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

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0.250.2Case 1 of Chemostatmotil cells (N)non−motil cells (M)nutrients (R)Population densities0.150.10.05010 −2 10 0 10 2Time (t)10 4 10 6Figure 2: Chemostat culture (Case 1): stable trivial equilibrium E 0 c .3.3.2 Stability Analysis - Case 2In the previous section we established the conditions for stability of both equilibriaEc 0 and Ec. ∗ Using those same conditions we can determine the stabilityof Ec 0 and E∗ c in Case 2. Condition (19) for stability of E0 c directly contradictsCondition (13), which is needed for feasibilty in Case 2. So for Case 2 of feasibilitythe equilibrium Ec 0 is unstable. For Case 2 of feasibility the equilibriumEc ∗ is stable, because Condition (13) implies Condition (23). This is shown inFigure 3.4 Discussion and ConclusionThe aim of this research was to establish a relationship between motile andnon-motile states. These transitions are characterized as certain rates (γ andδ) in which one state converts to the other. In addition, our model includedthe rate at which P. parvum cells are reproduced. Together these rates give abetter understanding of the total cell population. Both batch and chemostatmathematical models produced meaningful results and are discussed below.4.1 Batch CultureThe batch culture model was analyzed and gave two equilibrium solutions. Thetrivial equilibrium, which is unstable according to our stability analysis, had littlebiological significance because it represents a non-existing initial population.Having a zero initial population is not useful for our research. Our non-trivial11
10.90.8Case 2 of Chemostatmotil cells (N)non−motil cells (M)nutrients (R)Population densities0.70.60.50.40.30.20.1010 −1 10 0 10 1 10 2 10 3Time (t)10 4 10 5 10 6Figure 3: Chemostat culture (Case 2): stable non-trivial equilibrium E ∗ c .equilibrium did provide biologically meaningful results. According to our stabilityanalysis, if there is an initial cell population, it will approach our non-trivialequilibrium over time. We can see that populations N and M monotonicallyapproached EB. ∗ We observed that both cell populations approached equilibriumas nutrient was depleted to near zero values. In a typical batch culture,one would observe a plateau where the sample reaches its unique maximumconsumption rate, maintains the rate, and then begins the death phase of thegrowth cycle. Because our model does not account for death, it shows only thetime at which both P. parvum cell populations reach EB.∗The ratio of these equilibrium populations gives us an idea as to how thealgae will react in a limited nutrient environment. Adjusting the amount oftotal nutrient will only affect the total population size ofequilibrium(MB+N ∗ B).∗BecauseMB∗NB∗ = γ δ ,total nutrient has no effect on the M ∗ B and N ∗ B proportion. This means thatonly the conversion rates, δ and γ, govern the M and Nsteady state populationproportion. Our model demonstrates the impact of δ and γ on populationdynamics despite being given an elementary constant value. We make thisassumption for simplicity of the model although it may not represent the actualdynamics of the conversion rates.12
Page 1 and 2: Resting Stages and the PopulationDy
Page 3 and 4: (Baker et al., 2007). It was discov
Page 5 and 6: and therefore it is the summation o
Page 7 and 8: dMdt= µ maxRMK +R−DM −δM +γN
Page 9 and 10: The second condition for Case 1 is
Page 11: So if we are in Case 1 of feasibili
Page 15: [9] Brooks, B.W., James, S.V., Vale

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