Remark on the dissipativity of an N prey-one predator system

ELSEVIER

Remark on the Dissipativity of an N Prey-One Predator System ZDENEK POSPISIL

Department of Mathematical Analysis, Masaryk Unicersity, 662 95 Brno, Czech Republic Received 25 April 1994; recised 10 February 1995

ABSTRACT A sufficient condition for the dissipativity of an n prey-one predator system and an estimate for ultimate bounds of solution orbits are presented. An example illustrating the comparison with related results is also given.

1. INTRODUCTION The concept of permanence (or uniform persistence) has recently been suggested as the ,most suitable formalization of the vague but important notion "ecological stability," (See [1] and references therein.) A necessary condition for the permanence of an autonomous differential equations system is its dissipativity (ultimate boundedness). For verifying permanence, Gard [2] developed a technique that requires an estimate of the ultimate bounds of solution orbits. This paper deals with dissipativity and the ultimate bounds of solution orbits of an ODE system modeling one predator feeding on n competing prey. To the best of my knowledge, this particular problem has not yet been satisfactorily resolved. In [3], the dissipativity of the system under consideration was simply assumed. An estimate for orbit bounds by an examination of nullclines [4] is applicable for one prey-one predator systems only. The results for three-dimensional systems in [5] or [6] require an excessively strong assumption (see Example 3 below). The proof of the dissipativity of a two prey-two predator system in [7] contains a flaw (lines 7 and 8 on page 37). In what follows, hypotheses imposed on the system considered are formulated and their biological significance is indicated. Section 2 presents a proof of dissipativity and an estimate for ultimate bounds of solution orbits. All results are illustrated by one prey-one predator examples. MA THEMA TICAL BIOSCIENCES 131:173-183 (1996) © Elsevier Science Inc., 1996 655 Avenue of the Americas, New York, NY 10010

0025-5564/96/$15.00 SSD1 0025-5564(95)00045-F

174

ZDENI~K POSPISIL

Let us consider the general Kolmogorov-type system

xi=xiFi(x,y),

~=yG(x,y),

i = 1 , 2 ..... n,

(1)

y(0) = y ° > 0 ,

i = 1 , 2 ..... n,

(2)

with initial conditions

xi(O)=x°>O,

where x = ( X l , X 2 . . . . . Xn) , x, and y denote the size (biomass, population density, number of individuals) of the ith prey species and the predator, respectively. We assume that the functions F1, F 2..... Fn,G (specific growth rates) are continuously differentiable on II~n+ ..+ 1 = ((Xl ..... Xn,Y ) ~n + 1 : xl >/0 ..... xn >/O, y >/O} and that the initial value problem (1), (2) has a unique solution for each x ° = (x ° ..... xO), y0, (xO, y 0 ) ~ int R~_+ 1. Put

(3)

f / ( x , ) = F i ( 0 , . . . , 0 , x i , 0 ..... 0),

where fi is the specific growth rate of the ith prey species in the absence of any other prey and predator. We impose three standard assumptions on the functions F 1,F2..... Fn,G: (A1)

There are constants K i > 0 such that

(xi-Ki)fi(xi)
for each xi >~O,

where fi(xi)_are defined by (3). ap; (A2) -ff~(x,0)~<0 f o r e a c h x ~ i n t R + , J

(A3)

oFi

--~-(x,y)~<0

foreachx~intN+,

xi~Ki;

i = 1 , 2 .... ,n,

i , j = l , 2 ..... n. y>0;

i = 1 , 2 ..... n.

The assumptions have the following ecological meaning. (A1) restricts the evolution of an isolated prey population. A small population grows in time, but there is a carrying capacity of the environment beyond which the prey population will decline. In other words, the prey populations are self-supporting. According to (A2), the prey species may exhibit intra- and interspecific competition while predators are not present. Neither commensalism nor mutualism occurs among the prey populations in such a case. B y (A3), the predator does not support the growth of any prey species. Since the inequalities are not strict, model (1) admits the effect of a prey's refuge.

