Positive nonlinear difference equations

Nonlinear Analysis, Theory, Methods dr Applications, Vol. 30, No. 1, pp. 301-308, 1997 Proc. 2nd World Congress of Nonlinear Analysts

0 1997 Elwier Science Ltd

PII: SO362-546X(%)00274-X

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POSITIVE NONLINEAR DIFFERENCE EQUATIONS

ULRICH KRAUSE

Fachbereich Mathematik und Informatik, Universitgt Bremen, 28334 Bremen

Key words and phrases: Nonlinear difference equations, global asymptotic stability, stability trichotomy, non-autonomous equations, path stability, Poincare’s Theorem.

1. INTRODUCTION

As is well-known, even very simple nonlinear difference equations can show a complicated asymptotic behavior which is analytically often very difficult to handle. It has been recog- nized, however, for difference equations as well aa for discrete dynamical systems, that being confronted with nonlinearity it pays analytically to exploit properties of possitivity at hand (cf. [8, 91). To illustrate, consider a first order difference equation (Ae)

u(t + 1) = f(t+)), t E lW = {0,1,2,. . .} (1-l)

where f is a selfmapping of the cone of nonnegative real numbers R+ and ~(0) E W+. As shown by the logistic diflerence equation given by f(z) = Xs(1 - z)+ (X > 0, T+ = max{r, 0)) the asymptotic behavior still can be complicated. It has been argued by biologists [13] that, instead of using a logistic Ae, population pressure may be modeled also by f(z) = Xz( 1 + z)-‘, the so called Pielou di$erence equation [2, 61. The two models have some similarity and they differ in that in the logistic case population pressure is modeled in an additive manner by (1 - z)+ whereas in the Pielou case it is modeled in a multiplicative manner by (1 + z)-‘. The asymptotic behavior in the two models is completely different, whereas the logistic model exhibits chaotic behavior for certain values of X the Pielou model has a unique strictly positive equilibrium which is globally asymptotic stable (for initial values u(O) # 0). The Pielou model generalizes to

f(x) = xz(l + a~)-~ + cz + d U-2) with parameters A, a, b, c, d E R+. By the generalized Pielou difference equation obtained from (1.1) and (1.2) one is able to model a strictly decreasing growth rate as it is the case in the logistic model (the reproduction function (1.2) may possess a “bump”). The generalized Pielou equation also shows a stable asymptotic behavior. Namely, if f satisfies some contractivity condition then all solutions tend either to 0 or to a unique (strictly) positive equilibrium or to infinity. This behavior is called stability trichotomy and it will be addressed in Section 2 for nonlinear difference equations of arbitrary order.

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If in (1.1) the law of motion f itself depends on time, i.e. for a non-autonomous Ae, stability trichotomy cannot longer be expected. Nevertheless, assuming contractivity, solutions show path stability in that any path, how irregular ever, comes after perturbation from the outside finally back to its original behavior. This phenomenon will be considered in more detail in Section 3. Sticking to the non-autonomous situation in Section 4, a nonlinear version of a theorem of PoincarC is presented which, under appropriate conditions, states that for all solutions the ratio $$I tends to the same limit for t approaching infinity.

In the paper various results from the literature are reviewed (without proofs) and examples/ counterexamples are discussed as well as relationships between the results.

2. STABILITY TRICHOTOMY

Generalizing equation (l.l), consider a nonlinear autonomous Ae of order n

U(t+“)=f(u(t),...,u(t+n-l))$EN (2.1)

where f: K + W+, K = Iw;: = {z = (21,. ..,~,)1z;~W+}andu(i)~lR+forO~i~n-1. This Ae is said to have stability trichotomy if for all solutions t w u(t) with initial conditions

u= u(O)+(l),..., ( u(n - 1)) E K exactly one of the following three cases holds:

(i) j&u(t) = 00 for all U # 0

(ii) &%21(t) = 0 for all U

and r = 0 is the unique root of f(r, . . . , r) = r

(iii) ji+z u(t) = r* > 0 for all U # 0;

T* is the unique positive root of f(r, . . . , r) = r and r* is globally asymptotic stable (for u # 0).

