A population model for limited food competition

J. theor. Biol. (1973) 42, 333-347

A Population Model for Limited Food Competitionf GEORGE G. Ross

Department of Computer Science, City College of New York, New York 10031, U.S.A. (Received 6 October 1972, and in revisedform

1 March 1973)

In this paper a deterministic differential equation system is proposed to model the population dynamics of a biological community in which two species on the same trophic level compete for a common food, taken to be in limited supply. Food limitation is assumed to be the only inhibition of the growth of the populations and food quantity is assumed to be only a&ted by consumption. The model is thus designed to mimic a closed experimental situation rather than a natural community. Analytical properties of the solution of the differential equation system are developed and corresponding biological interpretations suggested. Cited laboratory data on the experimental batch community consisting of the marine ciliates Euplotes vannus and Wonem mar&turn feeding on bacteria motivated the model and supported its analytic properties.

1. Backgrod The study of the ecological dynamics of marine microcosms has received increasing scienti& attention in recent years. A prime motivation for such study has been the growing challenge from water pollution in the urban areas. Various mathematical approaches have been suggested to deal with the problem, the most classical being the deterministic differential equations models proposed by Lotka (1956) and Volterra (1931). Later research has provided stochastic models and, most recently, the theory of automata has played an important role. Time delays, changing the differential deterministic system into a difference-differential one have also received recent attention (Wangersky & Cunningham, 1957; Caswell, 1972; Ross, 1972). An excellent historical account of the development of deterministic models with emphasis on the pioneering influence of Volterra is provided by Scudo (1971). t This work was supported by NSF Grant No. GA 28454 and by City University of New York Research Foundation Grant No. RF-0401. T.B.

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2. Discussion of Motivating

Experiments

Recent experimental work of Ross & Rubin (1972) discusses the results of a series of experiments conducted on two marine ciliates, Uronema marinum and Euplotes vannus. The organisms were grown separately and together in competition for a limited food supply. The data indicated the expected logistic growth curve for the separate organisms and a similar behavior for the competition experiments except that steady-state values were depressed for both organisms. We suggest that the experiments described above are representative of elementary marine communities and that the model considered in this paper may serve as a typical building block in future work on multispecies trophic webs, i.e. in interactive capacity with other building blocks on both the same and different trophic levels.

3. Model Selection and Discussion In the selection of a model we are concerned with three criteria: (i) biological realism-the parameters and solution of the model must be readily identifiable with laboratory measurements; (ii) tructddity of the mathematics-the equations should be solvable or at least submit to partial analysis so that properties of the solution can be developed which can be compared with experimental data; (iii) richness--there should be a body of theory available which can be applied to the mathematical model and, if tractability is satisfied, will serve to provide a substance of analytic properties, e.g. stability theory in ordinary differential equations. With these considerations in mind the following differential equation system was chosen to model a general closed system with two species competing for a common food in limited supply:

dU(t) = u1 U(t)F(t), dt dE dU @(O -I---dt dt dt’ where E(t) = total biomass of species 1 at time t, U(t) = total biomass of species 2 at time t,

(lb) UC)

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F(t) = total biomass of food at time t, E, = constant describing

the rate at which species 1 can convert food into its own biomass,

U1 = constant describing the rate at which species 2 can convert food into its own biomass. The labels E, and U and Fare intended to relate species 1 with Euplotes, species 2 with Uronema, and food with bacteria as in the motivating experiments. Equations (la) and (lb) are to be interpreted considering species 1 and species 2 as food converting machines which function at a rate proportional to the product of their populations and the quantity of food present. Equation (lc) is to be interpreted as a material balance, i.e. whatever is lost in food biomass represents an equal gain in the sum of the biomasses of species 1 and species 2. Initial conditions complementing the differential equation system are the initial biomasses E(O), U(0) and F(0). It is to be noted that the model depicts an isolated community. The growth of the two predators depends only on food supply and not on any “outside” influences, such as higher level predation, presence of a second food, or competition from other predators on the same trophic level. Environmental conditions are assumed to be constant, e.g. light, salinity, pH, and general chemical composition of media. The food is considered to be subject only to the pressure of consumption by species 1 and species 2. These restrictions would technically limit the scope of application of the model to carefully controhed batch experimental systems. However, we feel that in many natural communities the “outside” influences above may be of secondary importance. The lack of a death term in the equations of the model requires comment. It is evident that the initial absence of food (F(0) = 0) produces the static situation in which both populations remain at their initial values. We suggest that E and U represent predator species whose reproduction rate depends on food supply and which remain dormant in the absence of food. Remaining dormant could mean maintenance at subsistence level through some independent means, e.g. a continuous supply of a second food not effective for reproduction. The constants El and U, can be identified from experimental data by examining the separate growth curves, i.e. if U is not present we have the classical equation of logistic growth for E: dE z = E(O(E,CE(O)+F(O)l)

