non-linear systems in plankton population dynamics and nitrogen cycles

Mathematical Methods in the Applied SciencesMath. Meth. Appl. Sci., 23, 1491}1511 (2000)MOS subject classi"cation: 92; 92 B

Non-linear Systems in Plankton PopulationDynamics and Nitrogen Cycles

Hiroshi Morimoto*

Mathematical Sciences Group, School of Informatics and Sciences, Nagoya University,Nagoya 464-01, Japan

Communicated by B. Brosowski

In this paper we propose a new perspective of population dynamics of plankton, by considering some e!ectsof global ecological cycles, in which a mixed population of plankton is embedded. The propagation ofplankton is extremely in#uenced by various material cycles, such as Nitrogen cycles. Taking this globale!ect into consideration, we will construct a mathematical model of non-linear system.

Our model is a non-linear, non-equilibrium system based on a stochastic model realizing populationdynamics of a mixed population of two species of plankton which is placed in a global nitrogen cycle. Weshow, in this article, that our model gives a new mathematical foundation of phenomena such as waterblooms and the predominance of one type of plankton against the other. We calculate the probability of theoccurrence of the water bloom of a mixed population and that is where one type of plankton predominates.

We show, as a characteristic feature of our model, that the function of predominance has somediscontinuity and that there exists a threshold point among the initial values, with respect to the type ofplankton that predominates the other. In other words, there is a sort of phase transition in dynamic changesof plankton population, as a result of global ecological cycles. Copyright ( 2000 John Wiley & Sons, Ltd.

KEY WORDS: water bloom; plankton; nitrogen cycle; non-linear; stochastic process

1. Introduction

Nitrogen cycle is one of the material cycles involved in the birth and death of plantsand animals, the air environment, the accumulation process into soils (or oceans), andhuman activities resulting in industrial waste. After the death of plants and animals,ammonia NH

3is produced by decomposition. Ammonia is also generated from

wastes (or excreta) of many animals through decomposition. There are nitrite-formingbacteria, called nitrite bacteria in soils and oceans which translate NH~

3to NO~

2.

There are other kind of bacteria, called nitrate bacteria which produce NO~3

fromNO~

2. These NO~

3or NO~

2return to plants and plankton as nutrients. This results in

*Correspondence to: Prof. Dr. Hiroshi Morimoto, Nagoya University, School of Informatics and Sciences,Mathematical Sciences Group, Nagoya 464-01, Japan.E-mail: [email protected]

Received 30 May 1997Copyright ( 2000 John Wiley & Sons, Ltd. Revised 20 May 1999

one nitrogen cycle.

plants and animals P NH~3

C BNO~

3Q NO~

2

There are other bacteria found in soils and oceans interacting with nitrogen cycles.Denitrifying bacteria produce gaseous N

2or N

2O by the reduction of nitrite and

nitrate, the path of which is

NO~3PNO~

2PNOPN

2OPN

2

Losing the balance of nitrogen cycles has various unfavourable e!ects on environ-ment. For example, it will bring about the problem of enriched lakes. It may producetoo much N

2O, and may result in the destruction of ozone barrier. Water bloom can

be regarded as both a result and a cause of the loss of balance in nitrogen cycles.Finding out the mechanism of the formation of water bloom (or red tide) is becomingmore and more important nowadays.

Theoretical aspects of physical environment of water blooms were "rst studied bythe method of di!usion equations. For example, in [6], they compared the multiplica-tion rate of plankton and the di!usion rate. If the multiplication rate is strictly lessthan the di!usion rate then water blooms never occur. Thus they gave the criticaldi!usion rate by di!usion equations of Fick type.

But the di!usion rate in still oceans are normally small if there is no turbulencee!ect (such as windrows). To compensate this di$culty, the dynamic theory ofaccumulation points was invented using Langmuir circulation (see, [12, 2, 14, 5] ).Floating particles on the surface of water is accumulated to some points by windrows.Theoretically, this explains the formation of patches of plankton. Although it isdi$cult to observe and record precisely such circulation in oceans, it can be statedthat these theories assume that simple di!usion process is not su$cient to bring aboutwater blooms. Planktons surely need some turbulence e!ect other than di!usion toburst into blooms.

The ecological aspect of water blooms has been studied from experimental methodsusing cultivated plankton. ElbraK chter [3] experimented using mixed cultures of twospecies of plankton, Prorocentrum micans and Gymnodinium splendens. He observedthe existence of some nutrient competition relation among these two species. Besides,he noted the existence of obstruction by one species against the other due to metabolicproducts. G. splendens hindered the growth of P. micans.

