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2 Nonlinear Analysis: Real World Applications () Contents lists available at ScienceDirect Nonlinear Analysis: Real World Applications journal homepage: Dynamics of a model of allelopathy and bacteriocin with a single mutation Lan Zou a,b, Xingwu Chen a, Shigui Ruan b,, Weinian Zhang a a Yangtze Center of Mathematics and Department of Mathematics, Sichuan University, Chengdu, Sichuan 664, PR China b Department of Mathematics, University of Miami, Coral Gables, FL , USA a r t i c l e i n f o a b s t r a c t Article history: Received January 9 Accepted July Keywords: Population dynamics Saddle node Index of equilibrium Center manifold Bifurcation In this paper we discuss a model of allelopathy and bacteriocin in the chemostat with a wild-type organism and a single mutant. Dynamical properties of this model show the basic competition between two microorganisms. A qualitative analysis about the boundary equilibrium, a state that both microorganisms vanish, is carried out. The existence and uniqueness of the interior equilibrium are proved by a technical reduction to the singularity of a matrix. Its dynamical properties are given by using the index theory of equilibria. We further discuss its bifurcations. Our results are demonstrated by numerical simulations. Elsevier Ltd. All rights reserved.. Introduction Antibiotic resistance among bacteria has become a worldwide public health threat [,]. One of the limitations of using broad-spectrum antibiotics is that they kill almost any bacterial species not specifically resistant to the drug [3]. Frequent use of these antibiotics result in an intensive selection pressure for the evolution of antibiotic resistance in both pathogen and commensal bacteria [4,3]. Alternative methods of combating infection has been considered [5,3]. Allelopathy is the chemical inhibition of one species by another [6,7]. Bacteriocin, a particular type of allelopathy, is a toxin produced by bacteria that inhibits the growth of closely related species [8]. Bacteriocins provide an alternative solution with their relatively narrow spectrum of killing activity. Special examples of bacteriocin productions involving more than two competing organisms, such as the bacteriocins of Escherichia coli and Klebsiella pneumonia, are given in [5]. Various mathematical models have been proposed to examine the interaction between bacteriocin-producer and sensitive strains ([9,,8], etc.). Recently, Abell et al. [8] proposed chemostat-type competition models with mutation and toxin production. Via numerical simulations, they showed how the coexistence of competitors depends upon the growth rates and toxin sensitivity. A chemostat is a laboratory bio-reactor used to culture microorganisms []. Competition for single and multiple resources, the evolution of resource acquisition, and competition among organisms have been investigated in ecology and biology using chemostats [ 7]. Since 95 [5,6], chemostat models have been studied. We refer to the monograph of Smith and Waltman [], the surveys of Hsu and Waltman [8] and Ruan [9] and the references cited therein. In this paper, we study the dynamics of the allelopathy model in the chemostat with a wild-type organism and a single mutation proposed by Abell et al. [8]. Let S(t) be the concentration of the nutrient at time t, x(t) be the density of the wild-type organism X at time t, and y(t) be the density of a mutant Y at time t, respectively. It is assumed that the two Corresponding author. address: ruan@math.miami.edu (S. Ruan) /$ see front matter Elsevier Ltd. All rights reserved. doi:.6/j.nonrwa..7.8

3 L. Zou et al. / Nonlinear Analysis: Real World Applications () microorganisms X and Y compete for the nutrient, the wild-type organism X can mutate to the microorganism Y, and the single mutant of a parental species Y can revert to the wild-type X during reproduction. The model takes the following form: S (t) = D(S S(t)) γ f (S)x(t) γ f (S)y(t), x (.) (t) = x(t)[( α)f (S) D] + βf (S)y(t), y (t) = y(t)[( β)f (S) D] + αf (S)x(t), where D is the dilution rate, γ is the yield constant, α is the rate at which X mutates, β is the rate at which Y reverts to X during reproduction, and f i (S) = m i S/(a i + S) is the Michaelis Menten Monod function, in which m i > is the maximum growth rate and a i > is the Michaelis Menten Monod constant. After rescaling the variables and some parameters and using the same notation, the model can be written as follows [8]: S = S f (s)x f (S)y, x = x[( α)f (S) ] + βf (S)y, (.) y = y[( β)f (S) ] + αf (S)x. Let Σ = S x y. Then, Σ = Σ and system (.) can be replaced with Σ = Σ, x = x[( α)f ( Σ x y) ] + βf ( Σ x y)y, y = y[( β)f ( Σ x y) ] + αf ( Σ x y)x. (.3) Since solutions of Σ = Σ all tend to as t +, system (.) is asymptotic to the two-dimensional system [,] [ x = x ( α) m ] ( x y) a + x y + β m ( x y) a + x y y := f (x, y), [ y = y ( β) m ] ( x y) a + x y + α m (.4) ( x y) x := g(x, y) a + x y in the region G = {(x, y) : x, y, x + y }. When α = β =, system (.) has been discussed in [3]. Later, Hsu et al. [4] presented some results for the case β = and < α <, which is equivalent to the case α = and < β <. Therefore, we mainly consider the case < α, β in this paper. Based on the study on system (.) in [8], we first discuss the properties of the boundary equilibrium E. We then consider the existence and uniqueness of the interior equilibrium E by using a singular matrix. By employing the index of the equilibrium, we study the stability of E by its Jacobian matrix. The paper is organized as follows. In Section we consider the boundary equilibria. The existence of the interior equilibrium is addressed in Section 3 and the properties of the interior equilibrium are given in Section 4. Section 5 deals with possible bifurcations of the model. Numerical simulations and some remarks are given in Section 6.. The boundary equilibria To find the equalibria of system (.4), we find zeros of the coupled equations f (x, y) =, g(x, y) =, i.e., [ x ( α) m ] ( x y) a + x y + β m ( x y) a + x y y =, [ y ( β) m ] ( x y) a + x y + α m ( x y) a + x y x =. We can see that E = (, ) is the unique boundary equilibrium of system (.4) when < α and < β. In fact, when x =, (.) is equivalent to that β m [ ( y) a + y y = and y ( β) m ] ( y) a + y =, implying that y =. Similarly, if y =, (.) exists if and only if x =. Furthermore, on x + y =, (.) is equivalent to that y = and x =, which obviously do not exist on x + y =. So there is no other boundary equilibrium except E. In order to determine the qualitative properties of E, we calculate the Jacobian of system (.4) at E, i.e., ( α)m βm J(E ) = a + αm a + a + ( β)m a + (.)

