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Mathematical Biosciences 245 (2013) 126–136
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Mathematical Biosciences
journal homepage: www.elsevier .com/locate /mbs
Phytoplankton–zooplankton dynamics in periodic environments takinginto account eutrophication
0025-5564/$ - see front matter � 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.mbs.2013.06.002
E-mail address: [email protected]
Jinhuo LuoCollege of Information Technology, Shanghai Ocean University, 201306 Shanghai, China
a r t i c l e i n f o
Article history:Received 29 August 2012Received in revised form 28 May 2013Accepted 3 June 2013Available online 20 June 2013
Keywords:Holling IIIPredator–preyAlgal bloomEutrophicationPhytoplankton–zooplankton
a b s t r a c t
In this paper, we derive and analyze a mathematical model for the interactions between phytoplanktonand zooplankton in a periodic environment, in which the growth rate and the intrinsic carrying-capacityof phytoplankton are changing with respect to time and nutrient concentration. A threshold value: ‘‘Pred-ator’s average growth rate’’ is introduced and it is proved that the phytoplankton–zooplankton ecosystemis permanent (both populations survive cronically) and possesses a periodic solution if and only if thevalue is positive. We use TP (Total Phosphorus) concentration to mark the degree of eutrophication. Basedon experimental data, we fit the growth rate function and the environmental carrying capacity functionwith temperature and nutrient concentration as independent variables. Using measured data of temper-ature on water bodies we fit a periodic temperature function of time, and this leads the growth rate andintrinsic carrying-capacity of phytoplankton to be periodic functions of time. Thus we establish a periodicsystem with TP concentration as parameter. The simulation results reveal a high diversity of populationlevels of the ecosystem that are mainly sensitive to TP concentration and the death-rate of zooplankton. Itillustrates that the eruption of algal bloom is mainly resulted from the increasing of nutrient concentra-tion while zooplankton only plays a role to alleviate the scale of algal bloom, which might be used toexplain the mechanism of algal bloom occurrence in many natural waters. What is more, our results pro-vide a better understanding of the traditional manipulation method.
� 2013 Elsevier Inc. All rights reserved.
1. Introduction
Algae are actually the most well known of a group of organismscalled phytoplankton. Algae can best be described as small ormicroscopic plants. These organisms are photosynthetic, meaningthat they function as plants, producing their own food from sun-light. Phytoplankton are the basis of most aquatic food chains,and are one of the primary producers of the oxygen we breath.Zooplankton are the heterotrophic (sometimes detritivorous) typeof plankton. Plankton are organisms drifting in the water column ofoceans, seas, and lakes. The name of zooplankton is derived fromthe Greek zoon, meaning ‘‘animal’’, and planktos, meaning‘‘wanderer’’ or ‘‘drifter’’. Many zooplankton are too small to be seenindividually with the naked eye. Zooplankton feed on bacterio-plankton, phytoplankton, other zooplankton (sometimes cannibal-istically), detritus (or marine snow) and even nektonic organisms,and zooplankton are primarily found in surface waters where foodresources (phytoplankton or other zooplankton) are most abun-dant. Through their consumption and processing of phytoplankton(and other food sources), zooplankton play an important role inaquatic food webs, both as a resource for consumers on higher tro-
phic levels (including fish), and as a conduit for packaging the or-ganic material in the biological pump. Since they are typically ofsmall size, zooplankton can respond relatively rapidly to increasesin phytoplankton abundance, for instance, during the springbloom. The enrichment of waters by inorganic plant nutrients iscalled eutrophication.
It is the consensus of experts that eutrophication has relevanteffects on water bodies: the main are algal blooming, excessiveaquatic macrophyte growth and oxygen depletion. Further conse-quences for human activities are: the decrease of water quality,aesthetic flow and navigation water problems and extinction insome water bodies of some oxygen depending organisms or ani-mals. To the phytoplankton ecologist, the role of eutrophicationin harmful algal bloom dynamics may seem to be primarily a ques-tion of understanding the effect of changing nutrient concentra-tions and nutrient ratios on the tendency of various toxic orotherwise harmful phytoplankton species to form blooms [25].Some experts examined the relationship between Total Phospho-rus (TP) concentration and algal bloom frequencies by observeddata [13,14,32]. A better understanding of the mechanisms thattrigger the occurrence or explain the absence of a phytoplanktonbloom is of considerable interest. A description of plankton popu-lations in the framework of an excitable model was first proposedby Truscott and Brindley [28]. It combines a logistic growth of
J. Luo / Mathematical Biosciences 245 (2013) 126–136 127
Holling-type III [15] and a standard linear zooplankton mortality.The model possesses a non-trivial stable fixed point (besidesextinction of zooplankton or of both populations). The stability ofthe non-trivial fixed point means that there is no possibility tooccur an oscillation of populations. But Truscott and Brindleynoticed that a rapid increase in the phytoplankton birth-rate mayalso trigger a bloom. So in their work they considered a linear increaseof the birth rate (up to a saturation) and from numerical simula-tions found a critical value for the rate of increase, i.e., the slope,beyond which blooms occurred. However, they never extendedtheir discussion to an explicit investigation of the seasonal forcingincluding aspects of timing or effects of fluctuations. Later Freund[11] fill the gap, presenting and discussing simulation results of theseasonally forced Truscott–Brindley (TB) model, and found dynam-ical bistability (bloom or non-bloom models), importance of timing(asymptotic behavior dependent on the phase of the annual cycle),and a noise-induced switching between both modes.
But another critical factor, the nutrient concentration, thatinfluence the growth rate and carrying capacity has been ignored.It is many researcher’s common sense that high nutrient levels andfavorable temperature conditions play a key role in rapid or mas-sive growth of algae and low nutrient concentration as well asunfavorable temperature conditions inevitably limit their growth[9,10]. The rates of phytoplankton growth may often be well bal-anced in oligotrophic, open ocean waters, leading to relativelysmall changes in phytoplankton biomass over time. This may beespecially true in the subarctic Pacific and other high-nutrient,low-chlorophyll regions of the world’s oceans [22,20]. But in otherareas such as temperate coastal waters and lakes, phytoplanktongrowth rate and the environmental biomass may present vast scalefluctuation in different season and nutrient concentration.Although many experts have judged that the excessive nutrientconcentration caused by eutrophication of water-body is the mainreason of algal bloom, few of them have an explanation from theaspect of evolutionary mechanism. The influence of phosphorusand temperature on algal growth rate and environmental biomasswere examined in laboratory [12]. Results from this experimentshow that both of the growth rate and environmental biomasschange with temperature and nutrient concentration, thus bothof them can be seen as function of two variables: time and nutrientconcentration.
The use of biomanipulation as tool for water managementstarted in the 1970s in small lakes and because of its success itwas also applied in large lakes in the 1990s. Shapiro and Wright[23] originally defined biomanipulation as ‘‘the deliberate exploi-tation of the interactions between the components of the aquaticecosystem in order to reduce the algal biomass’’. It was knownfrom work by Hrbcek et al. [17] and Brooks & Dodson [2] thatplanktivorous fish affect the zooplankton community and indi-rectly influence the phytoplankton community. Until the 1980secosystem thinking was largely in terms of bottom-up effects inthe food chain, particularly as in the 1970s the phosphates werefound to play a major role in the process of eutrophication. Lakeand reservoir managers concluded that the only way back wasthe reduction of the phosphate loads.
The present paper aims to establish a time-varying model to re-flect the dynamic evolution of phytoplankton and zooplanktonwhile taking into account nutrient concentration. The remainderof this paper is organized as follows: in Section 2, we briefly recallthe formulation of the TB model and some results of previous stud-ies. In Section 3 we fit the curved surfaces of growth rate and car-rying capacity based on previous experimental data and build aperiodic system with TP (Total Phosphorus) concentration asparameter, and then we present a theoretical analysis on the sys-tem. In Section 4, for different parameter values, simulation resultsof the corresponding system are given and discussed. The simula-
tion results reveal that excessive nutrient concentration is an arch-criminal of algal bloom, and they give an explanation of the tradi-tional manipulation. We close with a summary and some out looksin Section 5.
2. The T–B model
The phytoplankton–zooplankton model introduced by Truscottand Brindley [28] consists of two differential equations:
dPdt¼ rP 1� P
K
� �� RmZ
P2
a2 þ P2 ; ð1Þ
dZdt¼ cRmZ
P2
a2 þ P2 � lZ; ð2Þ
where P and Z represents the populations of phytoplankton andzooplankton respectively.
The first term in the right side of Eq. (1) is the logistic growthfunction,with a maximum/nature growth rate r, and an environ-mental carrying capacity K.The second term in the right side ofEq. (1) is called Hollings Type-III grazing function [15,16], whereRm represents the maximum specific predation rate and a governshow quickly that maximum is attained as prey densities increase.The constant c in Eq. (2) represents the ratio of biomass consumedto biomass of new herbivores produced.
As a predator–prey model, it was discussed by many research-ers [7,8,19,26,29]. We conclude some results as follows:
System (1) and (2) has two equilibrium points (more exactlysaddle points), E0ð0;0Þ and E1ðK;0Þ. In case
l < cRm and K > P� , affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
lcRm � l
r; ð3Þ
the third equilibrium EðP�; Z�Þ appears in the regionfðx; yÞjx > 0; y > 0g. Condition l < cRm insures that Zooplanktonhave a chance to survive in case they have enough food, andK > P� makes sure the environmental carrying capacity of phyto-plankton population reaches the need of survival of zooplankton. If
ð2l� cRmÞK < 2lP� ð4Þ
holds, the positive equilibrium EðP�; Z�Þ is globally asymptoticallystable [26]. It means that both of phytoplankton and zooplanktonwould reach a stable population amount only if they have positiveinitial populations. In this circumstances, algal bloom would nothappen. We quote typical parameter values [28,30,31]:
K ¼ 108 lg N=l; r ¼ 0:3=day; Rm ¼ 0:7=day;a ¼ 5:7 lg N=l; l ¼ 0:012=day; and c ¼ 0:05: ð5Þ
System 1 and 2 is called a T–B model when parameter valuesare taken from these in (5). One can verify that condition (3) and(4) are satisfied by these parameter values, so the correspondingsystem is asymptotically stable. It means that with any positiveinitial population level, each of the populations of phytoplanktonand zooplankton will eventually tend to a stable value, namelyoscillation is impossible. However, Truscott and Brindley reflectedthat the natural growth rate of phytoplankton increases with risingtemperature. Based on these biological facts, they took r as
r ¼min r0 þdrdt
t; rmax
� �;
where r0 ¼ 0:4/day, rmax ¼ 0:6/day, and referring to Uye [30] tookdr=dt as
drdtw 0:06=ðday
�CÞ:
Such they got an oscillating picture (See Fig. 1).
