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Advances in Difference Equations

, 2012:230 | Cite as

On a delay population model with quadratic nonlinearity

  • Leonid Berezansky
  • Jaromír Baštinec
  • Josef DiblíkEmail author
  • Zdeněk Šmarda
Open Access
Research Article
Part of the following topical collections:
  1. Progress in Functional Differential and Difference Equations

Abstract

A nonlinear delay differential equation with quadratic nonlinearity,

x ˙ ( t ) = r ( t ) [ k = 1 m α k x ( h k ( t ) ) β x 2 ( t ) ] , t 0 , Open image in new window

is considered, where α k Open image in new window and β are positive constants, h k : [ 0 , ) R Open image in new window are continuous functions such that t τ h k ( t ) t Open image in new window, τ = const Open image in new window, τ > 0 Open image in new window, for any t > 0 Open image in new window the inequality h k ( t ) < t Open image in new window holds for at least one k, and r : [ 0 , ) ( 0 , ) Open image in new window is a continuous function satisfying the inequality r ( t ) r 0 = const Open image in new window for an r 0 > 0 Open image in new window. It is proved that the positive equilibrium is globally asymptotically stable without any further limitations on the parameters of this equation.

Keywords

Equilibrium Solution Global Attractor Delay Differential Equation Positive Equilibrium Global Asymptotic Stability 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Introduction

To include oscillation in population model systems, Hutchinson [[1], 1948] suggested the following delay logistic equation:
d N ( t ) d t = r N ( t ) [ 1 N ( t τ ) K ] , Open image in new window

where N ( t ) Open image in new window is the population size at time t, r > 0 Open image in new window is the intrinsic growth rate of the population, τ > 0 Open image in new window and K > 0 Open image in new window is the carrying capacity of the population.

There are many generalizations and modifications of Hutchinson’s equation [2, 3, 4, 5, 6, 7]. In particular, a delay logistic equation with several delays,
x ˙ ( t ) = x ( t ) [ α k = 1 m β k x ( t τ k ) ] , Open image in new window
(1)

where α, β k Open image in new window and τ k > 0 Open image in new window are positive constants, is considered in [[8], p.87]. Equation (1) can be viewed as one with quadratic nonlinearities of the unknown function x. For more work on the stability and boundedness of equations and systems related to (1), one can refer to [9, 10, 11, 12, 13].

In the monograph [[14], p.177], the author considers the following population model with quadratic nonlinearity:
x ˙ ( t ) = k = 1 m α k x ( t τ k ) β x 2 ( t ) , t 0 , Open image in new window
(2)
where α k > 0 Open image in new window, β > 0 Open image in new window, τ k > 0 Open image in new window, and with the initial condition
x ( t ) = φ ( t ) , t [ τ , 0 ] , Open image in new window
(3)

where φ : [ τ , 0 ] R Open image in new window is a continuous function, τ = max k = 1 , , m τ k Open image in new window and φ ( t ) > 0 Open image in new window if t [ τ , 0 ] Open image in new window.

As can simply be verified, equation (2) has a unique positive equilibrium x ( t ) = K Open image in new window, t [ τ , ) Open image in new window, where
K = α β , α : = k = 1 m α k . Open image in new window
(4)

Theorem 1 [14], Corollary 3.2.2, p.177]

The positive equilibrium K is a global attractor of problem (2), (3).

This result is different from almost all known results on the stability for nonlinear delay differential equations since there are no limitations on the parameters of equation (2). The proof of Theorem 1 is based on the method of Lyapunov-Krasovskii functionals.

Consider the nonautonomous equation with quadratic nonlinearity
x ˙ ( t ) = r ( t ) [ k = 1 m α k x ( h k ( t ) ) β x 2 ( t ) ] , t 0 , Open image in new window
(5)

where α k > 0 Open image in new window, β > 0 Open image in new window, h k : [ 0 , ) R Open image in new window are continuous functions such that the inequalities t τ h k ( t ) t Open image in new window are true for a τ = const Open image in new window, τ > 0 Open image in new window, and r : [ 0 , ) ( 0 , ) Open image in new window is a continuous function satisfying the inequality r ( t ) r 0 = const Open image in new window for an r 0 > 0 Open image in new window. We suppose also that for any t > 0 Open image in new window the inequality h k ( t ) < t Open image in new window holds for at least one k.