The experiments in high-density cultured medium of algae was reported beforeElbraK chter [3] by Pratt [13]. The di!erence is that no growth retardant by metab-olites was found in Pratt's experiments. Using natural ocean water, he culturedSkeletonema costatum and Olisthodiscus luteus, and examined their growth interaction.He observed that these algae could not burst at the same time. In the culture mediumwith high density of O. luteus, the growth of S. costatum was curbed to a great extent.On the contrary, in the culture medium with low density of O. luteus, the growth ofS. costatum was promoted.

1492 H. Morimoto

Math. Meth. Appl. Sci., 23, 1491}1511 (2000)Copyright ( 2000 John Wiley & Sons, Ltd.

In view of the above experiments the following two points are to be noted (by Prattand ElbraK chter). Firstly, even under low-density population the burst can eventuallyhappen. Secondly, there is no need for growth retardant metabolite for one species ofalgae to reach ultimate "xation against the other. These two phenomena will beproved mathematically in the present paper. Our mathematical model is a non-linear,open and non-equilibrium model. Non-linearity implemented in our mathematicalmodel will turn out to be essential in proving the above two phenomena.

The predominance by only one type of algae was also studied by Shapiro [15]. Hetested the assumption that blue}green algae are more e$cient at obtaining CO

2from

low concentrations than green algae, and that under circumstances when the pH ishigh, as in the enriched lakes, blue}green algae should predominate. He subjecteda mixed population of algae in a series of polyethylene bags. The point is that Carbondioxide gas continuously bubbled at the bottom of the bags. This causes arti"cialcirculation in the bags. He observed a shift from blue}green algae to green algae, andvice versa. A slight change of the environment such as the density of CO

2and the pH

results in a rapid shift to the predominance by one type of algae.In the present paper, collecting all these experimental results and observations, we

make a mathematical model which underlines the mechanism of the occurrence ofwater bloom in a mixed population of plankton.

We describe, here, our model to a certain extent. We imagine an area in some lakeor ocean containing a mixed population of two types of algae or plankton. Wesuppose that there is circulation in the area. This assumption is natural as observedabove (cf. [15] ). There are many other reasons for the rationality of this assumption.At the mouth of the river there is a #ow from the lower ocean water. The #ow causedby wind results in rising water #ow. Convection of water causes stirring upside down.

We suppose the existence of a nitrogen cycle involving nitrite and nitrate bacteria,predatory animals or plankton, supply from industries, and the plankton in concern.At enriched lakes or the mouth of the river, there may be su$cient supply of nitrogenwhen water blooms really occur. Therefore, it is natural to suppose enriched property(of nitrogen) for the occurrence of water bloom.

In our model, these assumptions, the existence of circulation and a nitrogen cycle,are not independent. Circulation is supposed to stir the water and give a chance forthe plankton to meet NO~

3randomly. The assumption is natural for the following

reason. Imagine that the bottom of the area is not #at but curved (see Fig. 1(b) ).Nutrients will be accumulated at such bottom areas. Biological decomposition due tonitrite or nitrate bacteria produces gas and hence results in the stirring of the watervertically. See Fig. 1(a), where the bottom is #at. For such type of area, water bloomseldom occurs. The bottom of Fig.1(b) is rather round and the organic compounds canbe accumulated in the bottom. If the temperature increases then the organic com-pounds in the bottom decomposes resulting in various gases such as CO

2and N

2, as

in Fig. 1(c). Then we can say that this causes a random stirring of water. In sucha situation, water bloom is observed quite often. Thus this circulation will mix bothnitrogen and plankton randomly. In this sense the circulation in our model may becalled stirring or shu/ing.

In the present paper we will compute the probability of the predominance of onetype of plankton against the other according to our mathematical model. The

Non-linear Systems and Dynamic Cycles 1493


Fig. 1

probability depends on the initial balance of the two types of plankton. Actually, wewill determine completely this function (see section 4). We show that, as a result ofnon-linearity of the system, this function has a sort of discontinuity, which may becalled transition point (see Figs. 4 and 5).

2. Stochastic model and evolutional equation

Since a simple di!usion process in lakes or oceans is not su$cient to cause planktonblooms, many substitutes, such as Langmuir circulation, were investigated (see

1494 H. Morimoto


[12, 2, 14, 5] ). Even in experimental researches, some extra turbulence was needed tobe added in the cultured medium. For example, Shapiro [15] used CO

2gas to stir the

contents arti"cially. In the present paper, we suppose for this purpose that randomstirring (or random shu%ing) occurs in some area in the lakes or oceans. This type ofstirring takes place due to the gas produced by bacteria (such as nitrite or nitratebacteria) at the bottom by the decomposition of the accumulated organisms.

