《数学建模》美赛优秀论文：2009 C O Rebalancing Human-Influenced Ecosystems

Rebalancing 121 Rebalancing Human-Influenced Ecosystems YuanSi Zhang ShuoPeng Wang 1g Dept of Mathematics China university of Mining and Technology Advisor: Xingyong Zhang Summary populations. Then, based on the Analytic Hierarchy Process and a competi tionmodel, we obtain the ratio of different species in the second population, water quality satisfactory by adjusting the juality is nothigh, and make the predict that the steady-state level of wat numbers of six species In Task 2, when milkfish farming suppresses other animal species, we set up a logistic model, and predict that the water quality at steady-state is awful, the same as in the fish pens--insufficient for the continued healthy growth of coral species. When other species are not totally suppressed, with an improved predator-prey model we simulate the water quality of Bolinao(makingitmatch current quality), obtainpredictednumbers of pop ulations, and discuss changes to the predator-prey model aimed at making the numbers of the populations agree more closely with observations n Task 3, we establish a polyculture model that reflects an interdepend dent set of species, introduce mussels and seaweed growing on the sides of the pens, and obtain the numbers of populations in steady state and the outputs of or In Tasks 4 and 5, we differentiate the monetary values of different kind edible biomass and define the total value as the sum of the values of each species harvested, minus the cost of milkfish feed. Under circumstances The UMAP Jouna130(2)(2009)121-139. @Copyright 2009 by COMAP Inc. Allrights reserved Permission to make digital or hard copies of part or all of this work for personal or classroom use tt fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice. Abstracting with credit is permitted, but copyrights for components of this work owned by others than COMAP must be honored. To copy otherwise, to republish, to post on servers, or to redistribute to lists requires prior permission from COMAP

Rebalancing 121 Rebalancing Human-Influenced Ecosystems YuanSi Zhang ShuoPeng Wang Ning Cui Dept. of Mathematics China University of Mining and Technology Xuzhou, Jiangsu, China Advisor: Xingyong Zhang Summary In Task 1, we establish a Volterra predator-prey model with three bio￾logical populations, and we specify the steady-state numbers of the three populations. Then, based on the Analytic Hierarchy Process and a competi￾tion model, we obtain the ratio of different species in the second population, predict that the steady-state level of water quality is not high, and make the water quality satisfactory by adjusting the numbers of six species. In Task 2, when milkfish farming suppresses other animal species, we set up a logistic model, and predict that the water quality at steady-state is awful, the same as in the fish pens-insufficient for the continued healthy growth of coral species. When other species are not totally suppressed, with an improved predator-prey model we simulate the water quality of Bolinao (making it match current quality), obtain predicted numbers of pop￾ulations, and discuss changes to the predator-prey model aimed at making the numbers of the populations agree more closely with observations. In Task 3, we establish a polyculture model that reflects an interdepen￾dent set of species, introduce mussels and seaweed growing on the sides of the pens, and obtain the numbers of populations in steady state and the outputs of our model. In Tasks 4 and 5, we differentiate the monetary values of different kinds edible biomass and define the total value as the sum of the values of each species harvested, minus the cost of milkfish feed. Under circumstances The UMAPJournal30 (2) (2009)121-139. QCopyright2009byCOMAP, Inc. All rights reserved. Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice. Abstracting with credit is permitted, but copyrights for components of this work owned by others than COMAP must be honored. To copy otherwise, to republish, to post on servers, or to redistribute to lists requires prior permission from COMAR

122 The UMAP Journal 30.2(2009) model, from which we obtain an optimal strategy and harvest timization of acceptable water quality, we build anonlinear equilibrium of Bolinao. With the ratio between feed cost and net income as the inde In Task 6, we put forward a strategy to improve the water qualit the index value of the model is smaller than that of Bolinao area, which signifies the leverage of the strategy. Also, we analyze the polyculture system in terms of ecology. Introduction To improve the situation in Bolinao, we need to establish a practicable polyculture system and introduce it gradually. So our goal is pretty clear Model the original Bolinao coral reef ecosystem before fish farm intro Model the current bolinao milkfish monoculture. Model the remediation of Bolinao via polyculture. Discuss the outputs and economic values of species write a brief to the director of the Pacific Marine Fisheries council sum- marizing the relationship between biodiversity and water quality for coral growl Our approach is: Deeply analyze data in the problem, gradually establishing a model of he coral reef foodweb Withavailable data as evaluationcriteria, confirmthe water quality based on elements in the sediment Establish models, and interpret the actual situation with data, with the of . Do further discussion based on our work. Solutions Task 1 Aiming toward a coral reef foodweb model, we assume that all the species grow in the same fish pen. We divide the species into three popu- lations one alga species (Population 1);