0 100 200 300 400 500 600 700 8000
1000
2000
t days
phyt
opla
nkto
n
0 100 200 300 400 500 600 700 8000
200
400
t days
zoop
lank
ton
Fig. 1. Small simulation picture of the T–B model as r changes with temperature. Longitudinal unit: lg/l.
20
22
24
26temperature function
128 J. Luo / Mathematical Biosciences 245 (2013) 126–136
From Fig. 1, we see that the concentration of phytoplankton in-creases rapidly in a short time-interval and the peak value reaches3� 107 cells/l or so (since every algae stem cell weights approxi-mately 5� 10�8 mg; or 5� 10�5 lg). It exceeds the value1:5� 107 cells/l which marks the occurrence of algal bloom [1].This gives an explanation for the algal bloom.
0 50 100 150 200 250 300 350 4008
10
12
14
16
18
days
tem
pera
ture
Fig. 2. Small Temperature function.
3. A time-varying P–Z model
Traditionally when we talked about natural growth rate we of-ten neglect that it closely relates on temperature and nutriture. Weoften see that lakes near farmland or utility area have morechances to cause harmful algal blooms than those who are away,and that in spring and summer algal booms occur far more fre-quently then in winter. These phenomena tell us that seasonsand nutrient concentrations play an important role in algal blooms.So the two factors, time and nutrient concentration should be con-sidered in our model. Models Ignoring these factors could not re-flect the algal bloom phenomena objectively. In this section wewill reconstruct the T–B model to build a time varying P–Z modelwhich takes into account nutrient concentration.
3.1. Temperature changes with seasons
Temperature in nature environment varies as season changes,and year after year appears a periodic variation. Table 1 is anobservation of the temperature in QianDao lake in south China in2010. Using the average temperature data, we fit a temperaturefunction:
T ¼ 7:1798 � sinp
180� t þ 3:6824
� �þ 16:8641; ð6Þ
where t represents time (days) and TðtÞ represents temperature ontime t.
From Fig. 2 we see that the fitting is effective. Since naturallytemperature varies regularly with the season, we can view thefunction of both growth rate and carrying capacity as periodicfunction of time approximately.
Table 1Water temperature in Qiandao lake in 2010.
Month Jan Feb Mar Apr May Ju
Chishan 11.5 8.8 10.7 13.6 16.6 2Wenxindao 11.1 8.7 9.9 11.6 15.4 1Houdao 11.3 8.9 9.9 11.9 15.8 1Mushan 11.6 9.4 10.6 11.9 14.4 2Mishan 11.6 9.4 9.7 10.7 13.1 1Average 11.4 9.0 10.2 11.9 15.1 1
3.2. Curved surface of growth rate and carrying capacity
In the traditional Logistic model the growth rate r is often re-garded as a constant named intrinsic rate of natural increase. How-ever, an unassailable truth is that the growth rate r is often affectedby factors of temperature and nutriture, etc. For example, in springand summer most woody plants have a vigorous growth, while inautumn and winter they have a poor growth even die. Similarly,growth rate and carrying capacity can be affected greatly by nutri-ture. Many researchers are convinced of these, but few of themtake them together into a model. In 1994, Gao [12] did an experi-ment to detect how temperature and TP (Total Phosphorus) con-centration affect the growth rate and environmental biomass ofMicrocytic Aeruginosa. Data in Table 2 are taken from their exper-iment. We add an assumption that growth rate and environmentalbiomass are zeros when temperature is 0 �C or the TP concentra-tion is zero.
ne July Aug Sep Oct Nov Dec
2.3 23.8 25.4 24.7 22.6 19.5 15.69.1 22.1 24.2 23.5 21.7 19.6 15.89.1 22.2 24.2 23.5 21.9 19.7 15.80.2 23.6 22.8 23.7 22.1 20.9 16.47.1 18.1 22.2 19.7 18.4 18.1 15.69.6 22.0 23.8 23.0 21.3 19.6 15.8
Table 2Growth rate and carrying capacity in different temperature and TP concentration.
T �C s = 0 s = 0.0018 s = 0.018 s = 0.18 s = 1.8
r K r K r K r K r K
0 0 0 0 0 0 0 0 0 0 015 0 0 0.123 340 0.207 1060 0.282 1220 0.28 129820 0 0 0.151 342 0.211 2077 0.329 2660 0.331 226425 0 0 0.176 309 0.253 2187 0.341 4933 0.345 430530 0 0 0.176 319 0.247 2184 0.349 4120 0.347 397935 0 0 0 0 0 0 0 0 0 0
Where s is the TP concentration (unit: lg/ml), r is the growth rate (unit: /day), and Kis the carrying capacity (unit: � 102 cells=ml).