We suppose that there are two species of algae or plankton, say, type A and B. Thee$ciency of nitrogen metabolism by these plankton may be equal or almost equal.Plankton may be preyed by some predatory animals (or other plankton). But wesuppose that the area of lakes or oceans in concern are enriched and that the supply ofnitrogen is su$cient. For simplicity, the revenue and expenditure of nitrogen in thearea is balanced (i.e. equal). Let k denote the rate of the death of plankton of bothtypes due to natural or predatory cause. We denote by M the number of nitrogen inthe area of water in concern.

In order to make a relation of nitrogen metabolism and plankton propagation to bemathematically clear, we suppose the followings: Let M

1and M

2be the number of

nitrogens caught by the groups of plankton of type A and B, respectively (at sometime t ), and let P

1, P

2be the numbers of the new generations of plankton of each

type. Then we de"ne

a"P1

M1

, b"P

2M

2

(1)

which is supposed to be constant and called the birth rate. Since the number ofnitrogen is large compared with the amount of plankton, it is reasonable to countM by some unit, for example, by the amount one plankton needs for consumption.

Nitrogen will be caught by plankton if it is su$ciently close to the plankton.Therefore, there is some radius with the centre in the plankton such that if nitrogen isin the sphere of this radius then it will be taken in. This suggests that the randomcirculation or stirring may be regarded as discrete. Thus, we will consider a discreterandom displacement for both nitrogen and plankton.

In this regard, instead of the spheres of small radius, it will be convenient toconsider, approximately, small imaginary cubes of the same volume. Precisely speak-ing, let< denote the volume of the area in concern in lakes (or oceans) and let d denotethe volume of small cubes such that if nitrogen and plankton are placed in the samecube then the nitrogen will be absorbed by the plankton. Thus, we will considerdiscrete random displacement of plankton and nitrogen distributed over totallyR"</d cubes.

Let X and > be the total amount of plankton of type A and B, respectively, at time,say, t. Since our model is based on the in#uence of the nitrogen cycle, we will adopt asdensities of plankton those measured with respect to the amount of nitrogen. Let M bethe total amount of nitrogen in the area concerned. As M, we count both the amountof nitrogen in the area and that in the plankton. Note that the amount of nitrogen inthe area (this is the usual sense of amount of nitrogen) is not assumed to be constant. Itvaries according to the consumption by plankton. We only assume that M is keptconstant due to the nitrogen cycle. We measure M by some unit which is needed for



Fig. 2

a single plankton, because if we count it by a unit of one molecule then M will beextremely large compared to the plankton population (and not appropriate to thetheoretical treatment).

We de"ne the share of plankton by

x"X

M, y"

>

M. (2)

(Notice that x#y is not necessarily equal to 1.) Then x"x (t) and y"(t) changerandomly as the time goes. Thus, mathematically, x"x (t) and y"(t) are said to berandom variables for any t (where time t is considered to be discrete). Mathematically,our model can be said to form a Markov chain.

We begin our mathematical arguments with some observation of probabilisticbehaviour of the variables x and y as the time goes. Suppose that the shares of bothplankton are x"x (t) and y"y(t) at some time t, and "x them. Then the shares of thenext generations x (t#*t ), y (t#*t ) change randomly due to random shu%ing of thearea (here we suppose *t"1 for simplicity). Since we "x x and y, we can say that theincrease (or decrease) of shares

x(t#*t)!x, (3)

y(t#*t)!y (4)

are changing randomly. We focus our attention on the probabilistic behaviour ofthese values rather than x (t#*t ), y(t#*t ) themselves.

1496 H. Morimoto


Given t, x"x (t) and y"y(t) , We try to "nd the statistics of randomly changing (3)and (4). Some value of (3) (or (4)) may occur frequently and some other value mayoccur rarely. Therefore, we can mention on the probability that (3) (or (4)) takes somegiven value. Let *x be an arbitrary real number. Let us denote by

Px,y(*x), (5)

the probability that (3) takes the value *x:

x(t#*t)!x"*x . (6)

Here *x is nothing but a (random) variable and may be written simply like z, etc, butwe used * to emphasize its meaning, increase (or decrease).

Similarly,

Px,y (*y) (7)

is de"ned as the probability that (4) becomes *y, given t, x"x (t), y"y (t):

y(t#*t )!y"*y . (8)

The means of *x and *y, denoted by Mdx and Mdy , are de"ned as

Mdx"+*x

Px,y (*x)*x,

Mdy"+*y

Px,y (*y)*y.

Here we use the notation dx instead of *x, because we will later use continuousequations. Notice that Mdx , Mdy are functions of x, y and t.

