Dynamics of Two Phytoplankton Species Competing for Light and Nutrient with Internal Storage

We analyze a competition model of two phytoplankton species for a single nutrient with internal storage and light in a well mixed aquatic environment. We apply the theory of monotone dynamical system to determine the outcomes of competition: extinction of two species, competitive exclusion, stable coexistence and bistability of two species. We also present the graphical presentation to classify the competition outcomes and to compare outcome of models with and without internal storage. 1. Introduction. Nutrients and light are essential resource for growth of phy-toplankton. Competition for resources affects the composition of species in the ecosystem. When species compete for one single limiting nutrient, the species with the highest tolerance for nutrient conditions in the surrounding environment, hence the smallest break-even concentration, wins the competition [4, 5] and outcompetes the others. Similarly, when multiple species compete for light, the one with the highest tolerance for light environment, hence the lowest break-even light intensity, wins the competition [7]. These demonstrate competitive exclusion in ecosystems and preclude species coexistence. Nevertheless, competition for multiple resources may allow coexistence of species under certain conditions. For example, under the condition of trade-off in tolerance for different nutrients, two species coexist in the competition for two complementary nutrients [3]. Importance of trade-off in species coexistence was revealed in the competition model on resources of single nutrient and light [8]. The same conclusion has been demonstrated graphically by Tilman [14]. Resource competition model mentioned previously assumed constant yield for population dynamics. This implies no energy conservation from one stage to the next. Grover [2] suggested that storage of excess resource could lead to variable yield for converting nutrient into organism. Such storage results from surplus of one resource due to limited reproduction (population growth) restricted by the

1. Introduction.Nutrients and light are essential resource for growth of phytoplankton.Competition for resources affects the composition of species in the ecosystem.When species compete for one single limiting nutrient, the species with the highest tolerance for nutrient conditions in the surrounding environment, hence the smallest break-even concentration, wins the competition [4,5] and outcompetes the others.Similarly, when multiple species compete for light, the one with the highest tolerance for light environment, hence the lowest break-even light intensity, wins the competition [7].These demonstrate competitive exclusion in ecosystems and preclude species coexistence.Nevertheless, competition for multiple resources may allow coexistence of species under certain conditions.For example, under the condition of trade-off in tolerance for different nutrients, two species coexist in the competition for two complementary nutrients [3].Importance of trade-off in species coexistence was revealed in the competition model on resources of single nutrient and light [8].The same conclusion has been demonstrated graphically by Tilman [14].
Resource competition model mentioned previously assumed constant yield for population dynamics.This implies no energy conservation from one stage to the next.Grover [2] suggested that storage of excess resource could lead to variable yield for converting nutrient into organism.Such storage results from surplus of one resource due to limited reproduction (population growth) restricted by the 1260 SZE-BI HSU AND CHIU-JU LIN other resource.This level of resource storage is called cell quota [1,2].In the model proposed Passarge et al. [11], cell quota from previous stage constrains nutrient uptake rate of phytoplankton and affect population growth.This may alter dynamics of species in competition and lead to different equilibrium statement for species coexistence.Yet no analytical work was done for the model to the best of our knowledge.