THEOREM 2.1 ([8]) Supp ose that f is continuously differentiable on the interior i with f(z) >

0 for x # 0 and that f is contractiue on i in the sense that

~xil~(x)I < f(x) for all x &. i=l t

Then (2.1) has stability trichotomy.

EXAMPLES

(a) Generalized Pielou equation

u(t + 1) = f (u(t)), u(O) E W+, f(x) = Az(l + a~)-~ + cz + d with A > 0, a > 0,b > 0 and c 2 0,d 2 0. If b 5 max(2, f + 1) then f is contractive and stability trichotomy holds by Theorem 2.1.

If, e.g., a = 1,b = 2, X = 5,c = d = 0 then f(x) = 5x(1 + x)-” has a “bump” with a maximum at x* = 1 and an equilibrium r* = fi - 1 > x*. Thus, only case (iii) of the stability trichotomy survives and r’ is globally asymptotic stable (for u(0) > 0).

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(b) Bobwhite quail populations ([12]) The difference equation for those populations is given for n = 1 by f(z) = kr( 1 + zk)-l +

CIC + d with X,k > 0 and c,d 2 0. If k 2 2 + 4f then f is contractive and stability

trichotomy holds. If in addition 0 < c < 1 < X + c then the equation has a unique positive equilibrium which is globally asymptotic stable (for u(0) > 0).

From Theorem 2.1 a corollary is obtained which employs a weakening of contractivity, partial contractivity.

COROLLARY 2.2 Let g: K ----+ W+ be continuously differentiable on i with g(z) > 0 for x # 0. Suppose that g is partially contructiue, that is for arbitrary fixed jii E R+, i # j, the selfmappings Xj M g(Zr,. . .,Xj,.. . 2,) of R+ are contractive for all j E { 1,. . . , n}. Then

for f(xr,... ,x,) =g(xy,...,x; ) with (Y; such that 0 5 a; and 5 cri 5 1 the Ae (2.1) has i=l

stability trichotomy.

EXAMPLE. Rational recursive sequences ([5]). Th e mapping g: W: + W+ defined by

g(xl,xz) = (a + bxz)(A + x1)--l f or a, b, A > 0 is partially contractive. Hence, by Corollary 2.2, the mapping f( x1, x2) = (a + bzF)(A + x;l)-l with 0 5 oi and (~1 +crz 5 1 is contractive

and the second order Ae u(t + 1) = f(u(t),u(t + 1)) h as stability trichotomy. If err # 0 then only the third case of the stability trichotomy survives, that is the Ae has a unique positive equilibrium which is globally asymptotically stable. The case where or = oz = 1 is much more difficult to handle and has been intensively studied in the literature [5,6]. There, global asymptotic stability for the unique positive equilibrium has been proven for certain constellations of the parameters, e.g. a < bA, but is still a conjecture in the general case (cf.

PW Using Euler’s Theorem the following consequence is easily obtained from Theorem 2.1.

COROLLARY 2.3 Let f be continuously differentiable on i with f(x) > 0 for x # OSuppose f is non-decreasing in each component and positively homogeneous of degree 0 5 p < 1, that is I = #‘f(z) for all x E K, all X 2 0 where p < 1. Then the Ae (2.1) has stability trichotomy.

EXAMPLE. Consider the nonlinear Fibonacci difference equation of n-th order u(t+n) =

u(t)” + u(t + 1)” + . . . + u(t + n - 1)P where 0 < p < 1. Corollary 2.3 applies, and because

fb , . . . , r) = r has the unique positive root r* = n* with Q = (1 - p)-’ the equilibrium r* is globally asymptotically stable (for 2T # 0).

An interesting case not covered by Corollary 2.3 is p = 1. To this case we will come back in Section 4.


3. NON-AUTONOMOUS DIFFERENCE EQUATIONS: PATH STABILITY

Consider the non-autonomous nonlinear Ae of order n

U(t+n)=ft(u(t),...,u(t+n-l)),tEN (3.1)

where ft: K -+ W+ for all t E IV, K = IIP; and u(i) E W+ for 0 5 i 5 n - 1. The Ae (3.1) is said to possess path stability if for any two solutions t e u(t) and t H v(t)

with initial conditions u = ( u(O), . . . 7 u(n - 1,) and v= ( v(O),... ,v(n - 1)) it holds that

pim+ 1 u(t) - v(t) I= 0.