4 66 L. Zou et al. / Nonlinear Analysis: Real World Applications () and see that eigenvalues are zeros of the polynomial P(λ) := λ T λ + D, where T := tr J(E ) = ( α)m a + + ( β)m a +, D := det J(E ) = ( α β)m m (a + )(a + ) ( α)m ( β)m +. a + a + Since the discriminant ( := T α)m 4D = a + ( β)m + 4 a + It follows that E is neither a focus nor of the center type. αβm m >. (.) (a + )(a + ) Theorem. (i) E is a saddle if D <. Moreover, the trajectories starting from G all go far away from E, (ii) E is a stable node if D > and T <. (iii) E is an unstable node if D > and T >. (iv) If D =, then the equilibrium E is a saddle node and E is stable in G. Proof. In order to prove (i), we note that the stable manifold is tangent to the eigenvector (x s, y s ) at E and the unstable manifold is tangent to the eigenvector (x u, y u ) at E, where x s, y s, x u and y u satisfy ( α)m 4 ( β)m ( α)m ( β)m αm βm + 4 xu + βm a + a + a + a + (a + )(a + ) a + y u =, ( α)m 4 ( β)m ( α)m + ( β)m αm βm + 4 xs + βm a + a + a + a + (a + )(a + ) a + y s =, which means that x u y u > and x s y s <. Therefore, the intersection of the stable manifold and G is empty, while the intersection of the stable manifold and G is nonempty. This proves (i). When D >, we assure that T ; otherwise, = 4D <, a contradiction to (.). Thus we only need to discuss the case of T < and the case of T >. In the two cases the assumption D > implies results (ii) and (iii), respectively. In order to discuss the case (iv), we need to determine the qualitative properties in the degenerate case. Applying a timescaling transformation t = τ(a + x y)(a + x y), we can reduce system (.4) orbital-equivalently to the polynomial differential system x = f (x, y), y = g(x, y), (.3) where f (x, y) := [( α)m a ](a + )x + βm (a + )y + {a + (a + )[ ( α)m ]}x + [(a + )( ( α)m ) + a βm (a + )]xy βm (a + )y + [( α)m ]x 3 + [βm + m ( α)]x y + [βm + ( α)m ]xy + βm y 3, g(x, y) := αm (a + )x + [( β)m a ](a + )y m α(a + )x + [( ( β)m )(a + ) + a αm (a + )]xy + {a + (a + )[ ( β)m ]}y + αm x 3 + [m α + m ( β)]x y + [αm + ( β)m ]xy + [( β)m ]y 3. If D =, then {( α)m (a + )}{( β)m (a + )} = αβm m. (.4) Assume that T. Then, (a + )[( α)m (a + )] (a + )[( β)m (a + )]. Substituting (.4) in it, we get (a + )(( α)m (a + )) αβm m (a + ), which holds if and only if its both sides vanish. This contradicts to the inequality βm αm (a + ) >. In fact, β >, α >, m >, m >, and a >. It concludes that T <. Therefore, only one of eigenvalues of the system vanishes but the other is equal to T <. Furthermore, with a transformation u = αm (a + )x + (( α)m a )(a + )y, v = αm (a + )x (( β)m a )(a + )y,

5 L. Zou et al. / Nonlinear Analysis: Real World Applications () we change system (.3) into the form u = U(u, v), v = µv + V(u, v), (.5) where µ = ( α)m (a +)+( β)m (a +) (a +)(a +) and U, V are O( u + v ) and shown in Appendix A. By the implicit function theorem, there is a unique function v = φ(u) such that φ() = and V(u, φ(u)) =. Actually, we can solve from µv + V(u, v) = that φ(u) = αm {(a + )[(a + )( ( α)m ) + a ] [( β)m (a + )](a + )(a + )}u /[( α)m (a + ) + ( β)m (a + ) (a + )(a + )] + O( u 3 ). Substituting v = φ(u) in the first equation of (.5), we get U(u, φ(u)) = a αm (a + ) u + O( u 3 ), which implies that E is a saddle node of system (.5). i.e., E is a saddle node of system (.4). Moreover, since a αm (a + ) >, the two hyperbolic sectors lie on the left but the parabolic sector lies on the right. So E is stable in G. 3. Existence of interior equilibria It is much more difficult to determine the interior equilibria because much more complex computation is needed. In this section we consider (.4) in the interior of the region G, denoted by intg. Theorem. System (.4) has at most one interior equilibrium. Furthermore, system (.4) has exactly one interior equilibrium E = (x, y ) if and only if E is unstable in G, where βm z (a + z )( z ) x = (a + z )βm z (a + z )(K z a ), y = (a K + a K ) +, αβm m K K, z = (αβm m K K ) a a, αβm m = K K, a K + a K = (a K a K ) + 4αβm m a a, K = ( α)m, K = ( β)m. (a + z )(K z a )( z ) (a + z )βm z (a + z )(K z a ), Proof. The system (.) of determining equilibria is equivalent to the system x[( α)m ( x y) a + x + y](a + x y) + βm ( x y)(a + x y)y =, y[( β)m ( x y) a + x + y](a + x y) + αm ( x y)(a + x y)x =, which can be rewritten as [ ] [ ] (K z a )(a + z) βm z(a + z) x = αm z(a + z) (K z a )(a + z) y [ ], (3.) where z := x y. The system has an interior equilibrium if and only if there is a pair of nonzero x and y such that (3.) holds. It follows that the determinant of coefficient matrix of (3.) is equal to zero, i.e., Υ (z) := (αβm m K K )z + (a K + a K )z a a =. (3.) Since the region intg requires < z <, we need to discuss positive roots of the quadratic equation (3.). Case. αβm m K K >. In this case the quadratic equation (3.) surely has a positive root, which must be because z = (a K + a K ) +, (αβm m K K ) (3.3) := (a K a K ) + 4αβm m a a > (3.4)

