Persistence in food chains with general interactions

Persistence in Food Chains with General Interactions THOMAS C. GARD Department of Mathmatics, University of Georgia: Athem, Georgia 3OtSO2 Received 15 November 197% revised 31 March 1980

ABSTRACT Sufficient conditions for persistence in n-link food chain models with interaction functions gj are given: for g, nondecreasing and n=3, and for gr positive in so.me compact interval to the right of zero aud general n. The results generalize those for LotlcaWolterra type models. Necessary conditions for persistence are also given.

1. INTRODUCTION That persistence is one of the most relevant of the stability properties of ecological models is well documented (Goodman [4], Helling, [Sj, May [6], Maynard Smith [7], and Paine [9], for example). Attempts to cbtain persistence for such models generally involve showing the existence of an attractor in the interior of the feasible region (McGehee and Armstrong [g]). This approach can be difficult even for elementary models. 0n the other hand, we have had some success in determining‘ persistence by verifying that the invariant sets in the boundary of the feasible region are not attractors. We employ Liapunov-like functions to calculate persistence criteria as algebraic expressions in terms of model parameters. This paper represents the continuation of our study, which began with our work with Hallam [3], to obtain such criteria for food chain models. The system of differential equations

Cl) XL_ * =xn-

*[ -a,_

,,()+up-

,,n-Zgn-ZC;in--Z)xn-2

-a*-,,ngn-I(Xn-l)Xnl, xl:=x,[

MATHEMATICAL

-Q,.O+~n,n-,~,_l~X,-I~~~-tl

BIOSCXENCES

51: 16% 174 (1980)

OElsevier North Holland, Inc., 1980 52 Vanderbilt Ave., New York, NY 10017

‘165 5.5564/80/070 165+ lOW2.25

THOMAS

166

C. GARD

repments a food chain model if the ujJ are positive constants and if the real valued functions gd are nonnegative on 10, 00). We will also assume the g1are continuous. ‘Tl& system (I) is defined on RI: = {(AT*,.. ., xn)lxI > 0, 1 0 with trajectory in R”, . Of interest, ptict&uIy, are solutions with initial values in the interior of the positive cone, R”;O= ((x I,..g,~n)l~I>O, 1 O. If, on the other hand, lim,_,x,(t)=O I+oox,( t)= 0. This c8fl be verified as in the Lotkaforsome~,theniim Volterra case (C&d and Hallam [3D. 2. THE MAIlU RESULTS

Suppose gr is differentiable, non&creasing, and such that gi(0) >O. Let xr &ttdx$ be the solution of thesystem

(3) Z&n the wstem (1) with n = 3 i&persistent if both

xr<-QlO a11

(9 The system (1) with n= 3 fails to be persistent if either inequality (4) or (5) is rmrsed. At this point, a few words about the biological interpretation of the conditions (4) and (5) in Theorem 1 seem appropriate. In this model there are three equilibria, jFo(O,O,O),El(alo/~ll,O,O), and ZZ2(x;)&,0), that correspond to extinction of at least some component of the systm. S&e alo > 0, the first differen*;-’ _..a equation in the system (I) indica$a that the first component, representing the population density of the food source for the chain+ has positive net growth rate near Eo. The conditions (4) and (5) can

PERSISTENCE IN FOOD CHAINS

167

be viewed similarly as requirements that the components representing tke population densities of first and second level predators have: positive net growth rates near the equilibria El and Ez respectively. (This can be seen easily by considering the second and third differential equations of the model and the last five sentences of the proof of Theorem 1 below.) Systems modeled by (1) have the property that failure of persistence manifests itself by population densities approaching one of the three equihbria. To require positive net growth of the corresponding species near these e@libria is to say that the combined effect of all factors on the species growth-death rate is a positive one, at least when that species is in danger of being the first [in the trophic order) to go extinct. The persistence conditions given here can also be understood as stating that for persistence the growth Irate al0 of the food source must balance a certain linear combination of the death rates of the predators, where the coefficients of that linear combination involve lthe trophic level interaction rates. This interpretation is transparent in tie statement of the next result. Let

&Xnfilak+,

.

mm aj,,o,

m=

&,

l
a11

k-1

’

I

.

if

j=l.

-1

if

2Gja;n-- 1.

if j=n.

-4ssume gi>O on [OyXi], 1 Q i
cLp=a10-

-

61

a213)21

1

i-l

%J+ i

i-4 (i euen)

i-

II

I

’ a~-,J+,

j-2

(j e-0

aj+ll

II j-3

ai-l,i

Qj+ l,jmj

3

a.190

jmj

I

M j-1

a.190 J

J

THOMAS

1168

C. GARD

and i- 1

n

c

c(s-=10-

i-4 0 -0

77zeqstem (I) is persistent if pP> 0, and fails to persist if p, < 0. Before giving the proofs, we make a few observations on these results. First, we note that both theorems are generalizations of the Lotka-Volterra interactions case (gi--1, all i). Indeed, in this case, pP=p,=p, with the latter parameter as defined in Gard and Hallam [3], and Theorem 2 is identical to the main result in that paper: Furthermore, Theorem 1 includes Ivlev interactions [gi( xi) = 1 -e-b~x~] as a special case, while Theorem 2 generalizes Gard and Hallam’s result to include Helling type dynamics [g,( x,) = I/( b,+x,)]. Along these lines, we compare the application of Theorem 2 to the model x;=x,

x$=x2

4=x3(

( (

a12 b+xX2

qQ-Qllxl-

1

)

a21 -am+ b+xq-ag3

9

, 1

1

(8)