Similarly, we can de"ne the variance of *x and *y as follows:

<dx"+*x

Px,y(*x) (*x!Mdx)2 ,

<dy"+*y

Px,y (*y) (*y!Mdy)2 .

These are nothing but the means of (*x!Mdx)2 and (*x!Mdx)2, respectively.Since these random variables dx, dy are not independent to each other, their

covariance also makes sense, which is de"ned by

=dxdy"+*x

Px,y (*x) (*x!Mdx) (*y!Mdy) .

This is equal to the mean of the product (*x!Mdx) (*x!Mdx).We have thus de"ned Mdx , Mdy , <dx , <dy ,=dxdy with respect to the distribution of

*x and *y. All of these are functions of x and y. We will compute these functions in thefollowing lemma. Its proof is purely combinatorial and is based on counting all theinstances of the plankton and Nitrogen meeting each other caused by random shift,taking into consideration the death rate and the birth rate.



Lemma 2.1

Mdx"(a#k)Mx (1!x!y)

R!kx,

Mdy"(b#k)My (1!x!y)

R!ky,

<dx"M2 (!1#x#y) (x#y!1#R/M) (a#k)2x (x!R/M)

R2(R!1),

<dy"M2 (!1#x#y) (x#y!1#R/M) (b#k)2y (y!R/M)

R2(R!1),

=dxdy"M2 (!1#x#y) (x#y!1#R/M) (a#k) (b#k)xy

R2 (R!1).

Proof. The proof will be accomplished by purely combinatorial arguments, countingthe number of instances of the plankton and nitrogen meeting each others. Let X and> be the number of plankton of type A and B, respectively, at some time, say, t ("xed).After a random stirring (or shu%ing), some plankton absorbs nitrogen and some donot. Suppose for instance that the number of plankton of type A and B succeeds inmeeting nitrogen are Z

1and Z

2. Then from the de"nition of birth rate, the number of

plankton of type A (or B) which are newly born will be c1Z

1(or c

2Z

2). Then the

increase of X and > are

*X"c1Z

1!k (X!Z

1) ,

*>"c2Z

2!k(X!Z

2) ,

where k is the death rate.As mentioned before, the statement of the lemma, *x and *y are random variables,

is, they vary due to random stirring. Therefore *X and *> are also random variables.The randomness comes from the randomness of Z

1and Z

2. Hence we can question

the probability of the meeting numbers becoming Z1

and Z2, which will be denoted

by

fX,Y

(Z1, Z

2) ,

where X, > are considered to be "xed.Let us denote the means of *X and *> by

M*X , M*Y .

By de"nition, we can write

MdX (X, >)" +Z

1, Z

2

(c1Z

1!k (X!Z

1) ) f

X,Y(Z

1, Z

2) ,

MdY (X, >)" +Z

1,Z

2

(c2Z

2!k (X!Z

2) ) f

X,Y(Z

1, Z

2) .

1498 H. Morimoto


We will compute the probability fX,Y

(Z1, Z

2) by purely combinatorial way. Note

that the number of all the possible displacement of X plankton into R sectionsamounts to (R

X)"R!/X! (R!X)!. For each of these displacements, the possible

number of Z1

plankton of type A selected which happens to meet Nitrogen is equal to(XZ

1). After selecting X chambers for plankton of type A, only R!X chambers are left

to place plankton of type B. Therefore, the number of type B selected becomes (R~XY

).For each selected > plankton, the number of Z

2plankton selected which can meet

with Nitrogen becomes ( YZ2

). Then R!(X#>) chambers are left open and will beselected for the remaining Nitrogen. The number of Nitrogen is M!(X#>#Z

1#Z

2). (Here we count the number of Nitrogen by units, which each plankton need

to survive. Therefore, we must subtract X#> from the total number of Nitrogenwhich are already taken by plankton.)

Note that when we count the possible above selection, total number (or basenumber) of all the possible instances is equal to

AR

XBAR!X

> BAR

M!(X#>B(namely, all the possible selection of plankton and Nitrogen). Thus, we get thefollowing formula:

fX,Y

(Z1, Z

2)"A

R

XBAX

Z1BA

R!X

> BA>

Z2BA

R!(X#> )

M!(X#Z1#>#Z

2)B

AR

XBAR!X

> BAR

M!(X#>)B

"

AX

Z1BA>

Z2BA

R!X!>

M!X!Z1!>!Z

2B

AR

M!X!>BTherefore, in order to compute the means MdX(X, >) and MdY (X, >), it su$ces to

focus on the factors depending on Z1

and Z2, namely,

+Z

1,Z

2

Z1A

X

Z1BA>

Z2BA

R!X!>

M!X!>!Z1!Z

2B,

+Z

1,Z

2

Z2A

X

Z1BA>

Z2BA

R!X!>

M!X!>!Z1!Z

2B.

