Effects of Discrete Time Delays and Parameters Variation on Dynamical Systems

Delay Differential Equations (DDE’s) have received considerable attention in recent years. While most of these articles focused on the effects of the time delays on the stability of the equilibrium points and on the bifurcation that they may raised, very few papers address the key roles that system parameters play on if and how the discrete delays induce stability changes of the equilibria and produce bifurcations near such equilibria. In this article we focus on that question in a general setting, that is, if one has a system of DDE’s with one or multiple discrete time delays, what are the results of changing the system parameters values on the effects of the discrete time delays on the dynamic of the system. We present general results for one equation with one and two delays and study a specific example of one equation with one delay. We then establish the procedure for n equations with multiple delays and do a specific example for two equations with two delays. We compute the steady states and analyze their stability as both chosen bifurcation parameters, the discrete time delay τ and a local equation parameter μ, cross critical values. Our analysis shows that while changes in both parameters can destabilize the steady state, the discrete time delay can only cause stability switches of the steady state for certain values of μ, while the effects of the local equation parameter on the steady state do not necessarily depend on the value of τ . While μ may cause the system to go through different type of bifurcations, the discrete time delay can only introduce a Hopf bifurcation for certain values of μ. Keywords-delay differential equations; bifurcation; predator-prey.


I. INTRODUCTION
It is well known, that the values of the parameters play a crucial role in the behavior of dynamical systems and that changes in the values can change the behavior significantly.It has also been shown by many researchers (Perelson [1],Allen [2],Bellen [3]) that there is a need to incorporate discrete time delays in dynamical systems (biological systems, physical systems,...) as studied.
Models that incorporate such delays are referred to as delay differential equations (DDE's).DDE's have been extensively studied by many researchers including pioneers Bellman [4], Driver [5], and in more recent years by Culshaw [6], Gakkhar [7], Bellen [3], and a superb monograph on the subject by Gopalsamy [8].While most of these research papers focus on issue of the stability changes caused by the delay(s), the main motivation of this paper is to study how a local bifurcation parameter of the system may affect the changes in stability caused by the delay (s).
Published papers have shown that the incorporation of discrete time delays can highly impact the dynamics of the system, since they can switch the stability of a steady state point, and can also cause the system to go through a Hopf bifurcation near that steady state point (Culshaw[6], Gakkhar [7], Bellen [3]).In this paper we consider a system of n delay differential equations (DDE's) with one parameter µ as the bifurcation parameter and also with one or more discrete time delays, τ , which can also behave as bifurcation parameters.We are interested in investigating how the parameters µ and τ affect the stability of the steady state points of the system, and, more important, how their effects on the system are correlated to each other.We present general results in the one dimensional case (propositions 1 to 3) for necessary and sufficient conditions for a stability switch and present a specific example to illustrate these conditions.For the n dimensional case (n ≥ 2) we establish the main ideas, but since there are multiple possible cases, we consider only a specific example.We present a non-Kolmogorov type of predator-prey model similar to the model presented by Ruan [9].In this model we introduce two delays, τ 1 > 0 and τ 2 > 0, to represent the time lag in the growth to maturity of the prey, and the time lag in the growth to maturity of the predator, respectively.We show how the dynamics of the system change depending on certain conditions on τ 1 and on another bifurcation parameter R. We also point out conditions for the system to go through stability changes when both delays τ 1 and τ 2 are non-zero.We present necessary conditions for the system to go through a Hopf bifurcation for τ 1 > 0 and τ 2 = 0. Finally we show numerical results illustrating the theoretical results.

A. One Equation with One Delay
Consider the one dimensional delay differential equation with the time delay τ , and the parameter µ as bifurcation parameters: where f is assumed to be smooth enough to guarantee the existence and uniqueness of solutions to (1) under the initial condition (R. Bellman and K. L. Cooke [4]) Unfortunately equation ( 1) is too general to analyze.Therefore we will consider a more special form: This form has the advantage that it simplifies the analytical work and also it is the form present in many population dynamical models involving delays [6], [7], [9], [10].The DDE (2) may or may not have equilibrium points (or steady states) and these will depend on the values of µ.Let ,that is µ * is in the range of values of µ for which the DDE has an equilibrium point X * , i.e., f (X * , X * , µ * ) = 0. We are interested in studying the stability of such equilibrium point.In particular, in studying the effect of the parameter µ and of the discrete time τ on its stability.To do this we linearize the DDE around the equilibrium point.The characteristic equation is : and the stability of the equilibrium point (X * , µ * ) is determined by the sign of the real part of the eigenvalues λ of equation (3).

