Mathematical analysis of some three-species food-chain models

Mathematical analysis of some three-species food-chain models

Mathematical Analysis Three-Species Food-Chain H. I. FREEDMAN* Department of Mathematics, of Some Models University of Alberta, Edmonton, Alb...

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Mathematical

Analysis

Three-Species

Food-Chain

H. I. FREEDMAN* Department of Mathematics,

of Some Models

University

of Alberta,

Edmonton,

Alberta,

Canada

AND

PAUL WALTMAN Department of Mathematics,

The University

of Iowa, Iowa City, Iowa

ABSTRACT The paper is basically concerned with the question of persistence of all species in a three-level food chain. A general model is introduced and the equilibria analyzed. Boundedness and stability criteria are established. Three special cases of the model are analyzed, showing the applicability of the theory, and in certain cases extensions are given. The special cases include Lotka-Volterra (where we are able to give necessary and sufficient conditions for persistence), Lotka-Volterra predation with a carrying capacity at the lowest level, and a mixed Lotka-Volterra and Holling predation (at different levels) with a carrying capacity at the lowest level.

1.

INTRODUCTION

Recently there has been considerable interest in mathematical models simulating interactions between three species. In this paper we are concerned with the case that the three species form a food chain. Our main purpose is to examine the questions of survivability of all species, utilizing a relatively general model. The most general Kolmogorov-type model is looked at by Rescigno and Jones [24], who give some hypotheses and geometrical interpretationsof the equilibrium. Haussman [ 121, in discussing general food webs, also considers a specific food-chain model similar to ours. He analyzes, for his model, some of the equilibria, but not the one interior to the first octant. Hausrath [l l] considers a food-chain model which is basically a perturbed Lotka-

*Research for this paper was partially Canada, Grant No. NRC A-4823. MATHEMATICAL

BIOSCIENCES

0 Elsevier North-Holland,

Inc., 1977

supported

by the National

33, 257-276 (1977)

Research

Council

of

257

H. 1. FREEDMAN

258

AND PAUL WALTMAN

Volterra model, but one in which the highest trophic level is not solely limited by the middle level. For this model he shows the existence of a perturbed asymptotically stable equilibrium. Some analysis of competition models involving two species and a resource in a chemostat have been given by Hsu [15], Hsu, Hubbell and Waltman [16], and Saunders and Bazin [27]. Here, of course, there is a nutrient being constantly added, and species are continuously washed out of the system. These are not strictly food chains, but involve similar questions. A model involving networks was introduced by Yorke and Anderson [29]. Hubbell energy filters. General

[ 17,181 and May [21] considered food webs as discussions on how food webs ought to behave

ecologically were given by Conrad [4], Gallopin [9], Kerner [19], and Rosenzweig [25]. Barclay and van den Driessche [3] have introduced a time-lag food-web model. De Angelis [5] has considered the relation between food webs and stability. He has indicated that complexity could decrease stability. Here we show by example that the reverse could happen. In Sets. 2, 3, and 4 we treat the model in some generality and in Sets. 5, 6, and 7 deal with special cases where the general criteria can be applied and in some cases extended. In Sec. 2 the general model is described, the equilibria examined, and stability of the two simplest cases treated. In Sec. 3 the critical point in the xy plane is examined and conditions for its stability determined. In Sec. 4 the possibility of an equilibrium in the interior of the first octant is discussed and its stability determined. Section 5 assumes straight Lotka-Volterra dynamics, and there we are able to provide a necessary and sufficient condition for the persistence of all three species. In Sec. 6 we introduce a carrying capacity into the prey species via a logistic function while retaining Lotka-Volterra predation dynamics. The existence and stability of equilibria are investigated. The possibility of limit cycles here precludes a complete analysis. In Sec. 7 the Lotka-Volterra dynamics are replaced by Holling-type predation at the level of the first predator. Sufficient conditions for persistence of all three species are given. The analysis also shows that introduction of a third level can introduce an interior stable equilibrium point where the two-dimensional system had an unstable interior equilibrium point (interior, of course, changing as the problem moves from R * to R 3). In what follows, by persistence of a species we mean continued existence in the deterministic sense, i.e., limsup,,, N (t) > 0, where N (t) is the population of species N at time t. 2.

