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A mathematical model of facultative mutualism with populations interacting in a food chain
A Mathematical Model of Facultative Mutualism with Populations Interacting In a Food Chain RAVINDER
KUMAR*
AND
Applied Mathematics Institute, Edmonton, Canada TfiC 2GI Received
3 November
H. I. FREEDMAN+ Department
of Mathematics,
University of Alberta,
1988; revised 12 July 1989
ABSTRACT Facultative mutualism with populations interacting in a food chain is modeled by a system of four autonomous ordinary differential equations. Two cases are considered: mutualism with the prey and mutual&m with the first predator. In both cases persistence
and extinction criteria are developed in terms of the invariant flows on the boundaries.
1.
INTRODUCTION
This paper is mainly concerned with modeling facultative gate) mutualism with populations interacting in a food chain.
(i.e., nonobli-
Such mutualism may occur with any of the prey populations. For example, the situation described by Monteith [26] in a study conducted on a food chain in western Ontario found that the parasitism of sawfly larvae by the parasitoid B. harveyi on the host plant Lurix laricina is much lower in the presence of associated plants, which mask the odor of the sawfly larvae and their food plant. Facultative mutualism with populations at various trophic levels of a food chain have been observed in nature. Vance [32] describes a mutualistic interaction between clams and their epibionts that helps to protect the clams from their starfish predators. Fritz [20] observes that ants as mutualists decrease predation on treehoppers. For other mutualisms at the prey level, see [4] and [31].
*Research is partly based on a doctoral thesis at the University of Alberta. +Research is partly supported by the Natural Sciences and Engineering Council of Canada, grant NSERC A4823.
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236
RAVINDER KUMAR AND H. I. FREEDMAN
A mutualism with a population at an intermediate trophic level is described by de Vries [8]. Giant tortoises on the Galapagos Islands, which prey on small aquatic animals and whose eggs are preyed upon by feral mammals (black rats, pigs, dogs, etc.), have evolved a symbiotic relationship with finches, who remove ticks while the tortoises stand on their hind legs, thereby increasing their life span. A similar behavior is described for iguanas [9], which feed on algae, grasshoppers, and so on, and are preyed upon by snakes, mockingbirds, and rats. A mutualism that has the effect of increasing the hunting ability of predators is described in [27]. Yellowsaddle goatfish, P. cyclostomus, and bird wrasse, Gomphosus caeruleus, tackle coral reefs from both sides so that prey may be driven toward and caught by each other. There are now many papers in the literature dealing with mutualistic systems. Most of these papers deal with two-dimensional systems modeling direct mutual&m between two populations and ignore all other population interactions (see, e.g., [l], [5], [lo], [ll], [23], [30]). In recent years, some papers have appeared that deal with cases where the mutualism is due to or influenced by the interaction with a third population (see, e.g. [2], [ll]-[15], [29], [34]). For general biological discussion of mutualisms and how they may occur, see, for example, [l], [2], [5], [28], and [33]. As previously mentioned, the main purpose of this paper is to model facultative mutualism with populations interacting in a food chain. The food chain models we utilize have been discussed in [12], and [29] and in some of the works cited therein. To the best of our knowledge this is the first attempt to model mutualism with populations in a food chain. We are interested in developing criteria for persistence and extinction for our models. Persistence in biological systems in a context related to this paper has been discussed in [6], [12]-1161 and 1181. We utilize the definitions of persistence developed in [6], 1181,and [19], namely that if N(t) is such that N(t) > 0, we say that N(t) persists if lim,, m inf N( t) > 0. Further, if N(t) E Q, where V is a certain class of functions, and there exists 6 > 0 such that lim,_m inf N( t) > 6 for all N(t) E V, then we say that N(t) is uniformly persistent (also known as permanence). A system is said to (uniformly) persist if each component (uniformly) persists. We say that N(t), N(0) > 0, exhibits extinction if lim, _ m N(t) = 0. We note that nonpersistence does not necessarily imply extinction for all initial values N(0). If lim, _m N(t) = 0 for N(0) > 0, we say that our system exhibits extinction with respect to the N(t) population. Throughout this paper we utilize the following notations: .C@z denotes the positive v axis and3 its closure for any variable v. %‘,+, denotes the positive VW plane and %‘,\ its closure, etc. In the next section we develop our models. In Section 3 we discuss the equilibria and their stability. Persistence criteria are developed in Section 4,
MUTUALISM
237
WITH POPULATIONS IN A FOOD CHAIN
followed by extinction criteria in Section 5. Some examples Section 6, and the final section contains a brief discussion. 2.
are given in
THE MODEL
In this section we describe a general system that models a mutualist interacting with populations in a food chain. The mathematical formulation of facultative relationships between the mutualist and two different trophic levels of the food chain are also described. Finally, we establish the existence of a region of attraction in each case, showing that the model is well-behaved. We consider the autonomous system du -=uh(u,x,JJ,t), dt dx _Eit=“xg(u,x)-YP,(u,x)-zP,(u,x), (2.1)
~=z[-~2(U,Z)+C2(U)PZ(U,*)tC?(U)q(U,Y)1, u(0) = ug2 0,
x(0) = xg2 0,
Y(0) = Yo 2 0,
z(0) = z() 2 0,
as a model of a mutualist-food chain interaction with continuous birth and death processes. The variable u(t) represents the density of the mutualist at time t, and x(l), y(t), z(t) denote the prey and predator densities, respectively. In the absence of the mutual&t, that is, when u = 0, (2.1) reduces to a food chain model. The function h(u, x, y, z) represents the specific growth rate of the mutualist. We assume that h(u, x, y, z) possesses the following properties [3,10,23,29]:
(HI) (H2) There
exists a unique function
++,Y,Z),X,Y,Z)
L( x, y, z) > 0 such that
=O.
The first condition implies that, independent of the x, y, z populations, u is capable of growing even when rare. Also, the growth rate is assumed to be density-dependent and decreases as the population increases. The second condition implies that L(x, y, z) is the mutualist’s carrying capacity and in
RAVINDER KUh4AR AND H. I. FREEDMAN
238
part specifies in what way the predators and prey become part of the mutualist’s environment. The function g( u, x) is the specific growth rate of the prey x in the absence of any predation. We assume that
J&,x) 10
(Gl) g(u,O)’ 0, (G2)
There exists a unique K(U) > 0 such that g( u, K(u))
= 0
Condition (Gl) implies that the prey population is capable of surviving in the presence or absence of the mutual&t and that the growth rate in the absence of predation is density-dependent. Condition (G2) implies that K(U) is the carrying capacity of the prey in the absence of predation. Next, the functions p;(u, x), i =1,2, and q(u, y) denote the predator’s functional response to the prey and mutualist densities. They represent the change in density of prey per predator per unit time as a function of the prey and mutualist densities. We assume that
$(u,x)>o,
p,( u,O) = 0;
(Pl)
4( KO) = 0,
%(U,Y)
i =1,2;
>o.
The above conditions imply that in the absence of prey there is no predation. Further, the predator functional response is assumed to be an increasing function of the prey population. The functions si(u, y) and s2(u, z) are the specific death rates of the predators y and z in the absence of predation. We make the standard assumption that the death rate is an increasing function of the population:
(S1)
%(u,y) ay
a44
>. 7
,. .
aZ
The nonnegative functions c,(u), i =1,2,3, are the rates of conversion of prey biomass to predator biomass. Also a > 0 is a bifurcation parameter. Finally, we assume that all the functions are smooth enough that existence and uniqueness of initial value problems hold and any required analysis can be carried out. 2.1.
FACULTATIVE
MUTUALISM
WITH
THE LOWEST
TROPHIC
LEVEL
In this section we specify the hypotheses for facultative mutualism between the mutualist u and the prey x. In addition to (Hl) and (H2), we
MUTUALISM
WITH POPULATIONS
assume the following ist:
IN A FOOD CHAIN
239
for the specific growth rate h (u, x, y, z) of the mutual-
(H3)
dh( u, x, y, r) ax ‘0,
(H4)
lim L(x,O,O) x+m
Jh( u, x, Y, r) Jy so0,
Jh( u, x, y, r)
az
I; 0.
-L
Condition (H3) implies that II derives benefit from the prey population and that there might be a cost to the mutual&t due to its interactions with the predators. Condition (H4) implies that the mutual&t has a finite carrying capacity. In the case that ag(u, x)/au > 0, then, even in the absence of predation, u acts as a mutualist with respect to x. If ag( U, x)/au < 0, then there is a cost to x for associating with u and u can be a mutualist of x only by its effect upon the predator(s). If ag( II, x)/au = 0, the relationship between u and x without predation is commensal. Further, we assume
This condition implies that the mutualist can benefit the prey by decreasing the predator’s response. The mutual&t can also benefit the prey by increasing the death rate of y or by decreasing the rate of conversion from prey biomass to predator biomass:
Also, in order to have a viable system we must have that (S3)
Thereexistsan
P suchthat
s,(O,O)/c,(O)
=p,(O,P)
and Z
Finally, the mutualist can also benefit the prey indirectly, by its effect on the dynamics of the top predator z, that is, by affecting its death rate s2( U, z), the predator response function q( I(, y), or the rate of conversion c~( u) of prey biomass into predator biomass. Now we show that under the above hypotheses a certain set &’ contains the region of attraction of (2.1). The proof is based on the ideas developed in [16] and (181.
RAVINDER KUh4AR AND H. I. FREEDMAN
240 THEOREM 2.1 Whenever
the hypotheses
(Hl)-(H4),
(Gl), (G2), (Pl), (P2), and (Sl)-(S3)
hold, the set d =
{(u,
x, y,
2) : 0 5 u I
I?,0 5
x
I
fz,0
5
q(O)x + y
~~,oIc,(o)x+~~,+zz~},
(2.2)
where
Q_=
CdOM ___ [ad+ ~,(O~O)l SI(O,O) Y
min s,(u,O),
;*=
(2.3)
OlUli
and
is positively invariant and attracts all solutions initiating with nonnegative initial conditions. Proof. Let (u(t), x(t), y(t), z(t)) be any solution tial condition (u,, x0, yO, z,,). Since
whenever
with nomegative
ini-
u(t) 2 t we get u’ I uh( L, x,0,0)
< 0.
From this, it follows that u(t) I L for all t 2 0 whenever uO I i. Further, if uO > t, then u(t) 5 L, for large t. Next, let g(x) = maxO ~ u ~ r. g(x). Then since all solutions of the initialvalue problem, x’=xg(x),
x=x,>0
7
satisfy lim sup, _ m x(t) I k (see [8]), it follows from a standard
comparison
MUTUALISM
WITH
POPULATIONS
IN A FOOD
241
CHAIN
theorem (e.g., see [21]) that if x,, < 2, then x(t) 5 2 for all 12 0, otherwise x(r) is bounded above by g for sufficiently large t. Now, we consider
Thus if c,(O)x, + y, I AI, we get
Id2
c,(O)x(r)+y(t)
Otherwise
forall
t20.
there exists T > 0 such that
q(O)x( t) + y(t) Similar arguments
I
La
for t 2 T.
can be given to show that the inequality C*(0)X(f)+~~y(t)+z(t)
115
(2.4
holds for all t 2 0 whenever c,(0)xO + ?sy,, + z,, I k. Otherwise (2.4) is true for sufficiently large values of t. 2.2.
FACULTATIVE
MUTUALISM
WITH THE SECOND
System (2.1) exhibits facultative following hypotheses: in addition
TROPHIC
LEVEL
mutualism between u and y under the to (Hl), (H2), (Gl), (G2), (Pl), and
RAVINDER KUMAR AND H. I. FREEDMAN
242 (Sl)-(S3),
tH3’)
we assume
ahb,
X, Y, ax
4
(H4’)
yli_mmL(O,y,O) =i
(G3’)
ag(,U;X)
a.4 u7 Y) I o au 7 c;(u)
c;(u)
ro,
h(u,z) au
>() - >
IO.
