2004 Conference on Diff. Eqns. and Appl. in Math. Biology, Nanaimo, BC, Canada. Electronic Journal of Differential Equations, Conference 12, 2005, pp. 21–27. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

OSCILLATION AND ASYMPTOTIC STABILITY OF A DELAY

DIFFERENTIAL EQUATION WITH RICHARD’S NONLINEARITY

LEONID BEREZANSKY, LEV IDELS

Abstract. We obtain sufficient conditions for oscillation of solutions, and for asymptotical stability of the positive equilibrium, of the scalar nonlinear delay differential equation

\frac{dN}{dt} = r(t)N(t)[a − ( ∑_{k=1}^{m} b_{k}N(g_{k}(t)))^{γ}],

where g_{k}(t) ≤ t.

1. Introduction

Consider the following logistic differential equation which is widely used in Population Dynamics

\frac{dN}{dt} = rN(1 − \frac{N}{K} ).

Here N(t) is the size of a population, r ≥ 0 is an intrinsic growth rate, K is a carrying capacity or a saturation level. A large variety of nonlinear differential equations, besides the one above, has been developed for models of Mathematical Biology; see for example [3, 9, 1].

To model processes in nature and engineering it is frequently required to know system states from the past. Depending on the phenomena under study the aftereffects represent duration of some hidden processes. In general, delay differential equations (DDE) exhibit much more complicated dynamics than ordinary differential equations (ODE) since a time lag can change a stable equilibrium into an unstable one and make populations fluctuate, they provide a richer mathematical framework (compared with ordinary differential equations) for the analysis of biosystems dynamics.

Models of Population Dynamics, based on nonlinear DDE’s, have attracted much attention in recent years. The application of delay equations to biomodelling in many cases is associated with studies of dynamic phenomena like oscillations, bifurcations, and chaotic behavior. Time delays represent an additional level of complexity that can be incorporated in a more detailed analysis of a particular system.

2000 Mathematics Subject Classification. 34K11, 34K20, 34K60.

Key words and phrases. Delay differential equations; Richard’s nonlinearity;

oscillation; stability.

\begin{matrix}c \\ ◯ \end{matrix}2005 Texas State University - San Marcos.

Published April 20, 2005.

The delay logistic equation

\frac{dN}{dt} = rN(1 − \frac{N_{τ}}{K} )  (1.1)

appeared in 1948 in Hutchinson’s paper [7], where N_{τ} = N(t − τ), τ > 0.

The autonomous equation (1.1) has been extensively investigated by numerous authors. The first paper on the oscillation of a non-autonomous logistic delay differential equation was published in [14]. Since this publication, the oscillation of the logistic DDE as well as its generalizations were studied by many mathematicians. Some of these results can be found in the monographs [6, 5, 4].

It is a well-known fact, that the traditional logistic model, in some cases, produces artificially complex dynamics. Therefore, it would be reasonable to get away from the specific logistic form in studying population dynamics and use more general classes of growth models.

For example, to drop an unnatural symmetry of the logistic curve, we consider the modified logistic form by Pella and Tomlinson [13, 12] or the Richards’ growth equation with delay

\frac{dN}{dt} = rN[1 − (\frac{N_{τ}}{K} )^{γ}].  (1.2)

According to [13], 0 < γ < 1 is used for invertebrate populations (examples of invertebrates are insects, worms, starfish, sponges, squid, plankton, crustaceans, and mollusks), and γ ≥ 1 is used for the vertebrate populations (these include amphibians, birds, fish, mammals, and reptiles).

In [11] the authors considered (1.2) with several delays. They obtained conditions for existence of positive solutions and studied so-called long time average stability. In this paper we obtain oscillation and local stability results for non-autonomous (1.2) with several delays.

2. Preliminaries

Our objective is to study the scalar nonlinear delay differential equation

Ṅ(t) = r(t)N(t)[a − ( ∑_{k=1}^{m} b_{k}N(g_{k}(t)))^{γ}], t ≥ 0  (2.1)

under the following conditions:

(A1) r(t) is Lebesgue measurable essentially bounded on [0, ∞) function, r(t) ≥

0. (A2) g_{k} : [0, ∞) → ℝ are Lebesgue measurable functions with g_{k}(t) ≤ t and

lim_{t→∞} g_{k}(t) = ∞, k = 1, …, m. (A3) a > 0, b_{k} > 0, γ > 0.