MODELS

SIMULATING

A FOOD

CHAIN

We consider the following system as a model simulating a food chain, where x is the number of lowest trophic species or prey, y is the number of

FOOD CHAINS

259

middle trophic level species or first predator, trophic level species or second predator:

x’= 4x)

and z is the number

of highest

-_w(x),

(!=$ >

.Yj=y[ -r+cp(x)]-zqb+

(2.1)

z’=z[ -s+dq(y)]. where r,s, c, d are positive constants. g(x) is the specific growth rate of the prey and is always assumed satisfy g,(x) < 0

g(O)=a>O;

for x > 0.

(Hl)

In those models in which the assumption is made that the environment a natural carrying capacity, we will also assume 3K>O p(x) is the first predator

g(K)=O.

response function p,(x)

p(O)=O; Similarly,

3

q(y), the second predator

> 0

q,(y)>0

4(0)=0;

has

U-W

and is assumed

to satisfy

for x > 0.

response function,

to

(H3) is assumed to satisfy

forya0.

(H4)

We note that under (H3) and (H4), the predation curves include the usual curves found in the literature (see, e.g., Haynes and Sisojevic [13], Helling [ 141, Rosenzweig and MacArthur [26]). We now consider the question of existence of equilibria for the system (2.1). First note that (O,O,0) is always an equilibrium.

Further,

(El)

if (H2) is satisfied, then clearly

(K,O,O)

W9

is also an equilibrium. To determine conditions which guarantee an equilibrium in the interior of the first quadrant of the xy plane, note that the condition which guarantees that -r+cp(x)=O

(2.2)

260

H. 1. FREEDMAN

has a solution

(and hence

a unique

_: Assume

(2.3) holds,

solution)

they

Then

is (2.3)

and let 1 be such that

value of the equilibrium

and _$ is positive

PAUL WALTMAN

lRangep(x).

p(2)= Then

AND

when

(H2) holds

;.

(2.4)

is given by

if 2 < K.

(2.6)

(Q, 0)

(E3)

in case such .C and _$ exist,

is an equilibrium. The question of whether or not there exists an equilibrium in the interior of the first octant is also of interest. If such exists, we will label it

(x*,y*,z*). The condition

054)

for y* to exist is clear from

the third

of the equations

5 ERangeq(y). Then

there

(2.7)

is a y * such that

q(Y*) = $. From that

the first of equations

(2.8)

(2.1) we can solve for x* in terms of y* provided

y* ERange-

in which equations

(2. l),

case there may be more (2.1), z* is given by

z*=

xdx) P(X)



x > 0,

than one such x*. From

_Y*[-'+cP(x*)] q(Y*)

.

(2.9)

the second

of the

(2.10)

FOOD

261

CHAINS

I* is positive provided

that - r + cp(x*) >O, or that x*>2.

(2.11)

Hence if (2.7) (2.9) and (2.11) are satisfied, (E4) exists. In order to compute the stability of the equilibria we need the respective variational matrices. Let V(x,y,z) denote the variational matrix of (2.1) for general x,y,z. Then

V(-~,Y,Z)

X&(X) + g(x) -VP,(x)

=

I

0

-P(X)

CYPX(x)

-4(Y)

-r+cp(x)-y,(y)

0

dzq, (v)

.

(2.12)

-s + &7(y)

The stability of (El) and (E2) will be considered here. (E3) and (E4) will be treated in the two subsequent sections. Let V, and V, denote V(x,y,z) at (El) and (E2) respectively. Then 0

a

VI=

0 0

[(E2) and

-r

0

I

03V2”

0

--s

r

Kg,(K) 0

0

-P(K) -r+cp(K) 0

0 0

.

--s !

V2 exist only if (H2) is satisfied.]

Clearly then, from (2.13) both (El) and (E2) are hyperbolic points, each having two negative eigenvalues and one positive eigenvalue. Near (El) the prey population grows while both predator populations decline. Near (E2), the prey population remains in a neighborhood of K, the first predator population increases (since _?< K), and the second predator population decreases. At any rate, because of the hyperbolic nature of both equilibria, and because x’ > 0 if y is sufficiently small and y’>O if x is sufficiently close to K and z is sufficiently small, neither equilibrium can be the limit of a trajectory initiating in the interior of the first octant. 3.