1) Stability of the Steady State
The stability of the steady state then depends only on values of µ * within D µ .We have two cases: (a) The steady state (X * , µ * ) is stable if Assume that condition (a) holds, namely the steady state (X * , µ * ) is stable when there is no delay (τ = 0).We want to know if there exists τ > 0 for which the steady state will lose stability.So for τ ≥ 0, let λ(τ ) = α(τ ) + iω(τ ).The characteristic equation (3) becomes: where, for clarity in the notation, we have not explicitly shown the dependence on τ .Separating the real and imaginary parts, we have: The steady state will lose stability when the real part of the eigenvalue λ crosses the zero axis from negative to positive as τ passes a critical value.By Rouche's Theorem (Dieudonne [11], Theorem 9.17.4) and by the continuity in τ , the transcendental equation (3) has roots with positive real parts if and only if it has pure imaginary roots.Therefore, we look at when the real part of the eigenvalue λ becomes zero.In other words, we want to find if there exists a τ c > 0 such that α(τ c ) = 0. Since and α(0) < 0 by assumption (a) , therefore if τ c > 0 exists such that α(τ c ) = 0 then by the continuity (Michael Y. Li and Hogying Shu [10]) of α we have: • α(τ ) < 0 for any 0 ≤ τ < τ c , • α(τ ) > 0 for any τ > τ c .Namely the steady state (X * , µ * ) will lose stability as the delay parameter τ crosses a critical value τ c .Such τ c exists if and only if α(τ c ) = 0 and ω(τ c ) = ω c satisfies : ) Squaring equations ( 7) and ( 8), and adding them up, we obtain: If equation ( 9) has at least a positive root ω c , then there exists a τ c > 0 such that α(τ ) > 0 whenever τ > τ c (see proof in Appendix A).An important question we want to address is, since equation ( 9) depends on the bifurcation parameter µ * , can one chose µ * within D µ so that equation (9) does not have a positive root ω c ?That is, are there values of µ * within D µ such that the delay does not have any effect on the stability of the steady state (X * , µ * )?This question motivates the following propositions (see Appendix B for the proof).
Proposition 1: Consider the one dimensional delay differential equation And assume that the steady state (X * , µ * ) is stable for τ = 0 then we have (i) If df1 dX | (X * ,µ * ) < 0 and df2 dX | (X * ,µ * ) > 0 then the steady state (X * , µ * ) remains stable for all τ ≥ 0. (ii) If df1 dX | (X * ,µ * ) > 0 and df2 dX | (X * ,µ * ) < 0 then there exists a critical value of the delay such that the steady state loses stability as the delay crosses its critical value.(iii) If df1 dX | (X * ,µ * ) < 0 and df2 dX | (X * ,µ * ) < 0 then: (a) the steady state remains stable for all τ ≥ 0 if there exists a τ c > 0 such that the steady state becomes unstable for all τ > τ c if Proposition 2: Consider the one dimensional delay differential equation And assume that the steady state (X * , µ * ) is stable for τ = 0, that conditions of proposition 1(iii) hold, and that further more for some µ * within D µ we have: then there exists a critical value for µ * within D µ such that the steady state (X * , µ * ) will stay stable for all τ ≥ 0 when µ * > µ c .
2) Example: Consider the one dimensional DDE where µ is a bifurcation parameter and τ ≥ 0 is a discrete time delay.For µ ∈ D µ = R, the equation has two non-negative equilibrium points: the trivial one Y * 0 = 0, and the positive equilibrium point . The characteristic equation is given as • For the trivial equilibrium point Y * = 0, its stability only depends on µ since equation ( 10) evaluated at Y * = 0 becomes λ = µ.The trivial equilibrium is unstable for µ > 0 and all τ ≥ 0.
The trivial equilibrium is stable for µ < 0 and all τ ≥ 0.
, equation ( 10) becomes: then the stability of Y * 1 depends on both µ and τ .

1)
If τ = 0 then equation ( 11) becomes Remark: To better understand the situation, the stability of both equilibria when there is no delay is shown in the following table: At the equilibrium (Y, µ) = (0, 0), there is an exchange of stability.This is a transcritical bifurcation (Guckenheimer [12]).Geometrically, there are two curves of equilibria which intersect at the origin and lie on both sides of µ = 0. Stability of the equilibrium changes along either curve on passing through µ = 0.
Assume that the steady state And also H(ω 2c ) → ∞ as ω 2c → ∞.Then the intermediate value theorem assures that equation H(ω 2c ) = 0 has at least a positive root.
We now extend our analysis to a system of ndelay differential equations with multiple discrete time delays τ 1 , τ 2 , ..., τ k , and a local bifurcation parameter µ.

