Limiting behavior for several interacting populations

limiting Behavior for Several Interacting Populations ANTHONY Department

LEUNG of Mathematics,

Communicated

University

of Cincinnati,

Cincinnati,

Ohio 45221

by C. Mode

ABSTRACT This article is concerned with the system ; = x(a - bx - 27, t cjyi), ii = aiyi(x - &), i=l ,..., n, where a, b, q, a,, pi are positive constants. The system is a model for n predators competing for 1 prey. The behavior as t+ + co is studied, for all solutions satisfying initial conditions x(0) >0, y,(O)>O, i = 1,. . .,n. Essentially, the sizes of the parameters & relative to a/b, determine the limiting situation.

1.

INTRODUCTION

This article is concerned solutions of the system

with the limiting

(

i(t)=x

a-bx-

i:

c$yi

behavior,

as t+ + co, for

)

i-l (A)

i=l ,.**, n,

.9i(t)=aiYj(x-Pi),

where a, b, ci, oli, pi, i = 1,. . . , n, are all positive constants. Only positive solutions [i.e., x(t) > 0, y,(t) > 0, i = 1, . . , , n] are analyzed, because they are of realistic interest. The system (A) is ‘a simple model for the biological phenomenon of n predators feeding on 1 prey, where yi, i = 1,. . . ,n, are the numbers of predators and x is the number of prey, The parameters a, b, cr,, j?,, ci are related to the growth rates, death rates, and interaction rates among prey and predators (cf. [l], [3]). The system (A) is also applicable to the economic study of n groups exploiting 1 single life resource (cf. [2], [4]). Results are limited to the case when there is a predator corresponding to both of the smallest values of ai and pi, i.e., when there is an integer k such that ak=min,
0 American

Elsevier

BIOSCIENCES Publishing

29, 85-98 (1976)

Company,

Inc., I976

85

ANTHONY

86

LEUNG

shown that only those predators corresponding to those i for which ,B,= & will survive; and they together with the prey will eventually tend to certain numbers dependent partially on their initial numbers (see Theorems 1 and 2). If, however, ,f$ > a/b, it is shown that no predators survive in the long run and the prey tends to the equilibrium size a/b (see Theorem 3). Certain interesting analytical situations are also found. Consider the three dimensional system in Theorem 1, when n =2. The points on the segment of the line {(x,_~~,,yJ:x=p. c,y,+c,y,=a-bp}, in the first octant, are all critical points. However, initial conditions of the solutions will continuously determine their limiting points along the line [cf. Eq. (l.l)]. Note also that there is no limit cycle for the system, from the results of the theorems in this article. Laboratory experimental simulation of host-parasite (or prey-predator) relations described by Eq. (A) is possible (cf. Lotka [6], Odum [7]). On the other hand, the results of this article serve to test biological assumptions concerning interactions between various parasites feeding on the same host (cf. Smith [8]). One simply observes the long run population proportions of the various parasites (or predators) and the host (or prey), and compare with the results predicted by Eq. (1.1) of Theorem 1, (2.1) of Theorem 2 or Theorem 3. The mathematical results in this article appear reasonably realistic as an expected outcome estimate for the cases where parasites do not interfere among themselves directly. 2.

THE SPECIAL

CASE

We will first analyze the special case where all the parameters fi,, i=l , . . . ,n, are the same. The result of this case will be used to study the general case in the next section. THEOREM

1.

In the system

(A), assume

o
for aNi=

l,...,n.

[Hl]

Let X0 = (xO,yIO,. . . ,yno), where x0, yiO, i = 1,. . . ,n, are arbitrary positive con, . . . ,y,(t)) be the solution of (A) and let X(t; &,X0)= (x(t),y,(t)

stunts;

satisfying the initial condition X (t,; tO, X,) = A’,. Then +( ,8J10,. . . Jnno)as t-+ + W. Here A =Jkko is the unique positive equation

where k is an integer such that ~=rnin,<~<~ i= 1,...,n.

