BIFURCATION AND SPATIAL PATTERN FORMATION IN SPREADING OF DISEASE WITH INCUBATION PERIOD IN A PHYTOPLANKTON DYNAMICS

Transcription

1 Electronic Journal of Differential Equations, Vol. 212 (212), No. 21, pp ISSN: URL: or ftp ejde.math.tstate.edu BIFURCATION AND SPATIAL PATTERN FORMATION IN SPREADING OF DISEASE WITH INCUBATION PERIOD IN A PHYTOPLANKTON DYNAMICS RANDHIR SINGH BAGHEL, JOYDIP DHAR, RENU JAIN Abstract. In this article, we propose a three dimensional mathematical model of phtoplankton dnamics with the help of reaction-diffusion equations that studies the bifurcation and pattern formation mechanism. We provide an analtical eplanation for understanding phtoplankton dnamics with three population classes: susceptible, incubated, and infected. This model has a Holling tpe II response function for the population transformation from susceptible to incubated class in an aquatic ecosstem. Our main goal is to provide a qualitative analsis of Hopf bifurcation mechanisms, taking death rate of infected phtoplankton as bifurcation parameter, and to stud further spatial patterns formation due to spatial diffusion. Here analtical findings are supported b the results of numerical eperiments. It is observed that the coeistence of all classes of population depends on the rate of diffusion. Also we obtained the time evaluation pattern formation of the spatial sstem. 1. Introduction It is well known that the phtoplankton and zooplankton are single cell organisms that drift with the currents on the surface of open oceans. Further, the phtoplanktons are the staple items for the food web and the are the reccler of most of the energ that flows through the ocean ecosstem. It has a major role in stabilizing the environment and survival of living population as it consumes half of the universal carbon-dioide and releases ogen. So far, there is a number of studies which show the presence of pathogenic viruses in the plankton communit [1, 11]. A good review of the nature of marine viruses and their ecological as well as their biological effects is given in [12]. Some researchers have shown using an electronic microscope that these viral diseases can affect bacteria and phtoplankton in coastal area and viruses are held responsible for the collapse of Emiliania hulei bloom in Mesocosms [2, 14]. Marine viruses infect not onl plankton but cultivated stocks of Crabs, Osters, Mussels, Clams shrimp, Salmon and Catfish, etc., are all susceptible to various kinds of viruses. We know that the viruses are nonliving organisms, in the sense, the have no metabolism when out side the host and the can reproduce onl b infecting the living organisms. Viral infection of Mathematics Subject Classification. 34C11, 34C23, 34D8, 34D2, 35Q92, 92B5, 92D4. Ke words and phrases. Phtoplankton dnamics; reaction-diffusion equation; local stabilit; Hopf-bifurcation; diffusion-driven instabilit; spatial pattern formation. c 212 Teas State Universit - San Marcos. Submitted Jul 25, 211. Published Februar 2,

2 2 R. S. BAGHEL, J. DHAR, R. JAIN EJDE-212/21 the phtoplankton cell is of two tpes, namel, Lsogenic and Ltic. Moreover from literature, in ltic viral infection, when a virus injects its DNA into a cell, it hijacks the cell s replication machiner and produces large a number of viruses. As a result, the rupture the host and are released into the environment. On the other hand, in lsogenic viral infection, the DNA of the viruses do not use the machiner of the host themselves, but their genes are duplicated each time as the host cell divides. Man papers have alread been developed which have used this kind of lsogenic viral infection [3, 6, 1, 17]. In this kind of infection, infected