In this paper, we study a model of two phytoplankton species competing for two complementary resources.The model assumes internal storage of the first resource, nutrient, and no internal storage of the second resource, light, in a wellmixed aquatic environment.Moreover, we will compare our results with those of competitions for two nutrients in graphical illustration.
The paper is organized as following.In section two, we show the model and study the single species model.We present results and analyze the model of the competition of two species for single-nutrient and light in section three and four, respectively.In section five we discuss the graphical presentation for the two species competition model (2) with n = 2 or the system (13).The last section is the discussion.We compare the experimental results in [11] with our analytic results.
2. Model and single species growth model.Assume that species consumes nutrient and stores it into cell quota, and that the nutrient uptake rate increases with nutrient and decreases with cell quota.It is the Droop model [1] or the variable yield model.Assume that the light intensity changes in accordance with the Lambert-Beer's law (see the fourth equation in (DE n ) below).When species consumes nutrient and light, its growth function is modeled by Von Liebig's "Law of minimum" (see the first equation in (DE n ) below).We shall analyze the following n-species competition model which was proposed by J. Passarge et al [11]: Here N i (t) is the population density of phytoplankton species i at time t, and Q i (t) is the intracellular nutrient content of species i, Q min i is the minimum cellular quota satisfying µ i (Q min i ) = 0, for i = 1, 2, ..., n.The variable R(t) is the nutrient concentration in the water column, and I out (t) is the light penetration to the bottom of the water column.The parameter D is the dilution rate, R in is the nutrient input concentration, and I in is the incident light intensity at the water surface, k bg is the background turbidity caused by water, k j is the light attenuation coefficient of phytoplankton species j, and z m is the total depth of the water column.
There are some properties for the related functions.The function µ i (Q i ) is the growth rate of species i under nutrient limitation with and is continuous for where µ i∞ is some specific parameter.The function η i (I out ) is the growth rate of species i under light limitation with η i (I out ) ≥ 0, η i (I out ) > 0 and is continuous for I out ≥ 0. For example, η i (I out ) may satisfy Holling type II functional response with half-saturation constant a i and maximal growth rate m i for i-th species, respectively, it takes the form The function ν i (R, Q i ) is the nutrient uptake rate of species i, and it satisfies Grover's paper [2] presented a classical nutrient uptake function.
By the scaling, we may assume z m = 1.Denote I(t) to be I out (t), We define the following positive numbers σ i , ξ i and λ i by the equilibrium analysis of (2) (3) For simplicity, we assume the following hypothesis: (H1) λ i = λ j for i = j and λ i = R in for all i. (H2) 0 < ξ 1 < ξ 2 < ξ 3 < ... < ξ n and ξ i = I 0 for all i. (H3) µ i (Q i ) = η i (I) at equilibria for all i.In this paper we restrict our attention to the case n = 2, the competition of two species for light and one single nutrient with internal storage.
To understand the system (2) with n = 2, the first step is to consider the single species model, which takes the form: We define the following positive numbers λ, ξ, σ based on the equilibrium analysis of the system (4), (