-=30+=32X2),

with the result for this system that we published recently [2) in this journal. The latter result guarantees persistence for (6) if a21-am>0

(9

and =10-

ball

+(a12/ba32)a30 a21 -

a20

Q20

- I\ ou)

a12

--a&&

ba32

Let us now see what application of Theorem 2 obtains. Here we have

-. (=10>’ where

q -.

P

mflll

1 81(x1) = B+xl

m=min{alO, am, ajO}: and

g2(x&z

1,

PERSISTENCE IN FOOD CHAINS

169

so that 1

1 M’=T;’

and

ml=~b+(n,o)2,ma**’

M2=m2= 1.

The persistence criterion of Theorem 2 is

-

aI2

-Q)>o.

(11)

baj2

This inequality implies a10-

(barfialo )a20-fka30>0,

which is equivalent to (10); indeed, the right hand side of (112)is the product of 1 -a20/a21 and the right hand side of (10). Thus (10) is a sharper result. However, the technique used in [2] to derive (10) does not extend to models

of the form x;=

x1

x$=x3

for example. And Theorem 2 does apply in this case. 3. PROOFS OF THE MAIN RESULTS

Proofof rrtteorem I. If for some solution x(t), with x(O)ER~$ we have x3( t)+O as t+lo, then the positive limit set of x(t) must contain a periodic orbit or a critical point in the set ((x1, x2,x3)1 x1> 0,x2> 0,x3= 0).We begin by ruling out the etistence of such a periodic orbit. 0n this set, the system can be written Xi

=x*[Q*o-allXl

xi =x2[

-a12gl(x~~-~2]=~l(x~~

-azo+a21gl(xl)x+E2(xb

x2)9

x2)9

We apply Dul.ac’s criterion (see f 11, for example) with auxiliary B(Xl, X2)=x;-%;+

THOMAS

170

C. GARD

for all x1 >O, x2 >0, since we are assuming g; > 0. [The statement of Dulac’s criterion is that if the divergence of B(F,, F2) is of one sign in a $irnp~y connected r@x~ hofthe plane, for some smooth function B, then the corresponding system has no periodic orbits.] Persistence is determined, therefore, by the trajectory structure of the system near equilibriumpoints in the x1x2 plane. These points are EO(O,O,O), El(a&zrl,O,O), and E2
and

[It is not hard to see that the equation xg,(x)=a20/u21 has a unique solution, given the assumptions we have here on gi.] Linearization theory indicates that the trajectory structure of the system is determined by the eigenvalues of the matrices of the linearized system at each of the equilibrium points, provided the eigenvalues have nonzro real part. At Eo, these eigenvalues are alo, -urn, and -am; E. is a saddle point, with directions of approach along the positive x2 and x3 axes. No trajectory originating in R?O can have E. in its positive limit set. The eigenvalues at El are -alo, -a=, and -am+a2,gl

( )

a10

-

a10

-.

Qll

a11

For persistence to hold., therefore, this last value must be positive. This is equkalent to ala/all > xl* [Eq. (4)]. Notice that this condition is necessary for EB(xr,x2+,0) t) be situated in the set (x+0, x2>0, x3=0}. In this case persistence holds if the x3 direction eigenvalue at E2 is positive:

(9 and fails if it is negative. Positivity of the x3 direction eigenvalue corresponds to the manifold of trajectories approaching E2 lying in the x1x2 plane, whereas negativity of this eigenvalue means that the manifold must intersect R’;O.This completes the proof of Theorem 1.

Proof of Xheorem 2. We begin by establishing a uniform asymptotic bound on solutions of (1). This is done in the same way as in Gard and

PERSISTENCE IN FOOD CHAINS

Hallam [3]. Define, for x =(x1,.

171 . . , xm),

n-l

u(x)=

n-l

l-I

ai, i-i- 1

k-l

IT

k-j

n-l +

II Qi,i+lxn*

i-1

Then for any solution x(t) of (I), u(x(t))

< u(x(O))e-“‘+

m= ,min& ( ajlo}

and

f

,

c

This implies that given x(0) and E> 0, there is T> 0 such that, for t > T, u(x(t))

a+;

(13)

Since xi(t) > 0 for each i, this inequality implies that Xi(t)<(E:+l)Zi

fort>Tand I
x(t)+y

as t+oo,

i-1

where 4 is the interval [0, ZJ Therefore, by continuity, given S >O sufficiently small, there is a T> 0 such that \ 0T, lgign-1. Now assume pp>O; we can choose &>O sufficiently small so that the expression (6) with mj replace by mi- 6 and Mj by M,+S is positive; this new expression we denote by cc&.We use pa> 0 %a construct a function p(x(t))=&!_,[,x,(t)]ri, ri positive constants, which i:s