These can be calculated by the standard argument of binomial or multinomialdistribution. In fact, by comparing the coe$cients of both the sides of the polynomial

xRRx (1#x)X (1#y)>0 (1#z)R~X~Y ,



we obtain

+Z

1,Z

2

Z1A

X

Z1BA>

Z2BA

R!X!>

M!X!>!Z1!Z

2B"XA

R!1

M!X!>!1B ,

+Z

1,Z

2

Z2A

X

Z1BA>

Z2BA

R!X!>

M!X!>!Z1!Z

2B">A

R!1

M!X!>!1B .

For the calculation of the variances and the covariance, it su$ces to use

CxRRxAx

RRx h (x, y, z)BD (1#x)X (1#y)>0 (1#z)R~X~Y .

This completes the proof of Lemma 2.1. K

Let t (p, q, x, y; t) denote the probability that the densities of plankton of type A andB become x and y, respectively, at time t, given that the initial densities are p and q (i.e.,at time 0). Then by the standard argument of Markov chain, we see that t (p, q, x, y; t)satis"es the following di!erential equation called Kolmogorov forward equation (seefor the forward equation, (3.23) in [8] and (15) of Section 25 in [10] ):

Rt(p, q, x, y; t)

Rt "

1

2

R2Rx2<dxt#

1

2

R2Ry2<dyt#

R2RxRy=dxyt

!

RRxMdxt!

RRy Mdyt.

In the above equation, p and q are considered to be constant, and x and y are tobe the variables. Reversing the direction backward, we can regard p, q as variables andx, y as constants. Then we have the following backward equation called theKolmogorov backward equation (or Kimura's equation). (See for the backwardequation those referred in Remark 3.3).

Rt(p, q, x, y; t)

Rt "

1

2<dpR2Rp2

t#

1

2<dqR2Rq2

t#=dpqR2RpRq t

#MdpRRp t#Mdq

RRqt . (9)

Kimura (for example [8, 9] ) considered mainly one-dimensional case of theabove backward equation, and deduced his neutral or almost neutral theory for geneevolution.

3. Probability of the occurrence of water blooms of mixed species

In this section we estimate the probability of the occurrence of water bloom. Herethe water bloom contains two species of plankton, and we do not want to argue, inthis section as to which type will dominate the bloom. Such problem of dominance by

1500 H. Morimoto


one type will be discussed in sections 4 and 5. Therefore, the water bloom consideredin this section is the bloom caused by the mixture of both types.

Since we are interested in the interaction between the Nitrogen cycle and plankton,we will mean, by the water bloom, the case that all the Nitrogen in the area concernedwill be consumed by plankton in the area. This is the case that the sum of the shares ofplankton, x#y, will become 1.

Denote by h (w, t) the probability that the water bloom of the mixture occurs at timet starting from the initial data w ("x#y) at t"0. We de"ne the probability of thewater bloom of the mixture (i.e. the probability that w becomes one) by

h(w)"limt?=

h (w, t) .

We have de"ned the birth rate a and b for plankton of type A and B respectively (seethe equation (1) for the de"nition). We suppose a)b in this section. Let us imagine,for the moment, the case where all the plankton are of type A. Let f (w, t) denote theprobability that the water bloom by only type A occurs at time t (namely x"1 at timet), starting from the initial share w ("x (0) ). Similar to h(w), we de"ne

f (w)"limt?=

f (w, t) .

Now, consider the case where both types of plankton exist. Because a)b, we have

h (w, t )*f (w, t) (10)

and

h(w)*f (w).

Therefore if we calculate f (w) then this will give a lower estimate for h (w). In theremainder of this section we compute f (w).

We "rst estimate the birth rate of a mixed species of (x, y) (at every time t):

ax#by

x#y A"aX#b>

M B*ax#ay

x#y*a .

If we consider the sole plankton (type A) then we should write its density by x. Butwe write it again by w ("x#0), keeping the virtual mixture in mind, because it is thesame as the case of the mixture of type A and B of the same birth rate a.

Recalling the formula for the mean of the mixture from the previous section:

Mdw"M (1!x!y) (ax#by#k (x#y) )

R!k (x#y) .

Then, analogously, we have

Mdw"M(1!w) (aw#kw)

R!kw

"

M(a#k)

RwA1!

Rk

M(a#k)!wB . (11)



Let the mean of w be denoted by wN . Then wN is approximated by the logistic equation

dwNdt

"

M(a#k)

RwN A1!