5)
Equilibrium analysis: There are two types of equilibria representing species extinction and survival.First, we consider the equilibrium of extinction type, For the survival type, From the first equation in (4), we have either µ(Q c ) = D or η(I c ) = D where I c = I 0 exp(−kN c ). From the basic assumption (H3), there are two possible cases.In the first case, we have µ(Q c ) = D and η(I c ) > D, and denote In this case, we called that the species is R-limited.From (5), we obtain that Q R = σ and In the second case, we have η(I c ) = D and µ(Q c ) > D, and denote E c = E I = (N I , Q I ), we say that the species is I-limited.From (5), we have that I c = ξ and N I = (ln I 0 − ln ξ)/k.In this case, we denote R = R I .From (1), there exists a unique Then we have R I > λ, and Hence E I exists if and only if From ( 7) and ( 8), we conclude that if R in > λ and I 0 > ξ, then E c exists and E c is either E R or E I exclusively.
Stability analysis of E 0 , E R and E I : Consider the Jacobian of the system (4) evaluated at E 0 = (0, Q 0 ), the washout equilibrium, The eigenvalues of J(E 0 ) are: From (1), ∂ ∂Q ν(R in , Q 0 ) < 0, it follows that E 0 is locally asymptotically stable if and only if min{µ(Q 0 ), η(I 0 )} < D, i.e.Q 0 < σ or I 0 < ξ.It is trivial that if I 0 ≥ ξ, then we have that Q 0 < σ if and only if R in < λ.Therefore, we conclude that E 0 is locally asymptotically stable if and only if I 0 < ξ or R in < λ.
If the species N is R-limited, at the neighborhood of E R , the system (4) becomes The Jacobian of the system (9) evaluated at E R is The eigenvalues ρ of J(E R ) satisfies From (1) and Roth-Hurwitz criteria, the equilibrium E R is locally asymptotically stable.
If the species N is I-limited, at the neighborhood of E I , the system (4) becomes The Jacobian of the system (10) evaluated at E I is The eigenvalues are −kξη (ξ)N I < 0 and −N I ∂ν ∂R (R I , Q I ) + ∂ν ∂Q (R I , Q I ) − η(ξ) < 0. Hence the equilibrium E I is locally asymptotically stable.
The following theorem describes the global dynamics of system (4).
Theorem 2.1.The following holds.(i) If R in < λ or I 0 < ξ, then E 0 is the only equilibrium and (ii) If R in > λ and I 0 > ξ, then E 0 is unstable and E c exists, and either Proof.Let U = N Q, then we convert system (4) into the following equations Note that U = 0 when N = 0 and the system (11) is dissipative.
The isoclines of N are N = 0 and Next, we shall describe the isocline of U .Let the function F (N, U ) be defined by Note that N = 0 implies U = 0 and Q = Q 0 , then F (0, 0) = 0.And By implicit function theorem, there exists Ũ (N ), a function of N , which satisfies Ũ (0) = 0, F (N, Ũ (N )) = 0 and from the equation ( 6) the inequality holds.Hence Ẽ0 := (0, 0) is the only equilibrium of system (11) and it is locally asymptotically stable.Hence there is no periodic solutions, by Poincare-Bendixson theorem, and all solutions converge to Ẽ0 .Therefore, all solutions of system (4) converge to E 0 .(ii) If R in > λ and I 0 > ξ, it implies that min{µ(Q 0 ), η(I 0 )} > D and Ũ (0) > σ.
The nontrivial intersection of isoclines is either ẼR := (N R , σN R ) or ẼI := (N I , Ũ (N I )) (see the Figure 1).From the phase plane analysis, we know that the region A and B are positively invariant, where Hence there is no limit cycles, and by Poincare-Bendixon theorem the omega limit set of (N (0), U (0)), ω(N (0), U (0)), is an equilibrium.We know that Ẽ0 is unstable under the assumption of (ii).Therefore ω(N (0), U (0)) is either ẼR or ẼI for N (0) > 0. Thus, for system (4), ω(N (0), 3. Two species model.In this section we consider a model of two species competing for light and a single nutrient with internal storage.The model takes the form Let the set Ω be First of all, we find all equilibria and classify the stability of them.
Equilibrium analysis: The extinction equilibrium is For the case of competitive exclusion, we have the following semi-trivial equilibria E Ri and E Ii , i = 1, 2, representing that the species i is R-limited and I-limited, respectively. (1) ( ( , where , where Species 2 is I-limited if and only if Next, we consider the coexistent equilibria.In this case, we must have that min{µ a contrary to (H2).From above discussion, we know that the resource can not attain to the break-even concentration of each species at the same time.Hence the coexistent equilibria exist if one species induces µ i (Q i ) = D and the other causes η j (I) = D. Therefore we have exactly two types of coexistent equilibria. ( We need assumption (H4): (H4) Any two of ln Then the solution (N RI 1 , N RI 2 ) of ( 15) exists and is unique if and only if or (18) Similarly, the existence of Under (H5) equation ( 18) has a unique positive solution ( We present the criterion of the existence of E RI c and E IR c in Table 1.Note that the assumption (H2) implies the nonexistence of E IR c .