Together with (5), we consider an initial problem
x ( t ) = φ ( t ) , t [ τ , 0 ] , Open image in new window
(6)

where φ : [ τ , 0 ] R Open image in new window is a continuous function and φ ( t ) > 0 Open image in new window if t [ τ , 0 ] Open image in new window.

It is obvious that equation (2) is a particular case of equation (5) if we set
r ( t ) : = 1 , h k ( t ) : = t τ k , τ : = τ . Open image in new window

Moreover, it is easy to see that the constant K defined by (4) defines the unique positive equilibrium x ( t ) = K Open image in new window, t [ τ , ) Open image in new window of equation (5) as well.

In this paper, we prove that there exists a unique positive global solution to the problem (5), (6). Let us recall that a function x : [ τ , ) R Open image in new window continuous on [ τ , ) Open image in new window and continuously differentiable on [ 0 , ) Open image in new window is called a global solution to the problem (5), (6) if it satisfies equation (5) on [ 0 , ) Open image in new window and initial condition (6). In addition to this, we prove that this solution is a bounded function isolated from zero. Finally, we will extend Theorem 1 to nonautonomous equation (5). Unfortunately, the construction of a Lyapunov-Krasovskii functional for autonomous equation (2) given in [14] to prove Theorem 1 is not applicable to nonautonomous equation (5). Therefore, we use another method based on a special quasi-linearization of a given nonlinear equation.

We will apply the following standard notions given in the definition below.

Definition 1 The equilibrium solution x ( t ) = K Open image in new window, t [ τ , ) Open image in new window of equation (5) is locally stable if, for any ε > 0 Open image in new window, there exists δ > 0 Open image in new window such that the inequality | φ ( t ) K | < δ Open image in new window, t [ τ , 0 ] Open image in new window implies | x ( t ) K | < ε Open image in new window, t 0 Open image in new window if x ( t ) = φ ( t ) Open image in new window, t [ τ , 0 ] Open image in new window. If, in addition, for any such solution, lim t x ( t ) = K Open image in new window, the equilibrium solution x ( t ) = K Open image in new window, t [ τ , ) Open image in new window of equation (5) is called locally asymptotically stable.

The equilibrium solution x ( t ) = K Open image in new window, t [ τ , ) Open image in new window of equation (5) is called a global attractor of equation (5) if lim t x ( t ) = K Open image in new window for all solutions x ( t ) Open image in new window of equation (5) defined by all initial functions described by (6).

The equilibrium solution x ( t ) = K Open image in new window, t [ τ , ) Open image in new window of equation (5) is globally asymptotically stable if it is a global attractor for all solutions x ( t ) Open image in new window of equation (5) defined by all initial functions described by (6) and if it is also locally stable.

Main results

In this section, we employ a simple stability result to the following linear equation:
x ˙ ( t ) + a ( t ) x ( t ) + k = 1 m b k ( t ) x ( h k ( t ) ) = 0 , t 0 , Open image in new window
(7)
with bounded continuous functions
a : [ t 0 , ) R , b k : [ t 0 , ) R , h k : [ t 0 , ) R , Open image in new window

where t τ h k ( t ) t Open image in new window, τ > 0 Open image in new window, and for any t > 0 Open image in new window, the inequality h k ( t ) < t Open image in new window holds for at least one k.

Lemma 1 Assume that there exists a constant a 0 > 0 Open image in new window such that
a ( t ) a 0 , t [ 0 , ) Open image in new window
and
lim sup t k = 1 m | b k ( t ) | a ( t ) < 1 . Open image in new window

Then equation (7) is asymptotically stable.

This result has a long history. The first stability conditions of this kind were obtained by Krasovskii [15] for an equation with a single delay. A weaker result (a corollary of Lemma 1) is given in [[16], p.154]. Lemma 1 itself is a consequence of [[17], Corollary 3.13].

Theorem 2 A solution of problem (5), (6) is positive and global.

Proof Since φ ( 0 ) > 0 Open image in new window, there exists a unique positive local solution of problem (5), (6). Suppose [ 0 , c ) Open image in new window with c > 0 Open image in new window is the maximum interval of existence of this problem and a point t 0 ( 0 , c ) Open image in new window is such that x ( t ) > 0 Open image in new window, t [ τ , t 0 ) Open image in new window, and x ( t 0 ) = 0 Open image in new window. Then x ˙ ( t 0 ) 0 Open image in new window. Directly from equation (5), we get
x ˙ ( t 0 ) = r ( t 0 ) k = 1 m α k x ( h k ( t 0 ) ) > 0 , Open image in new window

which is a contradiction.

