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## Application: Microalgae growth under phosphorus and light limitation.

Here we consider a periodic version of the light-limited Droop model proposed by Passarge and collaborators in [5] for describing microalgae growth under light and phosphorus limitation. The We assume that the nutrient supply sin and the dilution rate D are constant and positive. μP (q) = μmax is the specific growth rate as described by Droop [2] under nutrient limitation, and μI(t, x) = 1 the vertical average of the local specific growth rate p(I) = μmax microalgae is only limited by light. I(t, z, x) is the light intensity perceived by microalgae at a distance z from the surface of the culture vessel and is determined from the Lambert-Beer law:

I(t, x, z) = Iin(t)e−(kx+Kbg)z, z 2 [0, L], (4.27)

Co-limitation by light and substrate under periodic forcing with Iin(t) the incident light intensity, k > 0 the specific light extinction coefficient of microalgae, and Kbg 0 the background turbidity. A direct

integration shows that:

with Iout(t, x) = I(t, x,L) the light intensity at the bottom of the culture.

We consider that the incident light intensity varies periodically according to Iin(t) = Imax max{0, sin(2t/!)}2, with ! > 0 the length of a day and Imax the maximal incident light (at midday). The uptake rate function is given where max is the maximal uptake rate of phosphorus, qL is the hypothetical maximal quota, and Ks is a half-saturation constant. It is not difficult to see that (4.26) satisfies the Assumptions 4.3.1-4.3.8 presented in section 4.3 (see Appendix B for the properties of μI ). Thus, we can apply Theorems 4.5.1 and 4.5.2 to obtain the following result.

Theorem 4.6.1. Consider the system (4.26). admits only two periodic solutions, the ! periodic solution represented by x = 0 and q, and a positive !-periodic solution represented by x > 0 and q.

Any solution starting with a positive microalgae concentration approaches the positive !-periodic solution. In this case, x1, q1 and x2, q2 correspond to two different solutions of (4.26) with x1(0), x2(0) > 0 and q1(0) = q2(0). We note that the cell quota remains between q0 and qL. A. Microalgae population density. B. Cell quota.

B. Intracellular phosphorus content. C. External phosphorus concentration.

D. Light and phosphorus limitation.

Proof. We recall equation (4.14) to study the uniqueness of the washout.

We note that 0 F0(t, q(t))dt < 0 for any function q(t) 2 [qL,1). Thus, the quota associated to any washout must intersect the set [q0, qL]. Since μ(t, x, q) := min{μI(t, x), μP (q)} 0, we have that [q0, qL] is positively invariant with respect to (4.14). Thus, the quota associated to any washout stays on [q0, qL]. Since q 7−! (q, sin) is strictly decreasing on [q0, qL], we have that q 7−! F0(t, q)/q is also strictly decreasing on [q0, qL]. This implies the uniqueness of the washout and part a) is proved.

We note that for any q > q0 there is a t0 2 [0, !] such that Iin(t0) > 0 and μI(t0, x) μP (q) for all x 0 i.e. μ(t, x, q) = μI(t, x). Then we have that x 7−! μ(t0, x, q) is strictly decreasing (see Proposition B10 in Appendix B). If we note that s 7−! (q, s) is strictly increasing for any q 2 [q0, qL] and that (q0, s) > 0 for any s > 0, from Theorem 4.5.2, we conclude that (4.1) admits at most one !-periodic solution with positive x-component. Applying Theorem 4.5.1, we conclude the proof.