-E,E2W

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whose solution is

W9 + WV E(f) = [F(O)/E(O)] exp { -E,[E(O)+F(O)]t} Evamating

Er from the initial

+1'

(3)

slope of the growth curve dE/dt(O) we have

E = dEldt(O) (4) l E(O)F(O)’ A similar analysis identifies U,. To analyze the competitive properties of the model we integrate equation (lc) and eliminate F(r) from equations (la) and (lb) to obtain g

=

El

$

=

Ul

wo

-E(t)

-

u(t)],

wo-m-U(t)],

where Be = I;(O) +E(O) f U(0) is the total biomass initially present in the system. Although this system for arbitrary El, Ur defies solution in closed form, mathematical analysis provides substantial qualitative information on the nature of the solutions. 4. Analytic Properties of the Model-Biological

Interpretatioos

In this section we discuss the application of the theory of ordinary differential equations to the non-linear system (5). We will state and prove a number of properties of the solution which fall short of a closed solution and, in each case, supply a biological interpretation of the analytical property. Before listing the properties we will comment on the value of developing them even when numerical solutions to the modeling equations can easily be obtained by difference approximation and the use of a large computer. The theory of difference approximation of ordinary differential equation systems is sufficiently developed to provide accurate numerical solutions to most systems, certainly to those proposed in this paper. In fact, we have carried out a numerical simulation of our equations (5) for several values of the parameters E(O), U(O), E,, U, and B,. At the end of the paper we discuss this effort. However, we believe that analytical properties of the solutions play an important role in the interplay between a mathematical model and its real world counterpart, and that mathematical analysis has not been rendered obsolete by the advent of large scale computers. First, numerical solutions depend on a particular choice of the parameters involved in the system and,

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without an analytic estimate of the derivatives, provide no information as to the interpolation of the solution with respect to the parameters. On the other hand, analytic properties generally speak to questions involving a continuum of parameters. Furthermore, many qualitative characteristics of the model are evident from analytical results which are diflticult, if not impossible, to observe from a plethora of computer output, evenifpresented in graphic form. For example, the question of stability of an equilibrium point of an autonomous differential system is answered simply and elegantly by a study of the eigenvalues of the linerarization at the equilibrium point, whereas numerical solutions cannot produce a definitive answer to the question, especially if the number of independent variables were greater than two. As a second example we mention that the asymptotic behavior of the solution can often be expressed by analytical means as a function of the parameters. This would not be possible by purely numerical methods. Asymptotic results for systems having time as the independent variable seem to have special relevance to biology. It is important to have an explicit formula for the dependence of the steady state (and system approach to steady state) on the parameters and initial conditions. Such formulae can be checked against a variety of experimental results to provide an identitlcation of the model which does not depend on special choices of parameters. Finally, we entertain the personal bias that analytic formulae, when available, are the most compact vehicles for conveying technical information. We find it difficult to extract concise and meaningful statements from great reams of computer output. We have often been motivated, by inspection of numerical solutions, to search for a particular analytic characteristic. We look on the search as an attempt to classify and organize information gleaned from conjectures formulated in perusal of computer output. We turn now to a listing of the analytic properties we have developed, together with their biological interpretation. The mathematical detail involved in the proof of the properties is deferred to Appendix A. (i) The solutions are analytic functions of time, Analyticity of the growth curves implies a smoothness of the curve itself and of its various derivatives. This quality is biologically palatable since one would not expect abrupt discontinuous changes with respect to time in any of the system variables or their rates of change. (ii) (a) If E(0) = 0, then E(f) is identically 0. (b) If U(t) = 0, then U(t) is identically 0. If a population should vanish at the start, it will remain null for all time. The model forbids spontaneous generation.