6 66 L. Zou et al. / Nonlinear Analysis: Real World Applications () and a a <. Thus, the quadratic equation (3.) has at most one zero in the interval (, ) and system (.4) has at most one interior equilibrium. Furthermore, with z given by (3.3) we can determine the coordinates x, y of the corresponding equilibrium. Actually, from the first equation in (3.), where z is replaced by z, we get Let {(a + z )(K z a )}x = {(a + z )βm z }y. ξ = It implies that x (a + z)βm z. y = (a + z )(K z a )ξ. (3.5) (3.6) Noting that z = x y, from (3.5) and (3.6), we obtain z = [(a + z )βm z (a + z )(K z a )]ξ, which implies that ξ = z (a + z )βm z (a + z )(K z a ). Thus, by (3.5) and (3.6), we uniquely obtain βm z (a + z )( z ) x = (a + z )βm z (a + z )(K z a ), y = (a + z )(K z a )( z ) (a + z )βm z (a + z )(K z a ). Case. αβm m K K. In this case we have K K >, implying that a K + a K, the coefficient of the first degree term in Υ (z), does not vanish. Since Eq. (3.) has no positive roots when a K + a K <, we only need to discuss in the case that a K + a K >. When a K + a K >, we obviously have K >, K >. In the circumstance that αβm m K K =, Eq. (3.) reduces to a linear equation and has a unique root z = a a /(a K + a K ). It lies in (, ) if and only if a K + a K a a >. Such a z determines the coordinates x, y by the same (3.7) as in the discussion in Case. In the other circumstance, i.e., αβm m K K <, the quadratic equation (3.) has two positive roots z = (a K + a K ) + (αβm m K K ), z = (a K + a K ), (αβm m K K ) where is defined in (3.4). Clearly, z < z. We claim that system (.4) has at most one interior equilibrium, the same point E = (x, y ) as in Case, where x and y are given in (3.7). It suffices to prove that the equilibrium E = (x, y ), determined by z, locates outside the region intg. For an indirect proof, assume that x >, y >. It is clear that < z := (x + y ) <. Since x, y have the same expressions as x, y in (3.7) where z is replaced with z and the numerator of x, i.e., βm z (a + z )( z ), is positive, we see that the common denominator (a + z )βm z (a + z )(K z a ) is also positive. It follows from the numerator of y, i.e., (a + z )(K z a )( z ), that K z a <. It is equivalent to say K (a K + a K ) a (αβm m K K ) > K. Taking the square of both sides, we have 4a αβm m K (a K a K ) + 4a α β m m > 4K αβm m a a, which implies that αβm m K K >, a contradiction to the assumption that αβm m K K <. In summary, in all cases we considered above, system (.4) has at most one equilibrium in the region intg. Now we further prove that system (.4) has at least one equilibrium in this open region when E is unstable. Construct a closed curve Γ with line segments AB := {(x, y) : x + y =, x, y }, BB := {(x, y) : y =, < ϵ x }, B A := {(x, y) : x + y = ϵ, x, y ϵ} and A A := {(x, y) : x =, < ϵ y }. Restricted to AB, system (.4) reduces to the form dx/dt = x, dy/dt = y, implying that both x and y decrease and trajectories staring from AB all enter G. Restricted to A A, system (.4) implies that dx/dt = βm ( y)y/(a + y) >, i.e., x(t) increases, and therefore trajectories staring from A A all enter G. Similarly, trajectories staring from BB all enter G. Moreover, all trajectories starting from B A go away from E as ϵ is chosen sufficiently small when E is unstable. This proves that all trajectories starting (3.7) (3.8)

