ARTICLE IN PRESS Journal of Theoretical Biology 253 (2008) 289– 295 Contents lists available at ScienceDirect Journal of Theoretical Biology journal homepage: www.elsevier.com/locate/yjtbi Patchy agglomeration as a transition from monospecies to recurrent plankton blooms$ Joydev Chattopadhyay a, Samrat Chatterjee b, Ezio Venturino b,à a b Agricultural and Ecological Research Unit, Indian Statistical Institute, 203 B.T Road, Kolkata 700108, India Dipartimento di Matematica, Universita’ di Torino, via Carlo Alberto 10, 10123 Torino, Italy a r t i c l e in fo abstract Article history: Received October 2007 Received in revised form March 2008 Accepted 10 March 2008 Available online 14 March 2008 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 Keywords: Phytoplankton–zooplankton Toxic chemicals Patch Recurrent bloom Red tides Hopf-bifurcation Coexistence 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 interactions 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’ mechanism (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 ‘topdown’ 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 $ 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) à Corresponding author Tel.: +39 011 670 2833 E-mail addresses: joydev@isical.ac.in (J Chattopadhyay), samrat_ct@rediffmail.com (S Chatterjee), ezio.venturino@unito.it (E Venturino) 0022-5193/$ - see front matter & 2008 Elsevier Ltd All rights reserved doi:10.1016/j.jtbi.2008.03.008 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 foodparticles 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 organisms 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 ARTICLE IN PRESS 290 J Chattopadhyay et al / Journal of Theoretical Biology 253 (2008) 289–295 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 function taken as a monotonically increasing function 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 In Section 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 In Section we perform the numerical simulations with special emphasis on the recurrent bloom phenomenon A final discussion concludes the paper 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 phytoplankton 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 in Sarkar 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, in Bender 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 ẵ1=NịkP1=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 phenomenon occurs for phytoplanktons like Phaeocystis (Assmy et al., 2007) Thus, the surface of the patch results proportional to ẵ1=NịkP2=3 ẳ rP 2=3 , with r ðk=NÞ2=3 Finally, the predation rate of the zooplankton population on the f À k ‘free’ phytoplankton population is assumed to follow what is called Michaelis– Menten or Holling type II functional response In these hypotheses the model reads cfZP F P; Zị, P_ ẳ rP bP a ỵ gP efZP Z_ ẳ À mZ À erP 2=3 Z F ðP; Zị, a ỵ gP (1) 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 Some preliminary results 3.1 Positive invariance By setting X ẳ P; ZịT R2 and FXị ẳ ẵF Xị; F XịT , with F: C ỵ ! R2 and F C ðR2 Þ, Eq (1) becomes X_ ẳ FXị, (2) R2ỵ It is easy to check that whenever together with X0ị ẳ X choosing X0ị R2ỵ with X i ẳ 0, for i ẳ 1; 2, then F i xịjX i ¼0 X0 Due to the lemma of Nagumo (1942) any solution of Eq (2) with X R2ỵ , say Xtị ẳ Xt; X ị, is such that Xtị R2ỵ for all t40 3.2 Existence of equilibria System (1) has only three equilibria Ei ¼ ðP i ; Z i ị; i ẳ 0; 1; 2: the origin E0 , the boundary equilibrium point E1 ¼ ðr=b; 0Þ Another feasible non-boundary equilibrium E2 Its positive coordinates are found in the P2Z phase plane by solving the nonlinear system e1 kịP=a ỵ gPị m erP2=3 ¼ and r À bP À cð1 À kịZ= a ỵ gPị ẳ Solving these two equations we nd Z ẳ r bP ị a ỵ gP2 ị=c1 kị, where P2 is the positive real root of the following equation: fðPÞ g3 e3 r3 P5 ỵ 3ag2 e3 r3 P fðeð1 À kÞ À mgÞ3 À 3a2 ge3 r3 gP ỵ f3mae1 kị mgị2 ỵ a3 e3 r3 gP À 3m2 a2 feð1 À kÞ À mggP ỵ m3 a3 ẳ (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 ARTICLE IN PRESS J Chattopadhyay et al / Journal of Theoretical Biology 253 (2008) 289–295 Of course, PðtÞ is differentiable and P ðtÞ is uniformly continuous for t 0; ỵ1ị Thus, with Eq (7) all the assumptions of the Barbalat lemma are verified, so that Table A hypothetical set of parameter values Parameters Values Units r 0.27 day b 0.1 m3 gÀ1 day c 0.3 day e 0.09 m3 gÀ1 day m 0.1 day – – g mÀ3 g mÀ3 k g a r 0.75 0.1 0.1 0.0225 lim À1 À1 lim À1 t!1 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 in Table The parameter values are chosen in such a way that the number of patches becomes N ¼ With this parameter set, Eq (3) becomes fðPÞ 0:164  10À7 P ỵ 0:492 107 P 0:1904 10À5 P3 (4) Eq (4) has two positive roots 1.423 and 7.71 For P ¼ 1:423, we have Z ¼ 0:412 and for P ¼ 7:17, we have Z ¼ À5:822 Thus the value of parameters given in Table 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 Let g be a real valued differential function defined on some half line ẵa; ỵ1ị, a 1; ỵ1ị If (i) limt!ỵ1 gtị ẳ a; jajo ỵ 1, (ii) g tị is uniformly continuous for t4a, then limt!ỵ1 g tị ẳ Lemma Assume at rst that the initial condition of Eq (1) satisfies Pðt ÞXr=b Then either (i) PðtÞXr=b for all tX0 and thus as t ! ỵ1, Ptị; Ztịị ! E1 ẳ r=b; 0ị, or (ii) there exists t 40 such that PðtÞor=b for all t4t If instead Pðt Þor=b, then PðtÞor=b for all tX0 (5) (6) If Z4r=b, then by the Barbalat (1959) lemma, we have dPðtÞ c1 kịZtịPtị ẳ lim Ptịr bPtịị ẳ lim t!1 dt t!1 a ỵ gPtị p lim ẵPtịr bPtịị ẳ ẵZr bZịo0 t!1 This contradiction shows that Z ẳ r=b i.e., lim Ptị ẳ r b (9) Hence, Eqs (7)–(9) are in agreement if and only if limt!1 Ztị ẳ This completes the case (i) Suppose that assumption (i) is violated Then there exists t 40 at which for the first time Pt ị ẳ r=b From Eq (1) we have dPtị c1 kịZt ịPt ị o0 ẳ dt tẳt1 a ỵ gPt ị This implies that once a solution with P has entered into the interval ð0; r=bÞ then it remains bounded there for all t4t , i.e., PðtÞor=b for all t4t Finally, if Pðt Þor=b, then applying the previous argument it follows that PðtÞor=b for all t40, i.e (iii) holds true This completes the proof & Lemma Letting l ¼ r ỵ Zị2 =4b there is Z 0; 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 implies that for any ðPðt Þ; Zðt ÞÞ such that Pðt ÞXr=b, then either a time t 40 exists for which Ptịor=b for all t4t , or limt!1 Ptị ẳ r=b Furthermore: if Pðt Þor=b then PðtÞpr=b for all t40 Hence in any case a non-negative time, say t , exists such that Ptịor=b ỵ , for some 40 and for all t4t Set W ẳ Ptị þ ZðtÞ Calculating the derivative of W along the solution of system (1), we find for t4t à dW cð1 kịZP e1 kịZP ẳ rP bP ỵ mZ erP 2=3 Z dt a þ gP a þ gP since cXe Taking Z40 we get dW ỵ ZWpr bP ỵ ZịP ỵ Z mịZ dt dW r ỵ Zị2 ỵ ZWpr bP ỵ ZịPp l dt 4b Hence, for all tX0, we have that dPðtÞ=dtp0 Let t!1 dPðtÞ cð1 kịZtịPtị ẳ lim t!1 dt a ỵ gPðtÞ Now if we choose Zpm, then Proof We consider first the case PðtÞXr=b for all tX0 From the first equation of (1) we get lim Ptị ẳ Z (8) prP À bP À mZ, We shall prove the following key lemma t!1 dP ¼ dt À1 The units of P and Z are g mÀ3 and time t is measured in days dP c1 kịZP ẳ rP bP dt a ỵ gP t!1 Combining then (7) with (1) we have ỵ 0:4704 105 P 0:375 105 P ỵ 10À6 ¼ 291 (7) 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 now follows from the ultimate boundedness of P & Let us define the following subset of R20;ỵ : n o r (10) O ẳ P; Zị R20;ỵ : Pp ; ZpM b Theorem The set O is a global attractor in R20;ỵ and, of course, is positively invariant Proof Due to Lemmas and for all initial conditions in R2ỵ;0 such that Pt ị; Zt ịị does not belong to O, either there exists a positive time, say T, T ¼ maxft ; t à g, such that the corresponding solution ðPðtÞ; ZðtÞÞ int O for all t4T, or the corresponding solution