122 The UMAP journal 30.2 (2009) of acceptable water quality, we build a nonlinear equilibrium optimization model, from which we obtain an optimal strategy and harvest. In Task 6, we put forward a strategy to improve the water quality in Bolinao. With the ratio between feed cost and net income as the index, the index value of the model is smaller than that of Bolinao area, which signifies the leverage of the strategy. Also, we analyze the polyculture system in terms of ecology. Introduction To improve the situation in Bolinao, we need to establish a practicable polyculture system and introduce it gradually. So our goal is pretty dear: "* Model the original Bolinao coral reef ecosystem before fish farm intro￾duction. "* Model the current Bolinao milkfish monoculture. "* Model the remediation of Bolinao via polyculture. "* Discuss the outputs and economic values of species. "* Write a brief to the director of the Pacific Marine Fisheries Council sum￾marizing the relationship between biodiversity and water quality for coral growth. Our approach is: "* Deeply analyze data in the problem, gradually establishing a model of the coral reef foodweb. "* With available data as evaluation criteria, confirm the water quality based on elements in the sediment. "* Establish models, and interpret the actual situation with data, with the purpose of improving water quality. "* Do further discussion based on our work￾Solutions Task 1 Aiming toward a coral reef foodweb model, we assume that all the species grow in the same fish pen. We divide the species into three popu￾lations: e one alga species (Population 1);

rebalance 123 o one erbivorous fish, one mollusc species, one crustacean species, and one echinoderm species(Population 2); and the sole predator species, milkfish(Population 3) The interrelationships among the species are presented in Figure 1 crustacean species specles algae Population I among On this basis, we can establish a Volterra predator-prey model with three populations [Shan and Tang 2007 ]. Let the number of the ith population be t(t). If we do not take into consideration the restrictions of natural resources, the algae species of Population 1 growing in isolation will follow an exponential growthlaw with relative growthrate T1, so that i(t)=r101 However, species of Population 2 feeding on the alga species will decrease the growth rate of the algae, so the revised model of the alga species is 1(t)=x1(r1-1x2) where the proportionality coefficient A1 reflects the feeding capability of species in Population 2 for the alga species Assume that the death rate of the species in Population I is r2 when existing in isolation; then i2(t)=-T2at2, so based on the foodweb we con clude that 2(+)=x2(-T2+A2x1), where the proportionality coefficient A2 reflects the support capability of the alga species for Population 2-which in turn provide food for the milkfish. The milkfish reduce the growth rate of the species in Population 2, so we must subtract their feeding effect to get 2()=2(-T2+A2x1-az3) Likewise, the model for the milkfish is 示a(+)=x3(-T3+g3x2)

Rebalancing 123 "* one herbivorous fish, one mollusc species, one crustacean species, and one echinoderm species (Population 2); and "* the sole predator species, milkfish (Population 3). The interrelationships among the species are presented in Figure 1. Population IIi herbivorous mollusc crustacean eci Populat fsh species sp cies ion ii Population I Figure 1. Interrelationships among three populations. On this basis, we can establish a Volterra predator-prey model with three populations [Shan and Tang 2007]. Let the number of the ith population be xi(t). If we do not take into consideration the restrictions of natural resources, the algae species of Population 1 growing in isolation will follow an exponential growth law with relative growth rate rl, so that ib(t) = rixi. However, species of Population 2 feeding on the alga species will decrease the growth rate of the algae, so the revised model of the alga species is WO= xi(r, -. A1X 2 ), where the proportionality coefficient A1 reflects the feeding capability of species in Population 2 for the alga species. Assume that the death rate of the species in Population II is r 2 when existing in isolation; then i2 (t) = -r 2x 2, so based on the foodweb we con￾dude that :i() X2 (-r 2 + 1\2X1), where the proportionality coefficient A2 reflects the support capability of the alga species for Population 2-which in turn provide food for the milkfish. The milkfish reduce the growth rate of the species in Population 2, so we must subtract their feeding effect to get .2(t) = X2 (-r 2 + A2X1 - IX). Likewise, the model for the milkfish is b3(t) = X3 (-r3 + A3X2)