Together with (2.1), we consider for t_{0} ≥ 0, the initial-value problem

Ṅ(t) = r(t)N(t)[a − ( ∑_{k=1}^{m} b_{k}N(g_{k}(t)))^{γ}], t ≥ t_{0},  (2.2)
N(t) = φ(t), t < t_{0}, N(t_{0}) = N_{0}  (2.3)

under the following conditions (A4) φ : (−∞, t_{0}) → R is a Borel measurable bounded function, φ(t) ≥ 0,

N_{0} > 0. Definition. A locally absolutely continuous function x : R → R is called a solution of problem (2.2)–(2.3), if it satisfies (2.2) for almost all t ∈ [t_{0}, ∞) and (2.3) for t ≤ t_{0}.

Lemma 2.1 ([11]). Suppose Conditions (A1)–(A4) hold. Then problem (2.2)-(2.3) has a unique positive solution N(t), t ≥ t_{0}.

3. Oscillation Criteria

Definition. We say that a function y(t) is non-oscillatory about a number K if y(t) − K is eventually positive or eventually negative. Otherwise y(t) is oscillatory about K.

Note that (2.1) has a positive equilibrium,

N^{∗} = a^{1/γ}/ ∑_{k=1}^{m} b_{k}.

In this section we study oscillation of solutions of (2.1) about the value N^{∗}.

We will present here some lemmas which will be used in this section. Consider the linear delay differential equation

ẋ(t) + ∑_{k=1}^{l} r_{k}(t)x(h_{k}(t)) = 0, t ≥ 0,  (3.1)

and the differential inequalities

ẋ(t) + ∑_{k=1}^{l} r_{k}(t)x(h_{k}(t)) ≤ 0, t ≥ 0,  (3.2)

ẋ(t) + ∑_{k=1}^{l} r_{k}(t)x(h_{k}(t)) ≥ 0, t ≥ 0.  (3.3)

Lemma 3.1 ([6]). Let (A1)–(A2) hold for the parameters of (3.1). Then the following three statements are equivalent:

(1) There exists a non-oscillatory solution of equation (3.1).

(2) There exists an eventually positive solution of the inequality (3.2).

(3) There exists an eventually negative solution of the inequality (3.3).

Lemma 3.2 ([6]). Let (A1)–(A2) hold for the parameters of (3.1). If

lim inf_{t→∞} ∫_{max_{k} h_{k}(t)}^{t} ∑_{i=1}^{l} r_{i}(s)ds > 1/e,  (3.4)

then all solutions of (3.1) are oscillatory.

Theorem 3.3. Suppose (A1)-(A4) hold and

∫_{0}^{∞} r(s)ds = ∞.  (3.5)

Then for every non-oscillatory solution N(t) of (2.1) we have

lim_{t→∞} N(t) = N^{∗}.  (3.6)

Proof. After the substitution N(t) = N^{∗}(1 + x(t)), Equation (2.1) reduced to

ẋ(t) = −ar(t)(1 + x(t))[( ∑_{k=1}^{m} B_{k}(1 + x(g_{k}(t))))^{γ} − 1], t ≥ 0,  (3.7)

where

B_{k} = b_{k}/ ∑_{i=1}^{m} b_{i}.  (3.8)

Condition (A3) implies B_{k} > 0 and ∑_{k=1}^{m} B_{k} = 1.

The zero solution is an equilibrium of (3.7), which corresponds to the equilibrium N^{∗} of (2.1).

By Lemma 2.1 any solution of (2.1) is positive. Then for any solution of (3.7) we have 1 + x(t) > 0. To prove the theorem we have to show that for every non-oscillatory about zero solution of (3.7) we have

lim_{t→∞} x(t) = 0.  (3.9)

Suppose x(t) is a non-oscillatory solution of (3.7). Without loss of generality we can assume x(t) > 0, t ≥ 0. Hence

( ∑_{k=1}^{m} B_{k}(1 + x(g_{k}(t))))^{γ} − 1 ≥ ( ∑_{k=1}^{m} B_{k})^{γ} − 1 = 0.

Then ẋ(t) ≤ 0 and hence there exists lim_{t→∞} x(t) = l. Suppose l > 0. Equality (3.7) implies

x(t) = x(0) − a ∫_{0}^{t} r(s)(1 + x(s))[( ∑_{k=1}^{m} B_{k}(1 + x(g_{k}(s))))^{γ} − 1]ds.  (3.10)

If t → ∞ then the right hand side of (3.10) tends to −∞, the left hand side has a finite limit. This contradiction proves the theorem. □

Theorem 3.4. Suppose conditions (A1)–(A4) and (3.5) hold, γ > 1 and there exists ϵ > 0 such that all solutions of the linear differential equation

ẏ(t) = −aγr(t)(1 − ϵ) ∑_{k=1}^{m} B_{k}y(g_{k}(t))  (3.11)

are oscillatory, were B_{k} are denoted by (3.8). Then all solutions of (2.1) are oscillatory about N^{∗}.