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(iii) (a) If B, = 0, E(t) is identically equal to its initial value E(0). (b) If U, = 0, V(t) is identically equal to its initial value U(0). A vanishing growth rate demands that the corresponding population remain at its initial value for all time (in competition as well as in pure culture). If an organism cannot benefit from the food in a non-competitive situation, it does not benefit under competition either. (iv) If E(O)+ U(0) = B,, then E(t) and U(t) are identically equal to their initial values E(O), U(O), respectively. In the absence of initial food (F(0) = 0), both populations remain at their initial values. As discussed in section 3 the biological assumption here is that the food only enables the organisms to reproduce and that they are capable of dormant existence in its absence. (v) There is a constant relationship between log E(t)/E(O) and log U(t)/U(O), i.e. EIU’

= [yE’.

A qualitative interpretation of this quantitative result asserts that the species with the superior pure culture growth rate will maintain its superiority in the competitive situation. More specifically, if both species begin with equal populations, the larger part of the food will be secured by the species with the better pure culture performance and this qualitative advantage will prevail for all time. (vi) If E(O), U(O), El, U, and Be are all positive, and B, > E(O)+ U(O), then (a) E(t) is a monotonically increasing function of t, (b) U(t) is a monotonically increasing function of I, (c) there is at most one inflection point in the solution E(t) and in U(t), (d) as t grows large, both E(t) and U(t) approach asymptotes of constant value. These properties endow the solution with the logistic form expected of real growth curves in pure culture. It is biologically plausible to observe similar behavior in competition and the Euplotes-Uronema experiments bear this out (Ross & Rubin, 1972). The steady state observed in the experiments corresponds to the asymptotic values predicted by the model (cf. equation (A3) in Appendix A). It is further apparent from equation (A3) that as the exponent EJU, increases, the magnitude of the positive root decreases and hence the quantity U,/U, also decreases, where U, is the asymptotic

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value of U(r). Interpreted biologically, if the growth rate of U decreases relative to that of species E, then the ratio of its asymptotic value to its initial value must decrease for fixed E(O), B,. It can no longer compete as effectively and can thus secure relatively less of the available food. Finally, we mention that the presence of sufficient initial food guarantees the existence of an inflection point in both solutions, i.e. a change in sign of the curvature of the growth curve (cf. proof of (vi) (c) in Appendix A). A definition of an index of competition arises in a natural way from the asymptotic results. The asymptotic value of both species grown in the absence of the other is B,,, the total biomass initially present. Under competition, they compete for the excess food. If we set E, = asymptotic value of E(t) and U, = asymptotic value of V(t) computed as above, then we define the index of E(E,) with respect to U as

Similarly

E _ & --E(O) - wo> Cl- [B, -E(O) - U(O)]’ the index of U with respect to E is defined

u = UC0-wowo> E [B, -E(O) - U(O)]’ These indices so defined represent the fraction of the available food absorbed by species 1 in the presence of species 2. They depend on the initial quantities B,,, E(O), and U(0).

5. sped

case solutions

In this concluding section we provide the solutions to system (5) in three special cases. Case A: WJ, = 1 (growth rates equal). Case B: &PI = 2 (E grows twice as fast as U). Case C: Computer simulation of the system for typical sets of parameters E(O), U(O), B,,, El, U,. Case A If

then

-- E(O) u(o)”

1 “i,BOu&:E(0),u(O)] =ult* o-u

’

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340 Upon integrating,

U,t

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this gives

1

Bo-

uo ’ l+Eo u [ U(O) 1 U

=$log 0

and solving for U(t) gives U, B. eBoUzt

Clearly the asymptote is Bo 1 +EolUo which is indeed the positive root of Bo-U--

w-8 u = 0 U(O) -

Case B If , then U1 dt =

I”

dU

E(O) ’ [wo2 1 -

~(O)B~U-U~-U~

The relation can be integrated giving -- 1

u,t= I

x log

1

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where E. indicates E(O), U, indicates U(O), and the expression on the right is to be evaluated between the limits U and U(0). Simplitkation yields 2