phtoplankton do not grow like susceptible phtoplankton but their number grows and it is proportional to the number of susceptible phtoplankton and infected phtoplankton. Moreover, we do not look on the viruses as mere pathogens that destro others life rather, the produce the fuel, essential to the running of the marine engine b destroing phtoplankton; i.e., the produce the essential minerals which are required for the growth of phtoplankton. Secondl, we have introduced incubated class in between a susceptible class and an infected class, Unlike simple S-I-S models in mathematical epidemiolog. Generall, we have been using direct shifting from susceptible to infected class, but this process is not regular, rather, phtoplankton sta for some definite period in an incubated class after leaving the susceptible class and joining the infected class. The period for which the sta in incubated class is termed as the incubation period. The incubation period is defined as the time from the eposure to the onset of disease, when the are eposed to infection. The incubation period is useful not onl for making the rough guesses for finding the cause and source of infection, but also in developing treatment strategies to etend the incubation period, and for performing an earl projection of the disease prognosis [12, 16]. In the previous thirt ears, the pattern formation in predator-pre sstems has been studied b man researchers. The spread of diseases in human populations can ehibit large scale patterns, underlining the need for spatiall eplicit approaches. The spatial component of ecological interactions has been identified as an important factor in a spatial world and it is a natural phenomenon that a substance goes from high densit regions to low densit regions. Also, spatial patterns are ubiquitous in nature. These patterns modif the temporal dnamics and stabilit properties of population densities at a range of spatial scales and their effects must be incorporated in temporal ecological models that do not represent space eplicitl. The spatial component of ecological interactions has been identified as an important factor in how ecological communities are shaped [7, 9]. Patterns are present in the chemical and biological worlds and since Turing [19] first introduced his model of pattern formation, reaction-diffusion equations have been a primar means of predicting them. Similarl structured sstems of ordinar differential equations govern the spatiotemporal dnamics of ecological population models, et most of the simple models predict spatiall homogeneous population distributions [5]. It is important to note that the above model takes into account the invasion of the pre species b predators but does not include stochastic effects or an influences from the environment. Nevertheless, a reaction-diffusion equation modeling predatorpre interaction show a wide spectrum of ecologicall relevant behavior resulting from intrinsic factors alone, and is an intensive area of research. An introduction to research in the application of reaction-diffusion equations to population dnamics are available in [8, 15].

3 EJDE-212/21 BIFURCATION AND SPATIAL PATTERN FORMATION 3 These numerical simulations reveal that a large variet of different spatiotemporal dnamics can be found in this model. The numerical results are consistent with our theoretical analsis. It should be noted that, if considered in a somewhat broader ecological perspective, our results have an intuitivel clear meaning. There has been a growing understanding during the past ears regarding the dnamics of real ecosstems. It is important to reveal different spatial dnamical regimes arising as a result of perturbation