《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