Table 1
Equilibrium Existence criteria Stability analysis: The details of the local stability analysis of each equilibrium are presented in the Appendix.We denote J(E) be the variational matrix of (13) evaluated at equilibrium E. The eigenvalues of J(E 0 ) are three negative real values and min{µ Similarly, for equilibrium E R2 , there are three eigenvalues of J(E R2 ) with negative real part and it is locally stable if There are three eigenvalues of J(E I1 ) with negative real part, Similarly, for equilibrium E I2 , there are three eigenvalues of J(E I2 ) with negative real part and it is locally stable if We summarize the criteria for the existence and local stability of
The stability of coexistent equilibrium is more complicated.We note that from assumption (H2) and Table 1, which is not zero by (H4).We state the outcomes of competition in the following theorems, and the proofs are postponed to the next section.
For the rest of this section, we always assume that R in > λ i , I 0 > ξ i for i = 1, 2. We denote some important parameters as following: and , then E I2 is locally stable and there exists a saddle equilibrium E RI c .The outcomes depend on initial conditions.(b) If T * < C I2 , then E I2 is unstable, and for , then E R1 is locally stable and E I2 is unstable, and for , then E R1 and E I2 are unstable and E RI c exists, and for We note that , a function of R in and I 0 , satisfies equation (14).For convenience, we consider 4. The proofs.To prove the results in section 3, we need the following lemma and theorems.The lemma states the nonexistence of coexistent equilibrium.Note that assumption (H2), ξ 1 < ξ 2 , implies that E IR c does not exist.Proof.Assumption (H2) implies that E I1 is locally stable if it exists.
(1) For the case E R1 or E I1 is locally stable and E R2 is unstable, then the instability of E R2 implies λ 1 < λ 2 , and it follows that E RI c does not exist.(2) Assume E R1 is locally stable and E I2 is unstable, then the instability of , and λ 1 > R I2 , a contradiction.
Assume E RI c exists, then (16) holds.E I2 is locally stable means that , a contradiction to (16).
By changing variables, we transform the system (13) into a monotone system, and apply monotone dynamical theory to obtain the global asymptotic behavior of the solutions.Let U i = N i Q i , i = 1, 2, then we have the following equations.
Then Ω is positively invariant for (19), and the forward orbit of (19) have compact closure in Ω.
From now on, we assume that (H1)-(H4) hold and R in > λ i and I 0 > ξ i , for all i = 1, 2. From Table 2, E 0 is a repelling in system (13), so is E 0 in system (19).Note that , and Theorem 2.1 tells us that the omega limit set ω(x [6] or Theorem 2.4.1 in [16], we have the following results.
Then S is positively invariant and the omega limit set of every orbit in X + is contained in S and exactly one of the following holds: (i) There exists a positive equilibrium Finally, if (ii) or (iii) hold, x = (x 1 , x 2 ) ∈ X + \ S and with Proof of Theorem 3.1.The assumption R in < λ i or I 0 < ξ i , by Theorem 4.2 or Theorem 4.3, we know that N i (t) → 0 as t → ∞.
Proof of Theorem 3.2.The assumption implies that E R1 (or E I1 ) is locally asymptotically stable if it exists, and the coexistent equilibrium E RI c does not exist.Assume that E R2 (or E I2 ) is also locally asymptotically stable, from Theorem 4.7, E RI c exists, a contradiction.Thus E R2 (or E I2 ) is unstable if it exists.By Theorem 4.5, the conclusion of this theorem holds.
Proof of Theorem 3.3.Note that, under the assumption λ 2 < λ 1 and ξ 1 < ξ 2 , we have that Proof of Lemma 3.4.From ( 14), we have where Differentiating the equation ( 21) with respect to R in , then Hence Similarly, differentiating (21) with respective to ln I 0 , we have 5. Graphical presentation.In this section we demonstrate the graphical presentation of the two species competition for light and single nutrient with internal storage under the assumption λ i < R in , ξ i < I 0 for all i = 1, 2. First of all, we consider the identities for resources R and I in ( 13) that is Here we rewrite the identity of I(t) in ( 13) as equation ( 23).Note that ( 22), ( 23) imply that the supply resource is equal to all consumption.We can rewrite ( 22), (23) in the form We called the vector in the left hand side of (24) the supply vector, and called the right hand side the consumption vector.We can use graphical presentation to predict the outcomes of competition, the method is similar to [14].See Figure 2, if the slope of supply vector at the corner of isocline of species 1 is larger than Rin−λ1 > C 1 .On the other hand, if T 1 < C 1 then species 1 is I-limited (region B) and the slope of consumption vector is For the case of species 2, we have similar results, that is, when T 2 = ln I0−ln ξ2 Rin−λ2 < C 2 = k2 σ2 , then species 2 is R-limited; when T 2 < C 2 , then species 2 is I-limited and the slope of consumption vector is . From Lemma 3.4, we know that C I2 is strictly decreasing in R in and strictly increasing in ln I 0 , and so is C I1 .