Mu,
Y)
au
-<
0,
c;(u) 20,
10.
In this case the following set .4? contains proof is similar to that of Theorem 2.1. THEOREM
x, .Y, z)
aZ
20. ,
(S2’)
Jh(u,
9
the region of attraction.
The
2.2
Wheneuer the hypotheses (Hl), (H2), (H3’), (H4’), (Gl), (G2), (G3’), (Pl), (Pz’), (Sl), (S2’), and (S3) hold, the set
where L=
lim L(O,y,O), ,v-+oa
k=
max
K(u),
OlUSi
~=~[.g(~,O)+sl(i,O)], 1
(2.6)
9
and
is positively invariant and attracts all solutions initiating with nonnegative initial conditions.
MUTUALISM
3.
243
WITH POPULATIONS IN A FOOD CHAIN
EXISTENCE, NONEXISTENCE, OF EQUILIBRIA
AND STABILITY
In this section we shall establish criteria for the existence and nonexistence of equilibria of system (2.1) and discuss their local stability. This information, besides being helpful in understanding the system, will be needed to derive the persistence criteria in Section 4. 3.1.
CRITERIA
FOR THE EXISTENCE
AND NONEXISTENCE
It is clear that E,(O,O,O,O) is an equilibrium THEOREM
OF EQUILIBRIA
for system (2.1).
3.1
System (2.1) has exactly two one-dimensional equilibria, EI( LO,O,O,O) and E2(0, K,,O,O), where L, = L(O,O,O) and K0 = K(0). Proof:
The subsystem
in 9Yu+ is given by
u’= uh( u,O,O,O),
u(0) > 0.
(3.1)
Now, from hypothesis (H2) it follows that E1 is an equilibrium. Furthermore, condition (Hl) implies that lim,,, u(t) = L, (see [lo]), establishing the uniqueness of E1. Similar arguments hold for E,. The subsystem in 9-y’ and 91’Z’ have no equilibria since all of their solutions tend to zero exponentially. Remark 3.2. It follows from the above arguments that E, is locally stable in the y and I directions and locally unstable in the u and x directions. As shown in [15], the following result holds in 9?:X. THEOREM
3.3
The subsystem in %‘A has an equilibrium, E2 are unstable in E.
E3( ii, I,O, 0), provided E1 and
Remark 3.4. As will be seen below, because (Gl), E3 always exists. THEOREM
of hypotheses
and
3.5
A necessary and sufficient condition for an equilibrium E4(0, x1, yI, 0) to exist in gX?$s that the hypotheses
323O
be satisfied.
(Hl)
K(0)
and
s,(O,O) ___ = p1(0,:> Cl(O)
of the form
(S3)
RAVMDER
244 Proof.
The solutions
of the subsystem
x’ = axg(0, x) -
y’=
KUMAR AND H. I. FREEDMAN
yp1(0,x)
9
Y[ - s1(0,Y) + clu9Pdwl
(3.2)
in 9:” are bounded for positive time [lo]. If (S3) is violated, then from hypothesis (Sl), it follows that lim, ~ m y(t) = 0, for all solutions initiating in 9?zV, Thus E, is globally stable with respect to 9%‘.&. Now suppose (S3) holds, It follows from Theorem 3.1 that E0 is unstable in the x direction. Further, (S3) implies that E2 is unstable in e. The result now follows by an application of the Poincart-Bendixson theorem. Similarly, THEOREM
we have the following result.
3.6
A necessary and sufficient condition for equilibrium E,(O, x2,0, z2) to exist is that
3x, 3 0 -c xg <
K(O)
and
%(OfO) = p(0,
____ c2(0)
x0).
(3.3)
We note that if the food chain in 97& is simple, E, does not exist. The following results for existence of equilibria for the three-dimensional subsystems of (2.1) follow from [6]. THEOREM
3.7
Let the following hypotheses hold for the subsystem in <: (i) All solutions with nonnegative initial conditions are bounded in forward time. (ii) E, and E2 are hyperbolic saddle equilibria. (iii) If E3 and (or) E4 exist, they are unique in 9; and (or) 9&, respectively, and are unstable in the direction orthogonal to them. Further, let there exist no periodic solution in 9&. Then there exists an equilibrium, E6(u3, x3, y3,0) in 9,&. THEOREM
3.8
Let the hypotheses (i) and (ii) of Theorem 3.7 hold for the subsystem in a;,. Further, let E3 and (or) E,, if they exist, be unique and locally unstable in the y direction and (or) the u direction, respectively. If there exist no periodic solutions in SX:, then there exists an equilibrium E7 ( u4, x4, 0, z4) in
KYCZ.
MUTUALISM THEOREM
245
WITH POPULATIONS IN A FOOD CHAIN
3.9
the following hypotheses hold for the subsystem in e:
Let
(i) All solutions with nonnegative initial conditions are bounded in forward time, (ii) E, is a hyperbolic saddle point. (iii) If E4 and (or) Es exist(s), they are unique and locally unstable in the z and y directions, respectively. Further, let there exist no periodic solutions in 9?+ and sX>. Thin there exists an equilibrium E8(0, x5, y,, zs) in 9,&. The following result ensures the nonexistence for a three-dimensional subsystem. THEOREM
of an interior
equilibrium
3.10
If in system (2.1) a three-dimensional subsystem exhibits total extinction with respect to one of the variables, then it does not have an interior equilibrium. Proof: Since the subsystem exhibits total extinction, the omega limit sets of all solutions of the subsystem initiating with positive initial conditions converge to its boundary. This implies that there does not exist an interior equilibrium for the subsystem. The following result for the existence system (2.1) follows from [5]. THEOREM Let
of an interior
equilibrium
of the
3.11
the following hypotheses hold for system (2.1):
(i) All the solutions with nonnegative initial conditions are bounded in forward time. (ii) The system (2.1) is persistent. (iii) The subsystems of (2.1) are isolated and acyclic. Then an interior equilibrium, E*( u*, x*, y*, z*) exists for system (2.1). THEOREM
3.12
System tion.
(2.1) has no interior equilibrium whenever it exhibits total extinc-
3.2.
STABILITY
OF EQUILIBRIA
To determine the stability of the various equilibria, we compute the variational matrix of system (2.1). The signs of the real part of the eigenvalues of this matrix evaluated at a given equilibrium determine its stability.
RAVINDER KUh4AR AND H. I. FREEDMAN
246 The variational
matrix is given by
V(u, X, JJ,z) uh,+
=
h
uh,
uhx
uhz
\
Q%, a-% + ‘yg -PI - P2 - YP,u-=P,u - YP,,- ZP2, Y(- Slu+ 4P1f CIPlu) - SI+ "IPI YClPlX -9 - 29u - l’s, y - z9,z '(-Szu+tc;pz+'2p2U "C2P2x zc39, -s,+c,p, + c39- zs2, + cl39+ c39,)
(3.4)
/
The equilibrium E,(O,O,O,O) has diagonal and for which the eigenvalues and those in the y and z directions stable and unstable manifolds. For the equilibrium Et( L,,O,O,O), L(O,O,O), is ‘~,~,(~,,‘W.O)
&h,(&,,o,O,O)
0
v, = \
the variational matrix V, which is in the u and x directions are positive are negative. Thus E, has nonempty the variational
matrix VI, where L, =
G,h,(&,,‘U,O)
Y4&>0)
G,h,(L,,O,O,O)’
0
0
0
0
-%(‘%l>O)
0
0
0
0
-S2(L3rO)
There is a positive eigenvalue in the x direction and negative the u, y, and z directions. Thus El has nonempty stable manifolds. A similar analysis for the equilibrium E2(0, K,, ,O,O) yields value is positive in the u direction and negative in the x eigenvalues in the y and z directions are (Ti= - s;(O,O) + c;(O)p,(O, I&),
,
eigenvalues in and unstable that its eigendirection. The
i =1,2,
(3.5)
where K, = K(0). Thus cq is positive whenever (S3) holds. The eigenvahte in the z direction may be positive or negative. Remark 3.13. unstable in + Se,,.
From the above analysis, it follows that El and E2 are
From Theorem 3.3, E3 always exists. The variational equilibrium point E3( ii, Z,O,O) is given by iih”( ii, Z,O,O) iih,( ii, Z,O,O) &=
a~g,(a,~-)
aig,(Li,Z)
0
0
0
0
ti,(l,f,O,O) -Pl(ii.P)
matrix
V, at the
Ih,(J,i,O,O)
-p2(U,Z) 0 -sl(ii.o)+c,(ic)P,(ii,~) 0 -Sl(ii.O)+C*(P)P*(i(,f)
MUTUALISM
241
WITH POPULATIONS IN A FOOD CHAIN
The eigenvalues
of E3 are p;= -s,(ii,o)+ci(ii)pt(ii,2),
i =1,2,
(3.6)
and x _+, where
and all the functions are evaluated at (ii, f,O,O). Thus, whenever h,g, - g,h, > 0, eigenvalues x + have negative real parts and E3 is asymptotically stable in tBU,. If hug, - g,h, < 0, E3 has nonempty stable and unstable manifolds. Similarly we analyze E4 and E,. For E4 the eigenvalue of V, in the u direction, h(0, x1, yl,O), is positive. Thus the equilibrium E4 is unstable. The eigenvalue in the z direction is Y =
s2(O,OJ
The other two eigenvalues
+
C2P2(
0,
Xl)
+
40)
do7
Y1).
(3.7)
are the roots of the equation
Y%,P- Yl+-wx
~2-_(~xlgx+~g-YlPlx-
+ “ET- YlPlJ
+ PIYlcl(o)Plx = 0.
(3.8)
In the case as,(O, yi)/ay = 0 (i.e., there is no self-limitation on y), one can use the Rosenzweig and MacArthur criterion (see [lo]) to determine the stability of E4 in sex’;. In general, from the Routh-Hurwitz criteria, the roots of (3.8) have negative real parts if and only if
and
Yl)[~Wx(O9 -4 + ‘y&do, x1>- YlPlX(O~ Xl>1 - Ylcl(o)Pl(o~ Xl)PlX(O~ Xl>< 0.
Yl%,@?
(3.9)
For E,, the eigenvalue of V, in the u direction, h(0, x2,0, z,), is positive and thus Es is also unstable. The eigenvalue in the y direction is 6 = %(O,O) + C,(O)P,(O, x2) - W&(0,0). The other two eigenvalues “X&(0,
(3.10)
have negative real parts if and only if
x1) + @%(O,x2) - Z2P2AOI x2) - z2%z(0, r2) < 0
(3.11)
RAVINDER
248
KUMAR AND H. I. FREEDMAN
and W*;(O7 z2)kWx(O, - %(O)P,(O,
uh, “XL - YP,, Y-
Slu + 4P1
+ C,P,u)
matrix
uh,
uh,
- Pl
- P2
uhx + ag - YP,,
ax&
YClPlx
- YSl,
-4
0
0
-s2 +c2p2 +c34
0
1
(3.12)
X,)P,X(O~ x2) < 0.
E6(u3,x3,y3,0) has the variational
The equilibrium
I$=
x2) + %(OY x2) - z*P2x(0, %)I
where all the functions are evaluated at ( ug, x3, y,, 0). The eigenvalue in the z direction is
~E--*(u3~0)+C2(U3,X3)+C3(U3)q(U3tY3).(3.13) The other eigenvalues
are the zeros of the polynomial X3+ a,A2+azX+a3=0,
(3.14)
where al = GL
+ ax3gx + ag - Y3( plx + sly)
y
a2=u3h.((Yx3g,+~g-y3P1x-Y3S1Y)
Y3PlJ
+ Y4Cl Pl Plx -
Y3%,(~X3&
- U3hx(~X3&
- Y3Plu) - U3Y3hy( - %4 + ciPl+
a3 = u3Y3{ (~x3g,
- k[
+ag--
- Y3hJ(c1~,~1,
ClPlPlx
- kc,)
- %y(ax,&
+ ( - %u + 4Pl+ x [ h,(~X3&
ClPlu),
+ ai? - Y3Pd]
ClPlJ + "g - Y3Ph)
-
Plk] }.