J. Luo / Mathematical Biosciences 245 (2013) 126–136 129
We see the growth rate and environmental biomass as func-tions of two independent variables: time t and TP concentrations, and we try to fit these functions by data in Table 2. Since theTP concentration varies irregular, we use bezier function to processthe data to obtain gridded data. Then we use radial-basis-functionas interpolating function to fit the bivariate function rðT; sÞ; andKðT; sÞ. For lack of enough space for a detailed description of thefunctions, we only present the fitting surface to give the reader adirect visual effect (See Fig. 3). From Fig. 3 we see that growth rater and environmental biomass K almost achieves maximum valuewhen temperature t take values between 20 �C and 30 �C. In order
00
0.5
1
1.5
20
0.1
0.2
0.3
0.4
Growth ra
TP concentration(mg/l)
Gro
wth
rate
(/da
y)
00
0.5
1
1.5
0
1000
2000
3000
4000
5000
6000
Carrying Cap
TP Concentration ( mg/l )
Car
ryin
g C
apac
ity (
102 c
ells
/l )
Fig. 3. (i) The growth rate surface; (ii) env
to see the situation more clearly, we scaled-down the coordinate ofTP concentration near s ¼ 0 and get Fig. 4. From Fig. 4 we see thatboth r and K have a sharp rise when s increases from 0 to 0.02(every s in the following text takes unit lg/ml). When s exceeds0.02, both r and K have a slow change, and they almost reachmaximum.
3.3. Time varying P–Z dynamics taking into account nutrientconcentration
In order to have a better explanation of the algal bloom, wemodify the P–Z model by setting the function of both growth rateand carrying capacity as time varying ones. What is more, they arealso affected by nutrient concentration in the environment. How-ever in a certain place the nutrient concentration is almost a con-stant, So we view the nutrient concentration as a parameter of thefunctions. We extend Logistic model to a generalized form:
dPdt¼ rðt; sÞP 1� P
Kðt; sÞ
� �; ð7Þ
where s denotes the Nutrient concentration. we assume that rðt; sÞ;and Kðt; sÞ are both periodic in time t, and there exist positive con-stants rm; rM ; Km and KM such that
rm 6 rðt; sÞ 6 rM ; and Km 6 Kðt; sÞ 6 KM:
510
1520
2530
35
Temperature ( o C )
te surface
510
1520
2530
35
Temperature ( o C )
acity Surface
ironmental carrying capacity surface.
05
1015
2025
3035
0
0.002
0.02
0.2
20
0.1
0.2
0.3
0.4
T ( temperature unit:OC)
Growth rate surface
s (TP concentration unit: mg/l)
Gro
wth
rate
( /d
ay)
05
1015
2025
3035
0
0.002
0.02
0.2
20
1000
2000
3000
4000
5000
6000
T ( temperature unit: oC )
Carrying Capacity
s (TP concentration unit:mg/l)
Car
ryin
g ca
paci
ty (
102 c
ells
/l)
Fig. 4. (i) Growth rate surface in scaled nutrient concentration. (ii) Environment biomass surface in scaled-down nutrient concentration.
130 J. Luo / Mathematical Biosciences 245 (2013) 126–136
In view of that Rm; c and l may have a seasonal change, we useRðtÞ; cðtÞ and lðtÞ to replace them respectively, and we assume thatthey are all positive and periodic with a same period x that rðs; tÞand Kðs; tÞ have. Thus we obtain a generalized T–B model,
dPdt ¼ rðt; sÞP 1� P
Kðt;sÞ
� �� RðtÞ P2
a2þP2 Z;
dZdt ¼ cðtÞRðtÞ P2
a2þP2 Z � lðtÞZ:
8<: ð8Þ
In absence of zooplankton, system (8) degenerates to a Logisticperiodic system (7).
3.4. Behavior of system (8)
In order to analyze the behavior of system (8), firstly we con-sider a Logistic system
duðtÞdt¼ uðtÞðaðtÞ � bðtÞuðtÞÞ; ð9Þ
where aðtÞ and bðtÞ are periodic continuous functions on R withcommon period x > 0. From [27] we have the following lemma.
Lemma 3.1. If bðtÞP 0 for all t 2 R andRx
0 bðtÞdt > 0, then Eq. (9)has a unique nonnegative x-periodic solution u�ðtÞ which is globallyasymptotically stable with respect to the positive u- axis, that is
uðtÞ � u�ðtÞ ! 0; as t !1
for any positive solution u (t) of Eq. (9).
Moreover, ifRx
0 aðtÞdt > 0, then u�ðtÞ > 0 for all t 2 R; andifRx
0 aðtÞdt 6 0, then u�ðtÞ � 0.
Now consider the following system,dudt ¼ uðaðtÞ � bðtÞuÞ � RðtÞ un
aðtÞ2þun v ;dvdt ¼ cðtÞRðtÞ un
aðtÞ2þun v � lðtÞv ;
8<: ð10Þ
where aðtÞ; bðtÞ;RðtÞ; aðtÞ; cðtÞ and lðtÞ are nonnegative periodiccontinuous functions on R with common period x > 0; n P 1.
Denote
f ðt;uÞ ¼ aðtÞ � bðtÞu; gðt;uÞ ¼ RðtÞun�1
aðtÞ2 þ un
and taking hðtÞ � 0, then it is easy to verify that fand g satisfy theconditions (A1)–(A7) in [21] (See Section 1 [21]). By Theorem 2.2and Theorem 2.3 in [21], we have the following lemmas.