of the sstem parameters [18, 2]. The objective of the paper is to consider the Bifurcation and spatial pattern formation in a Phtoplankton dnamics. In section 2 and 3, we have developed mathematical model and analzed dnamical behavior of sstem. In section 4, studied the Hopf bifurcation mechanisms analtical and numericall, in section 5, studied the spatial pattern formation and numerical simulation of two dimensional sstems and finall in section 6, we summarize our results and discuss the relative importance of different mechanisms of bifurcation and spatial pattern formation. 2. Mathematical Model Now, we consider the case of viral infection of phtoplankton population, the shifting from susceptible to infected class is not regular. In fact, the susceptible phtoplankton sta for some definite period in the incubated class after leaving the susceptible class and joining the infected class. Taking the population densities of susceptible and infected phtoplankton as P s and P i respectivel, at an instant of time T. The population of susceptible phtoplankton is assumed to be growing logisticall with intrinsic growth rate r and carring capacit K. Now taking P in as the population densit of population in incubated class. Here, we will use nonlinear Holling Tpe II functional responses for disease spreading because the disease conversion rates become saturated as victim densities increase. Let α be the disease contact rate and it is volume-specific encounter rate between susceptible and infected phtoplankton, which is equivalent to the inverse of the average search time between successful spreading of disease. The coefficients δ and β are the total removal of phtoplankton from the infected and incubated class because of the death (including recovered) from disease and due to natural causes respectivel. Again, γ 1 be the fraction of the population recovered from infected phtoplankton population and joined in the susceptible phtoplankton population and β 1 is the fraction of the incubated class population which will move to the infected class. Therefore, quantitativel δ > γ 1 and β > β 1. Keeping in view the above, the mathematical model for a viral infected phtoplankton population with an incubated class is governed b the following set of differential equations: dp s dt = rp s dp in ( P s 1 K ) αp s P i P s γ 1P i, (2.1) dt = αp sp i P s + 1 βp in, (2.2) dp i dt = β 1P in δp i, (2.3) P s () >, P i () >, P in () >. (2.4) The Holling tpe-ii the functional response αpspi P s+1 is used [4] and man other researchers. In this section, we have studied the dnamical behavior of the sstem

4 4 R. S. BAGHEL, J. DHAR, R. JAIN EJDE-212/21 (2.1)-(2.3), with initial population; i.e., P s () >, P in () >, P i () > and the total population at an instant t is N(t) = P s (t) + P in (t) + P i (t). Now, on rescaling the above sstem (2.1)-(2.3) using the change of variables: = Ps K, = Pin K, z = Pi K, t = rt, we obtain d az = (1 ) + cz, (2.5) dt + 1 d dt = az d, (2.6) + 1 dz dt = d 1 ez, (2.7) where a = αk r, c = γ1 r, d = β r, d 1 = β1 r, e = δ r, () >, () > and z() >. 3. Boundedness of the sstem Now, we will stud the eistence of all possible stead states of the sstem and the boundedness of the solutions. There are three biologicall feasible equilibriums for the sstem (2.5)-(2.7), (i) E = (,, ) is the trivial stead state; (ii) E 1 = (1,, ) is the disease free stead state and (iii) E = (,, z ) is endemic equilibrium state, where = ed d 1 a ed, = e 2 d(ad 1 2ed) (ed d 1 c)(ad 1 ed) 2, z = edd 1 (ad 1 2ed) (ed d 1 c)(ad 1 ed) 2. Further, it is clear from the above epression that E R+, 3 if a > de d 1 > c. Now, we will show that all the solutions of the sstem (2.5)-(2.7) are bounded in a region B R+. 