III. N DIMENSIONAL FIELD
Consider the following system non-linear delay differential equations: where assumed to be smooth enough to guarantee existence and uniqueness of solutions to (21) under the initial value condition (R. Bellman and K. L. Cooke [4] and J. K. Hale and S. M. Verduyn Lunel [13]) where Suppose f (x * , x * , ..., x * , µ * ) = 0, that is, (x * , µ * ) is a steady state of system (21).We are interested in studying the stability of such equilibrium point.
In particular studying the effect of the parameter µ and the discrete time delays τ 1 , τ 2 , ..., τ k on its stability.The linearization of (21) at (x * , µ * ) has the form (Ruan [9]): where X ∈ R n , each A j (µ * ) (0 ≤ j ≤ k) is an n×n constant matrix that depends on values of µ * within D µ .The transcendental equation associated with system (21) is given as : Equation ( 23) has been studied by many researchers (Ruan [9], R. Bellman and K. L. Cooke [4] and J. K. Hale and S. M. Verduyn Lunel [13]).The following result, which was proved by Chin [14] for k = 1 and by Datko [15] and Hale et al. [13] for k ≥ 1, gives a necessary and sufficient condition for the absolute stability of system (22).Lemma 1: System ( 22) is stable for all delays τ j (1 ≤ j ≤ k) if and only if Clearly, the stability of the steady state (x * , µ * ) and the effects of the discrete times τ j on its stability depend on values of µ * within D µ .To further investigate the effects of µ, and the discrete time delays τ j on the stability of (x * , µ * ), the exact entries of the matrices A j (µ * ) are needed to avoid doing a large number of cases.Note that the difficulty of the analysis is not due to the number of delays but to the number of equations.Even in the case of two equations with one delay, one needs to consider: where So the stability depends on all the entries of the A i , i = 0, 1, we have many different cases.
Therefore to present the ideas we consider a specific example with n = 2, k = 2, that is a two dimensional delay differential equations with two discrete time delays, and a local bifurcation parameter.

A. Two Dimensional Field Example
Consider the non-Kolmogorov type (Holling) predator-prey model where the parameters are described in the following table: Proposition 4: If the basic reproductive ratio (Ameh [16]) R > 1, the system has two nonnegative steady states: (x * 0 , y * 0 ) = (0, 0), and We consider R, τ 1 and 2 as the bifurcation parameters for the system (24-25) since changes of them may affect the existence and stability of the equilibrium points.
) is unstable for 0 ≤ τ 1 < τ 1c and τ 2 = 0. (ii) The steady state (x * 1 , y * 1 ) is stable for τ 1 > τ 1c and τ 2 = 0. Note that τ 1 affects the stability of the positive equilibrium only for values of R such that conditions C(0) are satisfied.Remark: For our parameter values, we have Proposition 7: Consider system (24-25) with τ 1 in its unstable interval (0 ≤ τ 1 < τ 1c ).If a 1 ≥ 2, then there exists a critical τ 2 > 0, such that the positive equilibrium becomes stable for τ 2 > τ 2c .Note that the effect of τ 2 on the stability of the positive equilibrium does not depend on the values of R.

D. Numerical Results
To illustrate the effect of the parameter R and the discrete time delay on the stability of the steady state (x * , y * ), and to support the theoretical predictions discussed above, we conducted numerical simulations for the system (24-25).We used DDE-BIFTOOL (Engelborghs [17]) for the stability and bifurcation analysis and also used the Matlab solvers ode23 and dde23 (Shampine [18],Shampine [19]) to see the behavior of the predator and prey populations through time.All the parameter values are given in Table II.For the given parameters values we have R = 10 > 1, and a positive equilibrium exists and is given as (x * 1 , y * 1 ) = (0.111, 1.111).When there is no delay the prey and predator populations variation through time is shown on Figure 4.
For our parameter values we have: Then there exists a τ 1c = 6 such that the steady state remains unstable for 0 ≤ τ 1 < τ 1c and τ 2 = 0 (see Figure 5), it becomes stable as τ 1 crosses τ 1c and τ 2 = 0 as shown on Figure 6.We examine closely the stability switch introduces by τ 1 .We use DDE-BIFTOOL to compute the eigenvalues of the characteristic equation (38) for τ 2 = 0 and 0 ≤ τ 1 ≤ 10.In Figure 7 we plot the real parts versus the imaginary parts of these eigenvalues.
For the case of two non-zero delays, we use Matlab to compute numerical simulations illustrating the effects of the two delays.The analysis is summarized in Table III  For τ 1 = 7 and τ 2 = 3.1 the equilibrium becomes unstable again as shown in Figure 12.

