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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,
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 2O, 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
REFERENCES 1 2 3 4 5 6 7 8
9 10 11 12 13 14 15 16
17 18 19 20 21 22 23
F. Albrecht, H. Gatzke, A. Haddad and N. Wax, The dynamics of two interacting populations, J. Math. Anal. Appl. 46, 658-670 (1974). A. A. Andronov, E. A. Leontovich, I. I. Gordon and A. G. Maier, Qualitative Theory of Second-Order Dynamic Systems, Wiley, New York, 1973. H. Barclay and P. van*derfDriessche, Time lags in ecological systems, J. Theor. Biol. 51, 347-356 (1975). M. Conrad, Stability of foodwebs and its relation to species diversity, J. Theor. Biol. 34, 325-335 (1972). D. L. De Angelis, Stability and connectance in food web models, Ecology 56, 238-243 (1975). H. I. Freedman, Graphical stability, enrichment, and pest control by a natural enemy, Math. Biosci. 31, 207-225 (1976). H. I. Freedman and P. Waltman, Perturbation of two-dimensional predator-prey equations, SIAM J. Appl. Math. 28, l-10 (1975). H. I. Freedman and P. Waltman, Perturbation of two-dimensional predator-prey equations with an unperturbed critical point, SIAM J. Appl. Math. 29, 719-733 (1975). G. C. Gallopin, Trophic similarity between species in a food web, Am. Midwest. Nat. 81, 336-345 (1972). N. S. Goel, S. C. Maitra and E. W. Montrol, On the Volterra and other nonlinear models of interacting populations, Rev. Mod. Phys. 43, 231-276 (1971). A. R. Hausrath, Stability properties of a class of differential equations modeling predator-prey relationships, Math. Biosci. 26, 267-28 1 (1975). U. G. Haussman, Abstract food webs in ecology, Math. Biosci. 11, 291-316 (1971). D. L. Haynes and P. Sisojevic, Predatory behavior of Philodromus rufu Walckenaer (Araneae: Thomisidae), Can. Entomol. 98, 1133133 (1966). C. S. Holling, Some characteristics of simple types of predation and parasitism. Can. Entomol. 91, 385-398 (1959). S. B. Hsu, A Mathematical Analysis of Competition for a Single Resource, Ph.D. Thesis, Univ. of Iowa, 1976. S. B. Hsu, S. Hubbell and P. Waltman, A mathematical theory for single-nutrient competition in continuous cultures of micro-organisms, SIAM J. Appl. Math., to be published. S. P. Hubbell, Populations and simple food webs as energy filters. I. One-species systems, Am. Nat. 107, 94-121 (1973). S. P. Hubbell, Populations and simple food webs as energy filters. II. Two-species systems, Am. Nat. 107, 122-151 (1973). E. H. Kerner, On the Volterra-Lotka principle, BUN. Math. Biophys. 23, 141-157 (1961). R. M. May, Stability and Complexity in Model Ecosystems, Princeton U. P., Princeton, N.J., 1973. R. M. May, Mass and energy flow in closed ecosystems: a comment, J. Theor. Biol. 39, 155-163 (1973). J. Maynard Smith, Models in Ecology, Cambridge U. P., Cambridge, 1974. V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations, Princeton U. P., Princeton, N.J., 1960.
276 24
H. 1. FREEDMAN
AND
PAUL
WALTMAN
27
A. Rescigno and K. G. Jones, The struggle for life: III. A predator-prey chain, Bull. Math. Biophys. 34, 521-532 (1972). M. L. Rosenzweig, Exploitation in three tophic levels, Am. Nut. 107, 275-294 (1973). M. L. Rosenzweig and R. H. MacArthur, Graphical representation and stability conditions of predator-prey interactions, Am. Nat. 47. 209-223 (1963). P. T. Saunders and M. J. Bazin, On the stability of food chains, J. Theor. Biol. 52,
28
121-142 (1975). F. M. Scudo, Vito Volterra
29
(1971). J. A. Yorke
25 26
U.S.A.
and
and W. N. Anderson,
70, 2069-2071
(1973).
theoretical
ecology,
Jr., Predator-prey
Them-et. Pop/. patterns,
Bio/. 2, l-23
Proc. Nat. Acad. Sci.
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
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 p
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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17 18 19 20 21 22 23
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