X (t; t,,, X,) root for the

{cyi}, and~iio=yioyko4’“*~~~‘LI*,

LIMITING

BEHAVIOR

FOR INTERACTING

Proof. Let rik = CQ/CQ.Consider

POPULATIONS

the plane autonomous

87 system

(1.2)

If (ii(t),

G(t))

is the solution

of (1.2) satisfying

the initial

condition

(ii (to), 5 (la)) = (xo,Y,&, then (6 (z),Y LO~~rl* ii(r)“*, . . . ,ynoykord i;(t)‘*) is the solution X (t; t,,,Xa) described in the statement of the theorem. The function p (v) = b-‘[a -X1_, ciyiOy,%“*] has the property P’(Y) < 0 for all Y> 0. Therefore there exists a unique T > 0 such that P(P) = /3. Hence Eq. (1.1) has a unique positive root A = V=J~ Further, we find P”(Y) < 0 for v > 0. For convenience, let f(u, v) = a - bu -IX;_, cfli,, y~5~ vGk), g(u,u) = Q(U - j3). The two curves f(u,v) = 0, g(u,v) = 0 partition the first open quadrant, S’l, in the (u,v) plane into four open regions: !Z?,={(U,V)E D:f(u,v)O}; ~2,={(u,u)ED:f(u,v)O, g(u,v) ={(U,u)EQ:f(u,v)>O, g(u,v)O}, as indicated in Fig. 1. ( fiJkko) is a critical point of (1.2) at the

FIG. 1

ANTHONY

88 boundary of the four regions Jacobian for the transformation

LEUNG

Oi, i = 1,. . . ,4. The eigenvalues, p, of the (u, u)-t(uf(u, u), ug(u, u)) are given by

At (u, u> =(&J/co),

and

Hence ( ,kQko) is an asymptotically stable critical point for the system (1.2). The function (u,v)+( uf( U,u), ug( u, u)) satisfies the Lipschitz condition in the closure of a. From the direction of the vector field (uj, ug), one can readily show that solutions which are in Q, (or a,) for some t= t must either reach Q2,(or respectively a,) at some finite time later than ;, or tend to ( p, yko) within S12,(or a,) as r+ + co. Further, any solution which is in Q2 (or 52,) for some time, must enter 8, (or respectively St,). Note that the Lipschitz condition prevents the solutions from reaching the u or u axes. Another important observation is that one can use Green’s theorem to show that there can be no periodic solution for the system (1.2) contained in D (See Lemma 1.1 below). From the last paragraph, we see that if the solution (u”(t),u”(t)) does not tend to ( p,pko) as t-+ + co, there must exist t,, t, with t, > t, > to such that E(t,)=z?(t,)=p and v”(t2)>d(t,)>~ko [or _&o<6(t2)<5(~1)]. In the case where t?(t,)> fi(t,) >yko, from the smoothness of the vector field (uf,ug) in D and the asymptotic stability of ( fi,jjko) one concludes that there must be a number y with u”(t,)> y >yko such that the solution of (1.2) through the point ( p, y) is periodic. This contradicts Lemma 1.1. On the other hand, if yko < ij(t2) < u”(t,), then (ii(t), a(t)) must wind inward around a positive limit set inside 52 if (c(t), i; (2)) does not tend to ( /!Qko). By the PoincareBendixson theorem, this positive limit set is a periodic solution, contradicting Lemma 1.1 again. (11 (t), v”(t))-+ ( p,y;ro) as t-+ + cc . Consequently, Hence, X(t;to,Xo)=(~((t),y,Oy~‘~~~((f)r’*,...,~nO~~r~*~(i)rn*)~(P, t++a.

Y,o,...,.V~o) as

LIMITING BEHAVIOR FOR INTERACTING LEMMA

89

POPULATIONS

I.1

The system (1.2) has no periodic solution inside the first quadrant. Proof. The positive u and 0 axes are invariant sets. On the horizontal axis, ti=uf(u,O)>O for O
I [u(t)1

-‘+*‘“k[u(t)]-‘[zi(t)ti(t)-ti(t)ij(t)]dt

C

= /C

&‘/*u-‘g(u,tl)du-

u-l+b'""f(u,u)du.