124 The UMAP Journal 30.2 (2009) Altogether, we have an interdependent and mutually-restricting mathe- matical model of the three populations: 示1()=a1(r1-入12) 示2()=m2(72+21-03), () Since this system of differential equations has no analytic solution, weneed to use Matlab to get its numerical solution. Ecologists point out that a periodic solution cannot be observed in most balanced ecosystems; in a balanced ecosystem, there is an equilibrium In addition, some ecologists think that the long-existing and periodically changing balanced ecosystems in nature tend toward a stable equilibrium; that is, if the system diverges from the former periodic cycle because of disturbance, an internal control mechanism will restore it. However, the periodically-changingstate describedby the Volterramodelisnon-structured stability, and even subtle adjustments to the parameters will change the pe riodic solution So we improve the model by letting the alga species follow logistic growth if in isolation 立1()=nz1(1 N where N, is the maximum population of the alga species allowed by the Population 2, so the model for the algae species is i1()=m11 where N2 is the maximum capacity of the species in Population 2 and o refers to the quantity ofthealgae(compared to N)eatenby theunitquantity species in Population 2(compared to N2) without the algae, the species in Population 2 will perish; let its death rate be r2, so that in isolation we will have (+) The algae provide food for Population 2, so we should add that effect; and the growth of the species in Population 2 is also influenced by internal blocking action; so we get (=(1-+mR)

124 The UMAP Journal 30.2 (2009) Altogether, we have an interdependent and mutually-restricting mathe￾matical model of the three populations: ::l(t) = XI(T1 - AX) &2()= Xr2 (-r 2 + A2X1 - AX) &3 (t) = X3(-r 3 + A3X2)- Since this system of differential equations has no analytic solution, we need to use Matlab to get its numerical solution. Ecologists point out that a periodic solution cannot be observed in most balanced ecosystems; in a balanced ecosystem, there is an equilibrium. In addition, some ecologists think that the long-existing and periodically￾changing balanced ecosystems in nature tend toward a stable equilibrium; that is, if the system diverges from the former periodic cycle because of disturbance, an internal control mechanism will restore it. However, the periodically-changingstate describ edby the Volterramodelis non-structured stability, and even subtle adjustments to the parameters will change the pe￾riodic solution. So we improve the model by letting the alga species follow logistic growth if in isolation: =rix, 1 - where N1 is the maximum population of the alga species allowed by the environmental resources. The alga species provides food for the species of Population 2, so the model for the algae species is N1 X2' where N2 is the maximum capacity of the species in Population 2 and o1 refers to the quantity of the algae (compared to N1) eatenbythe unit quantity species in Population 2 (compared to N2). Without the algae, the species in Population 2 will perish; let its death rate be r 2, so that in isolation we will have: &2(t) = -r 2x 2. The algae provide food for Population 2, so we should add that effect; and the growth of the species in Population 2 is also influenced by internal blocking action; so we get &2 (t) = r2X2 (1_ -X2 + 02 X

Rebalancing 125 where o2 is analogous to 1. Analogously, we can get a full model of the species in Population 2 via (=2(1-n2+an Without the species in Population 2, milkfish will disappear; we set their death rate as T3. The species in Population 2 provide food for the milkfish and the growth of milkfish is also restricted by internal blocking action Here the model is i(t)=T33 +a4忑2 Summarizing, we have simultaneous equations constituting aninterdepen- dent mathematical model for the three populations i1(t)=x1r1 101 N, 十σ ()=T3a3( N34 N2 We obtain the values of some parameters in the model, and through onlinear data fitting of the original data of the local three populations [Shan and Tang 2007; Sumagaysay-Chavoso 1998; Chen and Chou 2001], we get their natural growth rates 05, a3=0.5 N1=150×103,N2=30×103,N3=22×103 According to the volume of local fish pens and relevant materials, we get the original numbers of the three populations z1(0)=1215×103,x2(0)=27×103,x3(0) Then we use Matlab to implement the model, with the results of Figure 2, where we see that can see that with the passage of time, the sit)tend to the steady-state values 69, 027, 27, 015, and 1, 760 The number 27, 015 of the species in Population 2 is made up of herbiv- orous fish, molluscs, crustaceans, and echinoderms. Now we confirm the numbers of all the species in Population 2, which stay at the same trophic level, coexisting and mutually competing