To illustrate Theorem 4.6.1, let us consider the kinetic parameters for Chlorella vulgaris provided by Passarge [5]. The rest of parameters are chosen as D = 0.02 h−1, Kbg = 6m−1, sin = 15 μmol /L, L = 0.4m, and Imax = 2000 μmolm−2 s−1. Figure 4.1 illustrates the microalgae population density and the cell quota associated to the periodic solutions of (4.26) and their attractiveness property. Figure 4.2 illustrates the positive periodic solution (x, q, s) and its evolution during one day. The shaded area corresponds to the night (i.e. Iin(t) = 0). Figure 4.2D shows that during the day (t 2 [0, 0.5]) microalgae growth is mainly limited by phosphorus, while during the night (t 2 [0.5, 1]), there is no growth due to the absence of light. Thus, microalgae population only grows during the day (see Figure 4.2A), and consequently the internal cell quota and external nutrient concentration decrease during the day (see Figures 4.2B and 4.2C).

### Discussion and conclusions

In this work, we study the asymptotic behavior of a single microalgae model accounting for nutrient and light limitation. We found conditions such that prolonged continuous periodic culture operation (periodic dilution rate and nutrient supply) under periodic fluctuations of environmental conditions (such as the light source or the medium temperature) allows periodic concentrations to be maintained in the culture. More precisely, if (4.1) admits only one washout periodic solution (0, q, s), then the following condition: ensures the existence of a positive periodic solution. If this solution is the only one positive periodic solution, then it is globally stable (Theorem 4.5.1).

The uniqueness of this positive periodic solution is assured under additional hypotheses over the monotony of the functions μ and (Theorem 4.5.2).

As an application of our results, we gave sufficient conditions for the existence of a unique positive globally stable periodic solution for a periodic version of the model proposed by Passarge and collaborators [5]. In this model the growth rate is represented by the law of minimum. It is not diffi- cult to obtain similar results if we describe the growth rate as a multiplicative function i.e. μ(t, x, q) = μI(t, x)(1 − q0/q).

**Table of contents :**

**1 Introduction**

**2 State of the art: Modeling microalgae growth in chemostats **

2.1 Introduction

2.2 Microalgae modeling

2.2.1 Light limitation

2.2.2 Photoacclimation

2.2.3 Nutrient limitation

2.2.4 Co-limitation by light and nutrient

2.3 Dynamics of microalgae models

2.3.1 Light limitation

2.3.2 Nutrient limitation

2.3.3 Co-limitation by light and nutrients

2.4 Discussion and conclusions

**3 Theory of turbid microalgae cultures **

3.1 Chapter presentation

3.2 Introduction

3.3 Average growth rate (AGR)

3.4 Properties of the AGR in flat-plate PBRs

3.5 Discussion

3.6 Conclusions

**4 Co-limitation by light and substrate under periodic forcing **

4.1 Chapter presentation

4.2 Introduction

4.3 Model description and basic properties

4.3.1 Model description

4.3.2 Existence, uniqueness and boundedness of solutions

4.4 Reduced system

4.5 Main results

4.6 Application: Microalgae growth under phosphorus and light limitation

4.7 Discussion and conclusions

**5 Co-limitation by light and substrate: Including photoinhibition.**

5.1 Chapter presentation

5.2 Introduction

5.3 Model description

5.4 Basic properties and persistence

5.5 Existence of non-trivial steady states

5.6 Numerical study of the steady states

5.7 Conclusions

**6 Optimization of microalgae processes under light limitation**

6.1 Chapter presentation

6.2 Introduction

6.3 Modeling light-limited growth of microalgae

6.4 Maximizing microalgae productivity: Indoor case

6.4.1 Dynamics of a light-limited chemostat

6.4.2 Maximizing biomass productivity

6.5 Maximizing microalgae productivity by shading outdoor cultures

6.5.1 Control strategy

6.5.2 Dynamics of the controlled culture

6.5.3 Simulations

6.6 Conclusions

**7 Optimization of microalgae processes under light and substrate limitation**

7.1 Chapter presentation

7.2 Introduction

7.3 Model description

7.4 Maximizing microalgae productivity

7.5 Conclusions

**8 Microalgae growth limited by light, nitrogen, and phosphorus**

8.1 Chapter presentation

8.2 Introduction

8.3 Model description

8.4 Fitting of the model

8.5 Removal capacity of the system

8.6 Conclusions

**9 Conclusions**