X

1

4Eo Bo - l+y “1

29%3+1 w)2

0

4Eo

2$+1+ 0

J

1+7

Bo

0

It is not possible to solve this equation for U(f) because of the algebraic complexity of the right-hand side. However, it can be observed that if U remains bounded and t increases without bound, then either =0 -U(t)+lmo2

or

BoWe identify the solution to the linear first relation above as the unique positive root of the second relation. Hence again in this special case the constant asymptote of the closed solution will be the single positive root of the right-hand side of equation (A3) in the Appendix. Case C A simulation of the differential equation system for several values of the parameters E(O), U(O), B,, El and U, was carried out on the Control Data 6600 with a FORTRAN program using a forward difference scheme with time increments OGOl. The graphs of the results were prepared by a plot routine and the curves shown in Figs 1 to 8 are direct computer output.

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2.00

0 70

I-00

0 64

I.60 I. 40 0 46

I, 20 s j&

I.00

2

0.60

0.34

0 60

0 28

~ 0 40

0.40 O-16

0.20 I cl i 0 -00

I I.2

I I 2.4 3 2 Fig. I

I 4 0

Ii 4.8

5 E Tune

0

00

16

I

24 32 Fig. 2

40

48

5

Eros 1 and 2. Plot of E(r) vs. t (Fig. 1) and U(t) vs. t (Fig. 2). E(0) = 0.05, u(O)=O.l, El = 0.5, B,, = 2.5.

r

O-70,

2.00

0.64

I.80

0.50

I.60

0.52

I 40

c 0.46 .o s 2 0.40 2 0.34

1.20 3 I.00

0.20

0.60

o-22

0.40

0.16

0.20

0.10

030

0 0

0.5

I.0

I.5 2.0 Fig. 3

2.5

3.0 Time

f

I

0

0.5

I

I.0

1

I

’

I.5 2.0 Fig. 4

I

2.5

3.0

3.5

FIGS 3 and 4. Plot of E(Z) vs. t (Fig. 3) and U(t) vs. t (Fig. 4). E(0) = O-1, U(0) =ii 0.1, El = 1.0, M = 1.5, B. = 2.5.

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0.2

El

0.4

0 6 O-8 Fig. 5

MODEL-LIMITED

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ROS. 5 and 6. Plot of E(t) VS. t (Fii. = 1.0, Ul = 2.0, & = 5.0.

I.2

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I.4

I? Time

f

1.4

Fig. 6

5) and U(t)

vs. t (Fig.

9.00

I

8-10

-

7.20

6). E(O) = O-2, U(0)

= 0.1,

i

6.30

-

5.40

-

> 4.50

-

3.60

-

2.70

-

I.80

-

0 90

-

0 Fig, 7

FmS. 7 and 8. Plot of E(t) El

= l-0,

Ul = 2.0, &

vs. t (Fig. = 10-o.

Time

7) and V(t)

0

t

0.16

0.32

0.48

Fig.8

vs. t (Fig.

8). E(O)

= O-2, U(0)

= O-1,

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REFERENCES CASWELL, H. (1973). Submitted CODDINGTON, E. & LNINSON, York: McGraw-Hill.

to J. theor. Biol. N. (1955). Theory

L~TKA, A. J. (1956). EIements of Mathematical Ross, G. G. (1972). J. theor. Biol. 37,471.

of Ordinary Dlfirential

of Biology. New York:

Ross, G. G. & RUBIN, H. (1973). Submitted to J. Protozool. SCUDO, F. M. (1971). Theor. Popul. Biol. 2, 1. VOLTERRA, V. (1931). Lecons sur la theorie tnathematique de la lutte Gauthier-Villar. WANGERSKY, P. J. & CUNNINGHAM, W. J. (1957). Cold Spring Harb.

Equations. New Dover.

pour la vie. Paris: Symp. quant. Biol.

22, 329.