7 L. Zou et al. / Nonlinear Analysis: Real World Applications () from Γ enter the region surrounded by Γ. Therefore, the winding number of the vector field (.4) along the curve Γ is equal to. As indicated in [, p. 33] and [, Chpater 3], there is at least one equilibrium in the interior of the region bounded by the curve Γ. As a consequence, system (.4) has exactly one interior equilibrium E when E is unstable. Furthermore, we claim that no equilibrium exists in intg when E is stable. In this case, by Theorem, we have T < and D, which respectively implies that ( ( α)m a )+( ( β)m + a ) < and ( ( α)m + a )( ( β)m + a ) αβm m + (a +)(a +) >. It means that ( α)m a < and ( β)m + a <. Hence, + K < a, K < a, Υ () = αβm m K K + a K + a K a a, (3.) where Υ is defined in (3.). In the case that αβm m K K >, the quadratic function Υ is convex and has no zeros in (, ) by (3.) and the fact Υ () = a a <. In the case that αβm m K K = the function Υ, linking two non-positive Υ () and Υ () linearly, also has no zeros in (, ). In the case that αβm m K K <, the quadratic function Υ is concave. If it has a positive zero then, as discussed in the above Case, (3.8) holds, i.e., K > and K >. By (3.9), < K K < a a. It follows that σ := a a αβm m K K >. If Υ () < then, by (3.) and (3.4), the function Υ has either no or two zeros in (, ), but (3.), where we note that σ is equal to the product of two zeros, implies that at least one zero of Υ lies outside [, ]. This proves that Υ has no zeros in (, ). If Υ () = then one zero of Υ is z = and the other zero is z = σ > by (3.). It also implies that Υ has no zeros in (, ). Consequently, system (.4) has no interior equilibria. Summarizing the above discussion we see that system (.4) has exactly one interior equilibrium if and only if E is unstable. The proof of Theorem also gives a computable condition for the existence of the interior equilibrium, i.e., system (.4) has exactly one interior equilibrium if and only if < z <, K z a <, where z, K, K are defined in Theorem. 4. Properties of the interior equilibrium By Theorem, we only need to discuss the unique interior equilibrium E in the region intg when the boundary equilibrium E is unstable. Theorem 3. E is asymptotically stable if it exists. Moreover, E is a stable node if where (3.9) (3.) (3.) (3.3) δ := m z (a + z ) 4 (a + z ) β + c β + c, (4.) c = (a + z ) (a + z )m z [m z ( + α)(a + z )(a + z ) z (a + z ) (a + z ) m a x (a + z ) m a y z (a + z )(a + z ) + m a y (a + z ) ], c = ( α) m z (a + z ) (a + z ) 4 ( α)m m z (a + z ) 3 (a + z ) 3 + m z (a + z ) 4 (a + z ) ( α)m a x z (a + z )(a + z ) + ( α)m m a x z (a + z ) (a + z ) 3 + ( α)m m a y z (a + z ) 3 (a + z ) m a y z (a + z ) 4 (a + z ) + m a x (a + z ) 4 + m m a a x x (a + z ) (a + z ) + m a y (a + z ) 4 and x, y, z are defined by parameters α, β, a, a, m, m as in (3.7) and (3.3). Proof. Qualitative properties of E are determined by the signs of the trace T, the determinant D and the discriminant of the Jacobian matrix of system (.4) at E. Simple computation shows that T = (K z a )(a + z ) m a x (a + z ) + (K z a )(a + z ) m a y (a + z ), D = ( α)m z a + z ( β)m z a + z ( α β)m m a z y ( α β)m m a z x, (a + z )(a + z ) (a + z ) (a + z ) + ( α β)m m z + m a x (a + z )(a + z ) (a + z ) + m a y (a + z )

8 664 L. Zou et al. / Nonlinear Analysis: Real World Applications () = ( α) m z + m m z (α( + β) ( β)) (a + z ) (a + z )(a + z ) + m m a x z ( α β) (a + z ) (a + z ) + ( β) m z (a + z ) + m m a y z ( α β) (a + z )(a + z ) a y + m a x (a + z ) + m m a a x x 4 (a + z ) (a + z ) + m (a + z ). 4 ( β)m ( α)m a x z (a + z ) 3 a y z (a + z ) 3 From the first equation in (3.) we see that K z a < m z (a + z )x /{(a + z )y } < since a, a, m, K, x, y, z are all positive. Similarly K z a < by the second equation in (3.). It follows that T <. We further claim that D. In fact, if D <, i.e., E is a saddle, then the index of E is equal to. Consider the closed curve Γ composed in the proof of Theorem. The fact that all trajectories starting from Γ enter the region surrounded by Γ implies that the winding number of the vector field (.4) along the curve Γ is equal to. This makes a contradiction to the Theorem on the sum of indices ([, p. 33], [, Chpater 3]) because of the uniqueness of the interior equilibrium (given in Theorem ). In what follows we discuss the case D > and the case D = separately. In the case D >, E is either a stable node or a stable focus because T <. Furthermore, E is a node if the inequality holds. This inequality can be simplified as the condition (4.) because a + z > and a + z >. In the case D =, the equilibrium E is degenerate. Clearly, = T > in this case, i.e., condition (4.) is satisfied naturally. Since T <, the two eigenvalues at E are λ = and λ = T <. This enables us to diagonalize the linear part of the cubic system (.3) so that the system is transformed into the form dx dt = G (x, y) + G 3 (x, y), (4.) dy dt = y + H (x, y) + H 3 (x, y), where G j, H j are homogeneous polynomials of degree j (j =, 3) and the interior equilibrium