Stability and persistence of facultative mutualism with populations interacting in a food chain

StabiSty and Persistence of Facultative Mutualism with Populations Interacting in a Food Chain Afaf A. S. Zaghrout

Muthna~ics Department Faculty of Science Al-A.&al- University Nasr City P.O. Box 9019 Cairo 11765, Egypt

Transmitted

1,~ John Casti

ABSTRACT

Facultative

mntualism

system

of

criteria

are investigated.

1.

four

with

autonomous

populations

interacting

in a food chain

differential

equations.

The

stability

is modeled and

by a

persistence

INTRODUCTION

In this paper we investigate a model of four interacting populations where one of them, a mutualist, interacts with the other three so as to benefit and receive benefit from one of those three. We are interested in establishing criteria for persistence and stability, which can be interpreted as the survival of all biological populations. There are now many papers in the literature dealing with mutualistic system. h/iost of these papers deal with two-dimensional systems, modeling direct mutualism between two populations and ignoring all other population interactions (see [l, 3, 51). Recently, some papers have appeared which deal with cases where the mutualism is due to or influenced by the interaction with a third population (see [3, 6-9, 12, 131). Persistence in biological systems in a context related to this paper has been discussed in [4, G-101. We utilize the definition of persistence developed in [4, 111.

APPLIED

MATHEMATICS

AND COMPUTATION

0 Eisevier Science Publishing Co., Inc., 1991 655 Avenue of the Americas, New York, NY 10010

45:1-l-5

(1991)

1

0096-3003/91/$03.50

AFAF A. S. ZAGHROUT

2

DEFINITION. Let N(t) be such that N(O) > 0. Then we say that N(t) persists if lim, em inf N(t) > 0. Further, we say that N(t) persists uniformly if there exists 6 > 0 such that lim t,,inf N(t) 2 6. Finally, a system in R” persists 2.

(uniformly)

if all components

persist (uniformly).

MODEL

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 is also described. We consider the autonomous

system ’ = d/dt,

ti = uh(u,x,y,z), i=axg(u,x)-yp,(u,x,y)-zp,(u,x,z),

u(0) =

u(J

2

r(0) =

0,

x0

2

0,

as a model of a mutualist-food-chain death process. u(t) represents the

y(O) = yo 2 0,

z(0) =

20

2

0,

interaction with continuous birth and density of the mutualist at time t;

x(t), y(t), z(t) d enote the prey and predator densities

respectively. If u = 0, reduces to a food-chain model. The functions h, g> P,, P,, si> s2, ci> c2> cs are sufficiently smooth so that the solutions of (2.1) exist uniquely and are continuable for all positive time. The function h represents the specific growth rate of the mutualist. We assume that h has the

system

(2.1)

the following properties

(see [5, 12, 13, 151):

@I,)

h(O,x,y,z)>O,

~bv,y,zkO.

(H,)

There

a

exists

h(Ur, y,.z), x, y, z) = 0. (H,) h,(u,x,y,z)>O,

unique h,(u,x,y,zkO,

function

L(x,

y, z) > 0

such

that

h,(u,x,y,z)
(H 1) implies that, independent of the populations, x, y, z u is capable of growing even when rare. Also, the growth rate is assumed to be densitydependent and to decrease as the population increases. (H,) implies that L(X, y, z) is the mutualist carrying capacity and in part specifies in what way

Stability and Persistence in Populations the

prey

and predator

act as part

3 of the

mutualist’s

environment.

(Ha)

implies that u derives benefit from the prey population and that there might be a cost to the mutualist due its interactions with the predators. Also (Hi) implies (H,)

I&,

0 < limx _m L(X, 0,O) = E < 03,

L/,(x, y, 2) > 0,

I&

y, z) Q 0,

y, 2) < 0,

which implies that the mutualist has a finite carrying capacity. The function g(u, x) is the specific growth rate of the prey in the absence of any predation. (G,) (G,)

g(u,O)> 0, g,(u,x> < 0. There exists a unique K(u) > 0 such that

Z
We assume that g satisfies:

g(u, K(u))

> 0, K(u) <

E = max,,.U.,K(u).