Lemma 3.2. System (10) is permanent provided thatZ x
0cðtÞ RðtÞðu�Þn
aðtÞ2 þ ðu�Þn� lðtÞ
!dt > 0;
where u� is the unique periodic solution of (9) given by Lemma (3.1).
Lemma 3.3. Suppose thatZ x
0cðtÞ RðtÞðu�Þn
aðtÞ2 þ ðu�Þn� lðtÞ
!dt 6 0;
J. Luo / Mathematical Biosciences 245 (2013) 126–136 131
then
ðiÞ limt!þ1 inf vðtÞ ¼ 0;
ðiiÞ limt!þ1 inf uðtÞ > 0
for any solution of system (10) with positive initial conditions, where u�
is the unique periodic solution of (9) given by Lemma (3.1).
If n P 2, it is easy to verify that gðt;uÞ ¼ RðtÞ uðn�1Þ
aðtÞ2þun satisfies alocal Lipschitz condition with respect to u, thus by Corollary (2.6)in [21], we have
Lemma 3.4. SupposeZ x
0cðtÞ RðtÞðu�Þn
aðtÞ2 þ ðu�Þn� lðtÞ
!dt > 0; n P 2;
then system (10) has a positive x-periodic solution.
Since rðt; sÞ > 0 and Kðt; sÞ > 0 (in natural water environmentboth of temperature and TP concentration are greater than zero),and they are all x-periodic, by lemma (3.1) we have the followingtheorem.
Theorem 3.5. System (7) possesses a unique positive x-periodicsolution which is globally asymptotically stable with respect to thepositive p-axis, that is, there is a positive x- periodic function P�ðt; sÞ,such that for any solution Pðt; sÞ of system (7) with positive initialvalue P (0), we have
limt!þ1ðPðt; sÞ � P�ðt; sÞÞ ! 0:
Since P� is the x-periodic solution of Eq. (7), we have
_P� ¼ rðt; sÞP� 1� P�
Kðt; sÞ
� �;
Integrate both sides of the above equation from 0 to x, we getZ x
0rðt; sÞP�ðt; sÞ 1� P�ðt; sÞ
Kðt; sÞ
� �dt ¼ 0:
Denote Dðt; sÞ ¼ Kðt; sÞ � P�ðt; sÞ, then we haveZ x
0rðt; sÞP�ðt; sÞDðt; sÞ
Kðt; sÞdt ¼ 0: ð11Þ
From (7) when Kðt; sÞ � KðconstantÞ, P� � K is the periodic solu-tion of Eq. (7).
Remark 3.6. When Kð�; sÞ is a non trivial periodic function,obviously it is not a solution of (7) (since _K X 0). By (7), ifP� > Kðt; sÞ then _P� < 0, and thus Pðt; sÞ decreases while tincreases. by (11), P� could not be greater than Kðt; sÞ forever,and there must be a time s in (0, x) such that P�ðs; sÞ < Kðs; sÞ.For the same reason, P�ðt; sÞ could not be smaller than Kðt; sÞforever. Thus P�ðt; sÞ, the periodic solution of equation (7) mustfluctuate around Kðt; sÞ.
System (7) is a Riccatti equation, we can integrate it andobtain the analytic expression of the unique x-periodic solution(See [27]),
P�ðt; sÞ ¼1� exp �
Rx0 rðh; sÞdh
� �Rx
0rðt�h;sÞkðt�h;sÞ exp �
R h0 rðt � s; sÞds
� �dh
By Lemmas 3.2 and 3.3, we have
Theorem 3.7. Let P�ðt; sÞ be the x-periodic solution of system (7),then system (8) is permanent provided that
1x
Z x
0cðtÞRðtÞ ðP�Þ2
a2 þ ðP�Þ2� lðtÞ
!dt > 0 ð12Þ
and in case
1x
Z x
0cðtÞRðtÞ ðP�Þ2
a2 þ ðP�Þ2� l
!dt 6 0; ð13Þ
the population of zooplankton tends to extinction in the future, moreprecisely we have
ðiÞ limt!1ZðtÞ ¼ 0;
ðiiÞ limt!1infPðt; sÞ > 0:
By Lemma (3.4), we have
Theorem 3.8. Under condition (12), system (8) has an x-periodicsolution.
3.5. Predator’s average growth rate
Theorems 3.7 and 3.8 show that the value
F ¼ 1x
Z x
0cðtÞRðtÞ ðP�Þ2
a2 þ ðP�Þ2� l
!dt ð14Þ
is closely related to the behavior of system (8). When F > 0, pop-ulations of both phytoplankton and zooplankton keep oscillatingabove some positive population level. Otherwise when F 6 0,population of zooplankton (the predator) would tend to extinc-tion in the future. Ecologically, P� indicates the population levelof prey species in absence of predator, namely the richness ofprey, and RðtÞ; cðtÞ reflects the predation rate and efficiency offood conversion respectively, and lðtÞ is a mortality rate, andthus F reflects survival condition of predator. The greater the va-lue F, the easier for predator to survive. F 6 0 means that theenvironment could not raise even a very few predator, for thegrowth rate provided by the prey could not offset the death rateof the predator. In addition We see that smaller value of l orgreater value of c would cause greater value of F, which wouldbenefit the survival of predator. We call F the predator’s averagegrowth rate.