This is equivalent to say that any solution t e u(t) when arbitrarily perturbated (within K) at points in time to, to + 1 , . . . , to + n - 1 for arbitrary to E M comes finally back to its original asymptotic behavior.

THEOREM 3.1 ([S]) Let for all t E N ft be continuously differentiable on & with ft(z) > 0 for

z # 0. Suppose the family (f ) t t is uniformly contractive on i-, that is

~z~I~(z)I 5 cft(z) for all 2 &‘, all t E N id t

(3.2)

with some constant 0 5 c < 1. Then

u(t) lim - = 1 for any two solutions with ii, 6 # 0, t-ma v(t)

and path stability holds provided there is at least one bounded solution.

EXAMPLE. Consider the following non-autonomous generalized Pielou-equation:

Ae (3.1) for n = 1 and ft(z) = X(t)z(l + a(t)z)-2 + d(t). f t is continuously differentiable

w4t) on the interior of W+ and ft(z) > 0 for 2 > 0. Suppose that (Y = inf - teN A(t)

> 0. This

t conditionUr)plies that (f ) t is uniformly contractive with c = (1 + 4a)-‘. Hence, by Theorem

3.1, lim - = 1 for any two non-trivial solutions. Furthermore, f+b~ v(t)

1 A(t) ft(z) 5 za(t) + d(t) for all 5 E HP+ all t E N.

Assuming, in addition, that ;x d(t) < 00 it follows by Theorem 3.1 that path stability holds.

In particular, path stability does hold if all three parameters are bounded in time with strictly positive lower bounds.


4. NON-AUTONOMOUS DIFFERENCE EQUATIONS: A NONLINEAR POINCARti THEOREM

Consider a linear difference equation of order n with time-dependent coefficients, that is

U(t + n) = po(t)u(t) + p1(t)u(t + 1) + . . * + pn-l(t)u(t + n - l), t E N* (4.1)

Suppose that /iz p; (t) = pi for all 1 2 i < n - 1. (4.2)

A well-known Theorem of Poincare’(cf. [l, 2, 4, 61) states that in case all roots of the limiting n-1

characteristicequation C PiX’ = Xn h i=O

ave distinct mod&, for every nontrivial solution of (4.1)

there exists a root X of the characteristic equation such that

lim ‘tt + ‘) - X ~+~ zl(t) . (4.3)

We shall consider this Theorem in a nonlinear but positive setting. Consider the nonlinear non-autonomous Ae

up + n) = f&l), . . . , u(t + n - l))$ E IV (4.4

where ft: K + R+,K = W; and u(i) E R+ for 1 2 i 5 n - 1. Suppose that for some continuous mapping f: K + W+

&z f*(z) = f(x) for all z E K (4.5)

with uniform convergence on (x E K 1 11 z II= 1) (11 . II an y norm on B” such that 0 5 x 2 y implies II 2 II I II Y II.>

THEOREM 4.1 Let for Ae (4.4) assumption (4.5) be satisfied and assume that for every t E W ft is non-decreasing in each component and positively homogeneous, i.e. fto(x) = Aft(x) for all x E K, all X E R+. Furthermore, suppose there exist nl, . . . , nr E { 1,. . . , n} with r 2 2, nr = 1’ and gcd{n - nr + 1,. . . , n-n,+l}=lsuchthat

0 L x I y, xmi < ymi for some i implies f (2) < f(y).

Then for every nontrivial solution of t C) u(t) of (4.4) it holds that

(4.6)

(4.7) where X* is the unique strictly positive root of f (1, X,X2,. . . , A’+‘) = X”.

Before presenting examples and counterexamples, respectively, some comments on Theorem 4.1 are in order.

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Second World Congress of Nonlinear Analysts

(i) For a special case of condition (4.6) namely ni = 1, n,. = n, a proof of the Theorem can be found in [lo].