BEHAVIOR

OF SOLUTIONS

NEAR THE xy PLANE

In this section we assume that (2.3) and (2.6) hold, and hence that (E3) exists. The behavior of solutions in the xy plane was analyzed by Rosenzweig and MacArthur [26] near (E3) and by Freedman [6] in general in the first quadrant. It was shown there that in the absence of a stable equilibrium, there must always be limit cycles. We break our analysis into two cases, the behavior of solutions for small z >0 near (E3) and the behavior near a periodic orbit, if one exists.

H. I. FREEDMAN

262

First we concern Then

ourselves with the equilibrium

~&(f)+g(~)-y^Px(~.)

I

v,=

AND PAUL WALTMAN

(E3). Let V3= V(_?;,$,O).

.I

0

-P(i)

Cy^PX(i)

0

-q(9)

0

0

-x+&(.9)

(3.1)

The eigenvalue governing the stability in the z direction is - s + dq(j). We know that 2 O, and that there are no nontrivial periodic orbits in the open positive quadrant of the xy plane. Since all of the equilibria in the xy plane are hyperbolic, no orbit in the interior of the first octant can approach the xy plane as t+oo. By the above, we have proved the following theorem. THEOREM

3.1

Let (2.1) be such that there are no nontrivial

periodic

plane. Then a necessaty condition for the persistence arbitrary positive initial populations is

solutions

in the xy

of all three species for

-s+dq();)>O, and a sufficient positive

(3.2)

condition for the persistence

initial populations

of all three species for arbitraty

is -s+dq(y^)>O.

Consider

(3.3)

now the case that there are nontrivial periodic solutions in the where x=+(t), y=#(t) is such a

xy plane. Let V3p(t)= V(+(t),J/(t),O), periodic solution. Then

[

V+(t)=

Consider

-

@(t)gx(+(t))+g(44t))

I

-p(+(t))

0

1

G(t)p,(+(t))

c+(t)p,(+(t))

now a solution

0

- r + cp (Ht)) 0

-q(#(t)) -s

+ dq(+(t))

I (3.4)

of (2.1) with positive initial conditions

(aI, (Y*,aa)

FOOD CHAINS

sufficiently

263

close to the periodic orbit. Ciz/&,

is a solution

of

z’= [ -J+dq($(t))]z, (3.5) z(O)= 1, or z(r)=exp(

-st+di’q($(s))ds).

(3.6)

Hence using Taylor’s theorem,

Since q(t) is periodic (of period T, say), z increases as

or decreases according

d T q(+(t))dt I T,

-s+

is positive or negative. Since these periodic orbits, together with (E3), are the only possible limits, in the xy plane, of trajectories with positive initial conditions, we have proved the following theorem. THEOREM

3.2

Either let (E3) be an unstable equilibrium in the x and y directions or let (3.3) hold. Further, for each periodic solution x=+(t), y = q(t) in the first quadrant of the plane which is stable in the plane on at least one side, let

-s+

$LTq(#(t))dt>O.

(3.7)

Then aI three species persist for all time. 4.

BEHAVIOR

OF SOLUTIONS

AWAY FROM

THE xy PLANE

In this section we will first show that if (H2) is satisfied, then all populations are bounded whether or not (E4) exists. Then in the case that (E4) exists we examine its stability. Suppose that (H2) does indeed hold; then by the well-known property of logistic growth, the prey is limited by its carrying capacity. Specifically, suppose we are given initial values (x,,y,,z,) of the system (2.1). Then since,

264

H. I. FREEDMAN

from the first of equations

AND

PAUL

WALTMAN

(2.1), x’ Q x&x),

we have by the usual comparison x(t) < I,

theorem that where

I = max( x,,, K )

Now we add d times the second of the equations obtain (dy+t)‘=

(4.1)

(4.2)

(2.1) to the third and

-dry--sz+cdp(x).

(4.3)

Let m = min(r, s). Then - dry - sz < - m(& + z), and using (4.3) with w = dy +z, we obtain w’< -mw+cdp(l),

(4.4)

which implies w < w,e-“‘*+

or, using the standard

comparison

cdp(l)/m,

(4.5)

theorem,

O
Thus y(t) and z(t) established. THEOREM

are both bounded.