4. Simulation results of system (8) and the explanation of algalbloom
We can see these results intuitively by numerical simulations.Given a specific value of TP concentration s, and considering (6),we can get periodic functions of growth rate and carrying capacitywhich have only one independent variable (time t). For example,taking s1 ¼ 0:0018 and s2 ¼ 0:18 respectively, we get correspond-ingr1ðtÞ ¼ rð0:0018; TðtÞÞ; K1ðtÞ ¼ kð0:0018; TðtÞÞ;
r2ðtÞ ¼ rð0:18; TðtÞÞ; K2ðtÞ ¼ kð0:18; TðtÞÞ,Their concrete forms are as following,
r1ðtÞ ¼ 0:01009848428ð0:0001000000000þ T2Þ1=2
� 0:2000000000� 0:00174172364ð1:þ ðT � 15Þ2Þ1=2
þ 0:00023367827ð1:þ ðT � 20Þ2Þ1=2
� 0:00097380076ð1þ ðT � 25Þ2Þ1=2
� 0:02207357553ð1:þ ðT � 30Þ2Þð1=2Þ
þ 0:02595269303ð0:0001000000000þ ðT � 35Þ2Þ1=2;
132 J. Luo / Mathematical Biosciences 245 (2013) 126–136
K1ðtÞ ¼ 2870:194621ð0:0001000000000þT2Þ1=2
�10:0042256ð1:þðT�15Þ2Þ1=2
�9:662916ð1þðT �20Þ2Þð1=2Þ
þ13:300552ð1þðT�25Þ2Þ1=2
�53:5920712ð1:þðT�30Þ2Þ1=2
þ2902:620271ð0:0001000000000þðT�35Þ2Þ1=2�100000;
r2ðtÞ ¼ 0:01527500449ð0:0001000000000þ T2Þ1=2
� 0:2000000000� 0:00514583948ð1:þ ðT � 15Þ2Þ1=2
� 0:00348453283ð1:þ ðT � 20Þ2Þ1=2
þ 0:00413247713ð1:þ ðT � 25Þ2Þ1=2
� 0:04336517197ð1:þ ðT � 30Þ2Þ1=2
þ 0:04415186716ð0:0001000000000þ ðT � 35Þ2Þ1=2;
K2ðtÞ¼2895:958485ð0:0001000000000þT2Þ1=2
þ107:5057045ð1:þðT�15Þ2Þ1=2
þ113:965494ð1þðT�20Þ2Þ1=2
�334:752110ð1þðT�25Þ2Þ1=2
�366:5860116ð1þðT�30Þ2Þ1=2
þ3299:316342ð0:0001000000000þðT�35Þ2Þ1=2�100000;
where T is the function T (t) defined by (6).With these functions substituting into system (7) and (8), we
get corresponding periodic dynamical systems. Since every algaestem cell weighs approximately 5� 10�5 lg; some parameter val-ues in (5) are equivalent to
RðtÞ � R ¼ 0:7=day; cðtÞ � c ¼ 0:05; a ¼ 200 lg=l;l ¼ 0:012=day: ð15Þ
0 100 200 300 40
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
Tim
Phyt
opla
nkto
n ce
ll( u
nit 1
0 2 c
ells
/ml)
Carrying Capacities in D
a solutionSystem(8 (s=0.18)
a solutionof system(8)(s=0.0018)
carrying capacity (s=0.18)
Fig. 5. Figures of environment biomass in different TP concentration (represented in s(represented in dash line).
4.1. The influence of TP concentration towards behaviors of System (7)and system (8)
From theorem (3.5) and remark (3.6), system (7) possesses aperiodic solution which is globally asymptotically stable in the po-sitive axis and it fluctuates around the carrying capacity. It meansthat the long term behaviors of solutions of system (7) with posi-tive initial values are almost determined by the environmental bio-mass. It shows us that, without zooplankton, the reproduction ofphytoplankton would reach the environmental biomass which isa saturation value determined by the environment.
Let parameters in system (7) and (8) take values as in (15). Gi-ven an initial value (say P(0)=1000) and a TP concentration, we canget a numerical solution of system (7). In Fig. 5, the solid curves areenvironmental biomass when TP concentration take values ofs ¼ 0:0018 and s ¼ 0:18; respectively. The dash curves are solu-tions of system (7) with initial value Pð0Þ ¼ 1000; whens ¼ 0:0018 and s ¼ 0:18; respectively.
Simulation results verified our conclusion. From Fig. 5 we seethat either s = 0.18 or s = 0.0018, solutions of system (7) cling tothe solid curves closely and they fluctuates up and down aroundthe solid curves. TP concentration has a decisive influence to theenvironmental biomass. When TP concentration is low, the corre-sponding environmental biomass is also in a low level. whens ¼ 0:0018, the maximum concentration of phytoplankton in waterbody is no more than 5� 104 cells/ml or 2:5 lg/ml, and this meansthat algal bloom could not happen. When s = 0.18, the peak valueof population concentration of phytoplankton is over4:5� 105 cells/ml or 22:5 lg/ml, which marks the occurrence of al-gal bloom.
Fig. 6 is the numerical solution of system (8) when the param-eters take values in (15). The dash curves represent environmentalbiomass. The blue curves and red curves represent populations ofphytoplankton and zooplankton in a solution of system (8) respec-tively. From Fig. 6 we see that the algal cell concentration appearsperiodic changes, and its peak and valley are almost synchronizewith that of the carrying capacity.