Rebalancing 125 where 0-2 is analogous to 01. Analogously, we can get a full model of the species in Population 2 via t2(t) = r2x2 -1 - f + 0*2f - 0'3 • Without the species in Population 2, milkfish will disappear; we set their death rate as r3 . The species in Population 2 provide food for the milkfish, and the growth of milkfish is also restricted by internal blocking action. Here the model is b3 (t) = r3x 3 X3 + Or4 • Summarizing, we have simultaneous equations constituting an interdepen￾dent mathematical model for the three populations: *ti(t) = x 1 r, 1 ( - 0-1 , X2(t) = r2X2 -1 - 2+ U2 - U3 , 'b3(t) = r3X3 - I - -X3 + U-4 X2 We obtain the values of some parameters in the model, and through nonlinear data fitting of the original data of the local three populations [Shan and Tang 2007; Sumagaysay-Chavoso 1998; Chen and Chou 2001], we get their natural growth rates: u,i = 0.6, u"2 = 0.5, o-3 0.5, o"4= 2; N, = 150 x 10 3, N 2 = 30 x 10 3 , N 3 = 2.2 x 10 3 . According to the volume of local fish pens and relevant materials, we get the original numbers of the three populations: xi(0) = 121.5 x 103, x 2 (0) = 27 x 103, x3 (0) = 2 x 103. Then we use Matlab to implement the model, with the results of Figure 2, where we see that can see that with the passage of time, the xi (t) tend to the steady-state values 69,027, 27,015, and 1,760. The number 27,015 of the species in Population 2 is made up of herbiv￾orous fish, molluscs, crustaceans, and echinoderms. Now we confirm the numbers of all the species in Population 2, which stay at the same trophic level, coexisting and mutually competing

126 The UMAP Journal 30.2(2009) 10° We apply expert system and group decision theory to determine the weights of the species in Population 2. We have a multi-attribute deci on problem, where the aim is to select the optimal solution from many lternatives or to sort the available alternatives. Assume that the finite solution set is y=fy1 ., yny, the attribute set is C=fcl,., cab, and the decision expert set is E=el,., em] Let S=[s1,.., g be a predefined set consisting of odd-chain elements Expert ek selects one element from S as the value of solution yi under attribute c; i let it be denoted as pi E S, and let p=(ping denote the judgment matrix of expert ek on all the solutions for all the attributes. The attribute weight vector in evaluating information given by ert ek is W=wi, where wf is the weight of attribute c selected by expert ek from set S, v∈S This theory can be actualized through the Analytical Hierarchy Process (AHP), first put forward by American operational researcher T.L. Saaty in the 1970s AhP is a method for decision-making analysis that combines qualitative and quantitative methods. Using this method, decision-makers

126 The UMAP Journal 30.2 (2009) 12..xle 0 5 10 Figure 2. Numexical solutions for xi(t). We apply expert system and group decision theory to determine the weights of the species in Population 2. We have a multi-attribute deci￾sion problem, where the aim is to select the optimal solution from many alternatives or to sort the available alternatives. Assume that the finite solution set is Y = {yi, .. . ,y,}, the attribute set is C = {ci, . . . , cq}, and the decision expert set is E = {ei, ... , el}. Let S = {si,..., s.} be a predefined set consisting of odd-chain elements. Expert ek selects one element from S as the value of solution yi under attribute cj; let it be denoted as pikj E S, and let = (Pkj)n×q denote the judgment matrix of expert ek on all the solutions for all the attributes. The attribute weight vector in evaluating information given by expert ek is where wk is the weight of attribute cj selected by expert ek from set S, Wk ES. This theory can be actualized through the Analytical Hierarchy Process (AHP), first put forward by American operational researcher T.L. Saaty in the 1970s. AHP is a method for decision-making analysis that combines qualitative and quantitative methods. Using this method, decision-makers -X 0 ------- - --- -- - -- -- -- -- -- ----------- 4-- ------------ 48----- -- ------------------------------------------- --- L--------------------------- 6------------------------------ -------------- 24---- ------------------------------------------------------------------- /I SI I - - -I I 2------------------I----------------I---------------- I