E is translated to the origin. By Theorem 7. in [, Chapter ], the origin of (4.) is either a stable node or a saddle or a saddle node. However, by Bendixson s formula (see [3], [, Chapter 3]) J = + e h, where J is the index of the equilibrium, h is the number of hyperbolic sectors near the equilibrium and e is the number of elliptic sectors, we get either J = when E is a saddle or J = when E is a saddle node, both of which contradict to the Theorem on the sum of indices ([, p.33], [, Chapter 3]) because the winding number of the vector field (.4) along the curve Γ is equal to, as proved in the paragraph above (3.). This implies that E is a stable node. 5. Bifurcations It is indicated in Sections 3 and 4 that system (.4) has either exact one equilibrium in G when D > and T < or exact two equilibria in G in the other case. The following theorem displays the mechanism for the new one to arise. Let T := T (a + )(a + ), D := D (a + ) (a + ), and µ := [ 3( α)m (a + ) 4(a + )(a + ) + ( β)m (a + ) [ T 4D ] (a + a + ) ( β)m (3a + 5) 3( α)m (a + ) + 4(a + )(a + ) + + 4αβm m (a + )(a + ) [a + (a + )( ( α)m )] [3( α)m (a + ) ] + ( β)m (a + ) 4(a + )(a + ) + T 4D, ν := (3 α)m (a + ) 4( + a )( + a ) + ( β)m (a + ) + T 4D where T and D are defined as in Theorem. ] T 4D Theorem 4. If ν µ, system (.4) experiences a transcritical bifurcation at E when D =. More concretely, for sufficiently small D, a stable equilibrium appears in the first quadrant when ν µ D > and an unstable equilibrium appears in the third quadrant when ν µ D <.

9 L. Zou et al. / Nonlinear Analysis: Real World Applications () Proof. As shown in Theorem, E is a saddle node as D =. Applying the transformation u = αm (a + )x + [( α)m a ](a + ) T + T 4D y, v = αm (a + )x + [( α)m a ](a + ) T T 4D y (5.) to diagonalize the linear part of system of (.3), we change system (.3) into the form [ ] u T + T v = 4D [ ] u T T + 4D v [ ] q (u, v), (5.) q (u, v) where q (u, v) and q (u, v) are composed of those terms of degree or 3. With a rescaling dτ = (T the system can be reduced to the following suspended system of (5.) u = D u + T T 4D q (u, v), v = T T 4D v + T T 4D q (u, v), D =. T 4D )ds, By the center manifold theory (see [4]), as D = system (5.3) has a two-dimensional center manifold W c : v = W(u, D ) near E. In order to obtain the second-order approximation of function W, let (5.3) W(u, D ) := ϕ(u, D ) + O( u, D 3 ), (5.4) where ϕ(u, D ) := ν u + ν uv + ν 3 v, and let (N ϕ)(u, D ) := ϕ (u, D ) D u + T T T 4D q (u, ϕ(u, D )) T 4D ϕ(u, D ) T T 4D q (u, ϕ(u, D )). By Theorem 3 in [4, Chapter ], from the requirement (N ϕ)(u, D ) = O( u, D 3 ), we can solve ν = ν 3 = and [ ν ν = 4αm (a + )(T 4D 3( α)m (a + ) 4(a + )(a + ) + ( β)m (a + ) ) ] [ + T 4D (a + a + ) ( β)m (3a + 5) 3( α)m (a + ) + 4(a + )(a + ) ] [ + T 4D + 4αβm m (a + )(a + ) [a + (a + )( ( α)m )] 3( α)m (a + ) ] + ( β)m (a + ) 4(a + )(a + ) + T 4D. Thus, we can substitute (5.4) in (5.3) and obtain the equation u = D u + µu + µ u 3 + O( u, D ) 4, (5.5) the restricted system on W c, where µ := ν µ 4αm (a + )(T 4D ), (5.6) µ is defined in the beginning of this section and µ is expressed in Appendix A. Clearly, the expression (5.5) shows that a transcritical bifurcation [5, p. ] occurs at E as D varies through the bifurcation value D = when µ. Since D has the same zeros with D, the transcritical bifurcation occurs at E as D varies through the bifurcation value D =. More concretely, when ν µ D <, the origin O is stable and the other equilibrium A appears on the negative u-axis but

10 666 L. Zou et al. / Nonlinear Analysis: Real World Applications () Fig.. Phase portraits in cases (S) (S3). is unstable; when D =, the two equilibria coincide at O; when ν µ D >, the origin O remains an equilibrium but is unstable while a stable equilibrium A arises on the positive u-axis. Finally, the transformation (5.) gives the correspondence between A, A and the equilibria B, B of system (.3), which lie in the first quadrant and the third one, respectively. Note that the third quadrant is of no practical interests. This proof shows that equilibrium B (actually the same as the interior equilibrium E ) arises from a transcritical bifurcation as D varies through the bifurcation value D =. This explains how E is produced. We ignore B since it does not appear in the first quadrant. In Theorem 4, the required nondegenerate condition ν µ appears in the expression (5.6) of µ. If this condition