounting

for poison release through the patch surface, and finally the

Holling type II function representing zooplankton’s feeding

saturation Both models, case 5 of Chattopadhyay et al (2002)

and the one presented here, account for the recurrent blooms

Together, they show the crucial role that the Holling type II

function plays in modelling this periodic phenomenon

But the present analysis has one extra feature It also shows the

occurrence of monospecies blooms, which is not observed in

Chattopadhyay et al (2002) The most relevant characteristics of

planktonic dynamics, namely coexistence, recurrent blooms and

monospecies blooms, are here shown to depend on two relevant

quantities, namely the amount of toxic chemical released by the

TPP population as well as on the fraction of TPP population which

aggregates to form patches These are adequately taken into

account in our model via suitable parameters For the stability of

the system around the interior equilibrium point the number of

patches N has to lie between certain critical values Also the level

of toxicity r is here shown to play an important role in plankton

dynamics, as different dynamics such as monospecies bloom,

recurrent bloom and coexistence of all species are observed by

varying the level of toxicity Thus the formation of plankton

colonies or patches, in particular of TPP, plays an important role in

the aquatic system, explaining at least qualitatively the field and

experimental data collections on recurrent blooms (Edna and

Turner, 2006; Hallegraeff, 1993; Lampert, 1987; Lynch, 1980;

Mayeli et al., 2004; Smayda and Shimizu, 1993)

This investigation shows instead in particular the role, also

observed experimentally (Hessen and Van Donk, 1993), that patch

formations may also possibly play in this context, as they may

turn off or trigger the blooms when the parameter f, or equivalently k, crosses certain thresholds In addition, the full spectrum of plankton blooms, ranging from the harmful red tides

to the recurrent plankton blooms, together with a coexistence state in between, can be modelled by (1) The major novelty of this contribution is thus represented by its bridging together the monospecies and the recurrent blooms, merging the previous results in a single more general picture encompassing all the previous findings

Acknowledgements

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