Rebalancing 127 can separate complex problemsinto severallevels and factors, and compare and find the weights for different solutions, and provide the basis for the optimum solution AHP first classifies the problem into different levels based on the nature and the purpose of the problem, constructing a multilevel structure model ranked as the lowest level (program for decision making, measures etc ) compared with the highest level(the highest purpose). Based on AHP, we can establish the stratification diagram shown in Figure 3 Competitive relationship teen the decision Excreta quality adapt to the degree Oxygen s crustacea fish Figure 3. AHP stratification diagram. At last, we make consistency check of the result, finding that the consis- tency ratio of each expert's judgment matrix is below 1, so the consistency of the judgment matrix is acceptable. Finally we figure out the weight of the numbers of all the species in Population 2, as shown in Table 1 Table 1 Weight of each species in Population 2 as measured by AHP. Herbivorous fish Crustaceans Echinoderms Here we adopt population competition model to confirm the weight of es in Population 2: N1=N1(e1+nN2), N2=N2(e2+mM)

Rebalancing 127 can separate complex problems into several levels and factors, and compare and find the weights for different solutions, and provide the basis for the optimum solution. AHP first classifies the problem into different levels based on the nature and the purpose of the problem, constructing a multilevel structure model ranked as the lowest level (program for decision making, measures etc.), compared with the highest level (the highest purpose). Based on AHF, we can establish the stratification diagram shown in Figure 3. ................ .Competitive relationship .. . . . . . . . . . . .. . between the decision .. . . . . . . . . . . . . . Food requirements . Environment to . . . of the algae Excxeta quality, • adapt to the degree . . . . ............... .......... T O Figure 3. AHP stratification diagram. At last, we make consistency check of the result, finding that the consis￾tency ratio of each expert's judgment matrix is below 1, so the consistency of the judgment matrix is acceptable. Finally we figure out the weight of the numbers of all the species in Population 2, as shown in Table 1: Table 1. Weight of each species in Population 2 as measured by AH-RP Species Weight Herbivorous fish .21 Crustaceans .23 Molluscs .31 Echinodermis .24 Here we adopt population competition model to confirm the weight of each species in Population 2: N1 = Ni(e1 +y71N2), N2. N2.(e2 + 'yNi)

128 The UMAP Journal 30.2(2009) where Ei are birthrates and yi are coefficients of species interaction. According to these equations, we find that the ratio between different pecies is almost consistent with that obtained by ahP, which also confirms he correctness of our method, l In this way, we find that herbivorous fish, crustaceans, molluscs, and echinoderms can coexist and also compete. So the number of each species can be figured out based on the data in the steady state from the previous models, as shown in Table 2 Table 2. Number per pen of each species in steady state. Organism 69027 Herbivorous fish 5,638 Molluscs 8483 Echinoderms 6589 Milkfish Now we use the model to check the water quality, and make clear whether it is suitable for the continued healthy growth of the coral. First, we calculate the current concentration of chlorophyll in a fish pen. With help of relevant references, we find the regression equation between the Imber of algae and chlorophyll: N=1.2785+07568C where the units are 104/ml for N(algae) and ug/L for C (chlorophyll).For N=6.9027(from Table 2), we getC=7.43, a concentration of chlorophyll that is far beyond 0.25 Pg/L, the highest suitable concentration for the growth of coral From the available data in the problem, we figure out the mass of organic particles in the fish pen, and then work out the mass of each elemer The dry weight of echinoderms in the pen is 45.5 kg, the dry weight of milkfish excrement is 0.4-0.9 kg, so the total dry weight of excrement in the pen is 1.0-1.4 kg · The pen is10m×10mx8m, for a volume of800m3=800×103L Based on the percentage of elementorganic particles is 1186-1738 ug/L. Finally, we get the concentration of venin the problem, we figure out thenconcentrationof carbon C(10%),nitrogen(0.4%), andph bosphorus P(0.6%)(Table 3) DITOR'S NOTE: The authors paper does not give further details of the AHP calculationnor e population competition modeL