implies

that

the

prey

population

is capable

of surviving

in the

presence or absence of the mutualist and that the growth rate in the absence of predation is density-dependent. (G,) im pl ies that K(u) is the carrying capacity of the prey in the absence of predation. In the case that g,(u, X) > 0, then, even in the absence

of predation,

u

acts as a mutualist with respect to X. If g, < 0, then there is a cost to x for associating with U, and u can be mutualist of x only by its effect upon the predator. If gU(u, X) = 0, the relationship between u and x without predation is commensal. The functions p,(u, x, y), p,(u, X, z), and q(u, y, z) denote functional response to the prey and mutualist densities.

the predator’s

We assume: (Pi)

p,(u,O,O)

= 0, i = 1,2, and

The functions p,, p, are the competition functions, and condition (Pi) states that competition between x and y and between x and .a occurs when both populations are present, and is an increasing function of their population. (P,) implies that the mutualism may decrease the competitive effect of y on x and may increase the competitive effect of x on z.

AFAF A. S. ZAGHROUT

4

Also we assume: (or>

9(~,0,0)=0,

9,(u,y,z)G

9&q/,4

(Qz) When

0, and

ao,

9z(u,Y,z)

>o.

y = 0, zy,- + 9 = 0, 9 f 0.

Furthermore: 6,) 6,)

s&J, y) > 0, s&G a) > 0. s,,(u,y)&O, c~(u>
The nonnegative Functions C&U), i = 1,2,3, are the rates of conversion of prey biomass to predator biomass; (Y is a bifurcation parameter. The functions sr(u, y), sa(u, z) are the specific death rates of the predators y and z in the absence of predation. We make the standard assumption in (S r), (S,) that the death rates are increasing functions of population. Also, in order to have a viable system we must have that: (S,)

There exists xr,yr

such that

s,(O,O)= c,(~)P,(o,x>o)>

Xl

< K,(O)

Yl

< K,(O).

and

s,(O,O)= Cl(O)P,(O,O>Y,), (S,)

There exists xz,.zz such that

sa(O,O) = ca(O)p,(O,x,,O),

0 < Xa < E,( 0)

s,(O,O) = ca(O)p,(O>O,z,)>

O
and

We will make use of the following

theorem.

Stability and Persistence in Populations 2.1.

THEOWM

Under the ubove assumptions,

A={(u,x,y,z):O
the set


L = lim L(.r,O,O), X’cz

S2=

Z=

min

max ogu
s2(u,0),

K(u),

+ C3y + z < N},

jj=

Ca =

max

(2.2)

max g(u,O), 0 < II < L

c2( u),

OCU
O
and

N=c,(O)K(s, +a~)+c,E[S, +cl(0)pl(O,r)], is positively

invariant

and attracts aEl soktions

initiating with nonnegative

initial conditions.

P~oor. the proof

3.

The proof can be carried out by following the same of Theorem 2.1 in [13] and so will be omitted.

EXISTENCE In this

section

steps

OF EQUILIBRIA we shall

establish

criteria

for the

existence

tence of equilibria of the system (2.1).It is clear that the origin an equilibrium point for the system (2.1). From (H,), (H,), and Freedman [5], we have the following

and nonexisE,(O, O,O, 0) is theorem:

THEOREM 3.1. The system (2.1)has exactly two one-dimensional ria, E,(L,,O,O,O)

as in

and E,(O, K,,O,O).

equilib-

6

AFAF A. S. ZAGHROUT

REMARK 3.1. The subsystems in Rl and Rf have no equilibria, since all of their solutions tend to zero exponeniially. Also, it follows that E, is locally stable in the y and z directions and locally unstable in the u and x directions. As shown in [9], the following result holds in R:,

THEOREM provided

3.2. The subsystem

E, and E,

in Rz,

.

has an equilibrium

E,(ii,

X,0,0)

are unstable in Rex .

REMARK 3.2. Under hypotheses It is easy to prove the following

(H ,> and (G,), theorems

E, exists.

(see [5, 131).

TIIEOREM 3.3. A necessary and suficient condition for an equilibrium the form E&O, x,, yl,O) to exist in R.:,, is that hypothesis 6,) is satisfied.

