Persistence in the chemostat

Persistence in the Chemostat H. EL-OWAIDY Department Received

AND 0. A. EL-LEITHY

of Mathematics,

Faculty of Science, Al-Azhar

University, Cairo, Egypt

13 February 1989; revised 3 October 1989

ABSTRACT Sufficient conditions are obtained for persistence in chemostat models for interactions of a limiting nutrient (or substrate) and two populations, and for two limiting complementary substrates and a single population. The results of Freedman and Waltman are extended for the three interacting predator-prey populations.

1.

INTRODUCTION

The chemostat is a laboratory apparatus used for the production and physiological study of microorganisms. It is a culture vessel into which growth medium for one or more species of microorganisms is continually added at a fixed rate and from which a mixture containing the medium, cells, waste products, and unused nutrients is continuously removed. The medium contains all nutrients (or substrates) needed for the growth of the species, in excess of demand except for one, which is the growth-limiting factor. The system also approximates conditions for plankton growth in lakes, with the input of limiting nutrient such as silica and phosphate from streams draining the surrounding watershed. For a more detailed biological background, readers are directed to [9] and the references therein. When two or more populations compete exploitatively for a single limiting substrate in a chemostat, that species will survive whose Michaelis-Menten constant is smallest in comparison with its intrinsic rate of natural increase [9]. Two different substrates are complementary if they are basic but metabolically independent requirements for growth. The growth, in the case of two limiting complementary substrates, is limited at any time by one substrate only but not by both of them at the same time. The rate of growth in this case is the minimum of that in the two substrates (see [ll], [8]). We shall refer to equilibria on the coordinate lines as axial equilibria and those in the interior of the coordinate planes as planar equilibria [4]. MATHEMATICAL

BIOSCIENCES

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H. EL-OWAIDY AND 0. A. EL-LEITHY

28

One question that arises is that of determining conditions that prevent solutions of the chemostat models, which are initially strictly positive, from approaching the boundary of the cone as time evolves. This is of paramount importance in the modeling of biological populations where such conditions rule out the possibility of one of the populations becoming arbitrarily close to zero in a deterministic model and therefore risking extinction in a more realistic interpretation of the model. Generally speaking, the term persistence is given to systems in which strictly positive solutions do not approach the boundary of the nonnegative cone as t +m. For various definitions of persistence, see [5] and [6] for a version of weak persistence, [3] and [4] for persistence, and [7] and [lo] for uniform persistence (also called cooperativeness or permanent coexistence). This paper studies persistence (in the sense of [3]) in two mathematical models for the chemostat, based on Michaelis-Menten kinetics. The first model is for one limiting substrate and two populations, and the second, for two limiting complementary substrates and a single species. Freedman and Waltman [3] studied persistence in models of three interacting predator-prey populations. We can apply their results in the chemostat if we consider the limiting substrate as a prey and the Fopulation as a predator. For convenience we state the results in [3] that we need in this paper before beginning the applications. Case 1. For a system modeling the interactions of two competing predator species living exclusively on a common prey, the equations take the form

x’= XF(X,Y,,Y,), yi = Y,G,(~,Y,>Y,)> Y;=Y,G,(~>Y,>Y,)> x(0) = xg > 0, (the prime signifies

d/d),

Y,(O)

= YLOa 0,

i=1,2

with the following assumptions

(1.1) (stated in [3]):

(B,) x is a prey population, and yi, i = 1,2, are competing population living exclusively on the prey, that is,

JE
~0,

aci

gp-0, i=1,2;

i = 1,2;

3, I

i,j=1,2.

predator

29

PERSISTENCE

(B,) The prey grows to carrying that is,

capacity

in the absence

of predation;

$.O,O) <0,

F(O,O,O) > 0, there exists K > 0 such that

F(K,O,O)=O. (B,) There are no equilibria on the Y, or y, coordinate axes and no planar equilibria in the y,y, plane. (B,) Each predator can survive on the prey; that is, there exist two planar equilibrium points E*: (a*, h*,O)and &S,O,~> in the xy, and ayz coordinate faces, respectively, with a*, b*, 6,; > 0, a* < K, a^< K.