128 The UMAP Journal 30.2 (2009) where e are birthrates and -y, are coefficients of species interaction. According to these equations, we find that the ratio between different species is almost consistent with that obtainedby AHP, which also confirms the correctness of our method.1 In this way, we find that herbivorous fish, crustaceans, molluscs, and echinoderms can coexist and also compete. So the number of each species can be figured out based on the data in the steady state from the previous models, as shown in Table 2. Table 2. Number per pen of each species in steady state. Organism Number Algae 69,027 Herbivorous fish 5,638 Crustaceans 6,305 Molluscs 8,483 Echinoderms 6,589 MNflUfsh 1,760 Now we use the model to check the water quality, and make clear whether it is suitable for the continued healthy growth of the coral. First, we calculate the current concentration of chlorophyll in a fish pen. With help of relevant references, we find the regression equation between the number of algae and chlorophyll: N = 1.2785 + 0.7568C, (1) where the units are 10'/ml for N (algae) and ptg/L for C (chlorophyll). For N = 6.9027 (from Table 2), we get C = 7.43, a concentration of chlorophyll that is far beyond 0.25 jug/L, the highest suitable concentration for the growth of coral. From the available data in the problem, we figure out the mass of organic particles in the fish pen, and then work out the mass of each element. "* The dry weight of echinoderms in the pen is 45.5 kg, the dry weight of milkfish excrement is 0.4-0.9 kg, so the total dry weight of excrement in the pen is 1.0-1.4 kg. " The pen is 10 m x 10 m x m, 8 for a volume of 800 m3 = 800 x 103 L. "* Finally, we get the concentration of organic particles is 1186-1738 /Lg/L. Based on the percentage of elements given in the problem, we figure out then concentration of carbon C (10%), nitrogen N (0.4%), and phosphorus P (0.6%) (Table 3). 1 EDITOR'S NOTE: The authors' paper does not give further details of the AHP calculation nor of the population competition model

Rebalancing 129 Concentrations of elements in a pen, Element Concentration(ug/L) c(10%) 119174 P06%) Comparing the water quality in Sites A, B, C, and D, we find that the concentration of organics is between A and B, which is suitable for the growth of coral (here the concentration of elements is calculated only based in the excrement of milkfish and echinoderm), so the concentration of the microbes meets the reproduction needs of the coral. But the concentration of chlorophylls seriously out oflimits So we have to adjust thenumbers of some species to make the concentration of chlorophyll reach the standard We reason backward from the desired concentration(0. 25 Ag/L) of chlorophyll suitable for the growth of the coral, using the regression equa tion(1). With the estimated steady-state value, we can assume the initial values as:(10000, 5500, 350). From relevant references, we get the maxi- mum volume of fish pens: (N1, N2, N3)=(30000, 6000, 400), and through resimulation finally find the positive revised results for the steady-state val ues:(13732,5432,320). We work out the estimated steady-state number of algae N= 1. 4677, nd then derive the numbers of the three populations: (14677, 5744, 350 After revision, we get the actual steady-state number of the algae: N 1.3732. Putting this value into the regression equation, we getC=0. 125, thatis, the concentration of chlorophylls 0. 125 ug/L, which means that the water quality after adjustment completely meets the standard demanded Moreover, the totalnumber of milkfish and echinoderm is smaller than that before revision,so the index of the organics can certainly reach the growing demands of the coral, as shown in Figure 4 mo o the retroregulation process, which is the feedback mechanism of this del, with known water quality, we reason backwards to the estimated steady-state numbers of all the species, make positive simulation after es- timating the initial introducing value of all the species, and get the revised steady-state values. With this mechanism, we can find out the steady-state number of each species based on water quality, which provides great con- venience to the solution to the following problems Task 2a: Establishment of Logistic Model In this task, with all the herbivorous fish, crustaceans, molluscs, and echinodermsexcluded, we are required to find out the changes to thespecies and the circumstances of water quality. Based on our analysis, we make