3 We consider the function and substituting the values from (2.5)-(2.7), we obtain w(τ) = (τ) + (τ) + z(τ), (3.1) dw dτ = (1 ) (d d 1) (e c)z. If we choose a positive real number η = min{d d 1, e c}, then dw(τ) dτ + ηw(τ) (1 + η) 2 = f(). Moreover, f() has maima at = (1 + η)/2 and f() (1 + η) 2 /4 = M (sa). Hence, ẇ(τ) + ηw(τ) M. Now, using comparison theorem, as τ, we obtain sup w(τ) = M η. Therefore, (τ) + (τ) + z(τ) M η. and let us consider the set B = {(,, z) R+ 3 : (τ)+(τ)+z(τ) M/η}, hence, The sstem (2.5)-(2.7) is uniforml bounded in the region B R Local stabilit of the sstem. We have alread established that the sstem (2.5)-(2.7) has three equilibrium points, namel, E = (,, ), E 1 = (1,, ) and E = (,, z ) in the previous section. The general variational matri corresponding to the sstem is given b 1 2 az ( +1) a 2 ( +1) + c J = az a ( +1) d 2 ( +1) d 1 e

5 EJDE-212/21 BIFURCATION AND SPATIAL PATTERN FORMATION 5 Now, corresponding to the trivial stead state E = (,, ) the Jacobian J has the following eigenvalues λ i = 1, d, e. Hence, E is repulsive in -direction and attracting in z plane. Clinicall, it means that when there is no susceptible population then, there will be no mass in incubated and in the infected class. Hence, E is a saddle point. The corresponding to the disease free equilibrium point E 1 = (1,, ), we have following λ 1 = 1 and λ 2,3 are the roots of the quadratic equation λ 2 + (d + e)λ + (de ad1 2 ) = when de > d 1a/2, then both the roots have a negative real parts and thus E 1 (1,, ) is locall stable in this case. Further, from the eistence of E and the stabilit condition of E 1, it is clear that the instabilit of the disease free equilibrium will lead to the eistence of the endemic equilibrium. Now, we will eamine the local behavior of the flow of the sstem around the endemic equilibrium E. The characteristic equation corresponding to the equilibrium is where P (λ) = λ 3 + A 1 λ 2 + A 2 λ + A 3 =, (3.2) A 1 = 2 + A 2 = 2 e + 2 d + ed e d A 3 = az ( + 1) 2 + d + e 1, ad 1 ( + 1) + aez ( + 1) 2 + adz ( + 1) 2, eadz ( + 1) 2 + ad 1 ( + 1) 2ad 1 2 ( + 1) acd 1z ( + 2ed ed. + 1) 2 Hence, using Routh-Hurwitz criteria E is locall asmptoticall stable, if A i >, i = 1, 2, 3 and A 1 A 2 > A Hopf-bifurcation analsis In this section, we obtain the Hopf-bifurcation criteria of above sstem (2.5)- (2.7), taking e (i.e., total removal of phtoplankton from the infected population ) as the bifurcation parameter. Now, the necessar and sufficient condition for the eistence of the Hopf-bifurcation can be obtained if there eists e = e, such that (i) A i (e ) >, i = 1, 2, 3, (ii) A 1 (e )A 2 (e ) A 3 (e ) = and (iii) d de (u i), i = 1, 2, 3, where u i is the real part of the eigenvalues of the characteristic equation (3.2). After substituting of the values of A i for i = 1, 2, 3 in equation A 1 A 2 A 3 = and solving it for e, it reduces to where e 7 B 1 + e 6 B 2 + e 5 B 3 + e 4 B 4 + e 3 B 5 + e 2 B 6 + eb 7 + B 8 =, (4.1) B 1 = (d 6 2d 5 d 1 a + 8d 6 d 1 a), B 2 = (2d 4 d 1 2 a 2 + 2d 4 d 1 2 ac 2d 6 d 1 a + 2d 5 d 1 a + d 7 2d 5 d 1 c 12d 5 d 1 2 a 2 16d 5 d 1 2 ac + 8d 2 d 1 a), B 3 = (2d 5 d 1 2 ac 2d 4 d 1 a + 9d 3 d 1 3 a 2 c + 2d 6 d 1 a 4d 4 d 1 2 ac 2d 4 d 1 2 a 2 + 4d 3 d 1 3 a 3 d 5 d 1 2 a 2 + d 4 d 1 2 c 2 + 4d 2 d 1 2 ac 2d 6 d 1 c + 4d 4 d 1 3 a 3 + 8d 4 d 1 3 ac d 4 d 1 3 a 2 c 12d 6 d 1 2 a 2 16d 6 d 1 2 ac),

