Absolute and DelayDependent Stability of Equations with a Distributed Delay: a Bridge from Nonlinear Differential to Difference Equations
Abstract
We study delayindependent stability in nonlinear models with a distributed delay which have a positive equilibrium. Such models frequently occur in population dynamics and other applications. In particular, we construct a relevant difference equation such that its stability implies stability of the equation with a distributed delay and a finite memory. This result is, generally speaking, incorrect for systems with infinite memory. If the relevant difference equation is unstable, we describe the general delayindependent attracting set and also demonstrate that the equation with a distributed delay is stable for small enough delays.
AMS Subject Classification: 34K20, 92D25, 34K60, 34K23
Keywords: equations with a distributed delay, global attractivity, permanent solutions, Nicholson’s blowflies equation, MackeyGlass equation.
1 Introduction
In models of population dynamics which are described by an autonomous differential equation
(1) 
where and are reproduction and mortality rates, respectively, , for and for , for ( is the carrying capacity of the environment), the positive equilibrium is stable: all positive solutions converge to and are monotone. It was argued that the observed data usually oscillates about the carrying capacity; in order to model this phenomenon, it was suggested to introduce delay in the production term
(2) 
the latter equation can have oscillatory solutions, and the delay incorporated in the right hand side can be interpreted as maturation, production or digestion effects. It is usually assumed that the mortality rate was proportional to the present population level
(3) 
The global behavior of solutions of (3) has been extensively studied in literature, in particular in the cases of negative and positive feedback (see, for example, [1, 2] and references therein), the chaotic behavior is impossible in the case of the monotone feedback [3]. However in the case when is a unimodal function, i.e., increases for and decreases for , there may be delay induced instability and complex dynamics [4, 5]. For a detailed overview of the literature on the dynamics of (3) see the recent papers [6, 7]. It is demonstrated in [6, 7] that if is a unimodal function and positive equilibrium of the equation is globally asymptotically stable, then all solutions of (3) tend to . In particular, if has a negative Schwarzian derivative, then local stability of the equilibrium of the difference equation implies its global attractivity [8]. To the best of our knowledge, the first delayindependent stability conditions were obtained in [9]. In the present paper we will try to answer the general question: what are intrinsic properties of the reproduction function which allow us to conclude that any solution of the equation with a finite memory converges to the equilibrium? Here we consider both general delays (including integral terms) and continuous functions which may have multiple extrema, tend to infinity at infinity etc.
As special cases, (3) includes the Nicholson’s blowflies equation [10, 11] and the MackeyGlass equation [4, 13]. The Nicholson’s blowflies equation
(4) 
was used in [11] to describe the periodic oscillation in Nicholson’s classic experiments [10] with the Australian sheep blowfly, Lucila cuprina. Equation (4) with a distributed delay was studied in [12], where comprehensive results were obtained for the case .
The MackeyGlass equation [4, 13]
(5) 
models white blood cells production. Local and global stability of the positive equilibrium for equation (5) with variable delays was studied in [14, 15, 16, 17, 18, 19]; to the best of our knowledge, there are no publications on (5) with a distributed delay.
To incorporate random environment influence, some authors included noise in (4) and studied attractivity conditions, see, for example, [20]. However, in applied problems not only the derivative but also the delay value can be perturbed. We assume that the production delay is not a constant but some distributed value which leads to the equation
(6) 
where is the probability that at time the maturation delay in the production function is between and , where . We will assume that very large delays are improbable, substituting in the lower bound with which tends to infinity as . In the present paper we consider a rather general form of , which includes unimodal functions, as well as functions with several extrema. The only requirement is that has the only positive fixed point. The main result claims that if this fixed point is a global attractor for all positive solutions of the difference equation
(7) 
then all solutions of (6) with positive initial conditions tend to this fixed point as well. To some extent this establishes a link between stable differential equation (1) and difference equation (7) which can undergo a series of bifurcations and even transition to chaos. If (7) is globally stable, so is (6). If the unique positive equilibrium of (7) is unstable, (6) can be stable or not, depending on the delay.
The paper is organized as follows. In Section 2 we prove that all solutions with positive initial conditions are positive and bounded and establish some estimates for the lower and the upper bounds. Section 3 presents sufficient conditions under which all positive solutions converge to the positive equilibrium. In Section 4 delaydependent stability is investigated. In particular, it is demonstrated that equations are globally attractive for delays small enough; if (7) is unstable, then we can find such delays that the positive equilibrium of (6) is not a global attractor. In Section 5 these results are applied to equations of population dynamics with a unimodal reproduction function and a distributed delay, in particular, to the Nicholson’s blowflies and MackeyGlass equations; some open problems are presented.
2 Boundedness and Estimates of Solutions
We consider the equation with a distributed delay
(8) 
and the initial condition
(9) 
As special cases, (8) includes

The integrodifferential equation
(10) corresponding to the absolutely continuous for any . Here
is defined almost everywhere.