Fig. 6-(i) shows the dynamical process of the phytoplankton–zooplankton population evolution by numerical simulation whens = 0.0018, though the phytoplankton population has a eruption,the environmental biomass is less than 4� 107 cells/l. Due to the
00 500 600 700 800e ( day )
ifferent TP Concentration
of)
CarryingCapacity ( s=0.0018)
olid line) and corresponding solutions of system (8) with initial value P(0)=1000
0 500 1000 1500 2000 2500 3000 3500 40000
50
100
150
200
250
300
350
400
450
Time ( day )
Popu
latio
n ( u
nit:
102 c
ells
/ml)
Populatin evolution when s=0.0018.
0 500 1000 1500 2000 2500 3000 3500 40000
500
1000
1500
2000
2500
Time ( day )
Popu
latio
n (u
nit:1
02 cel
ls/m
l)
Population evolution when s=0.018
Carrying Capacity of phytoplankton
Phytoplankton population Zooplankton population
0 500 1000 1500 2000 2500 3000 3500 4000 45000
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
Time ( day )
Popu
latio
n(un
it: 1
02 cel
ls/m
l)
Population evolution when s=0.18
Fig. 6. Algal, zooplankton evolution diagram. (i) s ¼ 0:0018;R ¼ 0:7; c ¼ 0:05;l ¼ 0:012;a ¼ 200. (ii) s ¼ 0:018;R ¼ 0:7; c ¼ 0:05;l ¼ 0:012;a ¼ 200. (iii)s ¼ 0:18;R ¼ 0:7; c ¼ 0:05;l ¼ 0:012;a ¼ 200. Dash lines represent algal cell biomass, blue lines and red lines represent cell concentrations of Blue-green algae andzooplankton in system (8). (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this article.)
J. Luo / Mathematical Biosciences 245 (2013) 126–136 133
existence of zooplankton, the algal cell concentration in waterbody is apparently lower than the environmental biomass. Its peakvalue is no more than 2:5� 104 cells/ml. In this case, the perni-ciousness is not plainly apparent.
Fig. 6-(ii) shows the evolutionary process of populations of phy-toplankton and zooplankton when TP concentration s = 0.018. Nowthe environmental biomass of phytoplankton increases nearly 10times, achieving 2� 105 cells/ml or so. But thanks to zooplankton,
134 J. Luo / Mathematical Biosciences 245 (2013) 126–136
the actual algal cell concentration in water body is apparentlymuch lower than the environmental biomass, in fact it is almostonly ten percent of that, and the peak is 3� 104 cells/ml or so,not much higher than that of the situation when s = 0.0018.
Fig. 6-(iii) shows the evolutionary process of populations ofphytoplankton and zooplankton when TP concentration s ¼ 0:18.Now the peak value of environmental biomass reaches4:5� 105 cells/ml or so, and the algal cell concentration reaches2:5� 105 cells/ml, which is almost ten times of that whens ¼ 0:018, and it surpasses 1:5� 105 cells/ml, which marks theoccurrence of algal bloom (See [1]).
Remark 4.1. Thanks to zooplankton’s phagotrophy the phyto-plankton population decline in a vast scale from a high lever withina short period. So to some extent the zooplankton has scaled downthe density of phytoplankton cells. By constructing the surfaces ofgrowth-rate and biomass our model can explain the basic reason of
0 200 400 600 800 1000 12000
1000
2000
3000
4000
5000µ =0.012
0 200 400 600 800 1000 12000
1000
2000
3000
4000
5000
µ =0.0116
0 500 1000 1500 20000
1000
2000
3000
µ= 0.008
0 500 1000 1500 20000
1000
2000
3000µ= 0.012
0 500 1000 1500 20000
1000
2000
3000µ= 0.016
Fig. 7. The influence of parameter l on the population evolution of phytoplankton and zthe longitudinal represents biomass of phytoplankton or zooplankton (unit:102 cells/ml).line and red line represents the concentration of phytoplankton and zooplankton respectis referred to the web version of this article.)
algal bloom. It indicates that lowering the nutrient concentration isthe radical way to suppress algal bloom. From simulation resultswe also found that the initial population level of both phytoplank-ton and zooplankton does not affect the simulation results muchly.It seems that the system possesses a periodic oscillation which isstable in the positive axis, though from theoretical analysis it ishard to give this conclusion.
4.2. The influence of Zooplankton mortality towards algaereproduction
From (14) we see that smaller value of l would cause greatervalue of F, and it would make the predator easier to survive.Fig. 7-(i) shows the population evolution of system (8) when lchanges from 0.012 to 0.0114 (in case s ¼ 0:18). Fig. 7-(ii) showsthe population evolution of system (8) when s ¼ 0:018. We see that
0 200 400 600 800 1000 12000
1000
2000
3000
4000
5000µ =0.0118
0 200 400 600 800 1000 12000
1000
2000
3000
4000
5000
µ =0.0114
0 500 1000 1500 20000
1000
2000
3000
µ= 0.01
0 500 1000 1500 20000
1000
2000
3000µ= 0.014
0 500 1000 1500 20000
1000
2000
3000µ= 0.018
ooplankton of system (8). In the diagram, abscissa represents time (unit: day), andThe dash line represents the environmental biomass of phytoplankton, and the blue
ively. (For interpretation of the references to colour in this figure caption, the reader
J. Luo / Mathematical Biosciences 245 (2013) 126–136 135
in both cases even a tiny change of l could cause a significantchange in the population evolution. It shows us that the reproduc-tion of phytoplankton might be greatly suppressed by decrease thedeath rate of zooplankton.