Thus from the Routh-Hurwitz criteria, the necessary and sufficient tions for E6 to be asymptotically stable are that 5‘< 0 and al > 0,
a,>O,
and
ala2-
a3> 0.
(3.15) condi-
(3.16)
E7 and Es canbe analyzed similarly. In the case of E,,the eigenvalue of V,
MUTUALISM
WITH POPULATIONS IN A FOOD CHAIN
in the y direction
249
is given by (3.17)
~~--s,(~,,O)+c,(u,)P,(u,,~,)-~,q,(~,~O). The other three eigenvalues
are the zeros of the polynomial
A3+ b,A2 + bJ + b3 = 0,
(3.18)
where bl=Uqhu+taXqgx+(Yg--Zq(P2x+SZz), b, = %h,(%& +
-
+ “g -
P2 P2r -
z4c2
-
Z4S21(~X4&
Z4S2r) +
‘yg
-
Z4P2x)
-Z4P2u)+%P2(-S2”+C~P2+C2P2u),
~4k(~w!L
b3 = wt{
‘4p2x
(awu
- ~4~2u)(c2hr~2x - b2r)
- hL[C2P2P2X - s2z(%&
+ ag - %Pz*)l
+(-%u+GP2+c2P2u) X[G%c3x
Thus E, will be asymptotically b, ’ 0,
+~g--‘lP2x)-P2~xl)~
(3.19)
stable if and only if 9 < 0 and blb2 - b3 > 0.
&>O,
(3.20)
Similarly, since for E, the eigenvalue of V, in the u-direction, h(0, x5, y5, zs), is positive, the equilibrium is unstable. The other eigenvalues are the zeros of the polynomial
A3+ dlA2 + d,X + d, = 0,
(3.21)
where
d,=~(X~gx+g)-Y~(Plx+Sly)-z5(P2x+qy+S2r)-S2+c1P1, d,
=
zs [ wq.,
-
+ (“X5&
~2r(
-
s1+
cl
PI - y5s1, - z5qy)]
+ at? - Y5Plx - Z5P2xH - s1+ ClPl - Ys%v - z5qy - z5s2z)
+(Y5c,P,P,,+Z5C2P2P2x), d3 = ~5{ (ysclp1,>(
~1s2z- pzc3qy)
+c2P2x[P1q-P2(-~,+c1P1-Y5%y -(“X5&
+ w-
- z54JJ
YSPlx - Z5P2x)
x [ c3qq.v- s2;( --x1+
ClPl_
YSSlJJ - WJ]}.
(3.22)
RAVINDER KUMAR AND H. I. FREEDMAN
250 The equilibrium
E8 will be asymptotically
4 ’ 0,
4’0,
stable in %‘&Z if and only if d,d, -d,
> 0.
(3.23)
We shall now consider the stability of the interior equilibrium E*( u*, x*, y*, z*). In what follows, all the functions are assumed to be evaluated at (u*, x*, JJ*, z*). The variational matrix at E* is given by (3.4) evaluated at E*. In general, it is not possible to determine the stability of E*. Thus the mutualist’s interaction with food chain populations can result in either stabilization or destabilization of the system, as has been noted in two- and three-species models (see [2], [13]). Finally, it shows that the populations feeding on more than one trophic level do not necessarily cause an unstable system, the possibility of which has been pointed out by Pimm and Lawton [28], and that mutualistic interactions can have a significant effect on stability, even in the case of complex systems. 4.
PERSISTENCE
We now address the question of persistence of the populations given by our system. Specifically we derive criteria that ensure the uniform persistence of the system (2.1) in the cases of facultative mutualism between the mutualist u and the prey x as well as between u and prey y. The system (2.1) exhibits facultative mutualism between the mutualist u and the prey x whenever the hypotheses (Hl)-(H4), (Gl), (G2), (Pl), (P2), and (Sl)-(S3) are satisfied. From Theorems 3.1 and 3.2 we obtain that the equilibria E,, E,, E2, and E3 always exist. Hypothesis (S3) and Theorem 3.3 together imply that E4 exists. The equilibrium E, exists whenever 0~~> 0. Finally as the system (2.1) has a compact region of attraction (see Theorem 2.1) we conclude that (2.1) is a dissipative system. 4.1.
UNIFORM
PERSISTENCE
In this section we give criteria for uniform persistence in the three-dimensional subsystems and system (2.1). For the sake of simplicity we always assume that the boundary equilibria (whenever they exist) are unique. Case I. Facultative mutualism between u and x. First we introduce following hypotheses: (H5) Let the equilibrium E4 be globally asymptotically to solutions initiating in 92&. (H6) Let E, (if it exists) be globally asymptotically solutions initiating in Se,:.
the
stable with respect
stable with respect to
MUTUALISM
The following result holds for the food chain in 9&. to the one given in [16] for a simple food chain. THEOREM
251
WITH POPULATIONS IN A FOOD CHAIN
Its proof is similar
4.1
Let hypotheses (Gl), (G2), (Pl), (P2), (Sl)-(S3), (H5), and (H6) hold. Then the subsystem in 9’& is uniformly persistent whenever y > 0 and 6 > 0.
In the absence of the top predator z, the system (2.1) reduces predator-prey-mutualist system, and we have the following theorem. THEOREM
to a
4.2
Let hypotheses (Hl)-(H5), (Gl), (G2), (Pl), (P2), and (Sl)-(S3) hold. Then the three-dimensional subsystem in %‘L,, is umformly persistent whenever
I4 ’ 0. The proof of Theorem 4.2 can be found in [13]. Similarly one can prove the following theorem. THEOREM
4.3
Let hypotheses (Hl)-(H4), (H6), (Gl), (G2), (Pl), (P2), (Sl), and (S2) hold. Then the three-dimensional subsystem in 9AZ is uniformly persistent whenever p2 > 0.
Hypotheses (H5) and (H6) exclude the possibility of periodic orbits for the predator-prey systems in 9& and 9A. In the case of a simple food chain, it was shown in [17] that “weak” persistence (defined there) can hold even when there exist a finite number of limit cycles. Below we obtain a criterion for uniform persistence for the food chain in 9TzYz when periodic solutions exist. THEOREM
4.4
Let hypotheses (Gl), (G2), (Pl), (PZ), and (Sl)-(S3) hold. Further, let there exist a finite number of periodic solutions x = &i(t), y = q;(t), i = 1,2 ,..., k, andx=t,(t), z=qj(t), j=1,2 ,..., I, in 9?zY and 92, respectively. Then the food chain in CR& is uniformly persistent provided y > 0, 6 > 0, and for each periodic solution of period T in 9cY, m, =
-S2(0,0)+~~1[C~(0)p~(O~~~(f))+
c3(o)q(o~Icli(t))l dt>of i=1,2
,..., k,
(4.1)
and for each periodic solution ( tj (t), q, (t)) of period w in 9’2,
j=1,2
,..., 1.
(4.2)
RAVINDER KUMAR AND H. I. FREEDMAN
252
Proof. For any x E 9&, let O(x) denote the orbit through x and Q(x) be its omega limit set. By arguments similar to those given in [16], we conclude that Q(x) is bounded. We claim that E, 4 9(x). If E, E 3(x), then by the Butler-McGehee lemma (see [18]) there exists a point P in 8(x) n W”( E,). Since O(P) lies in + Q(x) and W”(E,,) is 9YVL,we conclude that O(P) is unbounded, which is a contradiction. Next E2 c%Q(x), for otherwise, since E2 is a saddle point, by the Butler-McGehee lemma there exists a point P in Q(x)n W”( E2). Now W( E2) = 93’: implies that either EOE Cd(x) or an unbounded orbit lies in n(x). In either case the result is a contradiction. Now we show that no periodic orbit in 9YX:,or E4 belongs to Q(x). Let y, , i = 1,2,. . . , k, denote the closed orbit of the periodic solution (9, (t), $i (t)) in .9?:, such that y, lies inside y,_ 1. The variational matrix v($(t), $,(t),O) corresponding to y, is given by
g(0.k (r>) ++,(r)g,(o,tJlp,(l)) - 4,(t)Pl\(o.+!(t))
I
-
P,(o,%(t))
- P2(O,+t(t))
c,(o)~,(r)P,~(O,cp,(t))
Computing
the fundamental
matrix of the linear periodic system
X’=r/;(t)X,
X(0) = I,
(4.3)
we find that its Floquet multiplier in the z direction is em,‘. Suppose yi lies in Q(x). Then from (4.1) its Floquet multiplier in the z direction is greater than 1 and therefore by the Butler-McGhee lemma there exists a point PI E W(yl)n 8(x). If PI lies outside yi, then O(P) lies in Q(x) and is unbounded, which is a contradiction. If yi is stable from inside, then A(P,) must be y2 and is contained in 8(x). If y2 is unstable from the inside as well, we have a contradiction, for a repeller cannot be in !A(x). If y2 is stable from the inside, again from the Butler-McGhee lemma there exists a point Pz inside y2 such that Pz E kV(y,) n Q(x). Continuing this argument we get that either yr is unstable from both sides or it is stable from inside and there exists a point Pk in W”(y,) n Q(x). If such a Pk exists, then E4 must be a repeller and A( Pk) = E4 lies in 8(x), which is again a contradic-
MUTUALISM
WITH POPULATIONS IN A FOOD CHAIN
253
tion. Thus yI does not lie in Q(X). By similar arguments we conclude that no y, lies in Q(x). Repeating the above steps we also conclude that no closed orbit in 92 or E, lies in Q(x). Thus sZ(x) lies in 9.& and the subsystem in yex’,= is persistent. Finally, since only the closed orbits and the equilibria form the omega limit set of the solutions on the boundary of 9x&, by the main theorem of [5], this implies that the food chain is uniformly persistent. Remark 4.5. In the case that there are no periodic solutions in 92 or if E, does not exist, Theorem 4.4 holds without assuming inequality (4.2). Remark 4.2 holds Theorem multiplier u direction hypothesis
4.6. For the predator-prey-mutual&t system in %‘zX,, Theorem even without hypothesis (H5), and for the system in 9A,, 4.3 holds even without (H6). This is so because the Floquet of any closed orbit (+(f), q(t)) of period T in 9$, or Bz: in the is exp[ J,,r/r(O,c#J(~),+(t),O) dt] and is always greater than 1 by (Hl).
Next we derive a criterion for the uniform persistence of system (2.1) when all its three-dimensional subsystems possess an interior equilibrium. It follows from [6] that whenever a subsystem of (2.1) is uniformly persistent it has an interior equilibrium. First we introduce the following hypothesis: (H7) Let the equilibria E6, E7, and Es be globally and @&z >respectively.
stable in .%‘A,,, WA,,,
THEOR EM 4.7 Let hypotheses (Hl)-(H4), (H7), (Gl), (G2), (Pl), (P2), and (Sl)-(S3) hold. Further, let either (H5) and (H6) or inequalities (4.1) and (4.2) hold. Then system (2.1) is unqormiy persistent whenever 6 > 0 and 7 > 0.
Proof. Let O(x) be an orbit through any point x E 9~Xvz. Then from Theorem 2.1 its omega limit set e(x) is bounded. Now we show that no boundary equilibria belong to a(x). Suppose E,, belongs to Q(x). Since E0 is a saddle point, by the Butler-McGehee lemma there exists a point P in Q(x)n W(E,,). Further, as W( E,) is precisely %$, O(P) is unbounded and lies in Q(x), which is a contradiction. E1 G o(x). Otherwise by the Butler-McGehee lemma there exists at least one point P E n(x) fl W( El). However, as W( E1) is e/e if P E 9:) then either A(P) = E, belongs to Q(x) or O(P) is unbounded and lies in n(x). Both of these statements are contradictions. Otherwise P E yeu’, u and again 0(P) is unbounded and lies in Q(x), contradicting %ii u9&7 the boundedness of Q(x).