6 6 R. S. BAGHEL, J. DHAR, R. JAIN EJDE-212/21 B 4 = (2d 4 d 1 3 a 2 c + 2d 6 d 1 2 ac + 2d 6 d 1 2 a 2 4d 5 d 1 2 ac 8d 5 d 1 2 a 2 22d 3 d 1 3 a 2 c + 2d 4 d 1 3 ac 2 + 4d 4 d 1 3 a 3 2dd 1 4 a 3 c 3d 2 d 1 4 a 2 c 2 + 6d 5 d 1 2 a 2 2d 3 d 1 3 ac 2 + d 5 d 1 2 c 2 2d 3 d 1 3 a 3 + 4d 5 d 1 2 ac 12d 3 d 1 4 a 2 c 2 8d 3 d 1 4 a 3 c + 4d 5 d 1 3 a d 5 d 1 3 a 2 c + 8d 5 d 1 3 ac 2 ), B 5 = (9d 5 d 1 3 a 2 c 12d 4 d 1 3 a 2 c + 1d 4 d 1 3 a 3 12d 2 d 1 4 a 3 + 4dd 1 5 a 3 c 2 12d 3 d 1 4 a 3 c + 7d 2 d 1 4 a 2 c 2 8d 4 d 1 3 a 2 c 4d 4 d 1 3 a 3 + 8dd 1 4 a 3 c dd 1 4 a 4 3d 3 d 1 4 a 2 c 2 + dd 1 5 a 4 c 2d 4 d 1 3 ac 2 + d 2 d 1 4 a 4 + 4d 2 d 1 5 a 3 c 2 8d 4 d 1 4 a 3 c 12d 4 d 1 4 a 2 c 2 ), B 6 = (d 4 d 1 4 a 4 4d 3 d 1 4 a 3 c 2d 3 d 1 3 a 4 12d 2 d 1 5 a 3 c 2 2d 5 d 1 4 a 2 c 2 6dd 1 5 a 3 c 2 + d 2 d 1 4 a 3 c + 11d 3 d 1 4 a 2 c 2 + 4d 2 d 1 5 a 4 c d 4 d 1 4 a 2 c 2 d 6 d 1 4 c), B 7 = (4d 3 d 1 5 a 3 c 2 6d 2 d 1 5 a 3 c 2 dd 1 6 a 4 c 2 + d 3 d 1 5 a 4 c + d 1 6 a 4 c 2 ), B 8 = (dd 1 6 a 4 c 2 d 2 d 1 6 a 4 c 2 ). For better understanding of the above result, taking an eample values of the parameters c =.1, d =.11, d 1 =.1 and a = 7.15, we obtain a positive root e =.74 of the quadratic equation (4.1). Therefore, the eigenvalues of the characteristic equation (4.1) at e =.74 are of the form λ 1,2 = ±iv and λ 3 = w, where v and w are positive real numbers. Now, we will verif the condition (iii) of Hopf-bifurcation. Put λ = u + iv in (3.2), we obtain (u + iv) 3 + A 1 (u + iv) 2 + A 2 (u + iv) + A 3 =. (4.2) On separating the real and imaginar parts and eliminating v between real and imaginar parts, we obtain 8u 3 + 8A 1 u 2 + 2(A A 2 )u + A 1 A 2 A 3 =. (4.3) Now, we have u(e ) = as A 1 (e )A 2 (e ) A 3 (e ) =. Further, e = e, is the onl positive root of A 1 (e )A 2 (e ) A 3 (e ) =, and the discriminate of 8u 2 + 8A 1 u + 2(A A 2 ) = is 64A (A A 2 ) <. Here, differentiating (4.3) with respect to e, we have ( 24u A 1 u + 2(A A 2 ) ) du de + ( 8u 2 + 4A 1 u ) da 1 + d de (A 1A 2 A 3 ) =. Now, since at e = e, u(e ) =, we obtain 2u da2 de [du de ]e=e = d de (A 1A 2 A 3 ) 2(A A, 2) de + which ensures that the above sstem has a Hopf-bifurcation. It is shown graphicall in figure Spatiotemporal model Now, we consider the phtoplankton dnamics with movement (i.e., diffusion). Therefore the population densities; i.e., P s, P i and P in become space and time dependent. Keeping in view of the above, our mathematical model can stated b the reaction diffusion sstem u dt auw = u(1 u) u cw + D a 2 u, (5.1) v dt = auw u + 1 dv + D b 2 v, (5.2)

7 EJDE-212/21 BIFURCATION AND SPATIAL PATTERN FORMATION 7 (A) (B) Infected Population Infected Population Incubated Population Susceptible Population.1 Incubated Population Susceptible Population (C) (D) Infected Population Infected Population Incubated Population Susceptible Population.1 Incubated Population Susceptible Population Figure 1. Phase plane representation of three species around the endemic equilibrium, taking c =.1, d =.11, d 1 =.1, a = 7.15: (A) e =.73, (B) e =.74, (C) e =.75, (D) e =.76 w dt = d 1v ew + D c 2 w, (5.3) where (, ) is the position in space for two dimensional on a bounded domain and D a, D b and D c are diffusion coefficients for susceptible phtoplankton, incubated phtoplankton and infected phtoplankton population, respectivel. The no-flu boundar conditions were used. Now, we will eplore the possibilit of diffusion-driven instabilit with respect to the stead state solution, i.e., the spatiall homogenous solution (u, v, w ) of the reaction diffusion sstem. We assume that E is stable in the temporal sstem and unstable in spatiotemporal sstem, which means that the spatiall homogeneous equilibrium is unstable with respect to spatiall homogeneous perturbations. We obtain the conditions for the diffusion instabilit to occur in sstem (5.1)- (5.3), one should check how small heterogeneous perturbations of the homogeneous stead state develop in the large-time limit. For this purpose, we consider the perturbation u(,, t) = u + ɛ ep((k + k)i + λ k t), (5.4) v(,, t) = v + η ep((k + k)i + λ k t), (5.5) w(,, t) = w + ρ ep((k + k)i + λ k t), (5.6) where ɛ, η and ρ are chosen to be small and k = (k, k ) is the wave number. Substituting (5.4)-(5.6) into (5.1)-(5.3), linearizing the sstem around the interior equilibrium E, we obtain the characteristic equation as follows: J k λ k I 2 =, (5.7)