Rk

M(a#k)!wN B .

Thus,

wN P1!Rk

M(a#k).

As for the variance, we have similarly the following:

<dw"(k#a)2w (w!1) (Mw!M#R) (!R#Mw)

(R!1)R2. (12)

Then f (w, t) satis"es the following backward equation:

Rf (w, t)

Rt "

1

2<dwR2 f (w, t)

Rw2#Mdw

Rf (w, t)

Rw . (13)

See Remark 3.3 for the sign of the left-hand side of this equation.Put

'(w)"d2f/dw2

df/dw.

Then from (11)} (13) we obtain

'(w)"!2Mdw<dw

"

2R(R!1)

(k#c)2(Rk!Ma!Mk#Maw#Mkw)

(w!1) (Mw!M#R) (Mw!R).

We divide '(w) into partial fractions and obtain

' (w)"l1

k1!w

!

l2

w#k2

!

l3

k3!w

, (14)

wherek1"1,

k2"

R

M!1,

k3"

R

M,

l1"

2R(R!1)

(k#a)2k

R!M,

l2"

2R(R!1)

(k#a)2a

2R!M,

l3"

2R(R!1)

(k#a)2(2R!M)k#(R!M)a

(R!M) (2R!M).

1502 H. Morimoto


Since we may suppose that R*M we can see, in (14), that

k1!w'0, w#k

2'0, k

3!w'0

for 0)w)1.Hence by integrating (14) we get

df

dw"C

(k3!w)l3

(k1!w)l1 (w#k

2)l2

(15)

for some constant C.We will integrate (15) over w3[0, 1]. We "rst notice that the right-hand side of (15)

contains a singularity at w"1 if l1*1. Therefore if we suppose l

1(1 then we can

integrate (15).Normalizing the integral ( f (1)"1) we "nally obtain the probability of water bloom

of the mixture

f (w)"

:w0

(k3!w)l3

(k1!w)l1 (w#k

2)l2

dw

:10

(k3!w)l3

(k1!w)l1 (w#k

2) l

2dw

. (16)

Combining (10) and (16) we can give an estimate of the probability that the waterbloom of a mixed population of plankton occurs, in the following theorem (cf. [11]):

Theorem 3.1. ¸et w be the initial density of a mixed population of plankton. Suppose thatl1(1. ¹hen the probability, h (w), that the water bloom of a mixed population of

plankton occurs satis,es

h(w)*

: w0

(k3!w)l3

(k1!w)l1 (w#k

2)l2

dw

: 10

(k3!w)l3

(k1!w)l1(w#k

2)l2

dw

.

Remark 3.2. Note that the assumption l1(1 is satis"ed if k is su$ciently small. For

example, consider the case a"1.0, k"0.003, R"100, M"10. We "nd l1"0.656

and the assumption is satis"ed. For this case the probabilities of "xation aref (0.1)"0.88 and f (0.2)"0.99. For the case, a"1.0, k"0.00004, R"10 000,M"1000, we get l

1"0.89 and we "nd again that the assumption is satis"ed.

Remark 3.3. The Kolmogorov backward equations, such as (13) or (9), have the plussign in the left-hand sides. Some other equations of the same name may have theminus sign. Look for the plus sign, (3.27) in [8], (8) of section 4.25 in [10] and (1.7) ofsection 5.1 in [7]. Look for the minus sign, (7.21) in section 5.7 in [7]. The principle ofthe sign selection can be explained as follows. The Kolmogorov backward equationsare derived from Chapman}Kolmogorov equations concerning the transition prob-abilities. Let p (t

1, x; t

2, y) denote the transition probability from x at time t

1to y at

time t2. If we "x y and t

2and consider the derivatives (of the distribution) by x and t

1,



then we have the minus sign in the Kolmogorov backward equation. On the otherhand, if we "x y and t

1and consider the derivatives by x and the time di!erence

t"t2!t

1(i.e., t

2) then we have the minus sign. Notice that in [7] there are both types

of equations as referred above.

4. Probability of the predominance by one type of plankton

In the previous section we have estimated the probability that the water bloom ofa mixed population takes place. But experimentally, it is observed in many instancesthat only one type of plankton bursts and that the growth of the other type ofplankton is obstructed (see [3, 13, 15] ).

In this section we will study a mathematical background of this phenomenon ofpredominance, and show that a purely mathematical argument can prove the pre-dominance by only one type of plankton. It will be shown that a very subtle di!erenceof nitrogen metabolism is su$cient to lead the mixed population to the predominanceof only one type. This phenomenon is interesting because it is quite similar to theKimura's neutral and almost neutral theory of gene evolution [9]. This phenomenonis a re#ection of non-linearity and feedback e!ect inherent in our mathematicalmodel.