Case2. For a system modeling interactions between tions and one predator population (the prey populations compete), the equations take the form

two prey populamay or may not

XI = x,F,(x,,-Q,y), XI. = -QF2(X,>X2,Y)> y'= YG(~,,~,,Y), Xi(O) = X;,, > 0, (the prime again signifies

d/&I,

(A, > xi is a prey population

!!i(

X,,X,,Y)

i = 1,2;

(1.3) Y(O) = Yo 2 0

with the following assumptions: and y is a predator

population;

g( X,,X,,Y) >0,

< 0,

(A,) Each prey population predation and/or competition;

i=1,2.

grows to carrying capacity in the absence of that is,

there exist K,,K, > 0 such that

F,(K,,O,O)=F,(O,K,,O)=O. The predator

population

that is,

dies in the absence G(O,O,Y)

< 0

of prey; that is,

30

H. EL-OWAIDY

AND 0. A. EL-LEITHY

(A,) If X, and x2 compete, there may or may not exist a critical point interior to the x,x2 plane. If it exists, it is unique and unstable. If they do not compete, they are neutral; that is,

There may exist at most one point

4: (a,>a,,O),

Fj(Q,,Q2,0)= 0,

i=1,2.

a, > 0;

(A,) In the absence of each prey species the predator can survive on the other prey. The conditions for this to occur are well known [1,2]. There exist a unique E*: (u*,O, b*) and g(O,6,6> such that a*, b*, ~?,6 > 0, and F,(a* ,O,b*)=G(a*,O,b*)=G(O,a,b)=F,(O,a,b)=O. Note that by the properties a* < K,,

of F, and G in the previous

a^< K,,

G(K,,O,O)

hypothesis,

G(0, K,,O) > 0

> 0,

For more detailed discussion of the persistence see [3]. We now proceed to study the application of the cases 1 and 2 in the chemostat in Sections 2 and 3. 2.

ONE LIMITING

SUBSTRATE

If we assume that nutrient uptake kinetics, the model equations are

is described

m2S(f)T

S’(I)=[~o_S(I)]D-um~~~~t’) ($y 1

a2

m&t) x;(t)

=x,(t) i

S(0) =

by Michaelis-Menten

_

ai+&!?

s, > 0,

D I

X,(O)

=

’

+

S(t)

i

) 1

i = 1,2, Xi0 >, 0

(‘= d/dt, t = time); So is the constant input concentration of substrate; D is a positive constant representing the washout rate; yi, m,, uj are positive specific growth rate, and constants that are the yield, maximum Michaelis-Menten constant, respectively, for the ith competitor; i = 1,2. s(t) is the concentration of substrate at time t, x,(t) is the concentration of the ith competitor at time t, i = 1,2. For simplicity, the yield constants y, and y, can be incorporated into x,(t) and x,(t), respectively, the system

31

PERSISTENCE

takes the form

x,(t),

S(0) = LEMMA

s, > 0,

x,(O) =

i = 1,2.

0,

Xj() >

(2.1)

2.1

(i> The positive cone R: is positively invariant under the solution map of (2.1). (ii> Solutions of (2.1) are uniformly asymptotically bounded as t -tm; that is, the system is dissipative. In fact, lim (S(t)+x,(t)+.x,(t)) t-m and the convergence

=S”

is exponential.

Proof (i> and (ii). Since the arguments details, except to note that if Z(t) = s(t)+

are quite standard, x,(t)+ x,(t), then Dt)

Z=S”+(Z,,-S”)exp(since D is a positive constant.

we omit the

(2.2)

It follows from (2.2) that

S(t) + x,(t)

+ x2(t)

-3

so

exponentially as t +m. A consequence of Lemma 2.1 is that the solutions of (2.1) are positive and bounded, as can be shown by Lemma 3.1 of [9]. Now, let S be a prey and x,,x2 two competing predator populations, and apply case 1 of Section 1 to Equations (2.1) by putting (2.1) into the form

S’(t) = SF(S,x,,x,) xi’(t) =xiGj(S,x,x2), S(0) =

s, > 0,

i=1,2, X,(O) =

xi(~ Z

O,

(2.3)

H. EL-OWAIDY AND 0. A. EL-LEITHY

32 where

m2

ml qw’-

Gi(S,x,,x,)=-&D,

&T-xx2

i=1,2.