THOMASc. GARD

172

eventually nondecreasing. Construction of such a function contradicts the fact that x,(r)4 as t-+00, since the latter implies that p(x(t))+O as t-boo. Toward this construction, we choose

Q-

at1

K- azl[ml-S]’

1

I

i

II r,+1

(;%I)

r1

i

-S,

aj-1, j[ Mj-l

+‘I

%+ LJ‘r mi_S] aj-l,j[

a11

Q21[ml-q

Mj-l+6]

’

aj+l.

j[

odd, (17)

i even,

II j-2 (j e-1

i

mj-6]

’

so that r, >O for all i. We show that for t sufficiently large, b(x( t))= &(x(r))/& > 0. Suppressing the t variable momentarily, we have * % ri(x)= *z, =“I

i

n-l +

~i[-~i,0+~i,i-t~i-l(Xi-l)Xi-1-ai,i+18i(Xi)xi+11

C i-2

+G[-a,,o+Q,,n-*&-1(Xn-l)XtPll

(18)

’ I

We can rAewrite(18)

(a10mj2(

iO)=w(x)

z)"i,O

+X1

(

r2
+ nxl Xi i-3

_-call

r,+1 -ai+l,igi ( r1

1

+x2

(

~.upr(X2)-~12g*(X1))

‘s-1 -ai-*, r1

( xi ) -

i&-l(xi-l) )

r

n-1 --%-1,.&J-, r1

( X,-l

.

) X, I

By (15), (M), and the choice of the ri ‘s (17), the expressions mukipl~g the x+, 1 c;j Q n - 1, in (19) are all nonnegative. Therefore, from (19) we obtain b(x(g)) 3 r&(g)){

I-c&?~%-1..r

_,+6]x,(r)).

PERSISTENCE IN FOOD CHAINS

Our assumption that x#)+O

173

as t+ao implies

and so )(x(t)) > (q&2)p(x(t)) >O for sufficiently large t. This completes the proof of the first part of Theorem 2. The second part is proved similarly. We have p, ~0; so defining ps as the expression (7) with mj replaced by mj-6 and M1 replaced by Mi + S, and chasing r, as in (17) except that m/-=6 factors are switched with Mj+S factors, we obtain

b(x(*))

Grlp8t@(*))

(20)

for any solution x(t) with initial value x(0) ERR* and sufficiently large t. Since pa<0 in this case, (2) implies that &x(t))+0 as t+ao, and so xj(t)+o

as *+a,

for at least some&

4. REMARKS In the proof of the second part of Theorem 2, we actually showed that all trajectories exhibit extinction as t gets large, whereas to verify the failure of persistence it is only necessary to ascertain that some trajectory behaves in this way. The condition p,< 0 is therefore stronger than needed here. Weaker conditions can be obtained by requiring the x, direction eigenvalue at any equilibrium in the set ((x,,...,x,))x,=O, x+0, 1 gi
Let k be an integet with 2 Q k Q n. Assume the hypothesesof Theorem2 with n replaced by k, i.e., let gi>O on [O,Ei], mi=inft,,,gi, and&~+upl,,z,lgi, 1 G&k. Define

THOMASC. GARD

174 If bLp,k > 0, thenfor any

solution x( t ) with x(U) E K I;“‘, I’lm xj(t)>O,

1gi;gk.

This theorem states that if jhp,& >O, at least the subsystem of (1) consisting of the first k tropbk levels must persist. This work was supported in part, & the U.S. Environmental Protection Agency unab Grant R806161-01-l.

4 5 6 7 8 9 10

A. A Andronov, E. A. Leontovich, I. I. Gordon, and A. G. Maier, Qtditative Theoy of Smmd order Qymrical $ystem, Wiley, New York, 1973. T. C. Gard, Pezsismlce in food webs: Helling-type food chains, Math. Biosci. 49~61-67 (1980). T. C. Gard and T. G. Hallam, Persistence in food webs: 1. Lo&a-Volterra food chim, Bull.Ma&em&d Biology 41~877-891 (1979). D. Gcdman, The thmy of diversity-stability relationships in ecology, Quurt. &CL Bid. 50~237-266 (197.5). C. S. Helling, Re&lience and stabiliq of ecological systems, Anmr. Rev. Akol. @wt. 4: 1-23 (1973). R M. May, S&abi&vand Conplerrity in MO& Ecqystem, 2nd ed., Princeton U.P., P&uXto& NJ., 1975. J. Maynard Smith, Ma&& in 1Ecology,University Press, Cambridge, 1974. R McGehee and R A. Armstrong, Some mathematical problems concerning the ecological problem of competitive exclusion, J. Di&wntiaf Equations 23330-52 (1977). R T. Paine, Food web complexity and sp&es diversity, Amer. Nut. 100:65-75 (1966). J. So, A note on fhe global stability and bifurcation phenomenon of a Lotka~Volterra food chain, J. Ykorett, Biol. SO:185- 187 (1979).