We recall Kolmogorov backward equation (9) where x, y are constants and p, q arevariables. The function t (p, q, x, y; t) denotes the probability that the densities ofplankton of type A, B become x, y at time t, starting from p, q (at time t"0). Since weare interested in the probability of the predominance by one type, say, type B, againsttype A, we set

x"0,

x#y"1,

that is

x"0,

y"1.

From now on we set

u(p, q, t)"t (p, q, 1, 0; t ).

Replacing the coordinates to x and y from p and p by

pPx,

qPy,

we obtain from (9) the following di!usion equation of two variables x and y:

RuRt"

1

2<dxR2Rx2

u#1

2<dyR2Ry2

u#=dxyR2RxRy u#Mdx

RRx u#Mdy

RRy u .

1504 H. Morimoto


In order to show the predominance by type B it su$ces to prove that the di!erenceof populations of each type, y!x, tends to 1. Therefore the following co-ordinatechange will be useful.

w"x#y,

z"y!x. (4.1)

We now derive the backward equation for the random variable z, which will turnout to be time-inhomogeneous equation.

Lemma 4.1.

Mdz"M(1!w)

R Cb (w#z)

2!

a (w!z)

2#kzD!kz ,

<dz"M2(w!1) (w!1#R/M)

R2 (R!1) C(a#k)2 (w!z)

2 Aw!z

2!

R

MB#

(b#k)2 (w#z)

2 Aw#z

2!

R

MB!(a#k) (b#k) (w2!z2)

2 D .

Proof. From the co-ordinate transformation (4.1) we have

Mdz"Mdy!Mdx ,

<dz"<d(y~x)

"E ((dy!Mdy)!(dx!Mdx) )2

"E( (dy!Mdy)2)!2E( (dy!Mdy) (dx!Mdx) )#E( (dx!Mdx)2)

"(<dy!2=dxdy#<dx).

Then, from Lemma 2.1, it follows that

Mdz"M(1!x!y)

R(by!ax#k(y!x) )!k(y!x) ,

<dz"M2 (!1#x#y) (x#y!1#R/M)

R2(R!1)

]C(a#k)2x Ax!R

MB#(b#k)2y (y!R/M)!2 (a#k) (b#k)xyD .

The lemma 4.1 follows directly from the above equations.As per the arguments in the previous section, the water bloom of predominance,

u(z, t), must satisfy the equation

RuRt"

1

2<dzR2uRz2#Mdz

RuRz .

Note that the densities x, y, z, w are not those at time t but at the initial time, since theequation is the backward equation.



Putting

RuRt"0,

RuRz"/,

we obtain

/@/"!

2Mdz<dz

.

We compute

g(z)"!

2Mdz<dz

.

Observe that g (z) can be written as

g(z)"Kh(z)"Kn(z)

d(z)"K

Cz#D

Ez2#Fz#G,

where

K"

4R(R!1)

(w!1) (Mw!M#R),

C"!2Mk#2Rk!Ma!Mb#Mbw#2Mkw#Maw,

D"Maw!Mbw#Mbw2!Maw2 ,

E"2Mab#4Mak#4k2M#a2M#b2M#4bkM,

F"!2b2R#4akR!2a2Mw#2b2Mw#4bkMw

#2a2R!4bkR!4Makw,

G"!2b2Rw!4k2Rw#b2Mw2!2a2Rw!4bkRw

#a2Mw2!4akRw!2Mabw2 .

We now decompose g(z) into partial fractions. Note that the discriminant of d(z) isequal to

4R(a#2k#b)2 [ (a!b)2R#4abMw#4akMw#4k2Mw#4bkMw]'0.

Then we see that d(z) has two distinct real roots, say, a, b with a'b. These roots arecalculated as follows:

a"(b!a) (1!rw)#Ja2#4arbw#4rakw!2ab#4k2rw#4bkrw#b2

(a#2k#b) r,

b"(b!a) (1!rw)!Ja2#4arbw#4rakw!2ab#4k2rw#4bkrw#b2

(a#2k#b) r,

where r"M/R.

1506 H. Morimoto


Here we notice that

a'w, b(!w,

because

d(w)"d(!w)"4w(b#k)2 (Mw!R)(0

from M)R, w(1.Therefore g (z) can be represented as

g(z)"K

E

Cz#D

(z!a) (z!b).

Using the formula of partial fractions

Cz#D

(z!a) (z!b)"

Ca#D

(a!b) (z!a)!