(ii) For the special case ft = f for all t E N a stronger conclusion can be obtained namely so

u(t) called relative stability, that is lim ~ exists for every solution. (The limits depend on t-k00 j$*t the initial conditions ii; X’ as in Theorem 4.1. For relative stability see [3, 7, 81.) That in general relative stability cannot be expected for the non-autonomous case is illustrated by the counterexample below. It would be interesting to know under which additional assumptions relative stability holds also in the non-autonomous case.

(iii) Conclusion /i~z (u(t))+ = A* in Theorem 4.1 is a consequence of lim u(t + 1) A’ t-+coU(t)= *

(This is similar to Wimp’s method in [2].) As shown by Example (a) below‘the former

conclusion may hold without the latter to be true. Property j& (u(t))+ = A* can be

viewed also as saying that the Lyapunov-exponent ji+; j log u(t) exists for every solution

(for u(0) > 0) and is equal to log X*.

(iv) Theorem 4.1 does not comprise Poincar6’s Theorem because the former presupposes pos- itivity properties but the latter does not. Assuming in Poincare’s Theorem pi(t) 2 0 for all t and all i and the monotonicity assumption (4.6) Theorem 4.1 sharpens Poincare’s

Theorem in that lim u(t + 1) ~ yields the same root for all nontrivial solutions. Fur-

t-k‘3 u(t) thermore, in the positive setting the latter conclusion may be more easily obtained by Theorem 4.1 than by checking all the roots in the characteristic equation as in Poincare’s Theorem. Moreover, as will be illustrated by Example (a) below, it may happen that

u(t + 1) Theorem 4.1 yields lim -

t+ccJ u(t) = X‘ for all solutions althozlgh not all roots of the char-

acteristic equation have distinct moduli. Thus Theorem 4.1 may be applicable where Poincare’s Theorem is not.

COUNTEREXAMPLE. Consider Ae (4.4) for n = 2 and ft(zl, 2s) = &$(zi + 5s). Obviously,

&,mft(z) = f(z) = f(zi + ~2) with uniform convergence on {z E “9: ] 51 + 2s). ft is

non-decreasing in each component and positively homogeneous. Condition (4.6) is satisfied because f is strictly increasing in each component. Thus, all assumptions of Theorem 4.1 are satisfied. i(l + A) = f(l,x) = A2 has the unique positive solution X* = 1. By Theorem 4.1, therefore,

lim 21(t + ‘) - = piI (u(t))’ = 1. t+~ u(t)

Relative stability, however, does not hold. For the solution with initial conditions u(O) = 4t) 1, u(l) = 2 it follows by induction that u(t) 2 t + 1 for all t E N. Therefore, F = u(t)


does not converge for t -+ 00. Since the considered Ae is linear and the limiting characteristic equation f(1 +A) = A2 has +l and -i as its roots we might as well apply Poincare’s Theorem. Since for nonnegative solutions A = -f cannot occur we arrive at the same conclusion as by applying Theorem 4.1.

EXAMPLES.

(a) Consider the simple linear and autonomous Ae’s of order 4 given by f(si, 22,53,x4) = i(zi + xi) for i = 2,3,4 respectively. For i = 2 and i = 4 all assumptions for Theorem 4.1 are satisfied (ft = f f or all t). Since $(l+ Xi-‘) = f(1, X, A2, X3) = A4 has the unique

positive root X’ = 1 for i = 2,3,4 it follows from Theorem 4.1 that lim - ?dt + 1) = 1 for +hx u(t)

all nontrivial solutions . This conclusion cannot be obtained from Poincare’s Theorem because the limiting characteristic equation f(1 + X’-‘) = A4 possesses for i = 2 as well as for i = 4 two different complex conjugate roots and, hence, the roots do not have distinct moduli. Considering the case i = 3, condition (4.6) of Theorem 4.1 is not

u(t + 1) satisfied. Indeed, jiz - =

4) X’ = 1 does not hold in this case for all nontrivial

solutions. For, taking as initial conditions u = (1,2,1,2) it follows that

u(t) = 1, t even 2, t odd

and, hence, u(t + 1) - does not converge for t + 00.

u(t) For this solution, nevertheless,

&E (u(t))’ = A’ = 1.

