A PERIODIC-PARABOLIC DROOP MODEL FOR TWO SPECIES COMPETITION IN AN UNSTIRRED CHEMOSTAT

. We study a periodic-parabolic Droop model of two species competing for a single-limited nutrient in an unstirred chemostat, where the nutrient is added to the culture vessel by way of periodic forcing function in time. For the single species model, we establish a threshold type result on the extinc- tion/persistence of the species in terms of the sign of a principal eigenvalue associated with a nonlinear periodic eigenvalue problem. In particular, when diﬀusion rate is suﬃciently small or large, the sign can be determined. We then show that for the competition model, when diﬀusion rates for both species are small, there exists a coexistence periodic solution.


1.
Introduction. The Droop model, also known as the variable-yield model, plays a significant role in the study of resource competition theory in phytoplankton ecology. The growth of the phytoplankton species is assumed to be determined by the most basic limited nutrient(s) (e.g., nitrogen or phosphorus). The competition is purely exploitative in the sense that organisms simply consume the nutrient(s), thereby making them unavailable for other individuals. An earlier model for the growth of microorganisms proposed by Monod [26] in 1950 assumes that the growth rate is constantly proportional to the nutrient uptake rate. However, for many nutrients, yields of algal cells are not fixed but depend on the physiological state of the population. For instance, in spatially heterogeneous environments, cells can consume nutrients at rates which exceed immediate requirements for growth and store them as quota, which enhances the survival of the species since they can use the stored nutrient(s) when they travels to a poor zone [9,10]. Thus the ecological system of variable-yield model leads to a great deal of studies among phytoplankton ecologists in both experimental and theoretical analysis. See e.g., the book by Nisbet and Gurney [28] and papers such as those by Sommer [35], Morel [27], Grover [6,7,8,10] and the references therein. According to their study, the Droop model provides better predictions and performs better especially in nonequilibrium habitat than the Monod model. Consider the following model formulated by Droop [3,4] which describes the competition for a single nutrient which is stored within individuals of the phytoplankton in a well-mixed chemostat (see, e.g., [34]): Here S(t) is the concentration of nutrient, u i (t) (i = 1, 2) represents the population density for the i-th species of phytoplankton, and Q i (t) (i = 1, 2) is the average amount of stored nutrient per cell of the i-th population. The chemostat is supplied with nutrient at constant concentration S (0) from an external reservoir. A matching out flow is at dilution rate D whose reciprocal gives the residence time of a cell in the chemostat. µ i (Q i ) is the growth rate of the i-th population as a function of cell quota Q i , f i (S, Q i ) is the per capital nutrient uptake rate per cell of the i-th population as a function of nutrient concentration S and cell quota Q i , and Q min,i is the threshold cell quota below which no growth of the i-th population occurs.
Biologically, when the cell quota is above the minimal cell quota, the growth rate increases with cell quota and cells expressing a higher reproductive rate require a larger cell quota of resource. The nutrient uptake rate increases with nutrient concentration and decreases with cell quota. Typically, the following choices for the growth rate µ i (Q i ) are made ( [1,2,4]): or where (Q i − Q min,i ) + is the positive part of Q i − Q min,i , µ i,∞ is the maximal growth rate of the i-th species and a i is the relevant half-saturation constant. According to [8,27], the uptake rate f i (S, Q i ) usually takes the form: where K i is the relevant half-saturation constant. The function ρ max,i (Q i ) is defined as follows: or where Q min,i ≤ Q i ≤ Q max,i and Q max,i is the maximum cellular quota of the i-th species. In [1,2], Cunningham and Nisbet took ρ max,i (Q i ) to be a constant. Complete mathematical analysis of (1) was carried out in [33]. It was shown that for most cases, competition exclusion holds for the two competing species. Later, Hsu et. al. [15] extended this result to the n-species case. Another modification of the basic chemostat model is to remove the well-mixed hypothesis and incorporate spatial variations. The authors in [19] mathematically investigated the Monod model including spatial heterogeneity in an unstirred chemostat in which two microbial populations compete for a single-limited nutrient. Inspired by the setting in [19], there has been a sequence of papers (see, e.g., [9,10,17,16,18]) studying variable-yield models in spatially variable habitats (e. g., an unstirred chemostat or a water column in oceans).
In natural environments, nutrient levels can be expected to vary temporally as a result of diurnal or seasonal variations. Thus, motivated by the study of [32,37,20,29], we intend to study a Droop model of two species competing in an unstirred chemostat for a single-limited nutrient, in which flow enters at one boundary supplying nutrient by way of periodic forcing function S (0) (t), and exits at another, removing nutrients and species, with diffusive transport of nutrient and species across the habitat. Thus, we consider the following system of reactiondiffusion equations: with boundary conditions and initial conditions where the positive constant d in (6) is the diffusion coefficient and the positive constant γ in (7) represents the washout rate; are the total amount of stored nutrient for species 1 and 2, respectively; S (0) (t) is positive and varies periodically in time with given period T > 0, i.e., We assume that the initial data u 0 By naturally extending the above functions µ i (Q i ) and f i (S, Q i ) to be defined in R + and R 2 + respectively, we assume that µ i (Q i ) and f i (S, Q i ) satisfy the following assumptions for i = 1, 2 (see also [17]) : The problem of understanding competition for resources in a both temporally and spatially varying environment is challenging and has received not as much attention as the temporally homogeneous but spatially varying case. This is our first attempt to study nonlinear periodic-parabolic Droop model and we hope to pursue further in this direction in future.
The rest of this paper is organized as follows. In Section 2, we study the single species model. We establish conditions for existence of positive periodic solutions and extinction. Moreover, we study the effects of diffusion rate on the persistence/extinction of the species. In Section 3, we study the competition for two populations. Using results in Section 2, we show that when both diffusion rates of the two competing species are small, there exists a coexistence periodic solution.
2. Population dynamics of single species model.