is such that Ptị; Ztịị ! E1 r=b; 0ị as t ! ỵ1 But, E1 qO Hence the global attractivity of O in R20;ỵ has been proved & ARTICLE IN PRESS 292 J Chattopadhyay et al / Journal of Theoretical Biology 253 (2008) 289–295 Assume now that ðPðt Þ; Zðt ÞÞ intðOÞ Then Lemma implies that PðtÞor=b for all t40 and also by Lemma we know that ZðtÞoM for all large t Finally note that if ðPðt Þ; Zðt ÞÞ qO, because Pðt Þ ¼ r=b or Zðt Þ ¼ M 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 (11) ða þ gP2 Þ 8r3 27ð1 À kÞ3 (15) 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 fc ¼ ðr À 2bP 2ịa ỵ gP2 ị2 cZ (16) and if this critical value f c satisfies the condition where c1 kịZ i ; a12 a11 ẳ r 2bPi a ỵ gP i ị2 " # kịa 1=3 rP a21 ẳ e Zi , a ỵ gPi ị2 i a22 ẳ Theorem Suppose the interior equilibrium point E2 exists and f ¼ À k represents the number of phytoplanktons that are ‘free’ When f crosses a critical value, f c , given by Stability and Hopf-bifurcation analysis The Jacobian matrix of system (1) has the form ! a11 a12 , Ji a21 a22 P2 Àcð1 À kÞP i ẳ , a ỵ gPi f c4 2ra ỵ gP Þ2 1=3 3P , (17) then system (1) experiences a Hopf-bifurcation around the positive steady state eð1 À kÞPi 2=3 À m À erP i a þ gP i At the origin, the eigenvalues r, Àm are found showing its instability At E1 , we have the eigenvalues Àr, r 2=3 ð1 À kịr m r , e ab ỵ rg e b Proof Notice that trJ2 ịjf ẳf c ẳ and Q ẳ detJ ịjf ẳf c 40 if the condition (17) holds For f ¼ f c, the characteristic equation may be written as Z2 ỵ Q ẳ (18) pffiffiffiffi pffiffiffiffi Eq (18) has two roots Z1 ẳ ỵi Q and Z2 ẳ i Q For all f, the general roots are of the form 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 r À bP Àcð1 À kÞP r À 2bP2 À B a ỵ gP a ỵ gP2 C B " C # C J2 B (12) B C ð1 À kÞa À1=3 @e A À rP Z2 a ỵ gP ị Substituting Zj f ị ẳ b1 f ị ỵ ib2 f ị into Eq (18) and calculating the derivatives, we have Thus the eigenvalues in this case are obtained as roots of the quadratic l2 trJ2 ịl ỵ detJ ị ¼ 0, where 2b1 ðf Þb01 ðf Þ À 2b2 f ịb02 f ị ỵ Q f ị ẳ 0, 2b2 f ịb01 f ị ỵ b1 f ịb02 f ị ẳ trJ2 ị ẳ r 2bP ec1 kịP a ỵ gP ( ) 1=3 rP a ỵ gP2 Þ2 with P2 or=b and we find that the Routh–Hurwitz criterion for stability is satisfied if detðJ Þ40, i.e if 1Àk À1=3 rP a þ gP We can now summarize these findings in the following result Theorem In system (1), the trivial equilibrium point E0 is always unstable The axial equilibrium point E1 is stable iff (13) The positive equilibrium point E2 is locally asymptotically stable if the following conditions hold: ro bP2 ẵ2a ỵ gP ị , a ỵ gP d ReZj f ÞÞÞf ¼f c a0; df j ¼ 1; (19) where A ẳ bf ịQ f ị Hence, the transversality condition holds This implies that a Hopf-bifurcation occurs at f ¼ f c and the theorem follows & 1Àk r bP a ỵ gP r 2=3 kịr m o ỵr ab ỵ rg e b Now we shall verify the transversality condition d A ReZj f ịịịf ẳf c ẳ a0, df 2b21 ỵ b22 ị Now, trJ2 ịo0, iff ro2bP ỵ Z2 f ị ẳ b1 f ị ib2 ðf Þ Solving (19), we get r À bP2 , a ỵ gP2 and detJ ị ẳ Z1 f ị ẳ b1 f ị ỵ ib2 f ị; (14) Thus, we observe that when the fraction of TPP population which does not form the patch, i.e., f ¼ À 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 phenomenon To verify these results, we now perform numerical experiments Numerical simulation Theorem 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 in Table 1, because for these values system (1) is locally asymptotically stable around the interior equilibrium point E2 1:423; 0:412ị, see Fig First we take k ẳ 0:65, retaining the other parameter values fixed, and observe periodic solutions, see Fig This supports our analytical claim that this model can show the recurrent bloom phenomenon ARTICLE IN PRESS J Chattopadhyay et al / Journal of Theoretical Biology 253 (2008) 289–295 Populations TPP 1.5 Zooplanton 0.5 100 200 300 400 500 Time Fig The figure depicts the local stability of system (1) around the interior equilibrium