Hence, x ( t ) > 0 Open image in new window, t [ 0 , c ) Open image in new window. If c = Open image in new window, the theorem is proved. Suppose c < Open image in new window. Then, by [[16], Corollary 3.1, p.45], either lim sup t c x ( t ) = + Open image in new window or lim inf t c x ( t ) = 0 Open image in new window.

Let lim sup t c x ( t ) = + Open image in new window. Then we have
x ˙ ( t ) r ( t ) k = 1 m α k x ( h k ( t ) ) , t [ 0 , c ) . Open image in new window
Corollary 2.4 [[18], p.32] implies that x ( t ) y ( t ) Open image in new window, where y is a solution of the initial linear problem
y ˙ ( t ) = r ( t ) k = 1 m α k y ( h k ( t ) ) , t [ 0 , ) , y ( t ) = x ( t ) , t [ τ , 0 ] . Open image in new window

Since every solution y of a linear delay differential equation is bounded on any finite interval, x is also bounded on the interval [ 0 , c ) Open image in new window. We have a contradiction.

Suppose now lim inf t c x ( t ) = 0 Open image in new window. Then either x is a monotone decreasing function on [ t 0 , c ) Open image in new window for some t 0 [ 0 , c ) Open image in new window or there exists a sequence t n [ t 0 , c ) Open image in new window such that
t n < t n + 1 , lim n t n = c , lim n x ( t n ) = 0 , x ˙ ( t n ) = 0 Open image in new window
and
x ( t ) > x ( t n ) , t [ τ , t n ) . Open image in new window
In the first case (without loss of generality, we assume that x is a monotone decreasing function for t [ 0 , c ) Open image in new window), lim t c x ( t ) = 0 Open image in new window. Hence, x is a solution of (5), (6) on [ 0 , c ] Open image in new window with x ( c ) = 0 Open image in new window and as above, we obtain a contradiction because
0 x ˙ ( c ) = r ( c ) k = 1 m α k x ( h k ( c ) ) > 0 . Open image in new window
Consider now the second case. Then
0 = x ˙ ( t n ) = r ( t n ) [ k = 1 m α k x ( h k ( t n ) ) β x 2 ( t n ) ] . Open image in new window
Let n Open image in new window. Then t n c Open image in new window, hence
k = 1 m α k x ( h k ( c ) ) = 0 , Open image in new window

where for at least one k one has h k ( c ) < c Open image in new window, and so x ( h k ( c ) ) = 0 Open image in new window. We have a contradiction. The theorem is proved. □

Theorem 3 For a solution x of problem (5), (6),
lim sup t x ( t ) < , lim inf t x ( t ) > 0 . Open image in new window
Proof Substituting x ( t ) = y ( t ) + K Open image in new window, equation (5) takes the form
y ˙ ( t ) = r ( t ) [ k = 1 m α k y ( h k ( t ) ) 2 α y ( t ) β y 2 ( t ) ] . Open image in new window
(8)
Hence,
y ˙ ( t ) r ( t ) [ k = 1 m α k y ( h k ( t ) ) 2 α y ( t ) ] . Open image in new window
Consider the linear equation
z ˙ ( t ) = r ( t ) [ k = 1 m α k z ( h k ( t ) ) 2 α z ( t ) ] , Open image in new window
(9)
where
z ( t ) = y ( t ) , t [ τ , 0 ] . Open image in new window
Corollary 2.4 [[18], p.32] implies that y ( t ) z ( t ) Open image in new window for t > 0 Open image in new window. Now, we apply Lemma 1 to equation (9). Since
r ( t ) k = 1 m α k 2 r ( t ) α = r ( t ) α 2 r ( t ) α = 1 2 < 1 , Open image in new window

by Lemma 1, equation (9) is asymptotically stable. Hence, the function y is bounded from above and consequently is the function x.