The equation with several concentrated delays
(11) with , , where for any . This corresponds to , where is the characteristic function of interval .
Definition. An absolutely continuous in function is called a solution of the problem (8),(9) if it satisfies equation (8) for almost all and conditions (9) for .
The integral in the right hand side of (8) should exist almost everywhere. In particular, for (10) with a locally integrable kernel, can be any Lebesgue measurable essentially bounded function. For (11) should be a Borel measurable bounded function. For any distribution the integral exists if is bounded and continuous (here we assume is continuous). Besides, as is commonly set in population dynamics models, is nonnegative and the value at the initial point is positive.
Consider (8),(9) under the following assumptions.

(a1) is a continuous function satisfying Lipschitz condition , , , for and for ;

(a2) , is a Lebesgue measurable function, ,

(a3) is a Lebesgue measurable essentially bounded on function, for any , ;

(a4) is a left continuous nondecreasing function for any , is locally integrable for any , , , . Here is the right side limit of function at point .

(a5) is a continuous bounded function, , .
Denote by the space of Lebesgue measurable functions such that
by the space of continuous in functions with the norm.
We will use the following result from the book of Corduneanu [21] (Theorem 4.5, p. 95). We recall that operator is causal (or Volterra) if for any two functions and and each the fact that , , implies , .
Lemma 1
Proof. To reduce (8) to the equation with the zero initial function, for any we can present the integral as a sum of two integrals
(14) 
where
Here is arbitrary, so we begin with and proceed to a neighboring to prove the existence of a local solution. Then in (12)
where
and for any there is , such that
for , where was defined in (a1), here can be chosen small enough. By Lemma 1 this implies existence and uniqueness of a local solution for (8). This solution is either global or there exists such that either
(15) 
or
(16) 
The initial value is positive, so as far as , the solution is not less than the solution of the initial value problem , which is positive and the former case (15) is impossible. In addition, for any
which contradicts (16). Thus there exists a unique global solution, which completes the proof.
Proof. After the substitution
(17) 
equation (8) becomes
(18) 
Thus and as far as , , consequently, for any . Since the signs of and coincide, then for any .
In the following we prove permanence of all solutions of (8) with positive initial conditions; moreover, we establish bounds for solutions.
Proof. By Theorem 2 the solution is positive for . By (a2) there exists such that , . Since the solution is a continuous positive function, then we can define
(19) 
Without loss of generality we assume , ; otherwise, we can choose , , where , as and , respectively. By (a1) the following values are positive
(20) 
Define
(21) 
Since , and , , then there exists such that for and for . Let us demonstrate
(22) 
By the definition of we have . Suppose the contrary: or for some .
First, let for some . Then for some . Denote
Since then the set
is nonempty, denote . Then , ; we also have and in the interval , thus the derivative is nonpositive
which contradicts the assumption .
Similarly, let us assume that for some , and some . After introducing
we have , , and for , hence
This contradicts the assumption . Consequently (22) is valid for and thus for any , the bounds are positive, so the solution is permanent, which completes the proof.
Example 1. The statement of Theorem 3 is not valid if we omit the condition . Consider the equation
(23) 
which is equivalent to the initial value problem
(24) 
its solution tends to zero as and so is not permanent.
3 Absolute Global stability for Stable Difference Equations
One of the main steps in establishing global stability property is the proof of the fact that all nonoscillatory about the equilibrium solutions tend to this equilibrium (see, for example, [22]). For ordinary differential equations all solutions are nonoscillatory, for retarded equations it depends on the delay. Below we demonstrate that convergence of nonoscillatory solutions to the equilibrium is quite a common property which is valid for any reproduction function with a unique positive equilibrium. It can be interpreted as: “if nonoscillatory, solutions of delay equations behave asymptotically similar to ordinary differential equations”.
Definition. A solution of (8),(9) is nonoscillatory about if there exists such that either or for all . Otherwise, oscillates about .
Proof. First, let , . Without loss of generality we can assume . By (a2) there exists such that for . Denote as in (21). By Theorem 3 we obtain that for any . Since is continuous and for , then . There may be two cases: and . In the former case, since , we have
as far as , thus the solution of the delay differential equation is not less than the solution of , , which is increasing and by (a3) (the integral of diverges) tends to .
Consider the latter case . By the definition of and for we have . Taking any , and assuming for any , we obtain
