Vegetation-Erosion DynamicsVegetation-Erosion Dynamics ModelVegetation-Erosion Dynamics ModelSolving the EquationQualitative AnalysisVegetation-Erosion ChartHeishui River BasinHeishui: Qualitative AnalysisHeishui GraphsConclusion1/11 Model of the Vegetation-ErosionRelationship using DifferentialEquationsKeren Crum and Bethany Adams2/11 Vegetation-Erosion Dynamics• Vegetation-Erosion Dynamics study the relationship between theVegetation Coverage Density (VCD) of an area to its rate of Erosion• The re is an inverse relationship between VCD and erosion in thatthe higher the VCD the less erosion, while the higher the erosionthe harder for vegetation to take hold.• Vegetation-Erosion Dynamics also integrate the effects of naturalforces and human interactions on the change in VCD and Erosionof an area• Erosion is a se rious problem that affects the farmlands and naturalhabitats.• Models predict the long-term nature of watersheds• Models show the negative and positive influences humans have onVE dynamics and provide analysis to devise a plan of action toimprove the ecosystem.3/11 Vegetation-Erosion Dynamics ModelThe basic equation for Vegetation-Erosion dynamics is given bydVdt= aV − cEdEdt= dE − fV(1)where V and E represent the percentage of vegetation-coverage andthe rate of erosion in ton/km2respectively.The coefficients a,c,d, and f are the natural rates of VCD, erosion,erosion effects that further erosion, and vegetation coverage that re-duces erosion, which are specific to the area.4/11 Vegetation-Erosion Dynamics ModelNatural and human stresses on vegetation and erosion must also beevaluated to create a complete model giving us the linear differentialequation model with forcing terms of Vτand EτdVdt= aV − cE + VτdEdt= dE − fV + Eτ(2)where Eτrepresents human affects on erosion Vτrepresents naturaland human ecological stresses.5/11 Solving the EquationThe solution to the linear differential equations (2) is given by thehomogeneous solution plus the nonhomogeneous solutionV = c1eλ1t+ c2eλ2t+ V1(t)E = c1a − λ1ceλ1t+ c2a − λ2ceλ2t+ E1(t)(3)where c1and c2are constants determined from the initial conditionsand λ1and λ2are eigenvalues in terms of the coefficients a,c,d, and fgiven byλ1=(a + d) −p(a + d)2− 4(ad − cf)2λ2=(a + d) +p(a + d)2− 4(ad − cf)2.(4)6/11 Qualitative AnalysisNow that we have solved our equations we can use qualitative analy-sis to determine the current condition of the system and the likelihoodof the ecological system to recover or deteriorate.Using the equations (2) from our original system, we can find t henullclines of the system by setting dV/dt and dE/dt equal to zero andsolving for E.E =acV +1cVτE =fdV −1dEτ(5)7/11 Vegetation-Erosion ChartEvaluating the vegetation density versus the erosion produces aVeget ation-Erosion Chart.0 0.5 10.511.522.53Vegetation Cover Density (%)Erosion Level (1000 ton/km2year)Unstressed Vegetation−Erosion ChartABCE=aV/cE=fV/d0 0.5 10.511.522.53Vegetation Cover Density (%)Erosion Level (1000 ton/km2year)Stressed Vegetation−Erosion ChartABCE=aV/c+Vtau/cE=fV/d−Etau/d8/11 Heishui River BasinThe Heishui River Basin is a small tributary of the Yangtze riverthat was targeted as a study for erosion control due to its high erosionrates of 7,243 ton/km2per year and low vegetation coverage of 7.6%.Beginning in 1978 the government reforested the river basin at arate of 4% per year(Vτ0), while also installing erosion control damsthat reduced erosion by 650 ton/km2per year(Eτ0).The ecological stresses in this case are constant whereVτ(t) = Vτ0and Eτ(t) = Eτ0,(6)Evaluating the solution from ( 3) gives the equationsV (t) = c1eλ1t+ c2eλ2t−dVτ0+ cEτ0ad − cfE(t) = c1a − λ1ceλ1t+ c2a − λ2ceλ2t+ −fVτ0+ aEτ0ad − cf(7)9/11 Heishui: Qualitative AnalysisThe Vegetation-Erosion Chart of the Heishui model demonstrateshow improving vegetation and erosion conditions decrease the highrisk (A) and the medium risk (B) sections.0 0.5 10481216Vegetation Cover Density (%)Erosion Level (1000 tons/km2year)Vegetation−Erosion Dynamics Chartwithout Erosion ControlABCdV/dt=0dE/dt=00 0.5 10481216Vegetation Cover Density (%)Erosion Level (1000 tons/km2year)Vegetation−Erosion Dynamics Chartwith Erosion ControlABCdV/dt=0dE/dt=010/11 Heishui GraphsUsing the initial conditions of V (1978) = .076 E(1978) = 7, 243 wecan simulate the activity of the Vegetation Coverage and the Erosionin the Heishui Basin.1980 1985 1990 199500.20.40.60.81Vegetation Cover Density over TimeYearVegetation Cover (%)Model w/ Erosion ControlModel w/o Erosion ControlActual VCD1980 1985 1990 1995048121620Erosion over TimeYearErosion (1000 tons/km2yearModel w/ Erosion ControlModel w/o Erosion ControlActual ErosionWe simulated the model with and without erosion control along withthe measured to show the importance of erosion control.11/11 Conclusion• Vegetation-Erosion Dynamics is a relatively new area in sciencethat has shown the important interconnectedness of vegetation toerosion.• Human activities have a great impact on the environment andshould take responsibility to sustain a vibrant ecosystem.• Vegetation-Erosion models can help determine the current condi-tions and help develop effective means to reduce erosion.• Vegetation-Erosion analysis will eventually be incorporated int ocivil engineering projects to help minimize impacts to the