2.1.
Well-posedness and essential boundedness of single species model. In this section, we consider dynamics of the periodic Droop model of one species consuming one nutrient. Setting u 2 = U 2 = 0 and omitting all the subscripts in (6)-(8), we obtain the following system: Let Let The following result is a direct consequence of Lemma 2.1 and parabolic comparison principle.
Define Q * to be the unique positive number such that It is obvious that Q min < Q * < Q B , where Q min and Q B are given by (H1) and (H2) respectively. U (x, t) u(x, t) ≥ Q min for all t ∈ (0, τ ).
Since the proof of the above lemma follows essentially from that of [17,Lemma 4.2] with mild modifications, we omit it here. Next, we apply Lemma 2.3 to show the global well-posedness results and eventual boundedness of solutions of (11). Proposition 1. Suppose (H1) and (H2) hold. Then for each initial condition in Y, system (11) has a unique classical solution (S, u, U ) that exists for all t > 0. Moreover, the solution satisfies (S(·, t), u(·, t), U (·, t)) ∈ Y for all t > 0. There exists a constant C > 0 independent of initial conditions in Y such that for any solution (S, u, U ) of system (11), we have lim sup t→∞ (S(·, t), u(·, t), U (·, t)) ≤ C.
It only remains to show (21). By (12), we see that U (x, t) ≤ Z(x, t). Hence, it follows from Lemma 2.1 that Then it follows from Lemma 2.3 that This together with Lemma 2.2 finish the proof of (21).