I

Note that the functions F,G,,G, satisfy the conditions of the persistence theorem 2.1 in [3] and the usual predator-prey assumptions (see Freedman

[21X It is clear that the basic predator-prey

relationship

(B,) is satisfied,

because

C3F ---$-g
i=1,2,

miai

aGi _ aS

i = 1,2,

(ai+S)2>o’

i= 1,2,

Gi(O,xl,x2)=-D
_

ax,

(2.4)

i, j = 1,2.

Also, the carrying capacity condition (B,) holds with the constant input = 0, concentration S”, which is the carrying capacity, since F(S',O,O) S'>O, and

F(O,O,O)> 0,

$(S,O,O) = - $g <0.

The lack of equilibria on the coordinate axes (Bs) is satisfied. For validity of (BJ we make the following necessary assumption:

m, > D, Hence, ($,0,6)

i=1,2.

there exist two planar equilibrium points E*: (S*,b*,O) and l?: in the coordinate planes Sx, and Sx,, where s* =

a,D

m,-D’

s*=

a2D m,-D’

b* =

So - s*,

~=sO-s*.

33

PERSISTENCE

Then, by using (B,), one can show that S*, b*, s^, 6 > 0 and S* < So, s^ < So. Then the persistence of all solutions is not clearly possible. Hence, we make the following assumption: (B,) The Michaelis-Menten constant for the ith competitor its specific growth rate; that is, ai < m,, i = 1,2. More conditions are contained in the following theorem: THEOREM

is less than

2.2

Let (B,)-(B,)

hold. Then, if there are not limit cycles, and if $D

s^>D

ml>S-D’

and S*D

S*> D,

m2>S*-D’

(2.5)

system (2.1) persists. Prooj The boundedness of solutions is proved in Lemma 2.1. The equilibrium E,: (S’,O,O> is stable along the S-axis, and there are no other axial equilibria, A simple variational equation argument shows that the unstable manifold of E, is two-dimensional, since G,(S’,O,O) > 0 [using (B,)] is a necessary condition for the existence of E* and g. Finally, conditions (2.5) [using (B,)] are sufficient to make the two equilibria in the interior of the coordinate planes unstable in the orthogonal direction as follows. The variational matrix J of Equations (2.3) at (S, x1, x2> is

F+Sas

f?F

S-g

SE 1

J=

dG,

1 a(32

aG2

aG, /as, aF ==-S2+

dG, X’ax,

G, +x1%

xl=

i, j = 1,2, are defined

mlxl (al

(2.6)

2

and JG, /ax,, SOD

’

G, + ~~2

X2ax,

X2x

where aF/dx,

2

+

m2x2 S)”

+(

a2 + S)’ ’

in (2.4),

(2.7)

H. EL-OWAIDY

34 The evaluation

of J at l?: (s^,O, 6) yields [using F(S,O, &) = G,(S,O,h) m,&

SOD -7+ S

-7

?n,s^

a, + s ,. m,S 7-D a, +s

(u2+qZ 0

J&t,,,&, =

0

The characteristic

The eigenvalue

equation

-7

= 01

irl$ u2 + s 0

.

(2.8)

0

is

in the x1 direction

is

m,s^/(a,+ s^)-

A,, =

Using hypothesis

AND 0. A. EL-LEITHY

D

(2.10)

(B,) and the first part of (23, A,, =

m,i/(

> m&+q

a, +

s^)-

D

+ s^) - D

= (s^-~)m,-$D m,+SI = (s^-

D)[rv-iD/(s^-

D)]

m,+SI > 0.

(2.11)

Then, the equilibrium k: (S,O, 6) is unstable in the orthogonal direction of the coordinate plane Sx,. The same procedure can be followed for E*: (S*,b*,O) with S*, b*, a,, and a2 replacing s^, 6, a2, and a,, respectively, in (2.9). By using (B,) and the second part of (2.5), one can show that E*: (S*, b*,O) is unstable in the orthogonal direction of the coordinate plane Sx,. This completes the proof of the theorem. Remark 2.1.. The condition (B,) may be replaced by the condition: (Bf6) The Michaelis-Menten constant of each competitor xi, i = 1,2, is less than or equal to the washout rate D; that is, ui < D, i = 1,2.

