Persistence in food webs: holling-type food chains

Persistencein Food Webs: Holling-Type Food Chains THOMAS C. GARD Department of Mathematics,

University of Georgia, Athens, Georgia 30602

Received 5 Ju& 1979; mxised 25 September

1979

ABSTRACT In this paper

we obtain

sufficient

conditions

for the persistence

of food

chains

modeled by Helling- or Michaelis-Menten-type dynamics. The persist.ence conditions involve positivity of a certain parameter which is a weighted sum of the birth-death rates for the species in the chain. Comparison with a necessary condition for persistence indicates,

1.

to some extent,

the sharpness

of our results.

INTRODUCTION

Among the relevant ecosystem stability concepts, (Holling [4], May [5], Maynard Smith [6j), the persistence-extinction phenomenon is, without doubt, one of the most important (Paine [8], Goodman [3]). However, the determination of persistence, for all but the most elementary models, is a difficult task. For example, persistence is obtained as a result of showing the existence of a globally Liapunov asymptotically stable invariant set (McGhee and Armstrong [7]). The approach taken in this paper is to analyze persistence for some three-species models of food chains involving Holling-type interactions, by means of the construction of persistence functions. Persistence functions, which are similar to Liapunov functions, have been employed successfully to discuss the survival problem for LotkaVolterra food chains (Gard and Hallam [2]). The construction of such functions implies that certain recurrent trajectories (including equilibria) in the boundary of the positive cone are not attractors for model trajectories. So persistence is obtained directly by showing that model trajectories cannot approach the boundary of the positive cone, rather than by demonstrating that they must approach an interior equilibrium. The conditions derived are inequality constraints on the coefficients of the model, which permit the construction of the persistence function. To show that the conditions must be nearly “best possible” we cite a necessary condition which is obtained by investigating the behavior of trajectories MATHEMATICAL

BIOSCIENCES

QElsevier North Holland, Inc., 1980

49:61-67(1980)

61 0@25-5564/80/030061+7502.25

THOMAS C. CARD

62

near an equilibrium in the boundary. We begin by writing down the models and briefly stating some basic properties, analogous to those for the Lotka-Volterra situation, about the models and persistence functions.

2.

PRELIMINARIES We consider food-web

models of the following

X; = xl alo[

x;=x, [ 4=~4 -

ai2 allxl - b+x a21 b+x,

-a20+

types:

11, 1 x2

x1-a2+3

y

a30 + a324

and

xi =x,

a,,-

a,,x,

aI2 (b+x,)(c+x,)

-

[

x;=x,

[

x;=x3 [

--a,+

I

x2 ’

(3

a23

a2, b+x,-v

-x c+x,

a32 -a30+c+x2

2

3’1

1 )

where the coefficients ai], b, and c are positive constants. These systems of differential equations are defined on R: = {(x,,x2,x3)1xi > 0, i= 1,2,3,}; for any initial value (xp, xg, x3 in R :, a unique solution x(t)=(x,(t),x,(t),x,(t)) exists for all time t > 0 with trajectory in R 1. We are interested, particularly, in solutions with initial values in the interior of the positive cone, R?‘= {(x,,x2,x3)~xi>0, i= 1,2,3). For the latter, in the case of either of the models above, it is not difficult to show that all trajectories are bounded and cannot reach the boundary of R: in finite time. Persistence for a system of the form (1) or (2) means that no component of any trajectory of the system with initial value in R$’ can tend to zero as t gets large: i.e., for each i=1,2,3, lim ,+oo~i(t) >O. If, on the other hand, lim,,,xj(t)=O for somej, then limr__, x3(t) = 0. The verification of these facts is similar to that for the Lotka-Volterra case (Gard and Hallam [2]). Freedman and Waltman [I] have given a persistence criterion for (1) in case there are no nontrivial periodic trajectories in the x,x2 plane. Our condition is a more restrictive inequality on the coefficients of the system, but it has the advantage that it is applicable regardless of the trajectory structure in the x,x2 plane.

PERSISTENCE

IN FOOD

A nonnegative ferentiable in R:‘,

63

WEBS

function p(x,,x,,x,) defined on R:, continuously is called a persistence function for a system

dif-

x,!=fi(x,,x,,x,) of differential

equations

(3)

provided:

(i) p = 0 if and only if xI = 0, some j. (ii) b= 2 aPf i >O i=i axi R$Olx,
on S,, for some e>O,

where

$={(x,,x2,x3)E

Condition (ii) can be modified somewhat, in view of the application to models (1) and (2), to take into account that failure of persistence requires the highest trophic level to go extinct. So we will replace (ii) with (ii’) For any solution x( 1) =(x,(t), xZ(t),xj( t)) of (3) with (~,(O),X~(~),X~(~))=(X~,X~,X~)E R?O, there is an e >0 such that

fqx(t))=

x(0) =

5 *x(x(‘)) 2 0

+, a4

for x(t) satisfying x3(t) Q E. The pertinent

result, then, is the following,

which is not difficult

to prove.

THEOREM (Gard and Hal/am [2]) If a persistence function exists for a differential system (3) the system is persistent.

3.