Rebalancing 129 Table 3. Concentrations of elements in a pen. Element Concentration (psg/L) C (10%) 119-174 N (0.4%) 5- 7 P (0.6%) 7- 10 Comparing the water quality in Sites A, B, C, and D, we find that the concentration of organics is between A and B, which is suitable for the growth of coral (here the concentration of elements is calculated only based on the excrement of milkfish and echinoderm), so the concentration of the microbes meets the reproduction needs of the coral. But the concentration of chlorophyll is seriously out of limits. So we have to adjust the numbers of some species to make the concentration of chlorophyll reach the standard. We reason backward from the desired concentration (0.25 pg/L) of chlorophyll suitable for the growth of the coral, using the regression equa￾tion (1). With the estimated steady-state value, we can assume the initial values as: (10000, 5500, 350). From relevant references, we get the maxi￾mum volume of fish pens: (N1, N2 , N3) = (30000, 6000, 400), and through resimulation finally find the positive revised results for the steady-state val￾ues: (13732,5432,320). We work out the estimated steady-state number of algae N = 1.4677, and then derive the numbers of the three populations: (14677, 5744, 350). After revision, we get the actual steady-state number of the algae: N = 1.3732. Putting this value into the regression equation, we get C = 0.125, that is, the concentration of chlorophyll is 0.125 [g / L, which means that the water quality after adjustment completely meets the standard demanded. Moreover, the total number of milkfish and echinoderm is smaller than that before revision, so the index of the organics can certainly reach the growing demands of the coral, as shown in Figure 4. In the retroregulation process, which is the feedback mechanism of this model, with known water quality, we reason backwards to the estimated steady-state numbers of all the species, make positive simulation after es￾timating the initial introducing value of all the species, and get the revised steady-state values. With this mechanism, we can find out the steady-state number of each species based on water quality, which provides great con￾venience to the solution to the following problems. Task 2a: Establishment of Logistic Model In this task, with all the herbivorous fish, crustaceans, molluscs, and echinoderms excluded, we are required to find out the changes to the species and the circumstances of water quality. Based on our analysis, we make

130 The UMAP Journal 30.2(2009) *+X1 99份的636合66-86e9969ee93 Figure 4. The numbers of species meeting the demands after adjustment. clear the reasons why the growth rate will decrease after the milkfish in- crease. Factors such as natural resources and environmental conditions restrict the growth of milkfish; and with their growth, the blocking effect will become greater and greater. The blocking effect is expressed in terms of the influence on the growth rate r of milkfish, making r decrease with the increase in the number a of milkfish. If we express r asr(=), afunction f it should be a decreasing function, so we have =r(am),(0)=霆 The simplest assumption of thatr(a)is a linear function: r()=r-s(r>0,s>0), where r is the intrinsic growth rate. To confirm the meaning of the coeffi- cient s, we introduce the maximum quantity m that is allowed by natural resources and environmental conditions, which we regard as the milkfish capacity. When a =am, then s will stop increasing, that is, the growthrate r will be 0. That occurs fors=r/am, so that we have (2) Another interpretation of (2)is that the growthrate(a)is in direct pro- portion to the unsaturated part of the milkfish capacity a=(am-s)/am

130 The UMAP Journal 30.2(2009) x10 0 5 10 15 Figure 4. The numbers of species meeting the demands after adjust•ment clear the reasons why the growth rate will decrease after the milkfish in￾crease. Factors such as natural resources and environmental conditions restrict the growth of millcfish; and with their growth, the blodking effect will become greater and greater. The blocIdng effect is expressed in terms of the influence on the growth rate T of milkfish, maldng T decrease with the increase in the number x of milkfish. If we express T as T(x), a function of x, it should be a decreasing function, so we have: & = r() (0) = xo. The simplest assumption of that Tr(X) is a linear function: r'(x) = r -- sx (,r > 0, s > 0), where T is the intrinsic growth rate. To cornfirm the meaning of the coeffi￾dient s, we introduce the maximum quantity x..• that is allowed by natural resources and environmental conditions, which we regard as the milkfish capacity. When x = x,,, then x will stop increasing, that is, the growth rate r(x) will be 0. That occurs for s r /x,,,, so that we have r(x) =r (1-- -- )• (2) Another interpretation of (2) is that the growth rate r(x) is in direct pro￾portion to the unsaturated part of the milkfish capacity x = (x. - x)/x7n