Proof. It is sufficient to prove, that all solutions of (3.7) are oscillatory about zero. Suppose there exists a non-oscillatory solution x of (3.7). Without loss of generality we can assume, that x(t) > 0, t ≥ 0. Theorem 3.3 implies, that for some t_{0} > 0 and for t ≥ t_{0} we have 0 < x(t) < ϵ.

Consider the function

f(u_{1}, …, u_{m}) = ( ∑_{k=1}^{m} B_{k}(1 + u_{k}))^{γ} − 1 − γ ∑_{k=1}^{m} B_{k}u_{k}.

Then we have

\frac{∂f}{∂u_{k}} = γ( ∑_{k=1}^{m} B_{k}(1 + u_{k}))^{γ−1}B_{k} − γB_{k},

\frac{∂^{2}f}{∂u_{i}∂u_{j}} = γ(γ − 1)( ∑_{k=1}^{m} B_{k}(1 + u_{k}))^{γ−2}B_{i}B_{j}.

Hence

f(0, …, 0) = 0, \frac{∂f}{∂u_{k}} (0, …, 0) = 0, \frac{∂^{2}f}{∂u_{i}∂u_{j}} (0, …, 0) = γ(γ − 1)B_{i}B_{j}.

Taylor’s Formula implies

f(u_{1}, …, u_{m}) = γ(γ − 1) ∑_{i=1}^{m} ∑_{j=1}^{m} B_{i}B_{j}u_{i}u_{j} + o(Δu),

where

Δu = ( ∑_{k=1}^{m} u_{k}^{2})^{1/2}, lim_{t→0} \frac{o(t)}{t} = 0.

Then for u_{k} ≥ 0, k = 1, …, m and Δu sufficiently small f(u_{1}, …, u_{m}) ≥ 0. Hence for ϵ small enough we have

ẋ(t) ≤ −aγr(t)(1 − ϵ) ∑_{k=1}^{m} B_{k}x(g_{k}(s)), t ≥ 0.

Lemma 3.1 implies that (3.11)) has a non-oscillatory solution. We have a contradiction with our assumption. The theorem is proven. □

Corollary 3.5. Suppose conditions (A1)–(A4) and (3.5) hold, γ > 1,

lim inf_{t→∞} aγ ∫_{max_{k} g_{k}(t)}^{t} r(s)ds > 1/e.  (3.12)

Then all solutions of (2.1) are oscillatory about N^{∗}.

Proof. Inequality (3.12) implies, that for some ϵ > 0,

lim inf_{t→∞} aγ(1 − ϵ) ∫_{max_{k} g_{k}(t)}^{t} ∑_{i=1}^{m} B_{i}r(s)ds > 1/e.

Lemma 3.2 and Theorem 3.4 imply this corollary. □

4. Asymptotic Stability

Consider a general nonlinear delay differential equation

ẋ(t) = f(t, x(t), x(g_{1}(t)), …, x(g_{m}(t))), t ≥ 0,  (4.1)

with the initial function and the initial value

x(t) = φ(t), t < 0, x(0) = x_{0},  (4.2)

under the following conditions:

(B1) f(t, u_{0}, u_{1}, …, u_{m}) satisfies Caratheodory conditions: Lebesgue measurable in the first argument and continuous in other arguments, f(t, 0, …, 0) = K (B2) g_{k}(t) are Lebesgue measurable functions,

g_{k}(t) ≤ t, sup_{t≥0}[t − g_{k}(t)] < ∞;

(B3) φ : (−∞, 0) → R is a Borel measurable bounded function. We will assume that the initial-value problem (4.1)–(4.2) has a unique global solution x(t), t ≥ 0. Definition. We will say that the equilibrium K of (4.1) is (locally) stable, if for any ϵ > 0 there exists δ > 0 such that for every initial conditions |x(0)| < δ_{0}, |φ(t)| < δ_{0}, δ_{0} ≤ δ, for the solution x(t) of (4.1)–(4.2) we have |x(t)−K| < ϵ, t ≥ 0.