4.3. Explanation of the theory of biomanipulation
Biologists have long realized that zooplankton can suppress al-gae bloom. In 1972 Hurlbert [18] found that in artificial pools gam-busia affinis greatly reduced rotifer, crustacean, and insectpopulations and thus permitted extraordinary development ofphytoplankton populations. Shapiro [23,24] found the way to con-trol algal blooms in many lakes by cultivating zooplankton, and heused the word ‘biomanipulation’ firstly to define the way of con-trolling algal bloom. This is the early classic theory ofbiomanipulation.
From the discussion of Section 4.1, we see that by reducing thenutrient concentration, the environmental biomass of phytoplank-ton could be lowered down, So it is the fundamental way to controlalgal bloom. From the discussion of Section 4.2, we see that undercertain nutrient concentration, reducing the death rate of zoo-plankton would scale down the algal bloom. The truth is that quitea few of phytoplankton are eaten off by zooplankton.
Usually the real determining factor to leading to the decision toinitiate a biomanipulation is not quite clear. The most importantcriterion is the level of eutrophication. If the lake has a high con-centration of nutrients even after serious reductions in the loadof nutrients, and high concentrations of algae, in particular cyano-phytes, the lake is considered as a serious candidate for biomanip-ulation. If nutrient reduction has only little or no effect at all on therecovery of the lake, biomanipulation will speed up the process ofrecovery.
5. Summary and out look
Because of ignoring the time-varying nature of growth rate andenvironmental biomass of phytoplankton, traditional autonomousmodel could not explain why algal blooms usually occur in hightemperature seasons. Similarly ignoring the influence of nutrientconcentration towards growth rate and environmental biomassof phytoplankton, we could not explain why algal blooms often oc-cur in areas of eutrophication.
Our model provides a rational explanation to these questionssuccessfully. Firstly it has a breakthrough in the traditional conceptof growth rate and carrying capacity, and it is the key to improvethe traditional model into a more practical one. Whenever thenutrient concentration is low, the corresponding environmentalbiomass is also in a low level, and from the results of the theoret-ical analysis, any solution of the system with positive initial valuesis controlled by the environmental biomass. This explains that un-der low nutrient concentration algal bloom will not happen. Sec-ondly from the analysis of the nonautonomous system, weintroduce a threshold value:‘‘Predator’s average growth rate’’,which not only depends on the richness of food resources, but alsodepends on the viability of the predator. So it can be used to deter-mine whether the system is permanent, and it reveals how themortality rate of zooplankton affects the behavior of the system.Thirdly, from the results of our simulations, when the nutrient con-centration nears a dangerous value, say 0.02 lg/ml, whether a algalbloom would burst is highly sensitive to the mortality rate of thezooplankton, and this gives a better understanding of the tradi-tional bio manipulation method.
Although factors, such as light, nitrogen concentration etc., thataffect growth rate and carrying capacity are not considered in ourmodel, we think our model can reveal the nature of the evolutionmechanism of phytoplankton and zooplankton. Certainly, to build
a more perfect model we have a lot of things to do. Firstly in ourmodel the parameter a in Hollings Type-III function representsthe difficulty level for predator to reach the saturated grazing abil-ity. Smaller a value indicates that the zooplankton is easier to ap-proach its maximum grazing ability. How it affects the dynamicalbehavior of the system is some thing for further study. Make clearthe situation is helpful for the management of water bloom.
Secondly our model is a periodic system, thus it is easier for usto do some theoretical analysis. Even so, we have problems to besolved. Theorem (3.8) tell us that under condition (12) system(8) possesses a periodic solution. Is the periodic solution stable?under what condition it is stable? Though from numerical resultsthe answer seems to be yes, it is not satisfactory without a theoret-ical conclusion.
Thirdly to some extent our model is an idealized one, for exam-ple we see temperature as a periodic function of time, yet in realworld temperature has random changes. In addition, the TP con-centration in lakes are not the same in every year, it is affectedby human behavior. That is why our numerical solutions of themodel appear like periodic and obviously it is not the case in realworld. Thus it can be considered to introduce some stochastic vari-ables in our model, to carry on further theoretical or numericalanalysis.
Finally, some researchers have established the fact that toxicsubstances released by some phytoplankton species have a repul-sive effect on the zooplankton population [3–6]. As a result, thezooplankton may have a higher death rate in the area filled withthe toxic phytoplankton species, and this might lower the zoo-plankton’s inhibitory effect on algal bloom. This also is what ourmodel to be improved.
Acknowledgements
The author is grateful to Prof. Qigen Liu (Shanghai Ocean Uni-versity) for relevant knowledge of algal bloom and the observationdata of water temperature in Qiandao Lake. The author also thanksProf. Jiong Ruan (Fudan University), Dr. Donghua Zhao (Fudan Uni-versity) for bringing attention to the problem of the existence ofperiodic solutions of the model. Finally, I would like to give myvery special thanks to Dr. Rongsong Liu (University of Wyoming)for many useful suggestions, corrections and encouragement.
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