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

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76Batchmotil cells (M)non−motil cells (N)nutrients (R)Population densities5432100 50 100 150 200 250 300Time (t)Figure 1: Batch culture: stable non-trivial equilibrium E ∗ B .The Jacobian matrix evaluated at EB ∗ for the simple batch culture model is:⎛−δ − γTµ max(δ +γ)K γ − γTµ ⎞max(δ +γ)KJ =⎜⎝δ−γ⎟⎠E ∗ B. (5)The determinant for the Jacobian matrix (5) is γTµ max, and the trace is −δ−KγTµ max− γ. Thus the determinant is positive and the trace is negative.(δ +γ)KTherefore the equilibrium is stable. Figure 1 shows that EB 0 is an unstableequilibrium and that EB ∗ is a stable equilibrium.3 Chemostat Culture ModelThe same dynamics from the batch culture model are at work in the chemostatsetting. The only difference in the models is the inflows and outflows of thesystem. The rate of the dilution D, represents the inflows and outflows of thesystem. In the inflow only nutrients come into the system thus we include apositive term DR in in dR . We assume that populations and nutrients are welldtmixed so that the outflow of each population is according to its density in thesystem. From the biological assumptions mentioned we develop the system5
dMdt= µ maxRMK +R−DM −δM +γN,dNdtdRdt= δM −γN −DN,= D(R in −R)− µ maxRMqK +R . (6)3.1 Reduction of OrderWe obtain a reduction of order of the system, evaluated as time goes towardsinfinity. Simplification of the model, by reducing the system of three differentialequations to a system of two differential equations, helps ease the stabilityanalysis.As in the batch model, we first define T as the total nutrientsand therefore the rate of change of T isT = R+Mq +Nq,dTdt = dRdt + dM dt q + dN dt q.By direct substitution from System (6) we obtain:dTdt = D(R in −T).After solving the above differential equation we obtain T as a function of time:T(t) = (T(0)−R in )e −Dt +R in . (7)As t goes to infinity, in Equation (7), T approaches the nutrient supply concentrationR in . This implies that as t approaches infinityR in = R+Mq +Nq.We then eliminate the equation for R and replace R with its asymptotic valueR in −Mq −Nq. Thus System (6) is simplified todMdtdNdt= µ maxM(R in −Mq −Nq)K +R in −Mq −Nq= −DN +δM −γN.−δM +γN −DM,(8)6
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