The following

(4.6)

theorem

has been

4.1

Let hypotheses (Hl) and (H2) hold. Let p(x) > 0 for x > 0. Then alI solutions of system (2.1) initiating in the first octant are bounded. We remark that this agrees with biological intuition. If the prey species is resource limited, then both predator species are also limited regardless of their predation curve shapes. We can state a consequence of this in the case the hypotheses of Theorem 3.1 or 3.2 hold. COROLLARY

4.2

Let the hypotheses of Theorem 4.1 and either Theorem 3.1 or Theorem 3.2 hold. Then there exists a recurrent motion &ing in the first octant. Proof. Since all trajectories initiating in the first octant are bounded lie in that octant, by a well-known theorem on dynamical systems Nemytskii and Stepanov [23]) the corollary is proved.

and (see

26.5

FOOD CHAINS

We now V(x*,y*,z*),

suppose we have

(Hl)-(H4)

and

that

(E4)

exists.

Setting

V4=

where m II=x*g,(x*)+g(x*)-y*~,(x*), m2, =

cy*px(x*> >O,

m23=

-4(y*)
The characteristic

m12=

m22=

-r+ cp(x*)-

-p(x*)
(4.8)

m32=dz*qY(y*)>0.

polynomial

whose roots are the eigenvalues

of V4 is then

f(h) =A3- (ml1+ m22P2 + (41m22 - m,2m21- m23m32P + mllm23m32.

(4.9)

Since j(0) -

m11m23m2,

_f(m,d=

-m7111w2m21,

(4.10)

and since m23m32 < 0 and mlzm2, < 0, we have either m,,=O

or

(4.11)

j(O)j(m,,)
Hence there is a real root either at 0 or between 0 and m,,. As a consequence we see that if m,, > 0, (E4) is unstable. Suppose now that m,, < 0. Let p < 0, 0 < IpI < Im,,J be the negative real root deduced above. Then, upon dividing j(h) by X-p, we obtain the quadratic

fl(~)=~2+(p-mll-m22)~ +[m,,m22--m,2m21-m2~ma2+~2-~(m,,+m22)], the roots of which are the remaining

two eigenvalues.

(4.12)

Since p- m,, > 0, if

m22 Q 0, then

P-m,,--22>0,

(4.13)

which implies that the roots of j,(h) have negative real parts. Hence we have proved the following theorem.

266

H. I. FREEDMAN

THEOREM

AND

PAUL

WALTMAN

4.3

Let (Hl)-(H4) hold, and suppose (E4) exists. Zf m,, >O, then (E4) is unstable. If m,, < 0 and mzz < 0, then (E4) is stable. If further m,, < 0, then (E4) is asymptotically stable. We look at m,, definition of x*,

mZ2 in a little

and

more

detail.

From

&(x*)

m ,,=x*g,(x*)+g(x*)-y*p,(x*)=x*g(x*)

~

+

-$ln

Hence m,, <0 (>0) x*. Similarly

the

1

1 _ A-(x*)

g(x*) = x*g(x*)

(4.8) and

p(x*>

x*

xg(x)

( )I p(x)

x=x*

if and only if xg(x)/p(x)

is decreasing

m 22= -‘+Cp(X*)-z*qJy*)=[-r+cp(x*)]

[ l-

(increasing)

“bl;(i;’

at

]>

or m2*=[

-r+cp(x*)

ly*i+( &))I .

(4.14)

Y=Y*

Now x*>l, and so -r+cp(x*); y/q(y) is decreasing (increasing)

> 0; thus mz2 < 0 (>0) if and only if at y*. The condition for m,, is related to

the graphical method of Rosenzweig and MacArthur We note that if q(y) is a Holling-type predation,

[26]. then m22 > 0. However,

if q(y) is linear, as in the Lotka-Volterra case, then m22 =O. m22 will be negative for predations curves with learning effects such as those shown in Haynes 5.