is violated, i.e., ν µ =, system (5.5) turns into the form u = D u + µ u 3 + O( u, D ) 4 (5.7) and the coefficient µ becomes a decisive quantity. Unfortunately, µ =. If µ, a pitchfork bifurcation [5, p. ] occurs at E as D passing through the bifurcation value D =. It means that the origin is the unique equilibrium as D µ >, implying as known in Section 3 that the function Υ defined in (3.) has no real zeros, i.e., αβm m > K K, = (a K a K ) + 4αβm m a a <. (5.8) This is an obvious contradiction because all parameters are positive. For the same reason we assure that the first nonzero coefficient in the expansion (5.7) appears in an even term. The corresponding bifurcations can be discussed similarly to the proof of Theorem 4. Note that no bifurcations occur at the interior equilibrium E, although its degeneracy for D = is shown in the proof of Theorem 3. In fact, when D = the asymptotically stable equilibrium E does not coincide with E ; otherwise, their coincidence gives a saddle node, which contradicts to the stability of E as shown in Theorem 4. Moreover, ignoring E, there also does not occur a bifurcation at E, which lies in the interior of G for D ; otherwise, the only possibility is the pitchfork bifurcation as shown in Theorem 4, which produces three equilibria in the first quadrant as D > and makes a contradiction to the uniqueness of interior equilibria (given in Theorem ). 6. Numerical simulations and remarks We simulate orbits of the system (.4) to demonstrate our Theorems and 3. We consider the following cases (S) (S3): Parameter values (m, a, α), (m, a, β) (S) (5,.7,.), (,.4,.) (S) (3.,.7,.), (.4,.4,.) Equilibrium E i Eigenvalues λ, λ of J(E i ) Properties E = (, ) λ =.657, λ =.368 Saddle E = (.69,.9) λ = 3.7, λ =.7 Stable node E = (, ) λ =.754, λ =.47 Unstable node E = (.,.499) λ =.37, λ =.48 Stable node (S3) (.,.4,.), (.3,,.) E = (, ) λ =.85, λ =.899 Stable node The phase portraits in the cases (S) (S3) are plotted separately in Fig., showing that the wild-type organism X and the mutant Y coexist in both (S) and (S), E is a saddle in case (S) but an unstable node in case (S), and both X and Y finally go to extinction in case (S3) while E is a stable node.

11 L. Zou et al. / Nonlinear Analysis: Real World Applications () beta beta beta Fig.. The graph of D (β) when α =. (left),.4 (middle), and.9 (right), separately beta beta beta Fig. 3. The graph of δ (β) when α =. (left),.4 (middle), and.9 (right), separately. When a >, a > are small enough and m >, m > are large enough, we see that D < as α is small and D > as α is large and β is small, implying that the condition D = in Theorems and 4 is reasonable. However, it is still difficult to prove either D or D = for some parameters. Fixing a =.7, a =.4, m = 5 and m = for instance, consider the function D (β) := 5( α)z ( β)z 5( α β)z z.4 + z (.7 + z )(.4 + z ) + 3.5x (.7 + z ) +.4y (.4 + z ) ( α β)z y (.7 + z )(.4 + z ) 3.5( α β)z x (.7 + z ) (.4 + z ), parametrized by α, where x, y, z are shown in Appendix B. It is shown in Fig. that D (β) has no zeros when α =.,.4,.9. It seems natural to assert that, in contrast to the statement of Theorem 3, E is a stable focus if δ <, but it is hard to find a choice of parameters for the simulation of focus at E. It suggests a conjecture that E is a node. Furthermore, with the same parameters a =.7, a =.4, m = 5 and m =, we calculate the function δ (β) = ς β + ς β + ς, where ς, ς, ς are shown in Appendix B. Its graphs, plotted in Fig. 3 for the three choices α =.,.4 and.9, show that δ has no zeros in (, ). However, it is still difficult to rule out the possibility of the inequality δ <. We have studied the dynamics of an allelopathy model with a single mutant. It was shown that the model has a unique stable interior equilibrium under certain conditions, which differs from the typical dynamics of chemostat models. If x and y denote the populations of the wild-type and resistant bacteria, respectively, the model and results may be used to discuss population dynamics of antibiotic-resistant bacteria [6,7]. It will be interesting to study the dynamics of the general model () in [8] with multiple mutants. We leave this for future consideration. Acknowledgements The second author was supported by NSFC Tianyuan Fund 9645 and SRFDP 988. The third author was partially supported by NIH grant RGM8367 and NSF grant DMS The fourth author was supported by NSFC # 854 and China MOE Research Grant.