(1.3)

This integral clearly has to be zero. On the other hand, Green’s implies that the line integral is equal to

Ij-

Bbu-’ + b/ak,

- ’

d,,

theorem

&,,

interior of C

which is strictly positive, because C is in the first open quadrant. 3.

THE GENERAL

CASE

We next investigate the last section. Let

(A) with an assumption

min

less restrictive

than that in

{pi}<%.

[H2al

l
Without loss of generality, we may assume /?, = & = . . . = &,, = minlGi<,, {Pi),mPmfori>m.Letx,,yiO,i=l,...,n,X,andX(t;t,,X,) be as described in the last section. THEOREM

2.

Assume hypothesis [H2a], and let the notation be as described aboue. Zf (A) further satisfies

aq= ,$Fm tail

<

,T-,‘:,{ai}

for some integer q,

1 < q < m,

[H2b]

ANTHONY

90 Then X(t;

t,,,X,J-+(

p,,9,0,.

,li,,,.O,.

.,O) as t+ + cc. Here

A=.$@

LEUNG

is the

unique positive root for the equation

and 9io =~~~,yg)~/~$$l~~, i = 1,. . . , m.

As in the last section, let rz4= a,/+ system

ti,=w,

If (E,(t), then

(

a-bw,-

5 i=

ciyioy~“~zLI”-

of (2.2) satisfying

X(t; to,Xo) stated in the theorem.

h,(w,,z,,t)=

a-bw,i

and

the two dimensional

1

E,(t)) is the solution

is the solution

and consider

fj i=l

(a,( to), i,( to)) = (xo,y&,

For later convenience,

~~,~y;I’qz+-

let

LIMITING

BEHAVIOR

FOR INTERACTING

We will next prove two lemmas from which Theorem LEMMA

91

POPULATIONS

2 follows readily.

2.1

Given any 6 > 0, there exist corresponding small 8, > 0 and large T > 0 such that: if (w,(t), z,(t)) is a solution of (2.2) with

for some L> T, then (w,(t),z,(t))E a(6) for all t > F; further, if 8 above is sufficiently small, (w,(t),z,(t))+( /3,,yq0) as t+ + 00. Proof.

From

(w,,z,)

(2.1)

one

finds H( p4,~4,,, t)= 0 (e-“‘) where The Jacobian of H(w,,z,,t) with respect has the form A + M(t), where M(t)= O(e-of) and

{cti(~i-&)}.

=minm+,
at (&j+,)

A=

-

u to

bPq i-l

1.

Using Taylor’s theorem, write the system (2.2) in the form 6’=AV+M(t)V+Q(V,t)+H(&&,,,t),

(2.3)

where V(t)= (w,(t) - Pq,zl(t) -.9&), and Q (V, t) is a column vector satisfying 1Q (V, t)l< K,[ VI’, K, a positive constant, for all t E[ t,, + co) and 1V( sufficiently small. The fact that &, c,, yiO, y@, riq, g@, aq, b are all positive implies that both eigenvalues of A have negative real parts. Hence, by the variation of parameters formula, 1V(t)1 < K2e-e(‘-i)l + K2

I;

V(i)1

‘e- ““-“‘IM(~)V(~)+Q(~(S),S)+H(P~,~~,S)I~~

(2.4)

for t > i> t,,, where K2, 8 are some positive constants. Here we may assume 0 < (I. Next, choose any E such that 0 < e < 0. Let IH (/3,,9@, t)l < K3ee0’ for t > t,,; and let K,=max{ K,, K,, Kj}. There exist small 8 and large T, with O to, such that: \M(s)V(s)l<(~/(2K,)) IV(s)1 for SE [T, + co), and lQ(V(s),s)( <(e/(2K4)) IV(s)] for s e[T, +co), whenever (V(s)1

ANTHONY

92

LEUNG

< 8. Hence (2.4) implies that


(n-8)-‘+rJ’e

e(s- =)I V(s)lak

(2.5)

T

aslongas IV(s)l
I~(t)l~[K41V(T)I+K~e-~T(a-_)-1]e-(e-’)(’-T)