Suppose lim inf t x ( t ) = 0 Open image in new window. Then either x is an eventually monotone decreasing function or there exists a sequence t n [ τ , ) Open image in new window such that
t n < t n + 1 , lim n t n = , x ˙ ( t n ) = 0 Open image in new window
and
x ( t ) > x ( t n ) , t [ τ , t n ) . Open image in new window
In the first case (without loss of generality, we assume that x is a monotone decreasing function for t > 0 Open image in new window), there exists t > 0 Open image in new window such that x ( t ) < K Open image in new window. Hence,
x ˙ ( t ) = r ( t ) [ k = 1 m α k x ( h k ( t ) ) β x 2 ( t ) ] r ( t ) [ k = 1 m α k x ( t ) β x 2 ( t ) ] = r ( t ) [ α x ( t ) β x 2 ( t ) ] > 0 . Open image in new window

This is in contradiction to our assumption.

Consider now the second case. Then there exists a sufficiently large integer n such that x ( t n ) < K Open image in new window. Hence,
0 = x ˙ ( t n ) = r ( t n ) [ k = 1 m α k x ( h k ( t n ) ) β x 2 ( t n ) ] > r ( t n ) [ α x ( t n ) β x 2 ( t n ) ] > 0 . Open image in new window

We have a contradiction and the theorem is proved. □

In the following, when discussing the stability properties of solutions of equation (5), we will assume that initial conditions (6) hold.

Theorem 4 The positive equilibrium K of equation (5) is globally asymptotically stable.

Proof We have to prove that K is an attractor for all solutions of the equation and that it also is locally stable. To do this, it is sufficient to prove that the zero solution is an attractor for all solutions of equation (8) and that it also is locally stable.

Suppose that x is a fixed solution of equation (5). Then y ( t ) = x ( t ) K Open image in new window is a fixed solution of equation (8). After substituting y ( t ) = e λ t z ( t ) Open image in new window, where λ > 0 Open image in new window is a sufficiently small number satisfying λ < r 0 ( α + β m ) Open image in new window, we have an equation
z ˙ ( t ) = r ( t ) [ k = 1 m α k e λ ( t h k ( t ) ) z ( h k ( t ) ) ( 2 α λ r ( t ) ) z ( t ) β e λ t z 2 ( t ) ] . Open image in new window
(10)
Since y ( t ) = e λ t z ( t ) Open image in new window, we can rewrite equation (10) as
z ˙ ( t ) = r ( t ) [ k = 1 m α k e λ ( t h k ( t ) ) z ( h k ( t ) ) ( 2 α λ r ( t ) + β y ( t ) ) z ( t ) ] . Open image in new window
(11)
By Theorem 3, there exist constants m, M, 0 < m < M Open image in new window such that
m < x ( t ) < M Open image in new window
for t > 0 Open image in new window and
m K < y ( t ) < M K . Open image in new window
For t > 0 Open image in new window, we have
2 α λ r ( t ) + β y ( t ) > 2 α λ r ( t ) + β ( m K ) = α λ r ( t ) + β m Open image in new window
and
k = 1 m α k e λ ( t h k ( t ) ) < α e λ τ . Open image in new window
Hence,
k = 1 m α k e λ ( t h k ( t ) ) 2 α λ r ( t ) + β y ( t ) < α e λ τ α λ r ( t ) + β m α e λ τ α λ r 0 + β m , t [ 0 , ) . Open image in new window
Since
lim λ 0 α e λ τ α λ r 0 + β m = α α + β m < 1 , Open image in new window
for a sufficiently small λ > 0 Open image in new window,
lim sup t k = 1 m α k e λ ( t h k ( t ) ) 2 α λ r ( t ) + β y ( t ) < 1 . Open image in new window
We will now fix such λ. Suppose u is an arbitrary continuous function such that m K < u ( t ) < M K Open image in new window. By Lemma 1, all solutions of linear equation (11), where y is replaced by u, tend to zero. Then it is also true if u = y Open image in new window. Hence, z as a solution of nonlinear equation (11) is a bounded function. Then
lim sup t y ( t ) = lim sup t e λ t z ( t ) = 0 . Open image in new window
(12)
It means that the zero solution is a global attractor of equation (8). We need only to prove that this equation is locally stable. The linearized equation for (8) has the form
u ˙ ( t ) = r ( t ) [ k = 1 m α k u ( h k ( t ) ) 2 α u ( t ) ] . Open image in new window
(13)