THE MAIN RESULTS

Before stating the main results, we note that it is necessary for persistence in the models (1) and (2) for a2, >a,,. Indeed, if azl < azo, then the equilibrium (a,,/a,,,O,O) is asymptotically stable, and so trajectories with initial values near (a,o/al,,O,O) satisfy x,(r)+0 and xj(t)+O as t-soo. We also note here that if a2, >a,,, no solution trajectory with initial value in R$’ can satisfy x,(t)+0 as t--+-co. In addition, for the system (2) the condition a32 >a,, is necessary for persistence; otherwise the equilibrium (xr,xT,O) in the positive x,x* quadrant will attract some trajectories in R$O.

THOMAS C. GARD

64 THEOREM

I

The system (1) is persistent if az, - a2,, > 0 and

ball+ =a,,-

h

THEOREM

aI2

-a3o

ba32

a2l -

azo

- *aSo>O. a2o ba,,

2

The system (2) is persistent if az, - a2o > 0, a32- a3o > 0, and

baI I +

p2=alo-

me0

a21 - azo

a2o - ma3o >

0,

where m=max Before giving the proofs, we note that necessary conditions for persistence can be obtained by observing that the x,-direction eigenvalue for the linearized system at the equilibrium (x:,x:,0) in the positive quadrant of the x,x2 plane cannot be negative. This observation yields: THEOREM

3

A necessary condition for persistence in the systems (1) and (2) is

u=a,O-

--a

a21 -

4.

PROOFS

OF THEOREMS

Proof of Theorem 1.

a2o

a20

aI2

ba,,

30 > 0.

1 AND 2

We make use of the function

w.here the ri’s are constants to be determined in the process of verifying that p is a persistence function for (1). Let x(t)= (x,(t),x2(t),xj(t)) be a solution of (1) with initial value x(0) = (~7, xg, x3 in R 2”. We will show that if p1 > 0,

x1(t)

dx(t)> =

b

[

+

x,(t)

1

65

PERSISTENCE IN FOOD WEBS

satisfies conditions (i) and (ii’). Since x(t) is bounded, and +(t)+O is impossible, p(x(t)) will satisfy (i) for any choice of constants rl,rz, r3 with r, and r, positive. It remains to analyze p(x(t)), and for brevity we will suppress the dependence on t: a12 =lo--a,l~,-~~z I

+

r2

-

a20 +

azl box,

xl

-

+3x3 I

i

+

r3[ -

a30 +

.

u32x2] I

To facilitate choosing the r,‘s, it is helpful to rewrite b as

i)=&

$a,,-

6 a,,-

:a3o

1

I i[

+

~(a2~-a20)-~~ll-

XI

?a30

1

[

+

ba,2

z(b+x,)a,,-

x2

b+x,

[

1

1.

~(b+*,b,,x,

Now let

a12 ball 2 r1

=

-

aI2

r2

and

-

z

32

=

rl

4

+

a2l

-

a20

Taking into account z(b+x,)a,,-

2

I

a $ba,,-a,,=0

and -a20)-ba,,-‘3a30=0, rl

we have

b+;,(r)

h-

+23x3(1)

a3o

THOMAS

66

C. GARD

which is nonnegative for sufficiently small x3(t). Thus p is a persistence function for (I), and the result follows as a consequence of the theorem quoted in the preliminaries. Proof of Theorem 2.

Here we construct

The proof is the same as that for Theorem

p of the form

1 up to the computation aI2

“o-aa,lx’-

(b+x,)(c+Q

xz I

1 Again, to motivate b=

the choices for the r,‘s we rewrite b as

(b+x~‘~c+x2)

alo-

$a,,- :a301

Now let

‘3

42 ball -‘ba, a32- a30

=m=max

rl

and r2 _=

rl

ba,I+

ma30

a2, -a20

of i,:

PERSISTENCE

IN FOOD

67

WEBS

With these choices for the r,‘s, and using the assumption the estimate for i,

-_ r2 rl

which is nonnegative proof.

for sufficiently

p2 > 0, we obtain

ca23 X3(f) [c

+

x2(O12

9 1

small x3(t), and this completes

the

We wish to thank Thomas Hallam for pointing out some errors in the original manuscript. This research was supported by the U.S. Environmental Protection Agency under Grant R806161-01-I. REFERENCES 1 2

H. I. Freedman and P. Waltman, Mathematical, analysis of some three species food chain models., Math. Biosci. 33: 257-276 (1977). T. C. Card and T. G. Hallam, Persistence in food webs: I. Lotka-Volterra food chains, BUN. Math. Biology 41 (1979).

3

D. Goodman,

4

Biol. 50: 237-266 (1975). C. S. Holling, Resilience and stability l-23 (1973).

5 6 7 8

The theory

of diversity-stability

relationships

of ecological

in ecology,

Quart.

Reu.

systems, Annual Reo. Ecol. Sysf. 4:

R. M. May, Stability and Complexity in Model Ecosysremr, 2nd ed., Princeton U.P., Princeton, N.J., 1975. J. Maynard Smith, Models in Ecology, Cambridge U.P., Cambridge, 1974. R. McGhee and R. A. Armstrong, Some mathematical problems concerning the ecological problem of competitive exclusion, J. Differenfial Equalions 23: 30-52 (1977). R. T. Paine, Food web complexity and species diversity, Amer. Not. 100: 65-75 (1966).