254
RAVINDER
KUMAR AND H. I. FREEDMAN
Next we show that E3 cannot be in Q(x). Since there are periodic orbits in 9; and E3 is unstable in the y and z direction, E3 E Q(x) implies that P lies in 9”,. Thus A(P) is E, or E1 or E,, or 0(P) is unbounded, and each of them results in a contradiction. If (H5) holds, then by analogous arguments E4 does not belong to Q(x), and if inequalities (4.1) hold, then by arguments given in Theorem 4.4, E4 B Q(x). In the same way we can show that E,, if it exists, does not lie in Q(x). In the case Eb E 9(x), then as 5‘> 0, by the Butler-McGehee lemma and hypothesis (H7), there exists a point P in O(x)n W( E6) and the unbounded orbit through P lies in Q(x) n %‘~xy, which is a contradiction. Similarly we get that E, and Es do not belong to Q(x). Finally, no point on the boundary of 9&Z lies in D(x), otherwise Q(x) will contain one of the equilibria or closed orbits, resulting in a contradiction. This completes the proof of the theorem. Case ZZ. Facultative mutualism between u and y. The system (2.1) exhibits facultative mutualism with the predator y when the hypotheses (Hl), (H2), (H3’), (H4’), (Gl), (G2), (G3’), (Pl), (P2’), (Sl), (S2’), and (S3’) described in Section 2 hold. The set 9 given by Theorem 2.2 contains the region of attraction for (2.1). The criteria for the existence of boundary equilibria and for uniform persistence remain the same as in Case 1. In particular, the following result holds. THEOREM
4.8
Let hypotheses (Hl), (H2), (H3’), (H4’), (H5)-(H7), (Gl), (G2), (G3’), (Pl), (P2’), (Sl), (S2), and (S3) hold. Then system (2.1) is uniformly persistent
whenever 6 > 0 and q > 0.
Theorem 4.8 describes the circumstances in which the mutualistic interaction between u and y does not dominate the interactions of the food chain and all the species can coexist. 5.
EXTINCTION
In the previous section we obtained criteria for all populations to persist. In the present section, we consider the situation where the mutualism is so strong that one or both of the predator populations exhibit total extinction. To do this we consider two cases. In the first case, the mutualist (of the prey) forces both predators to total extinction. In the second case, the mutualist (of the bottom predator, y) forces the top predator, z, to total extinction. First we introduce some notation. If populations &, . . . , Nk exhibit total extinction in the space BUT,,,,,U,, we denote this by gN,,,,,, Nk_ 0. Here
MUTUALISM
WITH POPULATIONS IN A FOOD CHAIN
255
N,, . . . , Nk and ul,. . , u, are subsets of {u, x, y, z}. Case I. THEOREM 5.1 Let hypotheses Further, let
(Hl)-(H4),
(Gl),
(G2), (Pl), (P2), and (Sl)-(S3)
hold.
where
K, 1, and 2 are given by (2.3). Then 8z_0
in %‘A,,,.
Proof. It can be easily seen that L, 5 u(t) I L for sufficiently This, together with inequality (5.1), yields that Z’I - Zz + c2(L,)p,( + &j < 0, for large t. Thus lim, _ m z(t) = 0. Similarly,
large t. L,, 2)
we can prove the following theorem.
THEOREM 5.2 Let hypotheses (Hl)-(H4),
(Gl), (G2), (Pl), (P2), and (Sl)-(S3)
-s1tL,,o)+c,tL,)P,(L,,~) co, then f,, _ 0 in 5?u&z.
hold. If
(5.3)
If in addition
-~~+&&J2(L3Jq
(5.4)
where L, and & are given by (5.2), then cyz _ 0 in .@u:,,. Case II.
Define L, = L( I?,o, ti),
where k and i? are given by (2.6). Then the following result holds for the system (2.1).
RAVINDER KUMAR AND H. I. FREEDMAN
256 THEOREM
5.3
Let hypotheses (Hl), (H2), (H3’), (H4’), (Gl), (G2), (G3’), (Pl), (P2’), (Sl), (S2’), and (S3) hold. Zf
-~~(~1~0)+c*(~1)P2(~1~~(~1))+cC3(~l)q(~1~~)
EXAMPLES
We now illustrate our results with examples. The particular coefficients chosen do not necessarily have biological meaning or apply to specific populations. Example
6.1. u’=u x’=ax
Consider the system
(li
U L+x 1-x
i
1
)
-ysxy-80x2,
y’= y -s,o-sliu-ssIzY+~yox-~~z). i z’ = z [ - szO- SZlU- sz*z + CZ8~X+ ( $0 + CjlU) 6, y] .
(f-3.1)
When L = 3, (Y= K = 4, y0 = 8, =l, sIO =1/3, slr =1/6, sr2 =1/4, c1 = 3, m=O, 6,=1, szO= szl =1/3, c2 =l, cs,, =1/3, and csr = 0, the above system has the boundary equilibria E,(O,O,O,O), E,(3,0,0,0), E2(0,4,0,0), E3(7,4,0,0),
We now show that for system (6.1), E4(0, x1, y,,O) is always globally asymptotically stable in B’:,,. For this we consider the Lyapunov function
on BAT,,.Computing the derivative of V(x, y) along the solutions (6.1), after some simplifications we get
of system
MUTUALISM
257
WITH POPULATIONS IN A FOOD CHAIN
Now, as the set {(x, JJ) E ST&: p = 0} = { E4 } is invariant, we conclude that E4 is globally asymptotically stable in %‘A. Similarly one can see that E, is globally asymptotically stable in g&,. Further, as a, = 49/6, & = 4/3, y = 149/117, and S = 2/3, ah the three-dimensional subsystems are uniformly persistent. Next we consider the global asymptotic stability of the interior equilibria in the three-dimensional subsystems. Computing the symmetric matrix 9(u, x, y) for E6(63/16, 15/16, 49/16,0) we get 16 &,=a’
- 16~ “’ = 126( 3 + x) ’
b22=3, The region of attraction
b23= 0,
is contained
1 b,,=E’
b33=-&
in the set
It can be easily verified that the principal minors of .%‘(u, x, v) are positive in &i and therefore .S?(II, x, y) is positive-definite in &i. Next the symmetric matrix B(u, x, z) for E7(17/3,8/3,0,4/3) is given by 3 dii=fl’
The region of attraction .J5f2= {(u,x,z)
G=
3u 34(3+x)
for the subsystem
Eqj;,:
’
43=;,
in 5?,+,, is contained
in the set
Orur7,O~x~4,O~z~11},
and LS(u, x, z) is positive-definite in s$. For the equilibrium E8 the matrix 9(x, y, z) is diag(l,$, 4). Finally, as 5 = 15/48 and n = 3/18, from Theorem 4.7 we obtain uniform persistence for the system (6.1). Next we give an example in which the mutual& u provides the predator y with an alternate food supply and increases the death rate of the top predator.
RAVINDER KUMAR AND H. I. FREEDMAN
258 Example
6.2.
U L+ry
(
u’=u
l-
x’=ax
i
Consider the system
l-
1
)
K:ku ___
+(+I + CllU)(YO + YlU)X - EOZI)
y’= y[ -
s10
+
z’=z
s20
-~21~-~22z+j~)(~)x+~~],
-
SllU
80
) -(Yo+Y+P1+% s12y
-
(6.2)
where all the constants are assumed to be nonnegative. With L = 1, I = l/3, cu= K = 3, y. = So = 1, k = yl = 6, = cl1 = cal = cgl = 0, slo = s12 = l/3, sI1 = l/24, cl0 =1/4, s20 =l, s2r =1/3, sz2 = l/2, c20 = 7/8, &, =1/36, q. = 4, the boundary equilibria are
E,,(O,O,O,O), E,(l,O,O,O),
E&+,~,%,O),
Es(O,%O,E)r
E,(O,%:,O,O), E,(l,+?_,O,+$), The region of attraction
and is contained
E3(1,3,0,0),
&(0,3,0,0),
Es(O,%k+%t+%). in the set
Since & =11/24, f12= 41/24, y = 68/23, S = 35/396, we conclude that all of the three-dimensional subsystems involving x are uniformly persistent. The symmetric matrix .G@( u, x, y) corresponding to Eb is given by
b,, = ; Further
9
it is positive-definite
43 =o,
and
h,,=$.
in the region of attraction
The symmetric matrix g( u, x, z) corresponding to E7 is given by d,, =1/6, d,, = 7/8, d,, = 0, and d,, =1/2. Also the 9(x, y, z), corresponding to E,, is given by fil =l, fi2 = 0, fi3 fz2 = 4/3, j& = 0, and f3s =1/2. Since the principal minors of 9( d,, = 0,
d,, = 1, matrix =1/16, u, x, z)
MUTUALISM
WITH POPULATIONS IN A FOOD CHAIN
259
and 9(x, y, I) are positive, we conclude that ET and E, are also globally asymptotically stable in &‘& and 5$,, respectively. Finally 6 = 281/492 and 1)=103/792 imply that the system (4.5) is uniformly persistent. The interior equilibrium is
Now we consider an example in Case II to show that total extinction occur. Example 6.3.
Consider
(1-m
u’=u
can
the system U 1
,
x/=x(3-x)-xy-xz,
Its
y’=y
i -&+u+y)+;x
)-yz
ZfCZ
1 ___2_,_1,+1,+1 5 10 [ 2
6
boundary
equilibria
are
y+l’
E,(O,O,O,O),
Y 2 i y-cl
(6.3)
,1 .
E,(l,O,O,O),
E2(0,3,0,0),
E,(4,3,0,0), E4(0,$,$,0),
E&,&:,0),
and - 15a2 + 10a + 25
where a3 = 20/7. The region of attraction d=
{(u,x,Y,z)
is contained
in the set
ESe,+,yr :o~u~4,o~xI3,oIy~5,o_
Since j3, = 5/18, the subsystem in L%‘& is uniformly persistent. Further, as E, does not exist and y = 5/144, the food chain is also uniformly persistent. However, the system (6.3) satisfies inequality (5.5), and therefore the top predator will eventually become extinct. 7.
DISCUSSION
In this paper we have considered a model of facultative mutualism with populations of a food chain. This model is an outgrowth of work since the time of Kolmogorov [23] for two-dimensional models and Rai et al. [29] for three-dimensional models.
260
RAVINDER
KUMAR
AND H. I. FREEDMAN
We have considered two cases. In the first case, the mutualism is with the prey population, and in the second case, the mutualism occurs with the second trophic level predator. The models and their analyses are of interest to biologists, since, as pointed out in the introduction, the mutualisms described in this paper do occur in nature. The analyses of these models suggest possible observations for field workers. A possible outcome of mutualism is predator extinction. This may help explain the absence locally of “superpredators” in predator-prey systems where mutualism with either population is present. The model analysis also indicates that the carrying capacity of the mutual&t populations increases, possibly at the expense of the other food chain populations. We have modeled in this paper nonsimple food chains, that is, ones where the population at the third trophic level may feed on all populations of the first and second trophic levels. In a future paper we will consider the case of obligate mutual&m and reversal of outcome from extinction to persistence.
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of plants
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host
on
host-finding by tachinid parasites, Can. Entomol. 92:641-652 (1960). R. Ormond and A.J. Edwards, Red Sea fishes, in Red Sea, A.J. Edwards and SM. Head, eds., Pergamon, New York, pp. 251-287, 1987. S.L. Pimm and J.H. Lawton, On feeding on more than one trophic level, Nature 275:542-544 (1978). B. Rai, H.I. Freedman, and J.F. Addicott, Analysis of three species models of mutual&m in predator-prey and competitive systems, Math. Biosci. 65:13-50 (1983). A. Rescigno and I.W. Richardson, The struggle for life. I: Two species, BUN. Math. Bioph_ys. 29:377-388 (1967). CF. Thompson, S.M. Lanyon, and K.M. Thompson, The influence of foraging benefits on association of cattle egrets (Bubalcus ibis) with cattle, Oecolgia 52:167-170 (1982). R.R. Vance, A mutualistic
interaction
between
a sessile marine clam and its epibionts,
Ecologv 59:679-685 (1978). C.L. Wolin, The population dynamics of mutualistic systems, in The Biology of Mutuahsm, D.H. Boucher, ed., Croom Helm, London, pp. 248-269, 1985. S. Zheng, A reaction-diffusion system of a predator-prey-mutualist model, Math. Biosci. 78~217-245 (1986).