(2.6)

whenever (2.5) applies. We may assume that we have chosen T such that K:e-“=(a - 0)-’ < (J/2). Thus from (2.6), if ) V(T)1 < 8/(2K,), then 1V(t)1 < 8 for all t > T. We can therefore let t-+ + 03 in (2.6) to conclude that 1V(t)l+O as t-+ + M. Since we can replace every T in (2.5), (2.6) by ;I;> by choosing 6, < $/(2K,). LEMMA

T, the lemma is proved

2.2

There exists a compact set 3 properly contained in the first open quadrant L? in the (w,,z,)p/ane, such that the solution (G,(t),?,(t)), which has initial value (G,(t,),E,( to)) = (xO,yqo), wiN remain inside CR for aN t > t,. Proof: Assume that there exist a sufficiently large f >0 and a small s”>O, such that (G,(t),?,(t))@ Ci.3 (6) f or all t > F. Otherwise this lemma is trivial, because of Lemma 2.1. As in Theorem 1, the curves

f(w,,zl)c

a-bw,(

5 c,v,~~;~~z;‘~ =o 1

i=l

partition d into 4 regions bi, i = 1,. . . ,4, as defined for Cli, with Oi replaced by di, (u,u) replaced by ( w,,z,), and f,g replaced by f,g, respectively. The configuration is analogous to that of Fig. 1.

LIMITING BEHAVIOR FOR INTERACTING

POPULATIONS

Let (+,,i) be a point on the straight line ~~,=a/b, ($(t),&(t)) be the solution of the system

I&=

w1

(

a - bw,-

93 with i>i@

Let

5 ~y~~y,$~i~zp, i=l

(2.7)

iz= z*oq(w2- Pq) satisfying the initial condition (6z( t,), Zz( t,)) = (16,i), t, any real number. There must exist a first t = t, > t, such that (&(tZ),ZZ(tZ)) first reaches a point (a,;) vertically above (&~,r,,). For (a,j3) in the (w,,z,) plane, let a,/?), z,(t; t’,a,fi)) be the solution of (2.2) satisfyG,(t;t’,a,p)=(w,(t;tO, ing G,(t’; tO,cu,fi) = (a,P). Let T > to be as described in Lemma 2.1 corresponding to a sufficiently small 6, and T, > T. Consider G,(t; T,,i,.$. We have +,(t; T,,$,$
in 6. Also, the functions on the right of (2.2) are Lipschitz in (w,,z,) in the closure of d for t > to. Hence, G,(t; T,,;,.?) cannot reach the vertical axis w, =O; and there must be a first t= T,> T, such that G,(T,; T,,8,6) first reaches the curvef(w,,z,)=O in 6, on the left of the line w,= &. Continuing the solution G,, one sees from the signs of 3,, i, that G,(t; T,,$,$ must either (i) reach a point vertically below ( &,yo) at a first t = T3 > T2; or (ii) tend to (/3,,9& within 6, as t-+ + co, while never reaching the line w, = & for t > T2. By means of the orbits ($(t),lz(r)) and G,(t; T,,;,,?), we will construct a region which will contain the solution (9,( t),Z,(t)) for all large t. Consider case (i). Let 9 be the compact region in fi bounded by the four curves in the (w,,z,) plane: r,~{(lqt),Z,(f)):t, lY2s{ G,(t; T,$,?):

a-

Q t < t*},

T, < t Q T,},

2 ci~~~y~~qz~(T~;T,,5,5) i-l

1

,~,(T~;T,,i,i)
I .

ANTHONY

94

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FIG. 2

Refer to Fig. 2. The solution (I?, (l),.?, (t)) of (2.2) cannot stay in 5, for all t in an interval of the form k < t < co, where k is a large number, because it would have to reach the line wr = & above (&y& if it did not enter 53 (8) for large t. As in the last paragraph, one sees that (a, (l),Zr (t)) cannot reach the axes at finite time, and has to enter 6, after it enters f12. Hence, there must be some l=s,> T3 such that (3,(s,),~,(s,))~~,~~,u {( j3 ,zJ: 0< zr s, such that (~,(sz),tlI(s2)) reaches the curve f(w,,z,)=O to the right of the line w, = /3, in fl. Hence it will enter ci, for some t > s2. Using the fact that Zj,(t),i*(t) are independent of t, and k,(t)< tiL2(t) if (G,(t),E,(~))=(w~(t),z~(~)) in 6, [here, ( w2(l), z2 (t)) is a solution of (2.7)], one concludes that (Gr (t), Z, (t)) cannot cross T, to leave 9 at t > s2. Checking the signs of the derivatives of the first and second components respectively in T, and r3, one sees that (Gt (t),,?, (2)) cannot leave 9 crossing r4 or lY3 for t > s2. Further, if (n, (tL2,