Cb#D

(a!b) (z!b),

we obtain

g(z)"!

c1

a!z#

c2

z!b,

where

c1"

K

E

Ca#D

a!b,

c2"

K

E

C(!b)!D

a!b.

Since a'w and b(!w, we observe that

a!z'0, z!b'0

for !w)z)w.Since g (z)"/@// , we have

log/"c1log (a!z)#c

2log (z!b).

Hence we obtain

RuRz"C

0(a!z)c1 (z!b)c2 .

Integrating this equation and normalizing by : w~w

u (z) dz"1, we "nally obtain thefollowing theorem which gives the probability of the occurrence of water bloom byone type of plankton.

Theorem 4.2. ¸et the sum of the initial populations be w("x0#y

0) and let the initial

superiority of the population (of type B against A) be z("y0!x

0). ¹hen the probability



u(z) (DzD)w) of the predominance by plankton of type B is equal to

u(z)": z~w

(a!z)c1(z!b)c2 dz

: w~w

(a!z)c1 (z!b)c2 dz.

Remark 4.3. By a simple calculation, we see that the integrand in Theorem 4.2,

(a!z)c1 (z!b)c2, !w)z)w

takes its maximum at z"m,

m"

c2a#c

1b

c1#c

2

.

The values of the function are concentrated on the value z0

for large M, namely, thefunction takes always zero away from m. This will turn out to be a certain thresholdpoint (or phase transition point) in the next section.

5. Discussion and conclusions

The present article constructed a stochastic system of a mixed population ofplankton of two species. With respect to environmental conditions, we assumed thatthe system is contained in a nitrogen cycle and that the area is randomly stirred(mainly vertically). The "rst assumption represents the enriched property of lakes andoceans. The latter was observed frequently in the occurrence of water bloom and wasalso simulated in experiments (such as [15] ).

First, we have calculated the probability of the occurrence of water bloom ofa mixed population. Second, we have computed the probability of the predominanceby one type of plankton against the other when water bloom really occurs.

We will present here some examples to "gure out the meaning of the formulaeobtained in this article. We begin with a purely neutral case, where there is nodi!erence of superiority (or inferiority) among both the types of plankton, i.e., a"b.We set

a"0.5, b"0.5, k"0.00004, M"1000, R"10,000, w"0.5,

where we supposed that the initial population (as a mixture) is w"0.5 and that thewater bloom of the mixture occurs. In Fig. 3 the horizontal axis z represents howmuch the population of plankton of type B is larger than that of type A (z"y!x). Ifz"0 then both the populations are equal at t"0. The vertical axe u represents theprobability of the predominance by type B. Therefore u (0)"0.5 implies that theprobability that the plankton of type B dominates the whole area is 50 percent, giventhat both the populations are equal at the beginning (as easily imagined).

Second, we consider the case, where there is slight superiority (or inferiority) amongthe types of plankton, i.e., say, a(b.

a"0.6, b"0.9, k"0.00004, M"10,000, R"1,000,000, w"0.3.

1508 H. Morimoto


Fig. 3

This is the case that the growth rate of type B is superior to that of type A, namelyb'a. We can observe that discontinuity of predominance occurs starting from a verysmall amount of both types of plankton (in this example, the initial value w is setto 0.3).

More extreme case is the following third case, where we put

a"0.2, b"1.0, k"0.00004, M"10,000, R"100,000, w"0.3.

In this case we can observe that, even if the population of a mixture is small at thebeginning, the water bloom occurs, and that the water bloom is completely dominatedby a single plankton.

Moreover, in view of Figs. 3}5, it should be recognized that the predominanceprobability function u (z) is almost discontinuous and that there is a sort of thresholdfor z (the initial balance of populations) depending on the type of plankton thatpredominates the other. This threshold corresponds to the value m in Remark 4.3. Thevalue m becomes, for each example,

m"0,

m"!0.06,

m"!0.2.



Fig. 4

Fig. 5

1510 H. Morimoto


6. Conclusion

The discontinuity observed above suggests that there may be a rapid shift of thepredominance by one type of plankton against the other, when some environmentalparameters change. In fact, this explains the Shapiro's observation of a spectacularshift of the dominance by blue}green algae against green algae [15].

We can extend the material cycles to include predatory plankton or animals. It willbe very interesting to take such predatory animals into consideration and constructa theory of ecosystem involving both predatory relation and material cycles.

Moreover, an ecosystem including denitrifying bacteria will be a very importantproblem in future, in view of global ecological problems such as the destruction ofozone barrier, because of the dynamic nature of interaction between air and ocean. Itcan be expected that the non-linear, open systems, which have been discussed in thisarticle, will also be e!ective in such "elds.

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non-linear systems in plankton population dynamics and nitrogen cycles

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