PERSISTENCE

35

This can be shown as follows. From (2.10), A,, =

m,&‘(u, + i) -

D D

= D[ Da, +

[using the first part of (2.5)]

s^(D - u,)]/(i-

D)(u,

+ i)

(2.12)

Since s^> D [from (2..5)], then A, > 0 if D > a,, which is the same result as in (2.11).

3.

TWO LIMITING

COMPLEMENTARY

SUBSTRATES

Let S and R be two complementary substrates that are in limited supply. Growth is limited at any time by one substrate or the other, but not by both substrates simultaneously. The rate of growth in this case is the minimum of that in the first substrate and that in the second substrate (see [ll]). The mathematical model is the following:

(3.1) where g(S(t),R(t))=min s(o)

=

s,, a

0,

mG(t>

m,R(t)

u,+S(t)‘a,+R(t)

R(O) = 4,

a 0,

u(0) =

U(J >

0,

(see [S]). R(t) is the concentration of the second substrate at time t, and R” is the concentration of the supply. The different responses of growth with respect to each substrate alone are indicated by subscripts S and R on the constants m, a, and y. The microorganisms survive in the chemostat if and only if 0 < a,D/(m, - D) < S” and 0 < a,D/(m. - D) < R”; therefore, we use this condition in studying the persistence in the model (3.1). It is obvious that R: is positively invariant under the solution map of Equation (3.1). Using arguments similar to those in the proof of Lemma 2.1, we obtain an analogous result.

H. EL-OWAIDY

36 LEMMA

AND 0. A. EL-LEITHY

3.1

The system (3.1) is dissipative; in fact,

I’m

S(t)+-

lim

[

u(t) Ys

1 s”, =

= R’.

(3.3)

I As in Section 2, the solutions of (3.1) are positive and bounded. Let S and R be considered two prey populations population, and apply case 2 of Section 1. Set Equations of Equations (1.3):

and u a predator (3.1) into the form

S’(t)=SF,(S,R,u), R’(t)

= RF,(S,R,u),

u’(t) =uG(S,R,u), S(0) = S,, > 0,

(3.4)

R(O)= R,,>,O, u(O)= U">O,

where D-&g(S,R)u,

D-&g(S,R)u, G(S,R,u)=g(S,R)-D. Note that F,, F2, and G satisfy the conditions of the persistence theorem 2.1 in [3] and the usual predator-prey assumptions (see Freedman [2]). Since the function g(S, R) > 0, condition (A r) automatically holds. Hypothesis (A,) with carrying capacities K, = S” > 0 and K, = R” > 0 is satisfied, since &(O,O,O) > 0, F,(S”,O,O)

i=1,2;

= 0 = F,(O, R”,O);

So, R’> 0;

then,

gqS,R,O)=-$
PERSISTENCE

37

Similarly,

g$(S,R,O)=-q
u (predator)

becomes G(O,O,O)

zero in the absence

they are neutral,

It is obvious that (8g/aSXS, R,u) and functions. There may exist at most one SR coordinate plane. So the hypothesis For the validity of (A,) we make the (A,)

Hence, there exist unique critical such that S*, b*, l?, 6 > 0, where

since

(dg/dR)(S, R,u) are nonnegative critical point, E: (S”, R’,O) in the (A,) holds. following necessary condition:

R”>a,

So>=,

s*=_D,

D < 0.

= -

If the prey S and R do not compete,

of prey; that is,

D < min(m,,m,).

points

E*: (S*,O,b*)

and i:

(0, a,&>

1

%D

9

(in the absence of the limiting substrate

%D s^=-,

i=yR

mR

R)

1

(in the absence of the limiting substrate

S)

By using (A,), it is easy to show that S*, b*,i?,h > 0; S* < So, R < R”; G(SO,O, 0) > 0, and G(0, R’,O) > 0. In fact, in the absence of the limiting substrate

R, G( S',O,O) = m,S’/(

as + So) - D Da, _D m,

1 >

0

[by (A,)]

H. EL-OWAIDY

38

AND 0. A. EL-LEITHY

Similarly, G(0, R,,,O) > 0 in the absence of the limiting substrate (A,) holds. Moreover, the following conditions are satisfied:

F,(O,R,b)>o, G( S”, R”,O) > 0

S. Hence,

F2( s*,o, b*) > 0, (when E exists).