REMARK ON DISSIPATIVITY

175

Let us note that no assumption is imposed on OG/Sxi; an effect of a refuge or even an active defense of a prey species is allowed. Also, no assumption is imposed on OG/o~y; thus the predator may exhibit intraspecific competition or a cooperative hunting strategy. Let us execute a heuristic consideration before formulating the last assumption, which will be a substantial aid in proving the dissipativity of system (1). Put Xi

q~i(x,y) = -~" [ L ( x i ) - F / ( x , y ) ] ,

g(y)

(4)

= -G(0,...,0,y)

(5)

for x~>~0, y > 0 , i = 1 , 2 ..... n. Obviously, ~(0 ..... 0 , y ) = 0 i = 1, 2 .... , n. By (A3) and (A2), Xi

¢Pi(x,y) = 7 [ f / ( x i ) Xi

7 [L(x,) -

Xi

Fi( x I ..... x., y)] >I7

[f/(xi)-

F,(O,...,O,x,,O,...,O,O)]

for y > 0 ;

Fi( x I ..... x.,0)]

= o

for (x, y) ~ int R~_+ 1. Further, put K(x,y) =

G(x,y)+ g(y)

(6)

]~ ~Pi(X, y) i=1 for (x, y) such that ET= 1q~i(x, Y) > 0. System (1) can be rewritten in the form

.¢ci=xifi(xi)-y~oi(x,y),

i = 1 , 2 ..... n,

.9=Y(-g(Y)+ K(x,Y) ~ ~oi(x,Y)) • i=1

q~i(x,y) represents the size of the ith prey species destroyed by a predator of unit size in unit time; this quantity may depend on the sizes of both other prey and the predator, g(y) represents the intrinsic death rate of the predator in an environment without any prey involved in the model. Let us assume further that any partial resources of predator apart from the prey introduced into model (1) cannot provide for a large predator population. That is, the death rate g(y) of the predator is positive for large y. Stated more precisely, iiminfg(y) > 0. y--*~

176

ZDENEK POSPISIL

If x i and y denote the biomass of the ith prey species and of the predator, respectively, then K(x,y)will be interpreted as the efficiency of conversion of the prey biomass destroyed into predator biomass. In such a case, the mass preservation law will imply that K(x,y)~< 1, and consequently by (6), (5), and as ]2n,=l~oi(x,y) >/0, ~oi(x,y) - G ( x , y ) / > g ( y ) , i=1 for each x, y for which K(x, y) is defined. The considerations above suggest the formulation of the fourth assumption: (A4) For y > 0, define r ( y ) --inf

~i(x,n)-G(x,n):x

~R~,~ >y

,

i

where ~o~(x,77) are defined by (4), and assume that y -- limy _~~ F(y) > 0. Let us note that F(.) is a nondecreasing function. Thus limy_~ F(y) is defined, and we need not consider lira inf. Further, lira F ( y ) =

y --~ ec

sup{r(y):y > 0}.

A certain strengthened version of (A4) will be also useful: (A4*)

~oi(x,y)-G(x,y):x~N~+,y>O

Y0=inf

> 0 , where

i=

pi(x, y) are defined by (4). Obviously, if (A4*) is satisfied, then (A4) also holds. In particular, (A4*) is valid for predator-prey systems when the predator has no alternative resources of nutriment except prey species introduced into the model under consideration. Indeed, since g,i(0..... 0 , y ) = 0 , i = 1 .... ,n, there exists a constant 8 > 0 such that the death rate g ( y ) = - G ( 0 ..... 0,y) >/3 for each y > 0 by (A4*).

Example 1. Let us consider a standard intermediate Gause-type predator-prey system Yc=xf(x)-yp(x),

.9=y[-6+cp(x)],

where 3, c are positive constants, c ~<1 and f(-), p(.) are continuously differentiable functions satisfying (HI) f ( 0 ) > 0 , f ' ( c ) < 0 for all x > 0 and there exists a constant K > 0 such that (x - K ) f ( x ) < 0 for all x =~K, x > 0. (H2) p(0) = 0, p ' ( x ) > 0 for all x > 0.