By Lemma 1, equation (13) is asymptotically stable (see equation (9) in the proof of Theorem 3). The theorem is proved. □

Remark 1 The proof of Theorem 4 provides not only global asymptotic stability of the positive equilibrium K, but also exponential estimation of the rate of convergence of solutions. Tracing the proof carefully, we have
x ( t ) K = y ( t ) = e λ t z ( t ) , Open image in new window
where x is a fixed solution of equation (5), y ( t ) Open image in new window is a corresponding solution of equation (8) and z ( t ) Open image in new window is a corresponding solution of equation (11). Since z ( t ) Open image in new window is bounded and (12) holds, we state that for a given solution x = x ( t ) Open image in new window of equation (5), there exists a constant H x Open image in new window such that
| x ( t ) K | H x e λ t , Open image in new window

where λ is a sufficiently small positive number satisfying λ < r 0 ( α + β m ) Open image in new window and t [ τ , ) Open image in new window.

Conclusions, concluding remarks and open problems

It is well known that the positive equilibrium K of the delay logistic equation
x ˙ ( t ) = a x ( t ) ( 1 x ( t τ ) K ) Open image in new window

is globally asymptotically stable if a τ < 3 / 2 Open image in new window and locally asymptotically stable if a τ < π / 2 Open image in new window. Thus, there is a gap between sufficient conditions for global and for local stabilities. One of the old problems is to show that local asymptotic stability implies global asymptotic stability for this equation. This problem also remains open for most of known nonlinear delayed equations in mathematical biology.

By Theorem 4, local and global stability conditions for equation (8) coincide since this equation is stable for all positive coefficients. It would be interesting to find other equations with such a property.

In the proof of Theorem 4, we applied the substitution y ( t ) = z ( t ) e λ t Open image in new window with a small parameter λ > 0 Open image in new window. Such kind of substitutions are well known in the investigation of stability of linear equations. Maybe such a substitution is used here for a nonlinear equation for the first time.

The usefulness of asymptotic methods could be exploited to extend other powerful (deterministic) techniques such as the Laplace decomposition method or He’s polynomials to deal with difference-differential models (see references [19, 20] for instance).

A partial case of equation (8) was considered in [21] where also some other delay differential equations with quadratic nonlinearity were studied.

Finally, we outline some open problems and topics for further research.
  1. 1.

    Study the oscillation properties of equation (5).

     
  2. 2.
    Extend Theorems 2-4 to the equation
    x ˙ ( t ) = k = 1 m a k ( t ) x ( h k ( t ) ) b ( t ) x 2 ( t ) Open image in new window
    (14)
     
if k = 1 m a k ( t ) = b ( t ) Open image in new window.
  1. 3.

    Prove the existence of a periodic solution for equation (14) with periodic functions a k ( t ) Open image in new window, b ( t ) Open image in new window and study its local and global stability.

     
  2. 4.
    Study the existence, uniqueness, oscillation and stability properties of solutions of the following equations:
    x ˙ ( t ) = k = 1 m a k ( t ) x ( h k ( t ) ) b ( t ) x n ( t ) , n > 0 , Open image in new window
     
and
x ˙ ( t ) = k = 1 m a k ( t ) x ( h k ( t ) ) b ( t ) x 2 ( g ( t ) ) . Open image in new window

Notes

Acknowledgements

The second author was supported by Grant P201/10/1032 of the Czech Grant Agency (Prague). The third author was supported by the Operational Programme Research and Development for Innovations, No. CZ.1.05/2.1.00/03.0097, as an activity of the regional Centre AdMaS. The fourth author was supported by the Grant FEKT-S-11-2-921 of the Faculty of Electrical Engineering and Communication, Brno University of Technology. The authors would like to thank the referees for helpful suggestions incorporated into this paper.

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This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  • Leonid Berezansky
    • 1
  • Jaromír Baštinec
    • 2
  • Josef Diblík
    • 2
    • 3
    Email author
  • Zdeněk Šmarda
    • 2
  1. 1.Department of MathematicsBen-Gurion University of the NegevBeer-ShevaIsrael
  2. 2.Department of Mathematics, Faculty of Electrical Engineering and CommunicationBrno University of TechnologyBrnoCzech Republic
  3. 3.Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering and Department of Mathematics, Faculty of Electrical Engineering and CommunicationBrno University of TechnologyBrnoCzech Republic

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