and Sisojevic

A FOOD

CHAIN

[ 131. OF LOTKA-VOLTERRA

TYPE

In this section we assume that the functions g,p,q yield terra dynamics. More specifically, we consider the system

the Lotka-Vol-

x’= a,x - a,,xy, y’=

- a2y + a,,xy - a,,yz,

z’ = - a3z + a32yz, x(O)=cu,

>o,

y(O)=Ly2>0,

z(O)=a3>0,

(5.1)

FOOD CHAINS

where

267

all of the constants

a natural

carrying

are positive.

capacity,

i.e.,

Note

(H2)

that this system

is not

satisfied.

does not have

In this

analysis of the preceding sections may be completed to yield answer to the question of the persistence of all three species. THEOREM

case

the

an exact

5.1

A necessary

a dynamical

and sufficient

system

governed

Proof: We will first show

condition for the persistence by the system that

of a/l three species in

(5.1) is that p = a,aJ2-

the xy plane

is an attractor

a3a,* > 0.

or a repeller

according as p < 0 or p > 0. To do this we first examine solutions of (5.1) with LYEsmall, i.e., solutions close to the xy plane. For CY~ =O, we have z(t)-0, satisfy

and solutions of (5.1) are given by (+(t),$(t),O), the Lotka-Volterra equations

u; = a,u,

(+(t),+(t))

where

- alZu,uZ,

u2= -azu2+a2,u,u2, and hence

are periodic.

The variational

equation

(3.4) about

such a solution

becomes

0

- a,244t) a,244t> -a2ddt) - a2 + a2h (t)

I

aI -

Y’=

As noted

above,

#(t) is periodic

1 Y.

-a,+ a32d4t)

0

0

0

(say of period

T) and from

[8, Eqs. (3.8)

(3.9)1

(Note

that +2(t) of [8] is a transformed

-a,T+-=-

variable.)

a32alT aI2

Thus

(3.7) becomes

T

I4 aI2

which proves the assertion of the theorem for y >O, or for pO,i=1,2,3, limsup [+_,z(t)= F> 0. If there were a sequence {t,,}, In-co, such that z(t,J-+0, then by what we have proved above, z(t) would tend to zero (the solution would eventually come “too close” to the attractor z ~0). Thus we can assume z(t) > S >0 for some 6. From the first

H. I. FREEDMAN

268 equation

of the system

PAUL WALTMAN

(5.1)

5 ~x’(t)

y(t)=

a12

which

AND

can be put into the third

z’(t) -+z(t)

a,,x(t>

equation

a32 a12



of (5.1) to yield

x’(t>

, a3241

3 1

x(t)

a12

or

(5.2)

Since the right-hand side of (5.2) tends to zero, so does the left. Since z(t) > 6 and a32/u12 >O, it follows that x(t)-+O. Choose Then

for t 2 t,, x(t) < (a2 + a2,S)/2a2,. follows that y’(t)

v(t)

from the second

a2+ 6023

< -a2-c9a23+a21p

equation

t, so that in (5.1)

it

= -;(-a,+6a,,)
2a2,

for t > to or lim,,,

y = 0. In the same manner it now follows from the third This is a contradiction, so limsup,,,z(t) =O, if p < 0.

equation in (5.1) that z(t)+O. and z = 0 is a global attractor

Finally we must consider the case p =O. In this case we show that all solutions are periodic [and hence by uniqueness of solutions of initial-value problems, z(t) > 0 for all t]. In the xz plane we have - a32 + a32yz

dz -_= dx

U,,XY

a,x=

z(a,,Y-a,) z(a,

=

-

a,zY)

Z(a3a,*Y/a, x(a,

Thus it follows

-

-a3

z

aI

x

a31

U,,Y)

that (5.3)

FOOD CHAINS

z(t)

269

in the first

leaving

two equations

the two-dimensional

may

be replaced

by the above

expression,

system

x’(t) = x(a,

- Q,*Y) (5.4)

where y = a3/a, >0 and c, = a3ajaz3. The system (5.4) may be considered in the phase plane (the xy plane), where the variables separate, and one obtains (5.5)

~,(~)+~2(Y)=~,(x(O))+cpz(Y(O))>O~ where x -aa,+a*,S-c,S-Y $1 (x) =

=-

azln$

+a2,(x-x*)+

s Y

h(Y)’

dS

s

s x’

~(xey-xtpy),

+ a,,SdS

-a,

= -a,ln$

S

+u,~(Y-y*)dS,

Y’

and x*,y*

are the solutions

of aI - Ol2Y

*=o

- a,+ az,x* - c,x*-y=o

[the critical points of (5.4) in the interior of the first quadrant]. To see that (x*,y*) do indeed exist, note that if f(cf)

= -a,+

a2,OL- c,(Y-y,

then lim f(a)=

- 03,

LX-O+

lim f(a)= a--t+CC

+ 00.