KUMAR*
AND
Applied Mathematics Institute, Edmonton, Canada TfiC 2GI Received
3 November
H. I. FREEDMAN+ Department
of Mathematics,
University of Alberta,
1988; revised 12 July 1989
ABSTRACT Facultative mutualism with populations interacting in a food chain is modeled by a system of four autonomous ordinary differential equations. Two cases are considered: mutualism with the prey and mutual&m with the first predator. In both cases persistence
and extinction criteria are developed in terms of the invariant flows on the boundaries.
1.
INTRODUCTION
This paper is mainly concerned with modeling facultative gate) mutualism with populations interacting in a food chain.
(i.e., nonobli-
Such mutualism may occur with any of the prey populations. For example, the situation described by Monteith [26] in a study conducted on a food chain in western Ontario found that the parasitism of sawfly larvae by the parasitoid B. harveyi on the host plant Lurix laricina is much lower in the presence of associated plants, which mask the odor of the sawfly larvae and their food plant. Facultative mutualism with populations at various trophic levels of a food chain have been observed in nature. Vance [32] describes a mutualistic interaction between clams and their epibionts that helps to protect the clams from their starfish predators. Fritz [20] observes that ants as mutualists decrease predation on treehoppers. For other mutualisms at the prey level, see [4] and [31].
*Research is partly based on a doctoral thesis at the University of Alberta. +Research is partly supported by the Natural Sciences and Engineering Council of Canada, grant NSERC A4823.
MATHEMATICAL
BIOSCIENCES
OElsevier Science Publishing 655 Avenue of the Americas,
Research
235
97~235-261
Co., Inc., 1989 New York, NY 10010
00255564/89/$03.50
236
RAVINDER KUMAR AND H. I. FREEDMAN
A mutualism with a population at an intermediate trophic level is described by de Vries [8]. Giant tortoises on the Galapagos Islands, which prey on small aquatic animals and whose eggs are preyed upon by feral mammals (black rats, pigs, dogs, etc.), have evolved a symbiotic relationship with finches, who remove ticks while the tortoises stand on their hind legs, thereby increasing their life span. A similar behavior is described for iguanas [9], which feed on algae, grasshoppers, and so on, and are preyed upon by snakes, mockingbirds, and rats. A mutualism that has the effect of increasing the hunting ability of predators is described in [27]. Yellowsaddle goatfish, P. cyclostomus, and bird wrasse, Gomphosus caeruleus, tackle coral reefs from both sides so that prey may be driven toward and caught by each other. There are now many papers in the literature dealing with mutualistic systems. Most of these papers deal with two-dimensional systems modeling direct mutual&m between two populations and ignore all other population interactions (see, e.g., [l], [5], [lo], [ll], [23], [30]). In recent years, some papers have appeared that deal with cases where the mutualism is due to or influenced by the interaction with a third population (see, e.g. [2], [ll]-[15], [29], [34]). For general biological discussion of mutualisms and how they may occur, see, for example, [l], [2], [5], [28], and [33]. As previously mentioned, the main purpose of this paper is to model facultative mutualism with populations interacting in a food chain. The food chain models we utilize have been discussed in [12], and [29] and in some of the works cited therein. To the best of our knowledge this is the first attempt to model mutualism with populations in a food chain. We are interested in developing criteria for persistence and extinction for our models. Persistence in biological systems in a context related to this paper has been discussed in [6], [12]-1161 and 1181. We utilize the definitions of persistence developed in [6], 1181,and [19], namely that if N(t) is such that N(t) > 0, we say that N(t) persists if lim,, m inf N( t) > 0. Further, if N(t) E Q, where V is a certain class of functions, and there exists 6 > 0 such that lim,_m inf N( t) > 6 for all N(t) E V, then we say that N(t) is uniformly persistent (also known as permanence). A system is said to (uniformly) persist if each component (uniformly) persists. We say that N(t), N(0) > 0, exhibits extinction if lim, _ m N(t) = 0. We note that nonpersistence does not necessarily imply extinction for all initial values N(0). If lim, _m N(t) = 0 for N(0) > 0, we say that our system exhibits extinction with respect to the N(t) population. Throughout this paper we utilize the following notations: .C@z denotes the positive v axis and3 its closure for any variable v. %‘,+, denotes the positive VW plane and %‘,\ its closure, etc. In the next section we develop our models. In Section 3 we discuss the equilibria and their stability. Persistence criteria are developed in Section 4,
MUTUALISM
237
WITH POPULATIONS IN A FOOD CHAIN
followed by extinction criteria in Section 5. Some examples Section 6, and the final section contains a brief discussion. 2.
are given in
THE MODEL
In this section we describe a general system that models a mutualist interacting with populations in a food chain. The mathematical formulation of facultative relationships between the mutualist and two different trophic levels of the food chain are also described. Finally, we establish the existence of a region of attraction in each case, showing that the model is well-behaved. We consider the autonomous system du -=uh(u,x,JJ,t), dt dx _Eit=“xg(u,x)-YP,(u,x)-zP,(u,x), (2.1)
~=z[-~2(U,Z)+C2(U)PZ(U,*)tC?(U)q(U,Y)1, u(0) = ug2 0,
x(0) = xg2 0,
Y(0) = Yo 2 0,
z(0) = z() 2 0,
as a model of a mutualist-food chain interaction with continuous birth and death processes. The variable u(t) represents the density of the mutualist at time t, and x(l), y(t), z(t) denote the prey and predator densities, respectively. In the absence of the mutual&t, that is, when u = 0, (2.1) reduces to a food chain model. The function h(u, x, y, z) represents the specific growth rate of the mutualist. We assume that h(u, x, y, z) possesses the following properties [3,10,23,29]:
(HI) (H2) There
exists a unique function
++,Y,Z),X,Y,Z)
L( x, y, z) > 0 such that
=O.
The first condition implies that, independent of the x, y, z populations, u is capable of growing even when rare. Also, the growth rate is assumed to be density-dependent and decreases as the population increases. The second condition implies that L(x, y, z) is the mutualist’s carrying capacity and in
RAVINDER KUh4AR AND H. I. FREEDMAN
238
part specifies in what way the predators and prey become part of the mutualist’s environment. The function g( u, x) is the specific growth rate of the prey x in the absence of any predation. We assume that
J&,x) 10
(Gl) g(u,O)’ 0, (G2)
There exists a unique K(U) > 0 such that g( u, K(u))
= 0
Condition (Gl) implies that the prey population is capable of surviving in the presence or absence of the mutual&t and that the growth rate in the absence of predation is density-dependent. Condition (G2) implies that K(U) is the carrying capacity of the prey in the absence of predation. Next, the functions p;(u, x), i =1,2, and q(u, y) denote the predator’s functional response to the prey and mutualist densities. They represent the change in density of prey per predator per unit time as a function of the prey and mutualist densities. We assume that
$(u,x)>o,
p,( u,O) = 0;
(Pl)
4( KO) = 0,
%(U,Y)
i =1,2;
>o.
The above conditions imply that in the absence of prey there is no predation. Further, the predator functional response is assumed to be an increasing function of the prey population. The functions si(u, y) and s2(u, z) are the specific death rates of the predators y and z in the absence of predation. We make the standard assumption that the death rate is an increasing function of the population:
(S1)
%(u,y) ay
a44
>. 7
,. .
aZ
The nonnegative functions c,(u), i =1,2,3, are the rates of conversion of prey biomass to predator biomass. Also a > 0 is a bifurcation parameter. Finally, we assume that all the functions are smooth enough that existence and uniqueness of initial value problems hold and any required analysis can be carried out. 2.1.
FACULTATIVE
MUTUALISM
WITH
THE LOWEST
TROPHIC
LEVEL
In this section we specify the hypotheses for facultative mutualism between the mutualist u and the prey x. In addition to (Hl) and (H2), we
MUTUALISM
WITH POPULATIONS
assume the following ist:
IN A FOOD CHAIN
239
for the specific growth rate h (u, x, y, z) of the mutual-
(H3)
dh( u, x, y, r) ax ‘0,
(H4)
lim L(x,O,O) x+m
Jh( u, x, Y, r) Jy so0,
Jh( u, x, y, r)
az
I; 0.
-L
Condition (H3) implies that II derives benefit from the prey population and that there might be a cost to the mutual&t due to its interactions with the predators. Condition (H4) implies that the mutual&t has a finite carrying capacity. In the case that ag(u, x)/au > 0, then, even in the absence of predation, u acts as a mutualist with respect to x. If ag( U, x)/au < 0, then there is a cost to x for associating with u and u can be a mutualist of x only by its effect upon the predator(s). If ag( II, x)/au = 0, the relationship between u and x without predation is commensal. Further, we assume
This condition implies that the mutualist can benefit the prey by decreasing the predator’s response. The mutual&t can also benefit the prey by increasing the death rate of y or by decreasing the rate of conversion from prey biomass to predator biomass:
Also, in order to have a viable system we must have that (S3)
Thereexistsan
P suchthat
s,(O,O)/c,(O)
=p,(O,P)
and Z
Finally, the mutualist can also benefit the prey indirectly, by its effect on the dynamics of the top predator z, that is, by affecting its death rate s2( U, z), the predator response function q( I(, y), or the rate of conversion c~( u) of prey biomass into predator biomass. Now we show that under the above hypotheses a certain set &’ contains the region of attraction of (2.1). The proof is based on the ideas developed in [16] and (181.
RAVINDER KUh4AR AND H. I. FREEDMAN
240 THEOREM 2.1 Whenever
the hypotheses
(Hl)-(H4),
(Gl), (G2), (Pl), (P2), and (Sl)-(S3)
hold, the set d =
{(u,
x, y,
2) : 0 5 u I
I?,0 5
x
I
fz,0
5
q(O)x + y
~~,oIc,(o)x+~~,+zz~},
(2.2)
where
Q_=
CdOM ___ [ad+ ~,(O~O)l SI(O,O) Y
min s,(u,O),
;*=
(2.3)
OlUli
and
is positively invariant and attracts all solutions initiating with nonnegative initial conditions. Proof. Let (u(t), x(t), y(t), z(t)) be any solution tial condition (u,, x0, yO, z,,). Since
whenever
with nomegative
ini-
u(t) 2 t we get u’ I uh( L, x,0,0)
< 0.
From this, it follows that u(t) I L for all t 2 0 whenever uO I i. Further, if uO > t, then u(t) 5 L, for large t. Next, let g(x) = maxO ~ u ~ r. g(x). Then since all solutions of the initialvalue problem, x’=xg(x),
x=x,>0
7
satisfy lim sup, _ m x(t) I k (see [8]), it follows from a standard
comparison
MUTUALISM
WITH
POPULATIONS
IN A FOOD
241
CHAIN
theorem (e.g., see [21]) that if x,, < 2, then x(t) 5 2 for all 12 0, otherwise x(r) is bounded above by g for sufficiently large t. Now, we consider
Thus if c,(O)x, + y, I AI, we get
Id2
c,(O)x(r)+y(t)
Otherwise
forall
t20.
there exists T > 0 such that
q(O)x( t) + y(t) Similar arguments
I
La
for t 2 T.
can be given to show that the inequality C*(0)X(f)+~~y(t)+z(t)
115
(2.4
holds for all t 2 0 whenever c,(0)xO + ?sy,, + z,, I k. Otherwise (2.4) is true for sufficiently large values of t. 2.2.