Table 3
Case Condition States Stability (I)

G.A.S. means globally asymptotically stable.
Assume λ 1 > λ 2 and (H2) hold, then species 1 is better competitor in light and species 2 is better competitor in nutrient.There are four possibilities in the following (see also Table 3): (I) Species 1 is I-limited and species 2 is I-limited, that is T 1 < C 1 , T 2 < C 2 .Because species 1 is better competitor in light, then E I1 is locally asymptotically stable.
The fact that and species 1 can not invade when the resident species attains the amount Hence E I2 is locally asymptotically stable.On the other hand, if min{µ 1 (Q I2 1 ), η 1 (ξ 2 )} − D > 0 then species 1 invade successfully and E I2 is unstable.Note that, under assumption (H2), (25) means that we have Hence E I2 is locally asymptotically stable if and only if T * > C I2 .Therefore, we have the following two cases.(I.a)If T * > C I2 then E I1 and E I2 are locally stable, hence the outcome depends on initial population, it is a bistable case; (I.b)If T * < C I2 , then E I2 is unstable.Hence E I1 is globally stable and species 1 competitively excludes species 2. (II) Species 1 is I-limited and species 2 is R-limited, that is We know that species 1 is better competitor in light and species 2 is better competitor in nutrient, hence E I1 and E R2 are locally asymptotically stable.Therefore, it is bistable and the outcome depends on initial population.(III) Species 1 is R-limited and species 2 is R-limited, that is We know that species 2 is better competitor in nutrient, hence E R2 is locally asymptotically stable.Note that min{µ and species 2 can not invade when the resident species attains the amount N R1 1 with cell quota Q R1 1 .Hence E R1 is locally asymptotically stable.On the other hand, if min{µ 2 (Q R1 2 ), η 2 (I R1 )}−D > 0 then species 2 invade successfully and E R1 is unstable.Under the assumption λ 1 > λ 2 , the inequality (26 , and λ 1 ≤ λ 2 , a contradiction.Hence the assumption λ 1 > λ 2 implies that (26) holds if and only if I R1 < ξ 2 , that is i.e., Hence We summarize all possible outcomes in Table 3 and present the relations in Figure 3. From calculation in the Appendix, we know that the curve T * = C I2 is a straight line.
We also draw the graphs, Figure 4, of the model in [10], which is two species (x 1 , x 2 ) competition for two resources R and S with internal storage.We follow the notations and results in [10], and demonstrate the competition outcomes graphically.We assume 0 < λ R2 < λ R1 < R 0 , 0 < λ S1 < λ S2 < S 0 and denote From calculation in the Appendix, we know that T * = C R1 and T * = C S2 are two straight lines.We summarize all possible outcomes of competition in Table 4 and illustrate in Figure 4.

Table 4
Case Condition States Stability (I)  6. Discussion.We analyze a competition model of two phytoplankton species for a single nutrient with internal storage and light in a well mixed aquatic environment.The results show that extinction, coexistence and bistability are predictable in this model.First, while the input concentration of nutrient is less than the break-even amount of each species, or when the input light intensity is lower than the basic need of each species, both species extinct (Theorem 3.1).Secondly, when the input concentrations of nutrient and light are higher than break-even concentration of all species, the outcomes vary as following.If the species i, i ∈ {1, 2}, has lowest breakeven concentration of nutrient and light, then it competitively excludes the other (Theorem 3.2).If the semitrivial equilibrium E i , i ∈ {1, 2}, is locally stable and E j , j = i, is unstable, then coexistence is impossible (Lemma 3.4) and E i is globally stable (Theorem 4.5).If both E i and E j are unstable, then coexistence is possible, that the species with the lowest break-even concentration of nutrient happens to have the lowest break-even light intensity, and competitively excludes others as the case in Theorem 3.2.The analytical result does not match the outcome of the experiment in Passarge et al. [11].We owe this to the violation on the assumption of trade-off between competitive ability for phosphorus and light in the focal species in Passarge et al. [11].Without the trade-off, the competitive exclusion will occur.
In reality, a large number of phytoplankton species coexist in competing for multiple nutrients and light.In this paper we consider only two species compete for a limiting single-nutrient with internal storage and light and obtain the possible outcomes that are classified in graphical presentation.To explore the mechanism promoting biodiversity in more realistic conditions, we will expand the model for multiple species with the assumption of internal storage for our future work.In addition, we will consider heterogeneity for light environment as suggested by Yoshiyama et al. [15].We trust such expansion in the model shall bring fruitful and insightful knowledge to such system.

Figure 1 .
Figure 1.The phase plane (N, U ) of system (11) in case (ii).Solid lines indicate the isoclines of N .Dashed line indicates the isocline of U .The vectors indicate the vector field of system (11).The set A in case (a) and the set B in case (b) are positively invariant sets.
in , ln I 0 ) as a function of R in and ln I 0 .Therefore C I2 = C I2 (R in , ln I 0 ) is also a function of R in and ln I 0 .The following lemma describes the monotonicity of C I2 .Lemma 3.4.C I2 (R in , ln I 0 ) is strictly decreasing in R in and strictly increasing in ln I 0 .

Lemma 4 . 1 .
Assume the semi-trivial equilibria E L1 and E L2 exist, L ∈ {R, I}.If one of them is locally stable and the other is unstable, then E RI c does not exist.