(b) Consider the he (4.4) with ft(x) = ,z$r (ail(t)xl + . . . + ai,(t)x,) where aij(t) 2 0

and ii& aij(t) = oij for all 1 5 i {mandl <j 5 n(m,n E NY). ftisnon-

decreasing in each component and positively homogeneous for every t E N. For f(z) = ll$~m(ailX1 + *. * + ainxn) lim ft(z) = f(z) for all z E K = W; and the convergenceis t+ca

uniform on {Z E K 1 11 CC II= 1) (11 z )I= ,$i I xi I). Supp ose for the coefficients o;j that

there exist ni, . . . , n,. E (1,. . . ,n}r>2,ni=l, gcd{n-nr+l,..., n-n,+l}=l such that ain,, > 0 for all 1 5 i 5 m and 1 5 k 5 r. Then f satisfies assumptions (4.6) in Theorem 4.1. Therefore, by Theorem 4.1, it holds for every non-trivial solution of (4.4) that

u(t + 1) t%%~ = A‘ and /;z (u(t))’ = X*.

Thereby, X’ is the unique positive root of the equation ,rnnm ( o;i+o;2A*+. . .+o;,X*(‘+‘)) =

X’“. The same argument applies, under the assumptions-made, to the mappings ft(x) =

l~iym (aa( + - . - + ain(t)x,). Th e result obtained can be used to check persistence of --


the solutions of (4.4) by examining the root A* for the limiting mapping f only. If A* < 1 then all solutions tend to 0 and if X’ > 1 then all nontrivial solutions are unbounded.

Thus, a necessary condition for persistence of all solutions (i.e. for each solution t I-+ u(t) exist 0 < (Y 5 /3 such that a 5 u(t) 5 /3 for all t E N) is that A* = 1. This condition, however, is not sufficient as can be seen from the counterexample above.

REFERENCES

1. AGARWAL R.P., Di$erence Equations and Inequalities, Marcel Dekker, New York (1992).

2. ELAYDI S., An Introduction to Diflerence Equations, Springer, New York (1996).

3. FUJIMOTO T. & KRAUSE U., Strong ergodicity for strictly increasing nonlinear operators, Lin. Alg. Appl. 71, 101-112 (1985).

4. KELLEY W.G. & PETERSON A.C., Diflerence Equations, Academic Press, Boston (1991).

5. KOCIC V.L., LADAS G. & RODRIGUES I.W., On rational recursivesequences, JMuth. AnaLAppl. 173, 127-157 (1993).

6. KOCIC V.L. & LADAS G., Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht (1993).

7. KRAUSE U., Relative stability for ascending and positively homogeneous operators on Banach spaces, J. Math. Anal. Appl. 188, 182-202 (1994).

8. KRAUSE U., Stability trichotomy, path stability, and relative stability for positive nonlinear difference equations of higher order, J. Diflerence Equations and Applications 1, 323-346 (1995).

9. KRAUSE U., Positive nonlinear systems: some results and applications, in Proceedings of the First World Congress of Nonlinear Analysts 1992 (Edited by V. LAKSHMIKAN- THAM), pp. 1529-1539, W. de Gruyter, Berlin (1996).

10. KRAUSE U., A theorem of Poincare type for non-autonomous nonlinear difference equations, in Proceedings of the Second International Conference on Di$erence Equations 1995 (Edited by I. GYORY), VeszprCm (1995), to appear.

11. KRAUSE U. & NUSSBAUM R.D., A limit set trichotomy for self-mappings of normal cones in Banach spaces, Nonlinear Analysis 20, 855-870 (1993).

12. LADAS G., Open problems and conjectures, in Proceedings of the First International Conference on Dilference Equations 1994, (Edited by S.N. ELAYDI, J.R.GRAEF, G. LADAS, A.C. PETERSON), pp. 337-349, Gordon and Breach Publishers, Luxembourg (1995).

13. PIELOU E.C., An Introduction to Mathematical Ecology, Wiley Interscience, New York (1969).

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Positive nonlinear difference equations