FACULTATIVE
MUTUALISM
WITH THE SECOND
System (2.1) exhibits facultative following hypotheses: in addition
TROPHIC
LEVEL
mutualism between u and y under the to (Hl), (H2), (Gl), (G2), (Pl), and
RAVINDER KUMAR AND H. I. FREEDMAN
242 (Sl)-(S3),
tH3’)
we assume
ahb,
X, Y, ax
4
(H4’)
yli_mmL(O,y,O) =i
(G3’)
ag(,U;X)
a.4 u7 Y) I o au 7 c;(u)
c;(u)
ro,
h(u,z) au
>() - >
IO.
Mu,
Y)
au
-<
0,
c;(u) 20,
10.
In this case the following set .4? contains proof is similar to that of Theorem 2.1. THEOREM
x, .Y, z)
aZ
20. ,
(S2’)
Jh(u,
9
the region of attraction.
The
2.2
Wheneuer the hypotheses (Hl), (H2), (H3’), (H4’), (Gl), (G2), (G3’), (Pl), (Pz’), (Sl), (S2’), and (S3) hold, the set
where L=
lim L(O,y,O), ,v-+oa
k=
max
K(u),
OlUSi
~=~[.g(~,O)+sl(i,O)], 1
(2.6)
9
and
is positively invariant and attracts all solutions initiating with nonnegative initial conditions.
MUTUALISM
3.
243
WITH POPULATIONS IN A FOOD CHAIN
EXISTENCE, NONEXISTENCE, OF EQUILIBRIA
AND STABILITY
In this section we shall establish criteria for the existence and nonexistence of equilibria of system (2.1) and discuss their local stability. This information, besides being helpful in understanding the system, will be needed to derive the persistence criteria in Section 4. 3.1.
CRITERIA
FOR THE EXISTENCE
AND NONEXISTENCE
It is clear that E,(O,O,O,O) is an equilibrium THEOREM
OF EQUILIBRIA
for system (2.1).
3.1
System (2.1) has exactly two one-dimensional equilibria, EI( LO,O,O,O) and E2(0, K,,O,O), where L, = L(O,O,O) and K0 = K(0). Proof:
The subsystem
in 9Yu+ is given by
u’= uh( u,O,O,O),
u(0) > 0.
(3.1)
Now, from hypothesis (H2) it follows that E1 is an equilibrium. Furthermore, condition (Hl) implies that lim,,, u(t) = L, (see [lo]), establishing the uniqueness of E1. Similar arguments hold for E,. The subsystem in 9-y’ and 91’Z’ have no equilibria since all of their solutions tend to zero exponentially. Remark 3.2. It follows from the above arguments that E, is locally stable in the y and I directions and locally unstable in the u and x directions. As shown in [15], the following result holds in 9?:X. THEOREM
3.3
The subsystem in %‘A has an equilibrium, E2 are unstable in E.
E3( ii, I,O, 0), provided E1 and
Remark 3.4. As will be seen below, because (Gl), E3 always exists. THEOREM
of hypotheses
and
3.5
A necessary and sufficient condition for an equilibrium E4(0, x1, yI, 0) to exist in gX?$s that the hypotheses
323O
be satisfied.
(Hl)
K(0)
and
s,(O,O) ___ = p1(0,:> Cl(O)
of the form
(S3)
RAVMDER
244 Proof.
The solutions
of the subsystem
x’ = axg(0, x) -
y’=
KUMAR AND H. I. FREEDMAN
yp1(0,x)
9
Y[ - s1(0,Y) + clu9Pdwl
(3.2)
in 9:” are bounded for positive time [lo]. If (S3) is violated, then from hypothesis (Sl), it follows that lim, ~ m y(t) = 0, for all solutions initiating in 9?zV, Thus E, is globally stable with respect to 9%‘.&. Now suppose (S3) holds, It follows from Theorem 3.1 that E0 is unstable in the x direction. Further, (S3) implies that E2 is unstable in e. The result now follows by an application of the Poincart-Bendixson theorem. Similarly, THEOREM
we have the following result.
3.6
A necessary and sufficient condition for equilibrium E,(O, x2,0, z2) to exist is that
3x, 3 0 -c xg <
K(O)
and
%(OfO) = p(0,
____ c2(0)
x0).
(3.3)
We note that if the food chain in 97& is simple, E, does not exist. The following results for existence of equilibria for the three-dimensional subsystems of (2.1) follow from [6]. THEOREM
3.7
Let the following hypotheses hold for the subsystem in <: (i) All solutions with nonnegative initial conditions are bounded in forward time. (ii) E, and E2 are hyperbolic saddle equilibria. (iii) If E3 and (or) E4 exist, they are unique in 9; and (or) 9&, respectively, and are unstable in the direction orthogonal to them. Further, let there exist no periodic solution in 9&. Then there exists an equilibrium, E6(u3, x3, y3,0) in 9,&. THEOREM
3.8
Let the hypotheses (i) and (ii) of Theorem 3.7 hold for the subsystem in a;,. Further, let E3 and (or) E,, if they exist, be unique and locally unstable in the y direction and (or) the u direction, respectively. If there exist no periodic solutions in SX:, then there exists an equilibrium E7 ( u4, x4, 0, z4) in
KYCZ.
MUTUALISM THEOREM
245
WITH POPULATIONS IN A FOOD CHAIN
3.9
the following hypotheses hold for the subsystem in e:
Let
(i) All solutions with nonnegative initial conditions are bounded in forward time, (ii) E, is a hyperbolic saddle point. (iii) If E4 and (or) Es exist(s), they are unique and locally unstable in the z and y directions, respectively. Further, let there exist no periodic solutions in 9?+ and sX>. Thin there exists an equilibrium E8(0, x5, y,, zs) in 9,&. The following result ensures the nonexistence for a three-dimensional subsystem. THEOREM
of an interior
equilibrium
3.10
If in system (2.1) a three-dimensional subsystem exhibits total extinction with respect to one of the variables, then it does not have an interior equilibrium. Proof: Since the subsystem exhibits total extinction, the omega limit sets of all solutions of the subsystem initiating with positive initial conditions converge to its boundary. This implies that there does not exist an interior equilibrium for the subsystem. The following result for the existence system (2.1) follows from [5]. THEOREM Let
of an interior
equilibrium
of the
3.11
the following hypotheses hold for system (2.1):
(i) All the solutions with nonnegative initial conditions are bounded in forward time. (ii) The system (2.1) is persistent. (iii) The subsystems of (2.1) are isolated and acyclic. Then an interior equilibrium, E*( u*, x*, y*, z*) exists for system (2.1). THEOREM
3.12
System tion.
(2.1) has no interior equilibrium whenever it exhibits total extinc-
3.2.
STABILITY
OF EQUILIBRIA
To determine the stability of the various equilibria, we compute the variational matrix of system (2.1). The signs of the real part of the eigenvalues of this matrix evaluated at a given equilibrium determine its stability.
RAVINDER KUh4AR AND H. I. FREEDMAN
246 The variational
matrix is given by
V(u, X, JJ,z) uh,+
=
h
uh,
uhx
uhz
\
Q%, a-% + ‘yg -PI - P2 - YP,u-=P,u - YP,,- ZP2, Y(- Slu+ 4P1f CIPlu) - SI+ "IPI YClPlX -9 - 29u - l’s, y - z9,z '(-Szu+tc;pz+'2p2U "C2P2x zc39, -s,+c,p, + c39- zs2, + cl39+ c39,)
(3.4)
/
The equilibrium E,(O,O,O,O) has diagonal and for which the eigenvalues and those in the y and z directions stable and unstable manifolds. For the equilibrium Et( L,,O,O,O), L(O,O,O), is ‘~,~,(~,,‘W.O)
&h,(&,,o,O,O)
0
v, = \
the variational matrix V, which is in the u and x directions are positive are negative. Thus E, has nonempty the variational
matrix VI, where L, =
G,h,(&,,‘U,O)
Y4&>0)
G,h,(L,,O,O,O)’
0
0
0
0
-%(‘%l>O)
0
0
0
0
-S2(L3rO)
There is a positive eigenvalue in the x direction and negative the u, y, and z directions. Thus El has nonempty stable manifolds. A similar analysis for the equilibrium E2(0, K,, ,O,O) yields value is positive in the u direction and negative in the x eigenvalues in the y and z directions are (Ti= - s;(O,O) + c;(O)p,(O, I&),
,
eigenvalues in and unstable that its eigendirection. The
i =1,2,
(3.5)
where K, = K(0). Thus cq is positive whenever (S3) holds. The eigenvahte in the z direction may be positive or negative. Remark 3.13. unstable in + Se,,.
From the above analysis, it follows that El and E2 are
From Theorem 3.3, E3 always exists. The variational equilibrium point E3( ii, Z,O,O) is given by iih”( ii, Z,O,O) iih,( ii, Z,O,O) &=
a~g,(a,~-)
aig,(Li,Z)
0
0
0
0
ti,(l,f,O,O) -Pl(ii.P)
matrix
V, at the
Ih,(J,i,O,O)
-p2(U,Z) 0 -sl(ii.o)+c,(ic)P,(ii,~) 0 -Sl(ii.O)+C*(P)P*(i(,f)
MUTUALISM
241
WITH POPULATIONS IN A FOOD CHAIN
The eigenvalues
of E3 are p;= -s,(ii,o)+ci(ii)pt(ii,2),
i =1,2,
(3.6)
and x _+, where
and all the functions are evaluated at (ii, f,O,O). Thus, whenever h,g, - g,h, > 0, eigenvalues x + have negative real parts and E3 is asymptotically stable in tBU,. If hug, - g,h, < 0, E3 has nonempty stable and unstable manifolds. Similarly we analyze E4 and E,. For E4 the eigenvalue of V, in the u direction, h(0, x1, yl,O), is positive. Thus the equilibrium E4 is unstable. The eigenvalue in the z direction is Y =
s2(O,OJ
The other two eigenvalues
+
C2P2(
0,
Xl)
+
40)
do7
Y1).
(3.7)
are the roots of the equation
Y%,P- Yl+-wx
~2-_(~xlgx+~g-YlPlx-
+ “ET- YlPlJ
+ PIYlcl(o)Plx = 0.
(3.8)
In the case as,(O, yi)/ay = 0 (i.e., there is no self-limitation on y), one can use the Rosenzweig and MacArthur criterion (see [lo]) to determine the stability of E4 in sex’;. In general, from the Routh-Hurwitz criteria, the roots of (3.8) have negative real parts if and only if
and
Yl)[~Wx(O9 -4 + ‘y&do, x1>- YlPlX(O~ Xl>1 - Ylcl(o)Pl(o~ Xl)PlX(O~ Xl>< 0.
Yl%,@?
(3.9)
For E,, the eigenvalue of V, in the u direction, h(0, x2,0, z,), is positive and thus Es is also unstable. The eigenvalue in the y direction is 6 = %(O,O) + C,(O)P,(O, x2) - W&(0,0). The other two eigenvalues “X&(0,
(3.10)
have negative real parts if and only if
x1) + @%(O,x2) - Z2P2AOI x2) - z2%z(0, r2) < 0
(3.11)
RAVINDER
248
KUMAR AND H. I. FREEDMAN
and W*;(O7 z2)kWx(O, - %(O)P,(O,
uh, “XL - YP,, Y-
Slu + 4P1
+ C,P,u)
matrix
uh,
uh,
- Pl
- P2
uhx + ag - YP,,
ax&
YClPlx
- YSl,
-4
0
0
-s2 +c2p2 +c34
0
1
(3.12)
X,)P,X(O~ x2) < 0.
