Resting Stages and the Population Dynamics of Harmful Algae in ...

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Table 1: NotationVariables Meaning UnitsM Motile algae cells/mLN Non-motile algae cells/mLR Nutrient concentration µmol/LParameters MeaningUnitsK Half-saturation constant for algal growth µmol/Lµ max maximal growth rate of motile algae day −1δ Rate of conversion from motile to non-motile day −1γ Rate of conversion from non-motile to motile day −1q Nutrient quota of algae µmol/cellR in Nutrient supply µmol/LdMdt= µ maxRMK +R−δM +γN,dNdtdRdt= δM −γN,= − µ maxRMK +R q. (1)Each differential equation represents the factors we assume to impact thechange in populations and toxicity. We assume that only motile algae (M) cantake up the nutrient (R) and reproduce; therefore there exists a relationshipbetween the growth of the motile population and the nutrient. It has been wellestablished that the growthof the algae is limited by the nutrient concentration,and the Monod function,µ max RMK +R ,has repeatedly been used to represent the interaction between the taking up ofnutrient and the growth of P. parvum [3]. In our model µ max (day −1 ) is themaximal growth rate of the motile algae and K (µmol/L) is the half saturationconstant. The consumption of the nutrient represented in dR is proportional todtthe growth rate of the motile algae in dM with coefficient q (µmol/cell), whichdtconverts the units from cells to nutrients (µmol). We also assume that motilealgae converts to non-motile algae (N) at a constant rate, δ (day −1 ), and thatnon-motile algae converts to motile algae at a constant rate, γ (day −1 ).2.1 Reduction of OrderThe system is changed from three differential equations to a much simpler systemof only two differential equations. T is the total nutrients in the system3
and therefore it is the summation of R, Mq, and Nq. The units convert bothM and N into units of nutrients by multiplying by q. ThusandT = R+Mq +Nq,dTdt = dRdt + dM dt q + dN dt q.Direct substitution in System (1) yields dT = 0, which implies that T is constant.Therefore, usingdtthatR = T −Mq −Nqthe equation for R can be eliminated and System (1) is simplified todMdtdNdt= µ maxM(T −Mq −Nq)K +T −Mq −Nq= δM −γN.−δM +γN,(2)2.2 Equilibrium EvaluationWe set the right-hand sides of all differential equations in System (2) equal tozero and solve for the equilibrium values of M and N. We obtain the solutions:E 0 B = (M0 B ,N0 B ) = (0,0)E ∗ B( ) (3)γT= (M∗ B ,N∗ B ) = (δ +γ)q , δT(δ +γ)qAlthough R is not represented in the system, it is a population of interest,with R = R 0 B = T at E0 B , and R = R∗ B = 0 at E∗ B .2.3 Equilibrium AnalysisIn the batch culture model the equilibria are always feasible. This is becausethe equilibria represent populations, which cannot be negative. The Jacobianmatrix evaluated at EB 0 for the simple batch culture model is:⎛−δ + Tµ ⎞maxγ⎜J =K +T ⎟⎝ ⎠ . (4)δThe determinant of the Jacobian matrix (4) evaluated at E 0 B is −γTµ maxK +T .The outcome of the determinant alone, because it is negative, implies that E 0 Bis unstable.−γE 0 B4
Page 1 and 2: Resting Stages and the PopulationDy
Page 3: (Baker et al., 2007). It was discov
Page 7 and 8: dMdt= µ maxRMK +R−DM −δM +γN
Page 9 and 10: The second condition for Case 1 is
Page 11 and 12: So if we are in Case 1 of feasibili
Page 13 and 14: 10.90.8Case 2 of Chemostatmotil cel
Page 15: [9] Brooks, B.W., James, S.V., Vale

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