point E2 Zooplankton TPP 1.8 Populations 1.6 1.4 1.2 0.8 0.6 0.4 0.2 TPP 0 0.2 0.4 0.6 0.8 0.6 0.8 (1−k) 0.8 0.6 0.4 0.2 0 0.2 0.4 (1−k) Fig The figure depicting the bifurcation diagram with f ¼ À k as the bifurcation parameter 2.5 Zooplankton Our aim now is to observe the role played by toxic chemicals in the system We begin with the parameter f ¼ À k We have already observed from Theorem 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, see Fig To 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 À k as the bifurcation parameter, while the other parameters are kept fixed at the level given in Table 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 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 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 Analysing this bifurcation diagram we observe also that for lower values of f, there is a huge increase in the TPP population 293 100 200 300 400 500 time Fig The figure depicts coexistence of all the species through periodic oscillation 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 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 in Table If the TPP population forms a single patch it is very difficult for the zooplankton population to survive, Fig 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, see Fig 5, only for very low values of N, namely 2pNp4, while for larger ones, persistent oscillations occur, Fig 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 À 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 corresponding to Figs 1, and The phase plane diagram corresponding to Fig shows that the equilibrium points E0 and E1 are saddle points while the interior equilibrium point E2 is a spiral sink, see Fig The phase plane diagram of Fig shows 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, see Fig The phase plane diagram related to Fig shows that E0 is a saddle while E1 is a nodal sink, see Fig Notice that in this situation the interior equilibrium point does not exist ARTICLE IN PRESS 294 J Chattopadhyay et al / Journal of Theoretical Biology 253 (2008) 289–295 0.9 0.8 0.7 TPP Zooplankton populations 2.5 1.5 Zooplankton 0.6 0.4 0.3 E1 0.2 E0 0.1 0.5 0 E2 0.5 20 40 60 80 0.5 100 time 1.5 TPP 2.5 Fig Phase plane diagram corresponding to Fig Fig For N ¼ we show here the monospecies TPP bloom 0.9 0.8 0 10 N 15 20 zooplankton TPP Zooplankton 0.6 0.5 0.4 0.3 E2 0.2 0.8 E0 0.4 0.5 0.2 10 N 15 2.5 20 Zooplankton 1.5 TPP Fig Phase plane diagram corresponding to Fig Fig The bifurcation diagram for low values of N as the bifurcation parameter 50 100 150 N 200 250 300 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 E0 0.6 E1 0.5 0.8 Zooplankton E1 0.1 0.6 TPP 0.7 1.5 TPP 2.5 Fig Phase plane diagram corresponding to Fig 0.4 Conclusion 0.2 0 50 100 150 N 200 Fig The bifurcation diagram as function of N 250 300 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 experiments (Fehling et al., 2005) and investigated via mathematical models (Pitchford and Brindley, 1999; Robson and Hamilton, ARTICLE IN PRESS J Chattopadhyay et al / Journal of Theoretical Biology 253 (2008) 289–295 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 Chattopadhyay et al (2002) and Sarkar 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 of Chattopadhyay 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=3 accounting for poison release through the patch surface, and finally the Holling type II function representing zooplankton’s feeding saturation Both models, case 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 295 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 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modelling Several research papers have already appeared, see Chattopadhyay et al (2002) and Sarkar and Chattopadhyay (2003) and the literature cited therein, to establish the role of TPP... 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