E6(u3,x3,y3,0) has the variational
The equilibrium
I$=
x2) + %(OY x2) - z*P2x(0, %)I
where all the functions are evaluated at ( ug, x3, y,, 0). The eigenvalue in the z direction is
~E--*(u3~0)+C2(U3,X3)+C3(U3)q(U3tY3).(3.13) The other eigenvalues
are the zeros of the polynomial X3+ a,A2+azX+a3=0,
(3.14)
where al = GL
+ ax3gx + ag - Y3( plx + sly)
y
a2=u3h.((Yx3g,+~g-y3P1x-Y3S1Y)
Y3PlJ
+ Y4Cl Pl Plx -
Y3%,(~X3&
- U3hx(~X3&
- Y3Plu) - U3Y3hy( - %4 + ciPl+
a3 = u3Y3{ (~x3g,
- k[
+ag--
- Y3hJ(c1~,~1,
ClPlPlx
- kc,)
- %y(ax,&
+ ( - %u + 4Pl+ x [ h,(~X3&
ClPlu),
+ ai? - Y3Pd]
ClPlJ + "g - Y3Ph)
-
Plk] }.
Thus from the Routh-Hurwitz criteria, the necessary and sufficient tions for E6 to be asymptotically stable are that 5‘< 0 and al > 0,
a,>O,
and
ala2-
a3> 0.
(3.15) condi-
(3.16)
E7 and Es canbe analyzed similarly. In the case of E,,the eigenvalue of V,
MUTUALISM
WITH POPULATIONS IN A FOOD CHAIN
in the y direction
249
is given by (3.17)
~~--s,(~,,O)+c,(u,)P,(u,,~,)-~,q,(~,~O). The other three eigenvalues
are the zeros of the polynomial
A3+ b,A2 + bJ + b3 = 0,
(3.18)
where bl=Uqhu+taXqgx+(Yg--Zq(P2x+SZz), b, = %h,(%& +
-
+ “g -
P2 P2r -
z4c2
-
Z4S21(~X4&
Z4S2r) +
‘yg
-
Z4P2x)
-Z4P2u)+%P2(-S2”+C~P2+C2P2u),
~4k(~w!L
b3 = wt{
‘4p2x
(awu
- ~4~2u)(c2hr~2x - b2r)
- hL[C2P2P2X - s2z(%&
+ ag - %Pz*)l
+(-%u+GP2+c2P2u) X[G%c3x
Thus E, will be asymptotically b, ’ 0,
+~g--‘lP2x)-P2~xl)~
(3.19)
stable if and only if 9 < 0 and blb2 - b3 > 0.
&>O,
(3.20)
Similarly, since for E, the eigenvalue of V, in the u-direction, h(0, x5, y5, zs), is positive, the equilibrium is unstable. The other eigenvalues are the zeros of the polynomial
A3+ dlA2 + d,X + d, = 0,
(3.21)
where
d,=~(X~gx+g)-Y~(Plx+Sly)-z5(P2x+qy+S2r)-S2+c1P1, d,
=
zs [ wq.,
-
+ (“X5&
~2r(
-
s1+
cl
PI - y5s1, - z5qy)]
+ at? - Y5Plx - Z5P2xH - s1+ ClPl - Ys%v - z5qy - z5s2z)
+(Y5c,P,P,,+Z5C2P2P2x), d3 = ~5{ (ysclp1,>(
~1s2z- pzc3qy)
+c2P2x[P1q-P2(-~,+c1P1-Y5%y -(“X5&
+ w-
- z54JJ
YSPlx - Z5P2x)
x [ c3qq.v- s2;( --x1+
ClPl_
YSSlJJ - WJ]}.
(3.22)
RAVINDER KUMAR AND H. I. FREEDMAN
250 The equilibrium
E8 will be asymptotically
4 ’ 0,
4’0,
stable in %‘&Z if and only if d,d, -d,
> 0.
(3.23)
We shall now consider the stability of the interior equilibrium E*( u*, x*, y*, z*). In what follows, all the functions are assumed to be evaluated at (u*, x*, JJ*, z*). The variational matrix at E* is given by (3.4) evaluated at E*. In general, it is not possible to determine the stability of E*. Thus the mutualist’s interaction with food chain populations can result in either stabilization or destabilization of the system, as has been noted in two- and three-species models (see [2], [13]). Finally, it shows that the populations feeding on more than one trophic level do not necessarily cause an unstable system, the possibility of which has been pointed out by Pimm and Lawton [28], and that mutualistic interactions can have a significant effect on stability, even in the case of complex systems. 4.
PERSISTENCE
We now address the question of persistence of the populations given by our system. Specifically we derive criteria that ensure the uniform persistence of the system (2.1) in the cases of facultative mutualism between the mutualist u and the prey x as well as between u and prey y. The system (2.1) exhibits facultative mutualism between the mutualist u and the prey x whenever the hypotheses (Hl)-(H4), (Gl), (G2), (Pl), (P2), and (Sl)-(S3) are satisfied. From Theorems 3.1 and 3.2 we obtain that the equilibria E,, E,, E2, and E3 always exist. Hypothesis (S3) and Theorem 3.3 together imply that E4 exists. The equilibrium E, exists whenever 0~~> 0. Finally as the system (2.1) has a compact region of attraction (see Theorem 2.1) we conclude that (2.1) is a dissipative system. 4.1.
UNIFORM
PERSISTENCE
In this section we give criteria for uniform persistence in the three-dimensional subsystems and system (2.1). For the sake of simplicity we always assume that the boundary equilibria (whenever they exist) are unique. Case I. Facultative mutualism between u and x. First we introduce following hypotheses: (H5) Let the equilibrium E4 be globally asymptotically to solutions initiating in 92&. (H6) Let E, (if it exists) be globally asymptotically solutions initiating in Se,:.
the
stable with respect
stable with respect to
MUTUALISM
The following result holds for the food chain in 9&. to the one given in [16] for a simple food chain. THEOREM
251
WITH POPULATIONS IN A FOOD CHAIN
Its proof is similar
4.1
Let hypotheses (Gl), (G2), (Pl), (P2), (Sl)-(S3), (H5), and (H6) hold. Then the subsystem in 9’& is uniformly persistent whenever y > 0 and 6 > 0.
In the absence of the top predator z, the system (2.1) reduces predator-prey-mutualist system, and we have the following theorem. THEOREM
to a
4.2
Let hypotheses (Hl)-(H5), (Gl), (G2), (Pl), (P2), and (Sl)-(S3) hold. Then the three-dimensional subsystem in %‘L,, is umformly persistent whenever
I4 ’ 0. The proof of Theorem 4.2 can be found in [13]. Similarly one can prove the following theorem. THEOREM
4.3
Let hypotheses (Hl)-(H4), (H6), (Gl), (G2), (Pl), (P2), (Sl), and (S2) hold. Then the three-dimensional subsystem in 9AZ is uniformly persistent whenever p2 > 0.
Hypotheses (H5) and (H6) exclude the possibility of periodic orbits for the predator-prey systems in 9& and 9A. In the case of a simple food chain, it was shown in [17] that “weak” persistence (defined there) can hold even when there exist a finite number of limit cycles. Below we obtain a criterion for uniform persistence for the food chain in 9TzYz when periodic solutions exist. THEOREM
4.4
Let hypotheses (Gl), (G2), (Pl), (PZ), and (Sl)-(S3) hold. Further, let there exist a finite number of periodic solutions x = &i(t), y = q;(t), i = 1,2 ,..., k, andx=t,(t), z=qj(t), j=1,2 ,..., I, in 9?zY and 92, respectively. Then the food chain in CR& is uniformly persistent provided y > 0, 6 > 0, and for each periodic solution of period T in 9cY, m, =
-S2(0,0)+~~1[C~(0)p~(O~~~(f))+
c3(o)q(o~Icli(t))l dt>of i=1,2
,..., k,
(4.1)
and for each periodic solution ( tj (t), q, (t)) of period w in 9’2,
j=1,2
,..., 1.
(4.2)
RAVINDER KUMAR AND H. I. FREEDMAN
252
Proof. For any x E 9&, let O(x) denote the orbit through x and Q(x) be its omega limit set. By arguments similar to those given in [16], we conclude that Q(x) is bounded. We claim that E, 4 9(x). If E, E 3(x), then by the Butler-McGehee lemma (see [18]) there exists a point P in 8(x) n W”( E,). Since O(P) lies in + Q(x) and W”(E,,) is 9YVL,we conclude that O(P) is unbounded, which is a contradiction. Next E2 c%Q(x), for otherwise, since E2 is a saddle point, by the Butler-McGehee lemma there exists a point P in Q(x)n W”( E2). Now W( E2) = 93’: implies that either EOE Cd(x) or an unbounded orbit lies in n(x). In either case the result is a contradiction. Now we show that no periodic orbit in 9YX:,or E4 belongs to Q(x). Let y, , i = 1,2,. . . , k, denote the closed orbit of the periodic solution (9, (t), $i (t)) in .9?:, such that y, lies inside y,_ 1. The variational matrix v($(t), $,(t),O) corresponding to y, is given by
g(0.k (r>) ++,(r)g,(o,tJlp,(l)) - 4,(t)Pl\(o.+!(t))
I
-
P,(o,%(t))
- P2(O,+t(t))
c,(o)~,(r)P,~(O,cp,(t))
Computing
the fundamental
matrix of the linear periodic system
X’=r/;(t)X,
X(0) = I,
(4.3)
we find that its Floquet multiplier in the z direction is em,‘. Suppose yi lies in Q(x). Then from (4.1) its Floquet multiplier in the z direction is greater than 1 and therefore by the Butler-McGhee lemma there exists a point PI E W(yl)n 8(x). If PI lies outside yi, then O(P) lies in Q(x) and is unbounded, which is a contradiction. If yi is stable from inside, then A(P,) must be y2 and is contained in 8(x). If y2 is unstable from the inside as well, we have a contradiction, for a repeller cannot be in !A(x). If y2 is stable from the inside, again from the Butler-McGhee lemma there exists a point Pz inside y2 such that Pz E kV(y,) n Q(x). Continuing this argument we get that either yr is unstable from both sides or it is stable from inside and there exists a point Pk in W”(y,) n Q(x). If such a Pk exists, then E4 must be a repeller and A( Pk) = E4 lies in 8(x), which is again a contradic-
MUTUALISM
WITH POPULATIONS IN A FOOD CHAIN
253
tion. Thus yI does not lie in Q(X). By similar arguments we conclude that no y, lies in Q(x). Repeating the above steps we also conclude that no closed orbit in 92 or E, lies in Q(x). Thus sZ(x) lies in 9.& and the subsystem in yex’,= is persistent. Finally, since only the closed orbits and the equilibria form the omega limit set of the solutions on the boundary of 9x&, by the main theorem of [5], this implies that the food chain is uniformly persistent. Remark 4.5. In the case that there are no periodic solutions in 92 or if E, does not exist, Theorem 4.4 holds without assuming inequality (4.2). Remark 4.2 holds Theorem multiplier u direction hypothesis
4.6. For the predator-prey-mutual&t system in %‘zX,, Theorem even without hypothesis (H5), and for the system in 9A,, 4.3 holds even without (H6). This is so because the Floquet of any closed orbit (+(f), q(t)) of period T in 9$, or Bz: in the is exp[ J,,r/r(O,c#J(~),+(t),O) dt] and is always greater than 1 by (Hl).
Next we derive a criterion for the uniform persistence of system (2.1) when all its three-dimensional subsystems possess an interior equilibrium. It follows from [6] that whenever a subsystem of (2.1) is uniformly persistent it has an interior equilibrium. First we introduce the following hypothesis: (H7) Let the equilibria E6, E7, and Es be globally and @&z >respectively.
stable in .%‘A,,, WA,,,
THEOR EM 4.7 Let hypotheses (Hl)-(H4), (H7), (Gl), (G2), (Pl), (P2), and (Sl)-(S3) hold. Further, let either (H5) and (H6) or inequalities (4.1) and (4.2) hold. Then system (2.1) is unqormiy persistent whenever 6 > 0 and 7 > 0.