If, in addition, lim_{t→∞}(x(t)−K) = 0, then the equilibrium K of (4.1) is (locally) asymptotically stable.

Suppose there exist M > 0, γ > 0 such that

|x(t) − K| ≤ M exp{−γt}(|x(0)| + sup_{t<0} |φ(t)|)

for all x(0) and φ(t) such that |x(0)| + sup_{t<0} |φ(t)| is sufficiently small. Then we will say that the equilibrium K of (4.1) is exponentially stable.

Lemma 4.1 ([10]). Suppose (A1), (B2), (B3) hold for the linear equation (3.1) and

lim sup_{t→∞} ∑_{k=1}^{l} r_{k}(t)(t − h_{k}(t)) < 1.

Then (3.1) is exponentially stable.

Lemma 4.2 ([2], [8]). Suppose that (b1)-(b3) hold, and that for sufficiently small u if |u_{k}| ≤ u, k = 0, …, m then

|f(t, u_{0}, …, u_{m}) − ∑_{k=0}^{m} \frac{∂F}{∂u_{k}} (t, K, …, K)u_{k}| = o(u),

where lim_{u→0} o(u)/u = 0. If the linear equation

ẏ(t) = ∑_{k=0}^{m} \frac{∂F}{∂u_{k}} (t, 0, …, 0) y(g_{k}(t))

is exponentially stable, then the equilibrium K of (4.1) is locally asymptotically stable.

Theorem 4.3. Suppose that for equation (2.1) Conditions (A1), (A3), (B2), (B3) hold and

lim sup_{t→∞} aγr(t) ∑_{k=1}^{m} B_{k}(t − g_{k}(t)) < 1,  (4.3)

were B_{k} are denoted by (3.8). Then the equilibrium N^{∗} of (2.1) is asymptotically stable.

Proof. The substitution N(t) = N^{∗}(1 + x(t)) implies that the equilibrium N^{∗} of (2.1) is asymptotically stable if and only if the zero solution of (3.7) is asymptotically stable. Lemma 4.1 and inequality (4.3) imply that the linear equation

ẋ(t) = −aγr(t) ∑_{k=1}^{m} B_{k}x(g_{k}(t))

is exponentially stable. Lemma 4.2 implies now that the zero solution of (3.7) is asymptotically stable. □

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London 1963, 462 pp.

[3] F. Brauer and C. Castillo-Chavez; Mathematical Models in Population Biology and Epidemiology, Springer-Verlag, 2001.

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[5] K. Gopalsamy; Stability and Oscillation in Delay Differential Equations of Population Dynamics, 1992, Kluwer Academic Publishers, Dordrecht, Boston, London.

[6] I. Györi and G. Ladas; Oscillation Theory of Delay Differential Equations, 1991, Clarendon Press, Oxford.

[7] G.E. Hutchinson, Circular causal systems in ecology, Ann. N.Y. Acad. Sci. 50 (1948), pp. 221-246.

[8] V. Kolmanovskii and A. Myshkis; Introduction to the Theory and Applications of Functional-Differential Equations. Mathematics and its Applications, 463. Kluwer Academic Publishers, Dordrecht, 1999.

[9] M. Kot; Elements of Mathematical Ecology, Cambridge Univ. Press, 2001.

[10] T. Krisztin; On stability properties for one-dimensional functional-differential equations. Funkcial. Ekvac. 34(1991), no. 2, 241–256.

[11] J. J. Miguel, A. Ponosov, A. Shindiapin; On a delay equation with Richards’ nonlinearity. Proceedings of the Third World Congress of Nonlinear Analysts, Part 6 (Catania, 2000). Nonlinear Anal. 47 (2001), no. 6, 3919–3924.

[12] J. Pella and P. Tomlinson; A Generalized Stock-Production Model, Inter.-Am. Trop. Tuna Comm. Bull., 13,(1969), 421-496

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[14] B. G. Zhang and K. Gopalsamy; Oscillation and nonoscillation in a nonautonomous delaylogistic equation, Quart. Appl. Math. XLVI (1988), 267-273.

Leonid Berezansky Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel

E-mail address: brznsky@cs.bgu.ac.il  Phone 972-7-6461602  Fax 972-7-6281340

Lev Idels Mathematics Department, Malaspina University-College, 900 Fifth Street Nanaimo, BC V9R 5S5, Canada

E-mail address: lidels@shaw.ca  Phone 250-753-3245 ext.  2429  Fax 250-740-6482