2.2.
Threshold dynamics of single species model. To characterize the persistence/extinction of the single species model (11), we now consider the following system: where z * (x, t) is given by Lemma 2.1. Substituting u(x, t) = e −Λt φ(x, t) and (25), we obtain the following associated nonlinear eigenvalue problem: We will rigorously show that the persistence/extinction of the phytoplankton species in system (11) is determined by the principal eigenvalue of the above nonlinear periodic parabolic eigenvalue problem. Hence, to study the existence of principal eigenvalue to (26), we introduce some notations (see also [17]). Let (X, · ) be a normed vector space (or NLS) over R. We call a subset C ⊂X a cone if (i) C is convex, (ii) tC ⊂ C for all t ≥ 0, and (iii) C ∩ (−C) = {0}. A cone C is said to be solid if it has non-empty interior. If C is a cone and also a complete metric space in the metric induced by the norm onX, we call C a complete cone. A cone C in an NLS (X, · ) induces a partial order Here ≤ D is the partial order generated by the cone D. If D is a solid cone, we say that T is D-strongly-order-preserving if T(x) D T(y) whenever x, y ∈ C satisfy x ≤ D y and x = y. Recall also the Bonsall cone spectral radius (see [23,24,36]) where T m C := sup{ T m (x) : x ∈ C and x ≤ 1}. We assume that the following hold: (C) Let C ⊂ D be complete cones in an NLS (X, · ), D be normal, and T : C → C be (i) continuous, (ii) compact, (iii) homogeneous of degree one, and (iv) Dorder-preserving.
. Assume (C) holds. If, in addition, D is a solid cone and T is D-strongly-order-preserving, then (i)r =r C (T) > 0 and there is a non-zero eigenvectorx ∈ C ∩ Int D such that Tx =rx.
(ii) If x ∈ C is another eigenvector of T, then x ∈ span{x} and Tx =rx .
We now show the existence of the principal eigenvalue of the nonlinear periodicparabolic eigenvalue problem (26). From now on, let D = C([0, 1]; R 2 + ) and ≤ D be the partial order in C([0, 1]; R 2 + ) induced by the cone D, i.e., where Q * is given in (17). It is easy to see that both C and D are complete cones and that D is both normal and solid.
By T -periodicity of z * (x, t) in t and similar arguments as in the proofs of Proposition 1, we deduce that system (25) generates periodic solution maps Π(t) on C, i.e., Π(t) satisfies that Then we have the following result concerning the existence of the principal eigenvalue of system (26).
Proof. Recall that system (25) generates solution maps Π(t) on C. It is easy to see that for all t > 0, Π(t) is continuous, compact and homogeneous of degree one. By similar arguments as in the proof of [17, Lemma 5.1], we can show that for all t > 0, Π(t) : C → C is D-strongly-order-preserving. Hence, we may apply Proposition 2 to the operator Π(T ) : wherer is the Bonsall cone spectral radius of Π(T ) : C → C. Denote Λ 0 = − 1 T logr. Then we can verify that Λ 0 is the principal eigenvalue to (25) with being the associated eigenfunction. The uniqueness of Λ 0 follows from Proposition 2.
We next study dynamics of the limiting system of (11): The biologically feasible region for system (27) is defined by We now show that the set D(t) is positively invariant for the solution map associated with (27).
Let S(x, t) = z * (x, t)−U (x, t). Then it is easy to see that (S(x, t), u(x, t), U (x, t)) ∈ Y satisfies system (11), where Y is given in (15). Hence, the rest of the proof follows from similar arguments as in [17,Lemma 4.2] and Proposition 1 and is thus omitted.
We Remark 1. We note that the dynamics of system (27) is unclear when Λ 0 = 0.
Recall the definition of Y in (15). We now define

1]} and its complementary set
We now state the following result concerning global dynamics of (11).
From (44) and (45), we see that the sequence {χ i } n i=1 ⊂ I with χ 1 = a and χ n = b satisfies that This shows that I is a compact, invariant and internal chain transitive set for the Pincaré map P : D(0) → D(0), which finishes the proof of Claim 2.4.

2.3.
Effects of diffusion rate on single species model. In this subsection, we study the role of diffusion rate on the extinction/persistence of system (27). Let Then we consider the following two ODEs: and From [32, Proposition 1.1], we have the following result: Lemma 2.8. System (50) (resp. (51)) admits a unique T -periodic solution Q(t) (resp. Q(t) ) to which all solutions are attracted. Moreover, Q(t) > Q(t) > Q min for all t.

(56)
Obviously, the boundary conditions for u and U hold. The last equality also holds due the periodicity of φ(x, t) and Q(t) in t. Thus, it suffices to show that the first two inequalities in (56) holds.
For the first inequality of (56), where we have used the fact λ < 0. For the second inequality of (56), Thus the strongly order preserving property of Π(t) implies that This means Π(T )(u(x, 0), U (x, 0)) ≥ D k(u(x, 0), U (x, 0)) for some k > 1, whence the Bonsall cone spectral radiusr must be strictly greater than 1, and by definition, Λ 0 = − logr T < 0. From a biological point of view, roughly speaking, a smaller diffusion rate d and a smaller washout constant γ would be more advantageous for the survival of the species. Moreover, as pointed out in [18] that for a system defined on interval [0, L], the parameter d is equivalent to L −2 and γ is equivalent to L −1 . Hence, for a fixed diffusion rate and washout constant, a larger patch size of the domain would be more beneficial for the survival of the species. These are consistent with the results obtained for the constant input of nutrient rate considered in [16,18].
Let Ψ t : X → X be the solution map associated with (6)- (8). We denote the Poincaré map