Proof. Let O(x) be an orbit through any point x E 9~Xvz. Then from Theorem 2.1 its omega limit set e(x) is bounded. Now we show that no boundary equilibria belong to a(x). Suppose E,, belongs to Q(x). Since E0 is a saddle point, by the Butler-McGehee lemma there exists a point P in Q(x)n W(E,,). Further, as W( E,) is precisely %$, O(P) is unbounded and lies in Q(x), which is a contradiction. E1 G o(x). Otherwise by the Butler-McGehee lemma there exists at least one point P E n(x) fl W( El). However, as W( E1) is e/e if P E 9:) then either A(P) = E, belongs to Q(x) or O(P) is unbounded and lies in n(x). Both of these statements are contradictions. Otherwise P E yeu’, u and again 0(P) is unbounded and lies in Q(x), contradicting %ii u9&7 the boundedness of Q(x).
254
RAVINDER
KUMAR AND H. I. FREEDMAN
Next we show that E3 cannot be in Q(x). Since there are periodic orbits in 9; and E3 is unstable in the y and z direction, E3 E Q(x) implies that P lies in 9”,. Thus A(P) is E, or E1 or E,, or 0(P) is unbounded, and each of them results in a contradiction. If (H5) holds, then by analogous arguments E4 does not belong to Q(x), and if inequalities (4.1) hold, then by arguments given in Theorem 4.4, E4 B Q(x). In the same way we can show that E,, if it exists, does not lie in Q(x). In the case Eb E 9(x), then as 5‘> 0, by the Butler-McGehee lemma and hypothesis (H7), there exists a point P in O(x)n W( E6) and the unbounded orbit through P lies in Q(x) n %‘~xy, which is a contradiction. Similarly we get that E, and Es do not belong to Q(x). Finally, no point on the boundary of 9&Z lies in D(x), otherwise Q(x) will contain one of the equilibria or closed orbits, resulting in a contradiction. This completes the proof of the theorem. Case ZZ. Facultative mutualism between u and y. The system (2.1) exhibits facultative mutualism with the predator y when the hypotheses (Hl), (H2), (H3’), (H4’), (Gl), (G2), (G3’), (Pl), (P2’), (Sl), (S2’), and (S3’) described in Section 2 hold. The set 9 given by Theorem 2.2 contains the region of attraction for (2.1). The criteria for the existence of boundary equilibria and for uniform persistence remain the same as in Case 1. In particular, the following result holds. THEOREM
4.8
Let hypotheses (Hl), (H2), (H3’), (H4’), (H5)-(H7), (Gl), (G2), (G3’), (Pl), (P2’), (Sl), (S2), and (S3) hold. Then system (2.1) is uniformly persistent
whenever 6 > 0 and q > 0.
Theorem 4.8 describes the circumstances in which the mutualistic interaction between u and y does not dominate the interactions of the food chain and all the species can coexist. 5.
EXTINCTION
In the previous section we obtained criteria for all populations to persist. In the present section, we consider the situation where the mutualism is so strong that one or both of the predator populations exhibit total extinction. To do this we consider two cases. In the first case, the mutualist (of the prey) forces both predators to total extinction. In the second case, the mutualist (of the bottom predator, y) forces the top predator, z, to total extinction. First we introduce some notation. If populations &, . . . , Nk exhibit total extinction in the space BUT,,,,,U,, we denote this by gN,,,,,, Nk_ 0. Here
MUTUALISM
WITH POPULATIONS IN A FOOD CHAIN
255
N,, . . . , Nk and ul,. . , u, are subsets of {u, x, y, z}. Case I. THEOREM 5.1 Let hypotheses Further, let
(Hl)-(H4),
(Gl),
(G2), (Pl), (P2), and (Sl)-(S3)
hold.
where
K, 1, and 2 are given by (2.3). Then 8z_0
in %‘A,,,.
Proof. It can be easily seen that L, 5 u(t) I L for sufficiently This, together with inequality (5.1), yields that Z’I - Zz + c2(L,)p,( + &j < 0, for large t. Thus lim, _ m z(t) = 0. Similarly,
large t. L,, 2)
we can prove the following theorem.
THEOREM 5.2 Let hypotheses (Hl)-(H4),
(Gl), (G2), (Pl), (P2), and (Sl)-(S3)
-s1tL,,o)+c,tL,)P,(L,,~) co, then f,, _ 0 in 5?u&z.
hold. If
(5.3)
If in addition
-~~+&&J2(L3Jq
(5.4)
where L, and & are given by (5.2), then cyz _ 0 in .@u:,,. Case II.
Define L, = L( I?,o, ti),
where k and i? are given by (2.6). Then the following result holds for the system (2.1).
RAVINDER KUMAR AND H. I. FREEDMAN
256 THEOREM
5.3
Let hypotheses (Hl), (H2), (H3’), (H4’), (Gl), (G2), (G3’), (Pl), (P2’), (Sl), (S2’), and (S3) hold. Zf
-~~(~1~0)+c*(~1)P2(~1~~(~1))+cC3(~l)q(~1~~)
EXAMPLES
We now illustrate our results with examples. The particular coefficients chosen do not necessarily have biological meaning or apply to specific populations. Example
6.1. u’=u x’=ax
Consider the system
(li
U L+x 1-x
i
1
)
-ysxy-80x2,
y’= y -s,o-sliu-ssIzY+~yox-~~z). i z’ = z [ - szO- SZlU- sz*z + CZ8~X+ ( $0 + CjlU) 6, y] .
(f-3.1)
When L = 3, (Y= K = 4, y0 = 8, =l, sIO =1/3, slr =1/6, sr2 =1/4, c1 = 3, m=O, 6,=1, szO= szl =1/3, c2 =l, cs,, =1/3, and csr = 0, the above system has the boundary equilibria E,(O,O,O,O), E,(3,0,0,0), E2(0,4,0,0), E3(7,4,0,0),
We now show that for system (6.1), E4(0, x1, y,,O) is always globally asymptotically stable in B’:,,. For this we consider the Lyapunov function
on BAT,,.Computing the derivative of V(x, y) along the solutions (6.1), after some simplifications we get
of system
MUTUALISM
257
WITH POPULATIONS IN A FOOD CHAIN
Now, as the set {(x, JJ) E ST&: p = 0} = { E4 } is invariant, we conclude that E4 is globally asymptotically stable in %‘A. Similarly one can see that E, is globally asymptotically stable in g&,. Further, as a, = 49/6, & = 4/3, y = 149/117, and S = 2/3, ah the three-dimensional subsystems are uniformly persistent. Next we consider the global asymptotic stability of the interior equilibria in the three-dimensional subsystems. Computing the symmetric matrix 9(u, x, y) for E6(63/16, 15/16, 49/16,0) we get 16 &,=a’
- 16~ “’ = 126( 3 + x) ’
b22=3, The region of attraction
b23= 0,
is contained
1 b,,=E’
b33=-&
in the set
It can be easily verified that the principal minors of .%‘(u, x, v) are positive in &i and therefore .S?(II, x, y) is positive-definite in &i. Next the symmetric matrix B(u, x, z) for E7(17/3,8/3,0,4/3) is given by 3 dii=fl’
The region of attraction .J5f2= {(u,x,z)
G=
3u 34(3+x)
for the subsystem
Eqj;,:
’
43=;,
in 5?,+,, is contained
in the set
Orur7,O~x~4,O~z~11},
and LS(u, x, z) is positive-definite in s$. For the equilibrium E8 the matrix 9(x, y, z) is diag(l,$, 4). Finally, as 5 = 15/48 and n = 3/18, from Theorem 4.7 we obtain uniform persistence for the system (6.1). Next we give an example in which the mutual& u provides the predator y with an alternate food supply and increases the death rate of the top predator.
RAVINDER KUMAR AND H. I. FREEDMAN
258 Example
6.2.
U L+ry
(
u’=u
l-
x’=ax
i
Consider the system
l-
1
)
K:ku ___
+(+I + CllU)(YO + YlU)X - EOZI)
y’= y[ -
s10
+
z’=z
s20
-~21~-~22z+j~)(~)x+~~],
-
SllU
80
) -(Yo+Y+P1+% s12y
-
(6.2)
where all the constants are assumed to be nonnegative. With L = 1, I = l/3, cu= K = 3, y. = So = 1, k = yl = 6, = cl1 = cal = cgl = 0, slo = s12 = l/3, sI1 = l/24, cl0 =1/4, s20 =l, s2r =1/3, sz2 = l/2, c20 = 7/8, &, =1/36, q. = 4, the boundary equilibria are
E,,(O,O,O,O), E,(l,O,O,O),
E&+,~,%,O),
Es(O,%O,E)r
E,(O,%:,O,O), E,(l,+?_,O,+$), The region of attraction
and is contained
E3(1,3,0,0),
&(0,3,0,0),
Es(O,%k+%t+%). in the set
Since & =11/24, f12= 41/24, y = 68/23, S = 35/396, we conclude that all of the three-dimensional subsystems involving x are uniformly persistent. The symmetric matrix .G@( u, x, y) corresponding to Eb is given by
b,, = ; Further
9
it is positive-definite
43 =o,
and
h,,=$.
in the region of attraction
The symmetric matrix g( u, x, z) corresponding to E7 is given by d,, =1/6, d,, = 7/8, d,, = 0, and d,, =1/2. Also the 9(x, y, z), corresponding to E,, is given by fil =l, fi2 = 0, fi3 fz2 = 4/3, j& = 0, and f3s =1/2. Since the principal minors of 9( d,, = 0,
d,, = 1, matrix =1/16, u, x, z)
MUTUALISM
WITH POPULATIONS IN A FOOD CHAIN
259
and 9(x, y, I) are positive, we conclude that ET and E, are also globally asymptotically stable in &‘& and 5$,, respectively. Finally 6 = 281/492 and 1)=103/792 imply that the system (4.5) is uniformly persistent. The interior equilibrium is
Now we consider an example in Case II to show that total extinction occur. Example 6.3.
Consider
(1-m
u’=u
can
the system U 1
,
x/=x(3-x)-xy-xz,
Its
y’=y
i -&+u+y)+;x
)-yz
ZfCZ
1 ___2_,_1,+1,+1 5 10 [ 2
6
boundary
equilibria
are
y+l’
E,(O,O,O,O),
Y 2 i y-cl
(6.3)
,1 .
E,(l,O,O,O),
E2(0,3,0,0),
E,(4,3,0,0), E4(0,$,$,0),
E&,&:,0),
and - 15a2 + 10a + 25
where a3 = 20/7. The region of attraction d=
{(u,x,Y,z)
is contained
in the set
ESe,+,yr :o~u~4,o~xI3,oIy~5,o_
Since j3, = 5/18, the subsystem in L%‘& is uniformly persistent. Further, as E, does not exist and y = 5/144, the food chain is also uniformly persistent. However, the system (6.3) satisfies inequality (5.5), and therefore the top predator will eventually become extinct. 7.
DISCUSSION
In this paper we have considered a model of facultative mutualism with populations of a food chain. This model is an outgrowth of work since the time of Kolmogorov [23] for two-dimensional models and Rai et al. [29] for three-dimensional models.
260
RAVINDER
KUMAR
AND H. I. FREEDMAN
We have considered two cases. In the first case, the mutualism is with the prey population, and in the second case, the mutualism occurs with the second trophic level predator. The models and their analyses are of interest to biologists, since, as pointed out in the introduction, the mutualisms described in this paper do occur in nature. The analyses of these models suggest possible observations for field workers. A possible outcome of mutualism is predator extinction. This may help explain the absence locally of “superpredators” in predator-prey systems where mutualism with either population is present. The model analysis also indicates that the carrying capacity of the mutual&t populations increases, possibly at the expense of the other food chain populations. We have modeled in this paper nonsimple food chains, that is, ones where the population at the third trophic level may feed on all populations of the first and second trophic levels. In a future paper we will consider the case of obligate mutual&m and reversal of outcome from extinction to persistence.
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