IV. CONCLUSIONS AND DISCUSSION
It is well known that changes in the parameters play a crucial role in understanding dynamical systems.There is a need to incorporate discrete time delays in dynamical systems (Biological systems, physical systems,...) as has been shown and studied by many researchers (Perelson [1],Bellen [3],..).
Published papers have shown that the incorporation of discrete time delays can highly impact the dynamics of the system, since they can cause stability switches of a steady state point, and can also cause the system to go through a Hopf bifurcation near that steady state point (Culshaw[6], Bellen [3],...).The highlight of this paper is on how a local bifurcation parameter of the system may modify the stability changes caused by the delay(s).To understand the effects of discrete time delays and parameter variations on certain biological system models, we carried out a bifurcation analysis of a system of delay differential equations in detail for n=1 with specific examples, gave the procedure for higher n, and did a concrete example for n=2.We investigated the stability of the steady states as both bifurcation parameters, the discrete time delay τ and a local bifurcation parameter µ, cross critical values.Our analysis shows that while both parameters can destabilize the steady state, the discrete time delay can cause stability switches of the steady state only upon certain values of µ.The local bifurcation parameter effects on the stability of the steady state do not depend on the value of τ .We also showed that both parameters act differently in term of bifurcation.While the discrete time delay may only introduce a Hopf bifurcation, the parameter µ can introduce other type of bifurcations.
Before we prove the above theorem , let just first consider a much simpler case.Consider the analytic function h(λ, a) = λ + e −λτ + a, with τ ≥ 0, and a ∈ R.
We have h(λ, a) is an analystic function in λ, a.When τ = 2jπ + π 2 , the function h(λ, 0) has no zeros on the boundary of Ω, where Ω = {λ, |Re(λ) ≥ 0, |λ| ≤ ρ}.Thus, Rouche's theorem (Dieudonne [11], Theorem 9.17.4) implies that there exists a δ > 0 such that : (1) for any a < δ, h(λ, a) has no zero on the boundary of Ω (2) for any a < δ, h(λ, a) and h(λ, 0) have the same sum of the orders of zeros belonging to Ω.It follows from lemma 2 that when τ > π 2 , the sum of the orders of the zeros of h(λ, 0) belonging to Ω is at least 1.Thus when τ > ), the sum of the orders of the zeros of h(λ, A 0 ) belonging to Ω is at least 1.Thus when τ > τ c , τ = τ j c and A ∞ < δ then h(λ, A) has at least a root with strictly positive real part.

A. Proof of Proposition 1
The characteristic equation of the one dimensional DDE is given by (3) and because the steady state is assumed to be stable at τ = 0 then 9) has no positive root meaning the steady state remains stable for all τ ≥ 0. 9) has a positive root then there exists a τ c > 0 such that α(τ ) > 0 whenever τ > τ c .If (iii)(a) holds then again equation ( 9) has no solution therefore α(τ ) < 0 for all τ ≥ 0 meaning the steady state remains stable.

B. Proof of Proposition 2
If conditions of proposition 1(iii)(a) hold then there is nothing to prove.Assume that conditions of proposition 1(iii)(b) hold then equation ( 9) has a positive solution, therefore the delay can affect the stability of the equilibrium point.But if for some µ * in D µ we have the extra condition then one can rewrite equation (9) as Then there exists a critical value µ c ∈ D µ of µ such that h(X * ) µ * ≈ 0 as µ * → µ c .Therefore equation ( 9) becomes ω 2 c = −[g(X * )µ c ] 2 < 0, which has no real positive root ω c , therefore α(τ ) < 0 for all τ ≥ 0. This implies the delay does not have any effect on the stability of the equilibrium point when µ * > µ c .

D. Proof of Proposition 6
The characteristic equation of the system evaluating at (x * 1 , y * 1 ) is given by λ , f 1 = r 1 r 2 .

TABLE III STABILITY
REGIONS IN CASE OF TWO NON-ZERO DELAYS