Since the range off is all of R, there exists a value, x*, such that f(x*)=O. Trivially, y* = al/a,*. Further, both +I, and +2 are positive functions for x #x*. Now f’(a) = a2, + yc,ay-’ > 0; thus for x > x*, f(x) > 0, and for x
>0

for

x>O,

(5.6)

H. 1. FREEDMAN

270

AND PAUL WALTMAN

or +,(x)>O for x#x*. Similarly &(y)>O fory#y*. Further, the range of cpi is R. In view of the monotonicity given by (5.6) and the above fact about the range of +i, for every positive number p, there exist exactly two positive numbers xi,xz with x1
+l(xi)=PY

applies statement arguments (see,

A similar geometric defined

to +2, and of course for example,

2) yield

that

Simple the

curve

about (x*,JJ*). Thus solutions of the autonomous system Since x(t) is periodic, so is z(t) by (5.3). This completes

A LOTKA-VOLTERRA WITH

CARRYING

We modify capacity into consider

+,(x*)=&(y*)=O.

[7], Sec.

by

is a closed curve (2.2) are periodic. the proof. 6.

1,2.

FOOD

CHAIN

CAPACITY

the model of the previous section by introducing a carrying the dynamics of the lowest trophic level. Specifically, we

the system

x’=x(a,(l-

$)-%+

(6.1)

y’=y(u,+a,,x-a,,z), z’=z(-ua,+a,,y),

x(O)= a,,

Y (0)

=

a27

z (0)

=

a37

where all of the constants are positive. The carrying capacity is K. (Hl)-(H4) are satisfied, and from Theorem 4.1 we know that all solutions with the above initial conditions are bounded, i.e., the closure of any trajectory is compact. We first analyze the critical points. As noted in Sec. 2, (El) and (E2) exist and are hyperbolic. The interest then focuses on (E3) and (E4). For the system (6.1), (E3) is given by

a,

(a,,K-

u2> ,O

al2a2,K

FOOD CHAINS

where

271

for y^ to be positive

we must

assume

K>$

The variational

matrix

(3.1) takes

(6.2)

the form

(-al/K)1

0

=2,y

-a,G 0

-a23y

0

0

-a3+a32y

^

Viewed as a critical point in the xy plane, (Z,j) is asymptotically since (- a,/K)i < 0. Further, the critical point will be unstable direction if - a3 + a,,_$> 0. But, aI

-a3+a32y=-a3+a32

(ad-

=32al (a,,K-

1

a21

a,,a,,K

=

stable, in the z

4 - a3a2,a12K

a12a2,K This quantity

will be positive

if a32al - a3a12 > 0,

=2a32al

K>

(6.3) 021 (a32al

If u~~u,- a3a,2< 0 or if K strictly be asymptotically stable.

-

violates

a3a12>

.

(6.3), the critical

point

(I,j,O)

will

If z(0) =O, the remaining two-dimensional system (6.1) may have limit cycles, at least one of which must be semistable. For these periodic solutions we are unable to obtain any more specific information than already given in Theorem 3.2. We consider now the existence of (E4). Solving for the critical point, one obtains x*=

(ala32- a12a3)K a32

9

y*=a3, a32 z*=

=2, (al=32- =,24Ka23=32

=2a3

H. I. FREEDMAN

272 and to be interior

to the positive

octant

requires

ala32 - a12a3 >

AND PAUL WALTMAN

that

0,

a2a3

K> a2,

(w32-

(6.4) a12a3)