8 8 R. S. BAGHEL, J. DHAR, R. JAIN EJDE-212/21 with 1 D a k 2 2u + au w (1+u ) aw 2 1+u c au 1+u J k = au w (1+u ) + aw 2 1+u (d + D b k 2 au ) 1+u, d 1 (e + D c k 2 ) where I 3 and k are third order identit matri and wave number respectivel. The characteristic equation following form: λ 3 + P 2 λ 2 + P 1 λ + P =, (5.8) where P 2 = (1 d e D a k 2 D b k 2 D c k 2 2u + au w ) (1 + u ) 2 aw 1 + u, ( P 1 = d + e de dd a k 2 + D b k 2 + D c k 2 dd c k 2 D a ek 2 D b ek 2 P = D a D b k 4 D a D c k 4 D b D c k 4 2du 2eu 2D b k 2 u 2D c k 2 u + ad 1u (1 + u ) + adu w (1 + u ) 2 + aeu w (1 + u ) 2 + ad bk 2 u w (1 + u ) 2 + ad ck 2 u w (1 + u ) 2 adw 1 + u aew 1 + u ad bk 2 w 1 + u ad ck 2 w ) 1 + u, ( de dd c k 2 + dd a ek 2 D b ek 2 + dd a D c k 4 D b D c k 4 + D a D b ek 4 + D a D b D c k 6 + 2deu + 2dD c k 2 u + 2D b ek 2 u + 2D b D c k 4 u + ad 1u 1 + u ad 1D a k 2 u 1 + u 2ad 1u u + acd 1u w (1 + u ) 2 adeu w (1 + u ) 2 add ck 2 u w (1 + u ) 2 ad bek 2 u w (1 + u ) 2 ad bd c k 4 u w (1 + u ) 2 acd 1w 1 + u + adew 1 + u + add ck 2 w 1 + u + ad bek 2 w 1 + u + ad bd c k 4 w 1 + u ). According to the Routh-Hurwitz criterium all the eigenvalues have negative real parts if and onl if the following conditions hold: P 2 >, (5.9) P >, (5.1) Q = P P 2 P 1 <. (5.11) This is best understood in terms of the invariants of the matri and of its inverse matri M J M 12 M 13 k = M 21 M 22 M 23, det(j k ) M 31 M 32 M 33 where M 11 = de ad1u (u+1), M 12 = d 1 c ad1u, M 13 = d(c au u+1 ), M 21 = aew (u+1), 2 M 22 = e(1 2u aw (u+1) ), M 2 23 = ( au (u+1)(1 2u aw (u+1) ) aw 2 (u+1) (c au 2 (u+1) )), M 31 = ad1w (u+1), M 2 32 = d 1 (1 2u aw (u+1) ), M 2 33 = d(1 2u aw (u+1) ). Here, 2 matri M ij is the adjunct of J k.

9 EJDE-212/21 BIFURCATION AND SPATIAL PATTERN FORMATION 9 We obtain the following conditions of the stead-state stabilit (i.e. stabilit for an value of k): (i) All diagonal cofactors of matri J k must be positive. (ii) All diagonal elements of matri J k must be negative. The two above condition taken together are sufficient to ensure stabilit of a give stead state. It means that instabilit for some k > can onl be observed if at least one of them is violated. Thus we arrive at the following necessar condition for the Turing Instabilit [13]: (i) The largest diagonal element of matri J k must be positive and/or (ii) the smallest diagonal cofactor of matri J k must be negative. B the Routh-Hurwitz criteria, instabilit takes place if and onl if one of the conditions (5.9)-(5.11) is broken. We consider (5.1) for instabilit condition: P (k) = D a D b D c k 6 (D a D b a 33 + D b D c a 11 + D a D c a 22 )k 4 + (D a M 11 + D b M 22 + D 3 M 33 )k 2 det J. According to Routh-Hurwitz criterium P (k 2 ) < is sufficient condition for matri J k being unstable. Let us assume that M 33 <. If we choose D a =, D b = and P (k 2 ) = D c M 33 k 2 det(j k ) = [dd c k 2( ) 1 2u aw 1 (1 + u ) 2 + (1 + u ) 2 (1 + u )(2u 1) (ed + D c dk 2 + edu + D c du k 2 ad 1 u ) (5.12) ] + (edaw + D c daw k 2 cad 1 ) <. Hence, in this sstem diffusion-driven instabilit occur. Now, we obtain the eigenvalues of the characteristic equation (5.8) numericall of the spatial sstem (5.1)-(5.3). Here, we choose some parametric values of a =.4, c =.1, d =.11, d 1 =.1, e =.8. In this set of values P (k 2 ) <, for all k >, hence from (5.12), we can observe diffusion driven instabilit of the sstem (see Fig.2) Pattern formation. Now, we will stud numerical sstem (5.1)-(5.3) for the pattern formation of two dimensional space with zero-flu boundar conditions is used. We choose the initial spatial distributions of each species are random and the numerical results are obtained using finite difference method for figure 3-4 and we use the parametric values same as above section. Now, we obtain that the spatial distributions of phtoplankton dnamics in the time evaluation in figures 3-4. B varing coupling parameters, we observed that one parameter value change then spatial structure change over the times of the spatial sstem. In figure 3-4 have observed well organized structures for the spatial distribution population also observed that time T increase from 1 to 3 the densit of different classes of population become uniform throughout the space. Finall, all these figure are shown that the qualitative changes spatial densit distribution of the spatial sstem for the each species. Conclusion. In this paper, we have studied a phtoplankton dnamics with viral infection. We observed that in a three dimensional phtoplankton sstem with the Holling-II response function, there eit Hopf bifurcation with respect to remove rate including nature death of infected phtoplankton. In the qualitative analsis, we

10 1 R. S. BAGHEL, J. DHAR, R. JAIN EJDE-212/21.15 (a) 1.5 (b) D a =, D b =, D c = D a =.1, D b =, D c =.1 ma Re(λ(k)).5 ma Re(λ(k)).5 D a =.5, D b =, D c =.1 D a =.9, D b =, D c = k k 1.2 (c) 1.5 (d) 1 ma Re(λ(k)) D =, D =, D =.1 b a c D =.6, D =, D =.1 b a c D =1., D =, D =.1 b a c ma Re(λ(k)) 1.5 D =.3, D =, D = c a b D =.6, D =, D = c a b D =.9, D =, D = c a b k k Figure 2. Plot of ma Re(λ(k)) against k. The other parametric values are given in tet (a) 1 (d) 1 (g) (b) 1 (e) 1 (h) (c) 1 (f) 1 (i) Figure 3. Spatial distribution of susceptible phtoplankton [first column], incubated phtoplankton [second column] and infected phtoplankton [third column] population densit of the model sstem (5.1)-(5.3). The diffusivit coefficient values are: D a =.2, D b =, D c = 5. Spatial patterns are obtained at different time levels: for plot T = 1 (a, b, c), T = 4 (d, e, f), T = 8 (g, h, i) studied the boundedness, dnamical behavior and local stabilit of the sstem. It is established that the rate of total removal of phtoplankton from the infected class; i.e., e, crossed its threshold value, e = e, then susceptible, incubated and infected

11 EJDE-212/21 BIFURCATION AND SPATIAL PATTERN FORMATION 11 (a) (b) (c) 1 1 (d) 1 1 (e) 1 1 (f) (g) (h) (i) Figure 4. Spatial distribution of susceptible phtoplankton [first column], incubated phtoplankton [second column] and infected phtoplankton [third column] population densit of the model sstem (5.1)-(5.3). The diffusivit coefficient values are: D a =.2, D b =, D c = 5. Spatial patterns are obtained at different time levels: for plot T = (a, b, c), T = (d, e, f), T = 3 (g, h, i) phtoplankton population started oscillating around the endemic equilibrium. The above result has been shown numericall in figure 1 for different values of e. In particular, in figure 1(A), we observed that the endemic equilibrium was stable, when e <.74, but when it crossed the threshold value of e =.74, the above sstem showed Hopf-bifurcation, shown in figure 1(B, C, D). We have also observed spatiall ordered structures of patterns in spatial sstems and the solutions of the spatial sstem converges to its equilibrium as time T increase from 1 to 3 in the two-dimensional pattern formation, shown in figures 3-4. It is numericall