(t))

and

($,(t),:,(t))

are

two

solutions

of

(2.2)

with

LIMITING

BEHAVIOR

FOR INTERACTING

95

POPULATIONS

Y,>Y~, then k(~i)>6(~3

(~~(y,),z,(y,))=(iiil(y2).il(y2)),

and or

= .Zr(yJ. Hence, (%,(t),?, (1)) cannot cross I2 from inside 9 at t > s2(> TJ. Therefore (i?, (t),Z, (t)) cannot leave 9 after it enters it later than s2. This proves the lemma for case (i). Consider case (ii). Denote I1 as before. Let 8 > 0, Ti > 0 be respectively small and large numbers such that any solution of (2.2) with (w, (t),.z, (1)) E 93 (8) for t > Ti will tend to ( &$J as t+ + cc. We may assume (b,i^) 4 33 (8). Let r= F3 be the first r such that G,(r; T,,$,$ reach the boundary of 53 (S), and let F,s { G,(r; T,,t%,$ : T, d r < T3}. As in the last paragraph, we see that (+,(r),Zl(r)) will enter O,n {(w,,z,):w, < a/b} from 6, for r> F3. %,(r),?,(r)

cannot later cross over r, or r2, and will eventually enter 33 (8).

Hence, the lemma follows.

(cu,p)~b, let G,(r;s,,a,p)= of (2.7) satisfying the initial condition G2(s,;s0,a,P)= (cu,j3). The system (2.7) is the same type as (1.2) with w2,zz, m, q, rIq,crq,& respectively replacing u,v, n, k, r,, a,,j3. Hence, ( P,,F& is an asymptotically stable critical point for the system (2.7), and all solutions in the first open quadrant tends to it as r+ + 00. For each g>O there exists a 8; > 0 depending on ? such that G,( r; so, (Y,p) will remain in an c-ball of (/3,,$> once it enters the &-ball of (&y,rJ. Further, for each (c,d)E& there must exist a T(&,c,d) such that jG2(r,s0,c,d)-(&$,&I < 6;/2 whenever r-s,> T(i&c,d). Moreover, from the smoothness of the vector field in the first open quadrant, there exists a neighborhood V(c,d) of (c,d) in fi such that Proof

of

Theorem

2. For

each

( w2(r; s,,, (Y,/I ), z2 (r; s,,, a, p )) be the solution

IGz(so+T(6;,c,d);s,,c’,d)-(P,,~~)I<6; Consequently, there exists a finite number (q,,q2) m the compact set ‘3,

for all

(Z,d ) E V(c,d).

N = N (Z) such that for all points

IG2(s,+r;s0~q1~q2)-(P4,~~40)l
w3)

for all r > N(Z). Note that N(e) is independent of s@ Let G,(r;s,,,a,,8) be defined as in the last lemma. From the proofs of the previous lemmas, solutions of both systems (2.2) and (2.7) cannot leave a bounded subset of the first open quadrant if they initially start in 9. In such a bounded subset, H(w,,z,,r) satisfies the Lipschitz condition uniformly for all large r, with Lipschitz constant, say, L. From (2.2) and (2.7),

ANTHONY

96

LEUNG

we have

n +

I: Wz(S;Sg,q~,qJ

C~ioYq~r~~e-'('-8~)(S-'O)Z~(S;Sg,qI,q~),0

i-m+1

(2.10) for all (q,, q2) E 3, t > so, For so sufficiently large, the norm of the last term on the right of (2.10) is bounded by jf,C,e-osds < Cze-“o for some constants C,, C,. From (2.9), (2.10) one obtains