12 668 L. Zou et al. / Nonlinear Analysis: Real World Applications () Appendix A. Expressions of U(u, v), V (u, v) U(u, v) := a αm (a + ) u + { αm (a + )[( ( α)m )(a + ) + a βm (a + )] + [( α)m a ](a + )[ αm (a + ) + ( ( β)m )(a + ) + a ]}uv + (a + ){αβm m (a + ) + (( α)m a )[( ( β)m )(a + ) + a ]}v αa m (a + )u 3 + (a + ){ αm ( + ( α)m + βm ) + (( α)m a )[( β)m + αm ]}u v + (a + ){αβm m (a + ) + (( α)m a )[( ( β)m )(a + ) + a ]}uv + { αm (a + )βm + (( α)m a )(a + )(( β)m )}v 3, V(u, v) := αm { (a + )[( ( α)m )(a + ) + a ] + [( β)m a ](a + )(a + )}u + { αm (a + )[( ( α)m )(a + ) + a β(a + )] [( β)m a ](a + )( αm (a + ) + ( ( β)m )(a + ) + a )}uv + {αβm m (a + )(a + ) [( β)m a ](a + )[( ( β)m )(a + ) + a ]}v + { αm (a + )( + ( α)m ) [( β)m a ](a + )αm }u 3 + { αm (a + )( + ( α)m + βm ) [( β)m a ](a + )[( β)m + αm ]}u v + { αm (a + )( + ( α)m + βm ) [( β)m a ](a + )(( β)m + αm )}uv + { αm (a + )βm [( β)m a ](a + )[( β)m ]}v 3 µ = αm (a + ) [(( α)m (a + ) ( β)m (a + )) + 4αβm m (a + )(a + )] (( β)m a )(a + ) α βm (( α)m a αm ) m (a + )(a + ) + (a + (a + )( ( α)m ))(3( α)m (a + ) + ( β)m (a + ) 4( + a )( + a ) Θ)(3( α)m (a + ) + ( β)m (a + ) 4( + a )( + a ) + Θ) + αm (a + )(( ( α)m )(a + ) + a βm (a + ))(3( α)m (a + ) + ( β)m (a + ) + 4(a + )(a + )) (3( α)m (a + ) + ( β)m (a + ) 4( + a )( + a ) Θ) [ (a + ( ( β)m )(a + ))αm (a + ) + (3( α)m (a + ) + ( β)m (a + ) 4( + a )( + a ) Θ)(3( α)m (a + ) + ( β)m (a + ) 4( + a )( + a ) + Θ) (( ( β)m )(a + ) + a αm (a + ))(3( α)m (a + ) + ( β)m (a + ) ] + 4(a + )(a + )) + [ αm Θ 8 (( α)m )(3( α)m (a + ) + ( β)m (a + ) 4( + a )( + a ) + Θ) αm (a + )(βm + ( α)m )(3( α)m (a + ) + ( β)m (a + ) 4( + a )( + a ) + Θ) + α m (a + ) (βm + ( α)m )(3( α)m (a + ) + ( β)m (a + ) 4( + a )( + a ) + Θ) + α 3 βm 3 m (a + ) 3 (a + ) (3( α)m (a + ) + ( β)m (a + ) 4( + a )( + a ) Θ) 8 (3( α)m (a + ) + ( β)m (a + ) 4( + a )( + a ) + Θ) (αm + ( β)m )(a + )(3( α)m (a + ) + ( β)m (a + ) 4( + a )( + a ) + Θ) + αm (αm + ( β)m )(3( α)m (a + ) + ( β)m (a + ) 4( + a )( + a ) + Θ)(a + ) + (( β)m )α m (a + ) 3 ], where Θ := (( α)m (a + ) ( β)m (a + )) + 4αβm m (a + )(a + ).