established that with slight change in a time T parameter of the sstem (5.1)-(5.3), can lead to dramatic changes in the qualitative behavior of the sstem. References [1] B. Dube, B. Das, J. Hussain; A model for two competing species with self and cross-diffusion, Indian Journal of Pure and Applied Mathematics 33 (2), no. 6, [2] S.A. Gourle; Instabilit in a predator-pre sstem with dela and spatial averaging, IMA journal of applied mathematics 56 (1996), no. 2, [3] F. M. Hilker, H. Malchow, M. Langlais, S.V. Petrovskii; Oscillations and waves in a virall infected plankton sstem:: Part ii: Transition from lsogen to lsis, Ecological compleit 3 (6), no. 3, 28. [4] C. Holling; Some characteristics of simple tpes of predation and parasitism, Can. Entomol. 91 (1959),

12 12 R. S. BAGHEL, J. DHAR, R. JAIN EJDE-212/21 [5] P. P. Liu, Z. Jin, Pattern formation of a predator pre model, Nonlinear Analsis: Hbrid Sstems 3(3) (9), no. 3, [6] H. Malchow, F. M. Hilker, R. R. Sarkar, K. Brauer; Spatiotemporal patterns in an ecitable plankton sstem with lsogenic viral infection, Mathematical and computer modelling 42 (5), no. 9-1, [7] A. B. Medvinsk, S. V. Petrovskii, I. A. Tikhonova, H. Malchow, B.L. Li, Spatiotemporal compleit of plankton and fish dnamics, Siam Review 3(44) (2), [8] A. Morozov, S. Petrovskii, B. L. Li; Bifurcations and chaos in a predator-pre sstem with the allee effect, Proceedings of the Roal Societ of London. Series B: Biological Sciences 271 (4), no. 1546, 147. [9] J. D. Murra; Mathematical biolog ii: Spatial models and biomedical applications, 3rd ed., in: Biomathematics, Springer, New York, 3. [1] S. Petrovskii, H. Malchow; Wave of chos: new mechanism of pattern formation in spatiotemporal population dnamics., Theor. Pop. Bio. 59 (1), [11] S. V. Petrovskii, H. Malchow; Wave of chaos: new mechanism of pattern formation in spatiotemporal population dnamics, Theoretical Population Biolog 59 (1), no. 2, [12] M. A. Pozio; Behaviour of solutions of some abstract functional differential equations and application to predator-pre dnamics, Nonlinear Analsis 4 (198), no. 5, [13] H. Qian, J. D. Murra; A simple method of parameter space determination for diffusiondriven instabilit with three species, Appl. Math. Lett. 14 (3), [14] S. Ruan, X. Z. He; Global stabilit in chemostat-tpe competition models with nutrient reccling, SIAM Journal on Applied Mathematics 31 (1998), [15] H. K. Pak, S. H. Lee, H. S. Wi; Pattern dnamics of spatial domains in three-trophic population model, Journal Korean Phsical Societ 44(3) (4), no. 1, [16] R. A. Satnuoian and M. Menzinger; Non-turing stationar pattern in flow-distributed oscillators with general diffusion and flow rates, Phsical Review. E 62(1) (), [17] B. K. Singh, J. Chattopadha, S. Sinha; The role of virus infection in a simple phtoplankton zooplankton sstem, Journal of theoretical biolog 231 (4), no. 2, [18] G. Q. Sun, G. Zhang, Z. Jin, L. Li; Predator cannibalism can give rise to regular spatial pattern in a predator pre sstem, Nonlinear Dnamics 58 (9), no. 1, [19] A. M. Turing; The chemical basis of morphogenesis, Philos. Trans. R. Soc. London B 237 (1952), [2] J. Xiao, H. Li, J. Yang, G. Hu; Chaotic turing pattern formation in spatiotemporal sstems, Frontiers of Phsics in China 1 (6), no. 2, Randhir Singh Baghel School of Mathematics and Allied Science, Jiwaji Universit, Gwalior (M.P.)-47411, India address: randhirsng@gmail.com Jodip Dhar Department of Applied Sciences, ABV-Indian Institute of Information Technolog and Management, Gwalior-4741,India address: jdhar@iiitm.ac.in Renu Jain School of Mathematics and Allied Science, Jiwaji Universit, Gwalior (M.P.)-47411, India address: renujain3@rediffmail.com