which leads to a contradiction as since diverges. Thus, for some ; moreover, since as then for any . Let , for some . By the definition of and we have and as far as and the following inequality holds
Assuming for any we again obtain a contradiction. Thus, there exist and such that and for . Then for any and , .
Further, let , here since for . Similarly, we obtain whenever , for some .
We continue this process. It can be finite (for example, in Fig. 1 we have , where the process stops and we deduce as ) or infinite (see the branch of Fig. 1). In the infinite case we have an increasing sequence , which does not exceed , so this sequence has a limit . Since is continuous then . If then should attain its minimum in but , so this minimum is positive and the equality leads to a contradiction.
Further, let . Similarly, we define as in (21) and . There may be two cases: and . In the former case we obtain as . Consider the latter case. By Theorem 3 we have for any . By the definition of and , we have . Similar to the case we demonstrate that there exists such that and for any . Let be such that for . Further, we define . We continue this process, it can be finite or infinite (Fig. 1 illustrates an infinite process for ). Similar to the case we obtain as , which completes the proof.
Example 2. Let us note that in the case of infinite delays nonoscillatory solutions do not necessarily tend to the positive equilibrium. For example, the solution of the equation
which is , tends to while the positive equilibrium is , the monotone solution is nonoscillatory.
Thus for any reproduction function with a unique fixed point nonoscillatory solutions tend to the equilibrium; this is not, generally, true for oscillatory solutions.
Example 3. Consider the Nicholson’s blowflies equation
(25) 
Denote
(26) 
where is a positive equilibrium. If then the positive equilibrium is locally asymptotically stable for and is unstable (locally, thus it cannot be globally attractive) for , and (25) undergoes a Hopf bifurcation at when [23] for .
Now we prove that absolute (delayindependent) convergence holds in some special cases.
Lemma 2
Proof. 1. By Theorem 3 any solution is permanent: , where , , and for some we have , . Denote and .
If then the derivative is negative for any and the solution either eventually does not exceed or is decreasing for any . In the latter case the solution tends to the equilibrium; if it has a different limit, we obtain that the derivative is less than a negative number, which is a contradiction. In the former case, if there exists such that (otherwise, we have a nonoscillatory case where we have already proved convergence), then for any (assuming we obtain that the derivative of is negative almost everywhere, while the function changes from to ), which is again a nonoscillatory case, by Theorem 4 solution converges to .
Thus, we can consider only. Then we introduce
and similarly define and , . There exists such that for and such that , . Similar to the proof of Theorem 4 we obtain that there exists a sequence such that
We assume that all , otherwise we have an eventually monotone case. Since for any continuous all three sequences are monotone (nonincreasing and nondecreasing , ) and tend to (see the end of the proof of Theorem 4), then .
2. Now suppose that (27) holds for any . We begin with in Theorem 3, , where are defined in (21), , . Denote
(28) 
The case was considered in Part 1, thus we can restrict ourselves to the case , maximum is attained at
here is the greatest point where the maximum is attained. Let us demonstrate that there exists such that , .
If for all then we have a nonoscillatory case and convergence to , which is a contradiction. Thus, for some ; assuming there is such that we obtain that the value of the function at the end of the segment is higher than at the beginning point, while the derivative is nonpositive. So there exists such that for . If then . Similarly, we find such that for large enough. Since the sequence is nonincreasing and tends to then there exists such that , . Further, we will consider , where whenever , only.
If for all , where was defined in (28), then we have a nonoscillatory case and convergence to , which is a contradiction. As above, we prove that for , here . If , then we construct a sequence which tends to . If then some . There is such that , . Consequently, we have found such that , . Let , .
Now, if then for the solution increases as far as . Thus, either for any and this monotone solution converges to , or, if for some , , , and again we have a nonoscillatory solution which converges to .
Denote
Let us assume , and demonstrate that there is such that , , where . In fact, for any the solution is nonincreasing as far as , which gives an upper bound. Considering where the equation refers only to the values where this bound is valid, we obtain that the solution is nondecreasing if , which together with confirms the statement. We continue the induction process
where , . This process can be infinitely continued if all , (otherwise, at certain stage we have a “monotone” case which implies convergence), and there exists , such that , . Let us assume that there are infinite sequences and , both are monotone and bounded. Then there exist limits
Since by the assumptions of the theorem , then . If