THEOREM

3.4.

A necessary

and suficient

E,(O, x2,0, z2) to exist is that hypothesis The following result for the existence system (2.1)follows from [3, 131:

THEOREM 3.5. Let the following (a) All solutions with nonnegative

(S,)

condition for

of

an equilibrium

is satisfied.

of an interior

hypotheses

equilibrium

of the

hold for the system (2.1):

initial conditions

are bounded

in for-

ward time. (b) The system (2.1)ispersistent. (c) The subsystems of (2.1)are isolated and acyclic. Then an interior equilibrium

4.

STABILITY To determine

E*(u*,

x*, y*,z*)

exists for the system (2.1).

OF EQUILIBRIA the stability of the above equilibria,

we need to compute

the

variation matrix of the system (2.1). The signs of the real part of the eigenvalues of this matrix evaluated at these equilibrium determine its

Stability and Persistence

in Populations

7

stability:

(4.1) where all functions

are evaluated at (u, .r, y, z) and we have used the notation

F, to represent dF/dx, etc. The variational matrix V,, [i.e. V evaluated at E,,(O, 0,0, O)] is diagonal, and its eigenvalues in the u and x directions are positive; hence the equilibrium point E, is locally unstable in the u and x directions. The eigenvalues in the y and z directions are negative, and hence E,, is stable in the directions. Thus, E, has nonempty stable and unstable manifolds.

y and z

Next, for the equilibrium E,(L,,,O,O), L(O,O,O) = I,,,, the variational matrix V, has a positive eigenvalue cug(L,,,O) in the x-direction and a negative eigenvalues L,h,(L,,O,O,O) in the other directions. Thus, E, is unstable in the x-direction and stable in the U, y, and z directions. Hence, E, has nonempty stable and unstable manifolds. The variational matrix V, for E,(O, K,,,O,O), K,, = K(O), has the form

0 h(O,K,,,O,O) &,Z,<(RK,I) a&,R,(K 4,) = 0 0 0 0 L

0 0 -p,(O.K,,,O) - p*(O,K,,,(J) 0 - s,(O,O)+ c,(O)p,(O,K,,,O) 0 ~ sz(O,O)+c2r)z(0,K,,,O) 1.

The equilibrium E, yields its eigenvalue h(0, K,,,O,O), which is positive, in the u-direction, and its negative eigenvalue c~K~g,(0, K,,) in the x-direction. The eigenvalues in the y and z directions are (yi = - ‘i(O,O)

+ c,(o)pi(o>

K()>“)>

i = 1,2.

8

AFAF A. S. ZAGHROUT

From Theorem 3.3, the eigenvalues in the y-direction are positive, eigenvalues in the z-direction may be positive or negative.

REMARK 4.1. unstable in R:,

From

the

above

analysis,

it follows

that

E, and

but the

E,

are

A similar analysis for the equilibrium E&ii, X, O,O>yields its eigenvalue BT = - s,(U, 0) + c,(O)p,(U, X, 0) in the ,--direction, which is positive or negative, but the eigenvalue in the y-direction is B$ = - s,(U,O)+ - cr(O)p,(u, x,0>, which by Theorem 3.3 is positive, and thus E, is unstable in the y-direction. The other eigenvalues, in the IJ and x directions, are

Thus, where

> 0, eigenvalues B,, have negative real parts E, has nonempty stable and unstable mani-

whenever hug, - g,h, CuZg:, + Eh, < 0. Thus,

folds. Similarly, we analyze E,(O,z,, yl,O). The eigenvalue of V,(O, x1, yr,O) in the u-direction has the value h(O,x,, y,,O), which is positive, and hence E, is unstable in the u-direction. The eigenvalue in the z-direction is given by

and the other

two eigenvalues

Y2-(‘y”lgx + ag -

From

the Routh-Hurwitz

YlPlx

are the roots

-

criteria,

YISI,

of

+ C1YIPIY)Y

the roots

have

negative

real parts

iff

Stability and Persistence in Populations

and

Under these conditions we notice that E, is stable in R&. For E,(O, x2, 0, z,), the variational matrix has the form V~(O,x2,0, $)

0 L 0

h(O,r,,o>%d

1

G 0

=

H

0

0

- PLO,x.2 >0)

- Pz(O.XPi %) 0

1

z,c,(O,O)p,,(O,r,.a,)

%%w%#YZ>%)

5

1,

where

L = cug(O,r,) +

““2&(0,X,)

- ~zP2x(o~x2~~2)~

s(O,O)+ c~(o)P,(o>q,~o)- ~24,(W~,)?