Appendix A (i) The solutions are analytic functions of t. Proof(i): The right-hand sides of equations (5) are Lipschitz continuous in U and E for any kite interval and hence by Picard’s iteration theorem (Coddington & Levinson, 1955), the successive polynomial iterates converge to analytic functions of t. (ii) (a) If E(0) = 0, then E(t) is identically 0. (b) If U(t) = 0, then U(t) is identically zero. Proof( (a) The nth order derivative of E(t) is composed of terms each of which has a factor of E(t) or of some derivative of E(t) having order less than II. Since E(0) = 0 we can conclude inductively that all derivatives of E(t) vanish at the origin. The analyticity of E(t) then tells us that E(t) vanishes identically for all t. Proof of (b) is identical. (iii) (a) If E, = 0, E(t) is identically equal to its initial value E(0). (b) If U, = 0, U(t) is identically equal to its initial value U(0). Proof (iii): (a) The right side of equation (5a) vanishes and by an argument similar to (ii) E(t) will have as its expansion at r = 0 only the single non-zero term E(0). Similarly for (b). (iv) If E(0) + U(0) = B,, then E(i) and U(t) are identically equal to their initial values (E(O), U(O), respectively. Proof (iv) : Consider $ [B,-E(t)-U(t)]

= - $ - g = - (E,E+U1

u)[B,-E(f)-

U(f)].

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Inductively, the nth time derivative of both E(t) and U(r) contain a factor U(t). Since the factor vanishes, the expansion of of the quantity B, -E(r)E(t), U(t) at the origin will have as their only non-zero terms the constants E(O), U(0) and so will remain identically equal to E(O), U(0)) for all time. (v) There is a constant relationship between log [E(t)/&] and log [t&)/U,], i.e. ye)“’ Proof(v):

= (A!@)“.

We divide equation (5a) by equation (5b) to obtain dE E,E -=dU U,U

or dE = dU -=EE, U&J’ Integrating log [E”E1] = log [U1’ul]

+constant.

The constant evaluates at s = 0 to E(0)UL/U(O)E1. We exponentiate the above integrated equation and have E(O)‘l E(t)“‘ = U(t)el ~ U(O)E’ which yields the stated relation. 64 If WO, u(O), El, u1 and B,, are all positive, and B, > E(O)+ U(O), then (a) E(t) is a monotonically increasing function of t, (b) U(r) is a monotonically increasing function of t, (c) there is at most one inflection point in the solution E(t) and in W), (d) as t grows large, both E(t) and U(t) approach asymptotes of constant value. Proof (vi): (a) The initial slope of E(t) is positive by hypothesis. If the sign of this slope does not change as time increases the function will increase monotonically. If it does change sign there must be a point t’ where dE/dt(t’) = 0 because E(f) is known to be a continuous (analytic) function. But then at t’ either E(t’) = 0 or B,-E(t’)U(t’) = 0 must hold. If E(t’) = 0 then as in (ii) the power series expansion of E(t) at t = t’ will have only vanishing terms and E(t) must be identically zero. If II,-E(t’)U(t’) = 0, then as in (iv) the expansion at t’ vanishes except for the constant E(f ‘) and E(t) is identically equal to E(t’).

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Both of the above conditions produce constant growth functions and contradict the positivity of the initial slope. We conclude that E(t) grows strictly monotonically under the given hypothesis. Proof (vi): (b) This is the same as for (a). A consequence of the above analysis is that dU/dt does not vanish in any finite time interval if it starts positive. Proof (vi) : (c) We eliminate E(t) from the second equation of the system (5) by applying the relation of (v)

E(t)= E(O)[m&l

We”“‘~

The result is Eo -

’

-

u(o)EI/V~

UEIIVI

1 -

Then

(Ala)

If we are to have an inflection point the second derivative must vanish. This condition implies either dU/dt vanishes or u must be a root of 642)

We have already seen in the proof of (vi) (a), (b) that dU/dt # 0 unless U is constant. Furthermore, the algebraic equation (A2) can have at most one positive root since a root occurs only when a constant function (B,) intersects the monotonically increasing function 2u+uEJV The condition is

[u($!j,v,

(1 + Z)].

for the single inflection point (root of equation (A2) to exist) .

B, >2U(O)+S(O)

Proof (vi): (d) Equation (Ala) shows, under the condition that U increases as time increases and that the term B,, - U - $&,,

UE1”’

3, > E(O)+ U(O),

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approaches zero through positive values. The constant asymptotic value of U (when dU/dt = 0) is therefore a root of the algebraic equation E(O) UEIIUI = 0. 643) U(0)El’“I The asymptote is unique since the equation (A3) has at most one positive root (I&, is constant and B,-U-

is monotonically increasing). The existence of the positive root is guaranteed by the initial condition B, > E(O)+ U(0).