13 L. Zou et al. / Nonlinear Analysis: Real World Applications () Appendix B. Expressions of x, y, z, ς, ς, ς z := (. +.5α +.875β +.5ϖ ), β x :=.65(α 73β + ϖ 6)(α + 63β + ϖ 6)(α + 7β + ϖ 6)/(64ϖ α 6αϖ + 83β 8βϖ 48α + α ϖ 376αβ + 35αβϖ + 35β ϖ + α α β + 645αβ + 45β 3 ), y := β (5.6ϖ α 96αϖ 793.6β βϖ 384α + α ϖ + 8αβ 68αβϖ β 64β ϖ + 3α 3 5α 3 ϖ 48α β + 5α βϖ 798αβ αβ ϖ + 448β 3 α 4 + 5α 3 β + 7.5α β αβ 3 )/(64ϖ α 6αϖ + 83β 8βϖ 48α + α ϖ 376αβ + 35αβϖ + 35β ϖ + α α β + 645αβ + 45β 3 ), ς = 5z 8 α 4z 8 α + 6z8 + 75z7 α 7z 7 α z7 + z6 x α 8z 6 x + 3.z 6 y 4z 6 y α + 9.5z 6 α 39.8z 6 α z6 6.3z5 x + 44.z 5 x α + 9.4z 5 y.6z 5 y α z 5 α 87.4z 5 α z5 +.5z4 x +.8z4 x y 54.5z 4 x z 4 x α +.6z 4 y z4 y 3.4z 4 y α +.36z 4 α 3.4z 4 α +.65z z3 x 3.z3 x + 6.6z 3 x y z 3 x α +.448z 3 y z 3 y 7.4z 3 y α + 4.3z 3 α z 3 α z z +.336z α z x +.76z x.4768z α z x y +.68z x α +.474z y +.478z y.384z y α z x +.886z x α.4768z x z y +.748z x y.95z y α +.95z y +.336x +.95x y y, ς := z 8 α + 8z8 + 33z7 α + 5.8z z6 α 7z6 x 7.z 6 y + 34.z 6 8.z5 x.64z 5 y z 5 α z5 +.56z4 α 8.55z4 x 3.3z 4 y z z3 α +.764z z3 x.86z 3 y.736z x +.95z α z y z.95x z.368y z, ς := z z z z5 +.88z4 +.46z z, where ϖ := 56 64α + 4β + 4α + 8αβ + 49β. References [] H.C. Neu, The crisis in antibiotic resistance, Science 57 (99) [] S.B. Levy, B. Marshall, Antibacterial resistance worldwide: causes, challenges and responses, Nat. Med. (4) S S9. [3] M.A. Riley, J.E. Wertz, Bacteriocins: evolution, ecology, and application, Annu. Rev. Microbiol. 56 () [4] E.S. Walker, F. Levy, Genetic trends in a population evolving antibiotic resistance, Evolution 55 (). [5] M.A. Riley, Molecular mechanisms of bacteriocin evolution, Annu. Rev. Genet. 3 (998) [6] R. Durrett, S.A. Levin, Allelopathy in spatially distributed populations, J. Theor. Biol. 85 (997) [7] Y. Iwasa, M. Nakamaru, S.A. Levin, Allelopathy of bacteria in a lattice population: competition between colicin-sensitive and colicin-producing strains, Evol. Ecol. (998) [8] M. Abell, J. Braselton, L. Braselton, A model of allelopathy in the context of bacteriocin production, Appl. Math. Comput. 83 (6) [9] L. Chao, B.R. Levin, Structured habitats and the evolution of anticompetitor in bacteria, Proc. Natl. Acad. Sci. USA 78 (98) [] S.A. Frank, Spatial polymorphism of bacteriocins and other allelopathic traits, Evolution Ecol. 8 (994) [] H.L. Smith, P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, Cambridge, 99. [] D. Herbert, R. Elsworth, R.C. Telling, The continuous culture of bacteria: a theoretical and experimental study, J. Gen. Microbiol. 4 (956) 6 6. [3] S.-B. Hsu, S. Hubbell, P. Waltman, A mathematical theory for single-nutrient competition in continuous cultures of micro-organisms, SIAM J. Appl. Math. 3 (977) [4] S.-B. Hsu, P. Waltman, G.S.K. Wolkowicz, Global analysis of a model of plasmid-bearing, plasmid-free competition in a chemostat, J. Math. Biol. 3 (994) [5] J. Monod, la technique de la culture continuous: theorie et applications, Ann. Inst. Pasteur 79 (95) [6] A. Novick, L. Szilard, Description of the chemostat, Science (95) 5 6. [7] P.A. Taylor, P.J.L. Williams, Theoretical studies on the coexistence of competing species under continuous-flow conditions, Can. J. Microbiol. (975) [8] S.-B. Hsu, P. Waltman, A survey of mathematical model of competition with an inhibitor, Math. Biosci. 87 (4) [9] S. Ruan, The dynamics of chemostat models, J. Central China Normal Univ. Natur. Sci. 3 (997) (in Chinese). [] H.R. Thieme, Convergence results and a Poincaré Bendixon trichotomy for asymptotically autonomous differential equations, J. Math. Biol. 3 (99) [] V.I. Arnol d, Ordinary Differential Equations, Springer-Verlag, Berlin, Heidelberg, 99, Transl. Roger Cooke. [] Z.-F. Zhang, T.-R. Ding, W.-Z. Huang, Z.-X. Dong, Qualitative Theory of Differential Equations, in: Thansl. Math. Monographs, Amer. Math. Soc., Providence, 99. [3] I. Bendixson, Sur les courbes définies par des équations différentielles, Acta Math. 4 (9) 88.

14 67 L. Zou et al. / Nonlinear Analysis: Real World Applications () [4] J. Carr, Applications of Center Manifold Theory, Springer-Verlag, New-York, 98. [5] S.N. Chow, C.Z. Li, D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Cambridge University Press, 994. [6] R.E. Lenski, S.E. Hattingh, Coexistence of two competitors on one resource and one inhibitor: a chemostat model based on bacteria and antibiotics, J. Theor. Biol. (986) [7] S.L. Peck, Antibiotic and insecticide resistance modeling is it time to start talking? Trends Microbiol. 9 () 86 9.