7 =

-

5=

z2[- s,,(O,z,) + c2(0)P,,wJ~~2)1~

The eigenvalue of V, in the u-direction is h(0, x2,0, z,), and thus E, is unstable in the u-direction. The eigenvalue

which is positive, in the y-direction

is given by

8, =

SI(O,O)+ C~(O)P,(O>~,>O) - ~24y(oA~2)>

and the other eigenvalues

in the x and z directions

are obtained

equation

a2 - (5 + L)6

+ L‘$ + Z2C2p2p2r

= 0.

from the

10

AFAF A. S. ZAGHROUT

The eigenvalues

6 1,2 have negative

real parts iff

.L+t
Similarly,

for E,(u,,

x3, y3, O), the variational

matrix has the form

where A = -

~3~1,

c = -

Pl-

F = Y3( -

+ ax3gu>

B = ag

-

YSP,,,

D=Y,(-s,,+c;P,+c,P,,),

YBPlY’

Sly + ClPl,)

+ ax3g,

I

K=-s,+c,p,+c,g,

and all the above functions are evaluated at (u,, x3, y3, 0). The eigenvalue of in the z-direction is K. The other eigenvalues are the roots of the polynomial

E,

l3 +

a,p + a,.$ +

a3

=

0,

where a, =

u3h, + B + F,

a2 = y,BF - u,h,A

- y,u,h,D

u3 = y3u3hU( BF + clplx3C) Thus,

from the Routh-Hurwitz

+ y,c,p,,C

+ u,h,B

+ y3F,

- u3hxy( AF - CD) + u3y3h,( clplxA - FB). criteria,

the necessary

and sufficient

condi-

11

Stability and Persistence in Populations tions for E, to be asymptotically K CO, Also, for E,(u,,

stable in Ruxy are that

x4,0, z,),

and

a3 > 0,

a, > 0,

the eigenvalue

u1u2 - us > 0.

of V, in the y-direction

is given

by

The other eigenvalues

are the roots of the polynomial

q3 + bpf + b2q + b, = 0, where

b,=-(u,h.+J+N), b, = u4h,(]

+ N) + NJ - Z.z4c2p2, - u,h,M

b, = u4huZz4cZp2x

+ u4M]h,

- uqh_Mz4c2p21

] = z4c2p2; I=

- u4NJh,

- u4Qhz,

- u,ZQh,

+ u4QNh;,

- .z$SZ;,

N = ag + axz,gx M = ox4g,,

-(P,+z,P,J, Q = z4( - s,,

+ c;p,

z4p2x>

- z4pau >

+ czpeu + 49).

All the above functions are evaluated at (u,, x4,0, z4). E, will be asymptotically stable in RlxZ iff

q* Similarly,

for

co,

b, > 0,

E,(O, x5, yS, zs)

the

b, > 0, eigenvalue

Es is unstable h(0, x5, y5, .z5) > 0, and hence eigenvalues are the zeros of the polynomial

b,b, of

V, in

A3+ d,h” + &A + d, = 0,

- b, > 0. in the

u-direction

u-direction.