Figure 2 .
Figure 2. The isocline of N 1 is right-angled.In region A, T 1 > C 1 and species 1 is R-limited; in region B, T 1 < C 1 and species 1 is I-limited.

Figure 3 .
Figure 3. (a) and (b) are modified from Figure 1 in Passarge et al. [11].(c)-(e) illustrate outcomes of our model.The isoclines of N 1 and N 2 are right-angled and denoted by graphs 1 and 2, respectively.In dotted region, species 1 wins; species 2 wins in gray region; the feathered region indicates bistable case; the brick region represents species coexistence.The case C 2 < C 1 are (a) and (c); The case C 1 < C 2 are (b), (d), (e) where C I2 < C 1 in (d) and C I2 > C 1 in (e).
then E I1 , E R2 exist and both are locally stable and there exists a saddle equilibrium E RI c .The outcomes depend on initial conditions.(iii)If T 1 > C 1 , T 2 > C 2 , then E R1 , E R2 exist and E R2 is locally stable.(a)If T * < C 1 , then E R1 is locally stable and there exists a saddle equilibrium E RI c .The outcomes depend on initial conditions.
then E R1 and E I2 are both locally stable and there exists a saddle equilibrium E RI c .The outcomes depend on initial conditions.(d) If C 1 , C I2 < T * , then E R1 is unstable and E I2 is locally stable, and for Table 2 tells us that E R2 , E I1 are locally asymptotically stable if they exist; T * < C 1 if and only if E R1 is locally asymptotically stable; T * > C I2 if and only if E I2 is locally stable.(i) If T 1 < C 1 , T 2 < C 2 , then E I1 , E I2 exists and E I1 is locally stable.We know that the stability of E I2 affects the global behavior of system (13).Hence there are two subcases: (a) If T * > C I2 , then E I2 is locally stable.Hence, by Theorem 4.7, the outcomes depend on initial conditions.(b) If T * < C I2 , then E I2 is unstable.Hence, by Theorem 4.5,lim t→∞ and they are locally stable.Hence, by Theorem 4.7, the outcomes depend on initial conditions.(iii)IfT 1 > C 1 , T 2 > C 2 , then E R1 , E R2exist and E R2 is locally stable.Similarly, we will consider the following subcases.(a) If T * < C 1 , then E R1 is locally stable, and by Theorem 4.7, the outcomes depend on initial conditions.(b) If T * > C 1 , then E R1 is unstable, by Theorem 4.5, lim t→∞ (N 1 Similarly, we will consider the stability of E R1 , E I2 .
(a) If T * < C 1 , C I2, then E R1 is locally stable and E I2 is unstable.Hence, by Theorem 4.5, lim t→∞ (N 1 then E R1 and E I2 are unstable and E RI c exists, since Theorem 4.4.Hence, by Theorem 4.6, lim t→∞ then E R1 and E I2 are locally stable.Hence, by Theorem 4.7, the outcomes depend on initial conditions.(d) If C 1 , C I2 < T * , then E R1 is unstable and E I2 is locally stable.Hence, by Theorem 4.5, lim t→∞ (N 1 E R1 is locally asymptotically stable if and only if T * < C 1 .(III.a)IfT * < C 1 then E R1 and E R2 are locally asymptotically stable, hence the outcome depends on initial population, it is a bistable case; (III.b)IfT*>C 1 , then E R1 is unstable.Hence E R2 is globally stable and species 2 competitively excludes species 1.(IV) Species 1 is R-limited and species 2 is I-limited, that is T 1 > C 1 , T 2 < C 2 .From the stability of E R1 and E I2 which were discussed in (I) and (III), we have the following subcases:(IV.a)IfT * < C 1 , C I2 , then E I2 isunstable and E R1 is locally asymptotically stable.Hence E R1 is globally stable and species 1 competitively excludes species 2; (IV.b)If C 1 < T * < C I2 , then E R1 and E I2 are unstable, then there exists an coexistent equilibrium E RI c and it is globally stable; (IV.c)If C I2 < T * < C 1 , then E R1 and E I2 are locally asymptotically stable.Hence the outcome depends on initial population, it is a bistable case; (IV.d)If C 1 , C I2 < T * , then E R1 is unstable and E I2 is locally asymptotically stable.Hence E I2 is globally stable and species 2 competitively excludes species 1.