Comparing with (6.2) (6.3) we observe that if K satisfies (6.2) and (6.3) but (6.4) is violated (which is possible if a3 is sufficiently large), then there is no interior critical point in the first octant, and the critical point in the xy plane is unstable. Since the trajectory has compact closure, its w-limit-set contains a (nontrivial) recurrent trajectory. The variational

equation

(4.7) has entries

- a,x* m II= ~ K m2l=

m12=

a21_v*,

m22 *

m23

With

the above

-a2,y

=

conditions

a12x*,



g

making

--0,

m32 = a32z*.

x*,y*,z*

positive,

m,,
Theorem 4.3 implies that (E4) is asymptotically stable. Finally we note that the asymptotic stability criteria are local, not global. The possibility of limit cycles in the plane exists, as well as the possibility of more general three-dimensional limit sets (necessarily containing recurrent solutions). The analysis of the stability of such sets appears to be a very difficult 7.

problem.

MODELS

INCORPORATING

HOLLING-TYPE

PREDATION

We modify the model of the previous section to include a Holling-type predation of the first predator on the prey. Specifically we consider the system x.=x[o(l-;)-&],

y’=y

(

-r+&-YL

,

1

(7.1)

z’=z(-3+&y),

x(O)=a,, The equilibrium

Y (0)

=

a27

(E3) is given by XC_

_q=

r cp-ra’ ac[K(cp-ra)-r] K(c/3-ra)2

i=o.

z(0)

=

a3.

273

FOOD CHAINS

For 1 and_9 to be positive requires

cpK>‘.

ra > 0, c/3-ra

Rather than approach the variational equation (3.1) directly, we utilize known criteria for the stability of the two-dimensional system (see Rosenzweig and MacArthur [26], Freedman [6]). The isocline has the graph y=A(K-x)(l+ax). PK

Its maximum

occurs at aKX=2a.

1

Hence the critical point (&j) in the z = 0 plane will be unstable if _?< X and stable if f > X. This condition takes the form of a restriction on the carrying capacity as Kc;+--.

2r c/3--ra

(See also Hsu [ 151.) Hence if c/3 - ra > 0 and r c/3-ra


2r c/3--ra’

(7.2)

(Z$, 0) exists with 2 > 0, _$> 0 and is asymptotically stable in the xy plane. Hsu [15], using a theorem of Dulac [2, p. 2051, has shown that in this case the two-dimensional system has no limit cycles. Since there are no other critical points in the open positive quadrant, and since all solutions are bounded, the absence of limit cycles and the Poincare-Bendixson theorem allows one to conclude that a solution of (7.1) with as=0 satisfies lim x(t)=.? r-+m lim ~~(t)=j. r-+.x If K>;+-

2r c/3-ra’

then Hsu [15], using a result of Albrecht, Gatzke, Haddad, and Wax [l], has shown that there exists at least one periodic orbit (outermost, semistable, outside; innermost, semistable, inside).

H. 1. FREEDMAN

274

AND PAUL WALTMAN

Assuming (7.2) holds (and, of course, cp - ra > 0), Theorem all species will persist if y^> s/6y. For the equilibrium (E4) we note first that y*=s

du

3.1 says that

>o.

x* is given as a root of aaxZ-x(aK-l)cu-K(cr-py*)=o. For there to be a positive root, either

or Y* 2 a/P,

K>l/a is required. component

In the second case there are two critical points xr,x:. is z*_ (cP--ra)x*-r y(l+ax*)

The final



which is positive if r x*> ~ cp-ra Suppose there is an interior

=x.

critical point. In the notation

m 11=x *

(

_a

aPr*

I?+ (1 + ax*)2

of Sec. 4,

CO, 1

and by (4.14), mz2 =O, since y/q(y)= y. Thus the critical point (x*,y*,z*) is asymptotically stable by Theorem 4.3, provided the inequality apKs (1 + ax*>2 > holds. day The above discussion introduces the possibility of stabilization of the interior equilibrium point of a two-dimensional system by the addition of a third trophic level. Suppose for example that .C< F=(aK1)/2. Then for the two species system the interior equilibrium point (.C,j) is unstable. The introduction of a third trophic level with an interior equilibrium (E4), (x*,y*,z*) with x* > X, introduces an asymptotically stable interior equilibrium into the three-dimensional system.

FOOD

CHAINS

275

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9 10 11 12 13 14 15 16

17 18 19 20 21 22 23

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AND

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WALTMAN

27

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