REMARK ON DISSIPATIVITY

177

We have

p(x)

F(x,y)=f(x)-y

x

'

G(x,y)=-8+cp(x), = p(x),

aF(x,O) Ox = f ( )' t

aF(x,y) dy

X

---~

--

p(x) x

-

-

Assumptions (A1)-(A3) are obviously satisfied. Further, Y0 = i n f { ~ p ( x , y ) - G ( x , y ) : x = inf{8 + ( 1 -

c)p(x):x

>/0,y > 0} >10} = 6,

since p(.) is increasing. Thus (A4*) is satisfied too. It may happen that a particular system (1) does not satisfy assumption (A4). Sometimes in such a case, one can rescale y to get rid of the problem. The transformation y - - k ' q is diffeomorphism, and it doesn't influence the qualitative properties of the solution of system (1). Example 2.

Let us consider the Lotka-Volterra system ~ = x( A -

Bx-Cy),

~ = y( D + E x - Fy),

(7)

where all constants are positive, C < E. Equations (7) describe the evolution of a predator-prey system such that the prey under consideration is not the unique food resource for the predator. Assumptions (A1)-(A3) are obviously satisfied. ¢(x, y) = Cx, ~p(x, y ) - G(x, y) = - D +(C - E ) x + Fy, F(y) = - ~ for all y > 0 because C - E < 0. Let y = kT/. Then system (7) is transformed into .~ = x( A - Bx - CkT1),

• = 7( D + Ex - F k ~ ) ,

and q~(x,~)- G ( x , ~ ) = - D +(Ck - E ) x + Fk~l. If k is large enough that C k - E > 0 (that is, k > E / C ) , then F ( ~ ) = - D + Fk~! and 7 = lim n ~ o~F('q) = lira, _~~( - D + Fk~) = o~> 0. Let us note that Assumption (A4*) is violated for system (7) with D > 0 even if C > E. 2.

RESULTS

First, let us mention three simple consequences of assumptions (A1)-(A4). (i) f~(0)> 0, f ~ ( K i ) = 0 , i = 1,2,...,n. This fact is implied by (A1) and by the continuity of the functions Fi, i = 1, 2 .... , n.

178

ZDENEK POSPISIL

(ii)

For any i ~{1,2 ..... n}, the differential equation =

(8)

has the constant solution x(t) =- g i. Any solution x(.) of (8) with initial value x(0) > K i is strictly decreasing with lim t _~~ox(t) = g i because by (A1), f/(x) < 0 for x > K i. (iii) By (A4), there exists a constant L > 0 such that F(y) > 3,/2 for all y >~ L. THEOREM 1 Suppose that F/, i = 1,2 ..... n, and G in (1) satisfy (A1)-(A4). Then system (1)/s dissipative. Proof. Let xl(.) .... , xn('), y(') be a solution of (1), (2) defined on the maximal interval of existence [0, b). First we show that b = oo. By (A3) and (A2), Yci = xiFi(x,y ) <~xiFi(x,O) <<.xiFi(O ..... O, xi,O ..... 0,0) "~- x i f i ( x i )

,

(9)

i = 1,2 ..... n.

Denote /~i = max[Ki, x°] • Remark (ii) and the comparison theorem for differential equations (e.g., [8, II1.4.2]) imply xi(t)<~ Ki, i = 1,2 ..... n, for all t ~> 0. Further, set ~ / - min[-~,f,(0) ..... f , ( 0 ) ] , i = 1 , 2 ..... n,

C = ~[/~if,(O)+/xi],

S(')=Xl(')+'"+x.(')+y("

).

i=1

Let y(t)>~L for t ~ [ t o , r ) c [ O , b ) . [L is the constant introduced in remark (iii).] For t ~ [to, r), the following holds: S= ~..xifi(xi)-y i=1

i=1

~pi(x,y)+ y G ( x , Y ) i=1

i=1

REMARK ON DISSIPATIVITY

179

n

4-

Ex,L(O)-y-~+c i=1

<<.- ~,s + c . Let t ~ [to, T). By the comparison theorem mentioned previously,

S(t) ~ ~C+(s(to)-C)exp[~,(to-t)] Thus,

(10)

~=-+ Y

i

xi(to) + y ( t o ) -

exp[~/(to - 0 ] .