The

is

other

12

AFAF A. S. ZAGHROUT

i.e.,

h”+(M-N-L)h”

pfw+L) + AZ + YC,P,,(P,+ YPh,)+ =czPzAP2+ “P,;))

+ +(!a

+

%J(Y

- xg(zq,

+

ClPlX~C3Y!,

-

“Wd)

+

QFlPdP,

+

YP,!,)

+ z)M - MNL = 0,

where

N= (-

s1+

CIP,) + Y( - SI!, + CIPh,) - W,]

L = ( - s2 + czp2 + c+J) + z( - s2; + czp2: + C.+/_). The equilibrium

Es will be asymptotically

d, > 0, Now,

we

will

consider

d, > 0, the

E*(u*, x*, y*, z *). In what follows at (u*, x*, y*, z*). The variational

E*. In general mutualists’s

stable

and

in RI,_

iff

d,d, - d, > 0.

stability of the interior equilibrium all functions are assumed to be evaluated matrix

of E* is given

by (4.1) evaluated

stabilization or destabilization of the system, as has been noted three-species models (see [2, 71). Finally, we see that populations more than one trophic level do not necessarily cause an unstable possibility mutualistic complex

5.

at

[13], it is not possible to determine the stability of E*. Thus, interaction with food-chain populations can result in either in two- and feeding on system,

the

of which has been pointed out by Pimm and Lawton [14], and that interactions can have a significant effect on stability, even in systems.

PERSISTENCE

AND

UNIFORM

PERSISTENCE

In this section, we shall investigate the persistence given by the system (2.1). We shall derive criteria that persistence of the system (2.1) in the cases of facultative the mutualist

and the prey

x as well as between

of the populations ensure the uniform mutualism between

u and the prey

y.

13

Stability and Persistence in Po&ations (i)

Facultative

Mutualism

between

The system (2.1) exhibits

u and x

facultative

mutualism

between

the mutualist

u

and the prey x whenever the hypotheses (Hi)-(H,), (G,), (G,), (Pi>, (Pa), (S,), and (S,) are satisfied. Also, we shall assume the following hypotheses: (Pi) Let th e e q UI‘1i b rmm E, be globally asymptotic stable with respect solutions initiating in Rl,, . (PL) Let E, (if it exists) be globally asymptotic stable with respect solutions initiating in R:; . (P$ Let th e e q UI‘1’1 1 )ria E,, and R:,,, respectively. The following

TIIEOREM5.1. (S,)-(S,)

E,

THEOREM5.2.

TIIWREM 5.3.

to

be globally stable in Rex,, , Rlxz,

results hold for the food chain in RT!,; (see [lo] and [13]).

Let hypotheses

(Hi)-(H,),

hold. Then the system (1.1) persistsfor

(Ph> hold. Then y1> 0, 6, > 0.

(S ,)-(S,) persistent

E,,and

to

(Gil,

(G,),

(Pi),

Let hypotheses (G,), (G,), (Pi), (P,), (S,)-(S,), the subsystem in Rz!,; is uniformly persistent

Let hypotheses

(H,)-(H,),

(P,>,

and

CX~ > 0.

(Pi)-(Pi),

hold. Then the three-dimensional subsystem whenever Bz > 0, and RTf,; is uniformly

(P,),

(Pi) and whenever

(P,),

and

in Rixt, is unqormly persistent whenever

By > 0. It follows from [4] that whenever a subsystem persistent, it has an interior equilibrium.

of (2.1)

is uniformly

THEOREM 5.4. (S,)-(S,), whenever

PROOF. omitted.

Let hypotheses (H,)-(H,), (G,), (G,), (Pi), (P,), and (Pi)-(Pj) hold. Then the system (2.1) is un$ormZy persistent T* > 0 and K > 0.

The proof is similar to that of Theorem

4.4 in [13], and will be

14

AFAF A. S. ZAGHROUT

(ii)

Facultative Mutualism between u and y This case can be treated as case (i) with some suitable will be omitted. 6.

conditions

and so

EXAMPLE Consider

the system

h(l-&), x’=x(3-x)-xy-rz”u, y’=

The system (6.1)

(6.1) y)+%x)-z&y,

y(-&(1+

has the boundary

equilibria E,(6,3,6,6),

JG,(O,O,O,O), E,(LO,0,0),

E,(1,3,0,0),

Es(O,;,?,?)

E,(1,2,0,1),

After simple but long calculations, it is easy to see that E,, E, are globally subsystems are asymptotically stable in R*‘y and all the three-dimensional uniformly persistent. Using Theorem 5.4, we can easily prove that the system (6.1)

is uniformly

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