THEOREM 3.2

Let the hypotheses (A,)-(A,)

hold. Then if there are no limit cycles, the

system (3.4) persists. Proof. The proof of boundedness of solutions is supplied in Lemma 3.1. The critical points E,: (S”,O,O> and E,: (0, R”,O) are stable along their axis by (A,) and unstable in the u direction because G(S”,O,O) > 0, G(0, R’,O) > 0. By (A,), (A,), and (A,), there is at most one planar equilibrium in each positive coordinate plane. Finally the conditions (3.5) are exactly playing as quietly as needed to make the planar equilibria unstable in the direction orthogonal to the plane, as may be seen by computing the variational matrix of the system (3.4) at each of these equilibria as in the proof of Theorem 2.2.

DISCUSSION In this paper we have extended the mathematical analysis of Freedman and Waltman [3] for three interacting predator-prey populations to chemostat models for interactions of a limiting substrate and two species of microorganisms, and for two limiting complementary nutrients and a single species. The analysis has dealt principally with the parameters So, D of the resource part and parameters of the ith species; a,, the Michaelis-Menten (or half-saturation) constant; rn;, the maximum specific growth; and Da; /(m, - D). Since, in a chemostat, the ith species will survive if its Michaelis-Menten constant ai is smallest in comparison with its intrinsic rate of increase (rn; - D) [9], we set the parameter Da; /Cm; - D) sufficiently small such that So > Da, /Cm; - D), i = 1,2, in Section 2. Also, in Section 3, for So > Da, /(ms - D> and R > Da, /(mR - D>, various definitions of persistence of the ecological systems have been given [3,.5, lo], each of which conveys the idea that none of the component populations becomes extinct, that is, the “survival of all components of the system.” Hence we can conjecture that persistence, in a chemostat, means that the “concentrations of the substrates (or nutrients) and of the microorganisms still do not vanish.” Our analysis predicts that persistence occurs in a chemostat with interacting limiting nutrient and two species, if the input concentration So of the nutrient S is higher than the product of the Michaelis-Menten constant (ai of the ith species) times the ratio of the

39

PERSISTENCE

death (washout) rate to the intrinsic rate of natural increase (mj - D), the Michaelis-Menten constants of ith species is less than its maximum specific growth rate, and (2.5) satisfied. REFERENCES F. Albrecht, populations,

H. Gatzke, A. Haddad, and N. Wax, The dynamics J. Math. Anal. Appl. 46:658-670 (1974).

H. I. Freedman, Deterministic Dekker, New York, 1980.

Mathematical

Models

in Population

H. I. Freedman and P. Waltman, Persistence in models populations, Math. Biosci. 68:213-231 (1984). H. I. Freedman and P. Waltman, Persistence populations, Math. Eiosci. 73:89-101 (1985).

8

9

10 11

of two interacting Ecology,

of three interacting

in a model

of three

Marcel predator

competitive

T. C. Gard, Persistence in food chains with general interactions, Math. Biosci. 51:165-174 (1980). T. C. Gard, Top predator persistence in differential equations models of food chains. The effects of omnivory and external forcing of lower trophic levels, J. Math. Biol. 14:285-299 (19 ). J. Hofbauer, General cooperation theorem for hypercycles, Monatsh. Math. 91:233-240 (19 ). C. B. Hsu, K. S. Cheng, and S. P. Hubbel, Exploitative competition of microorganisms for two complementary nutrients in continuous culture, SLAM J. Appl. Math. 41:422-444 (1981). S. B. Hsu, competition 32:366-383

S. Hubbel, and P. Waltman, A mathematical in continuous cultures of microorganisms, (1977).

theory for single-nutrient SUM J. Appl. Math.

V. Hutson and G. T. Vickers, A criterion for permanent coexistence of species, with an application to a two-prey one-predator system, Math. Biosci. 63:253-269 (1983). J. A. Leon and D. B. Tumpson, Competition between two species for two complementary or substitutable resources, J. Theor. Biol. 50:185-201 (1975).