Since, by remark (i), ~/> 0, S is decreasing on [t0,z) and

y(t)<~C+

~.xi(to)+ y ( t o ) - -C~ ) exp[ Z/to] i=1

The solution Xl(.),x2(-) ..... xn(.),y(.) of (1), (2) is bounded, and the solutions prolongation theorem (e.g., [8, II.3.2]) yields that it is defined for all t >/0, that is, b = o¢. Let e > 0 be any number, and denote

K* = K i +

e,

i = 1,2 ..... n,

tl

c* = E [/Cf,(0) + ~,1, i=1

L = max[C*/-~,L],

L* = / ~ + e.

By (9) and remark (ii), there exists a T, possibly dependent on x/°, y0, i = 1 , 2 ..... n, such that & ( t ) < K* for all t >~T. Now, without loss of generality, we can suppose that x ° < K*, because (1) is autonomous. Suppose for contradiction that there is a T0 >/0 such that y(t) > L* for all t >/r 0. In a way similar to (10), we can show that

y(t) <~L +

xi(%) + y(T0)-- ~ i

exp[~(T 0 - t)]

for t >~r o.

180

ZDENl~K POSPiSIL

But the limit of the right-hand side of the last inequality is L < L* because 5' > 0. Thus there is a t 1 >/r I such that y(t~) ~ L*. If there is a r~ > tl such that y(~-a) > L*, then there exist a t 2 E[tl,7"l) and a t 3 > t 2 such that Y ( t 2 ) = y ( t 3 ) = L * and y(t)> L*> L for t C(tz,t 3) because y(.) is continuous. Again analogously to (10),

y(t)<~L*+

K*+Y(t2)--

~- e x p [ ~ ( t 2 - t ) ]

i

C* <~L* + (i=~ K ' + L * - --~- )exp[ {/t2 ] for t ~ It2, t3]. Since (1) is autonomous, n

y(t)~L*+

~K*+L*\i=1

=

2L*

-

--=-

~/

C*)

. exp[ y0]

T +

K*

i=1

for all t >~t I such that y(t)>~ L*. Thus the solution XI('),X2(" ) . . . . . xn(-), y(.) eventually enters and remains in the set (Xl, X2 . . . . . x . , y ) : O ~
~, +

K* .

(11)

i=1

An estimate of the ultimate bounds of solution orbits is given by (11). A more delicate estimate can be obtained using the strengthened assumption (A4*). THEOREM 2

Suppose that F/, i = 1,2 ..... n, and G in (1) satisfy (A1)-(A3), (A4*). Let e > 0 be arbitrarily small, and let al, a 2.... , a n be positive real numbers such that

/n ~=inf

Y'~ aiq~i(x 1..... x,, y ) - G ( x l , . . . , x , ,

/ y):O<~ x i ~0 >0.

i=1

(12)

REMARK ON DISSIPATIVITY

181

Let K*

#,=max[xfi(x):x>~O ],

= K i + o° ,

i = 1 , 2 ..... n, (13a)

y* = min[ f,(0), f2(0) ..... f,,(0), "~],

ai[K*fi(O)+tzi]+e.

M * = -~-

(13b) (13c)

i=1

If xl(.),Xz(.) ..... xn(-),y(.) is a solution of (1), then there exists a T>~0, possibly dependent on initial values, such that xi(t ) < K i,

y(t) + ~ aixi(t ) < M*

i = 1,2 ..... n,

i=1

for each t >~T. Proof. System (1) is dissipative by Theorem 1. It means that xl(t), x2(t) ..... xn(t), y(t) is defined for all t ~ [0,~) in particular. In a way similar to the proof of Theorem 1, we can show that there is a T* >/0 (dependent on initial values) such that xi(t) ~< K*, i = 1,2 ..... n, for each t >/T*. Define S(') = a,xl(. ) + a2x2(.) + -." + a,x,(.) + y(.). Then for t >/T*, n