IGl(t;so,ql,qd- Gdt;SO,ql,qJl < Cs-“‘o+

s‘LI(G,-

G2)(s;s0,ql,q2)lds

so

for so sufficiently large, t > so. Gronwall’s inequality gives

IGl(cso,ql,q2)--Wl;so,q,,q2)l Q Ge-““exp{~(~-.d),

(2.11)

which is true for t = s,,+ N (F). Choose so= s,,(q sufficiently large that C,e-OSO(~exp{ LN (q} < Z. Hence, (2.11) with t = sa(q + N (0 and (2.8) imply that (G,(so(Z)+N(Z);so(Z),q,,q,)-(P,,~~)I<2~

(2.12)

for all (q,, q2) E 3. Since ? is arbitrary, Lemmas 2.1 and 2.2 and the inequality (2.12) imply (a,(t),Z,(t))--+( /3,,y& as t+ + co. Therefore, Theorem 2 follows, in view of the paragraph immediately preceding Lemma 2.1. 4.

THE SIMPLE

To examine to the quantity

CASE

other possible distributions of the parameters pi in relation a/b in the system (A), the following theorem is included.

LIMITING THEOREM

BEHAVIOR

FOR INTERACTING

POPULATIONS

97

3

In the system (A), assume that there exists j, 1 < j < n, such that

{ pi } = pj > g

min l
[H3al

and min

{ai}=ali.

l
kI3bl

Let x0, yio, i- l,..., n, X0 and X (t; to,Xo) be as described in Theorem 1; then X(t; to,Xo)+(a/b,O ,..., 0) as t++ 00. Proof. Let riJ= ai/aj.

Consider

the system

(3.1) Let

J”(h,k)=a-bh-

2 cflioy,Grukru, iES

where S={i:i integer, 1 < ifn, fii=pj}; i(h,k)=oj(h-Pj). Hypothesis [H3a] implies that the curves f(h, k)=O and i(h, k)=O do not intersect in the first open quadrant, as indicated in Fig. 3. Let (i (t),k” (t)) be the solution of system (3.1) satisfying (h”(to), k”(to)) = (xo,ylo). Firstly, observe that (6 (t), L (t)) cannot leave the first open quadrant, has to enter the region left of the line h = pi and remains in it. Secondly, the curves for a-

bh-

2

c~,yj~'~e-'4(~~-8,)('-'~)k'u=0

i-l

in the first quadrant t++co. In the case parallel to the From the sign enter the open

become arbitrarily

close below the curve _f(h, k) = 0 as

pi > a/b, draw any small open rectangle whose sides are axe; with the point (a/b,O) as midpoint on the lower side. of h(t) and R(t) for large t, one sees that (h”(t), i (t)) has to rectangle and remain in it for all large t. In the case pi = a/b,

98

ANTHONY

LEUNG

FIG. 3

one sees that (&(t),k”( t)) has to enter and remain in the Lntersection of the above open rectangle with the region {(h, k) : h > 0, k > O,f(h, k) > 0}, for all sufficiently large f. Hence, in both cases, (h”(t),~(t))-+(a/6,0) as t++ 00. However, the solution of the given initial value problem, X (t; t,,,X,J, is

which ‘tends to (a/b.0

,..., 0) as t++

co.

REFERENCES M. Hirsh and S. Smale, Differential Equations, Dynamical Systems and Linear Algebra, Academic, New York, 1974, pp. 255-275. A. Leung and A. Wang, Analysis of models for commercial fishing: mathematical and economical aspects, Economerrica, to be published. A. Rescigno and I. Richardson, The struggle for life; I, Two species, Bull. Math. Biophys. 29, 377-388 (1967). V. Smith, Economics of production from natural resources, Am. Econ. Rew. 58, 409-43 1 (1968). F. Albrecht, H. Gatzke, A. Haddad and N. Wax, The dynamics of two interacting populations, J. Math. Anal. Appl. 46, 658470 (1974). A. Lotka, Elements of Physical Biology, Williams & Wilkins, Baltimore, 1925. E. Odum, Fundamentals of Ecology, 2nd ed., Saunders, Philadelphia, 1959. F. Smith, Population dynamics in Daphnia magna, Ecology 44, 651663 (1963).