S= L a i x i f i ( x i ) - Y ~_, aiq~i(x,y)+ yG(x,Y) i=1

i-1

i=1

i=1

n

+E

a,x,[L(o)+L(x,)]

i=1 n

<~-y*S+

Z ai[K*fi(O)+ I~i]. i=l

By the comparison theorem for differential equations,

1 Lai[K.fi(O)+lJ.i ] +( S(T*)- l~y*i=,Lai[K*fi(O)+ p'i]) exp[y*(T*-t)]' and since 3'* > 0, the proof is complete.

•

182

ZDENIEK POSPISIL

Let us note that constants a l , a 2.... ,a n exist by (A4*): in particular, a 1= a 2 . . . . . an = 1 satisfy condition (12). But another choice of these constants might improve the estimation of the solution bounds. The following example shows the efficiency of Theorem 2 in an estimate of the bounds of solution orbits in a special case of system (1).

Example 3. Let us consider the system 2x l+2xY,

~--x(1-x)-

(1 2x )~=y - ~ - + ~

)

(14)

.

This is a widely used predator-prey system with a type II functional response (see, e.g., [9]). The functions 2 l+2xY

F(x,y)=l-x-

1

2x

G ( x , y ) = - g + +2-------~ 1

and

satisfy (A1)-(A3), (A4*) with K = 1, 3'0 = 1/4. According to (13) and (12),

1

K*=l+e,

[

~=inf (a-l)

( x)l ~

/~ = ~-,

]

+~-,O~
where a is a positive real number and e denotes an arbitrarily small positive number. Now we have

l<~a

1/4,

;/= (2/3)(a-1)+1/4,

5/8
and "~<0, y*(a) =

0
1/4, (8a - 5 ) / 1 2 , 5 a + ~,

M*(a)=

15a/(8a-5)+ e,

l~
Thus min[M*(a):5/8
U={(x,y) ~[~Z:O ~
REMARK ON DISSIPATIVITY

183

Examination of the nullclines of (14) (see proofs of Theorems 1 and 2 in [4]) yields that each orbit of the system (14) eventually ends in the set

V={(x,y)

E ~ 2 : 0 ~ X ~ 1 + e,0~< y ~< 6.77}.

Since U c V, Theorem 2 gives in this case an even better estimation of the solution bounds than the delicate method developed particularly for two-dimensional systems. The dissipativity of system (14) cannot be proved by methods from [5] or [6] because

OF ax(X,y)

4y (1 + 2 x ) 2

1

is positive for large y. So the assumption that prey populations exhibit inter- and intraspecific competition in an environment with arbitrarily large predator populations is not satisfied even in a simple and widely used model. •

This work represents part of my doctoral thesis. It was done during my visit to the University of Bologna, Italy, supported by the TEMPUS 92 JEP 4547 grant. I am grateful to Professor E. Lanconelli, Bologna, for his kind hospitality and to Professor J. Kalas, Brno, for useful discussions during the preparation of the manuscript. Dr. V. Hutson, School of Mathematics and Statistics, The University of Sheffield, UK, read the first version of this paper, and his suggestions led to essential improvements. REFERENCES 1 V. Hutson and K. Schmitt, Permanence and the dynamics of biological systems, Math. Biosci. 111:1-71 (1992). 2 T. C. Gard, Uniform persistence in multispecies population models, Math. Biosci. 85:93-104 (1987). 3 V. Hutson and R. Law, Permanent coexistence in general models of three interacting species, J. Math. Biol. 21:285-298 (1985). 4 R. R. Vance, A general dynamical model of one consumer-one resource interactions, J. Math. Biol. 28:645-669 (1990). 5 H. I. Freedman and P. Waltman, Persistence in models of three interacting predator-prey populations, Math. Biosci. 68:213-231 (1984). 6 L. Turyn, Remarks on "Persistence in models of three interacting predator-prey populations," Math. Biosci. 110:125-130 (1992). 7 D. Mukherjee and A. B. Roy, Uniform persistence and global stability of two prey-predator pairs linked by competition, Math. Biosci. 99:31-45 (1990). 8 P. Hartman, OrdinaryDifferentialEquations, Wiley, New York, 1964. 9 J.D. Murray, MathematicalBiology, Springer, Berlin, 1990.