10.1 Introduction to Delay Differential Equations (DDEs)

Delay differential equations (DDEs) are a fascinating extension of ordinary differential equations. They include a time delay, making them perfect for modeling real-world systems where past states influence the present. DDEs pop up in biology, engineering, and economics.

In this section, we'll dive into the basics of DDEs. We'll look at their definition, characteristics, and how they differ from regular ODEs. We'll also explore some cool applications and learn how to classify different types of DDEs.

Delay Differential Equations

Definition and Characteristics

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• Delay differential equations (DDEs) incorporate a time delay or lag term into the equation, representing the dependence of the current state on past states
  • The general form of a DDE is $x'(t) = f(t, x(t), x(t-τ))$, where $τ$ is the delay term, which can be constant, time-dependent, or state-dependent
  • DDEs are infinite-dimensional systems because the initial condition requires specifying a function over an interval rather than just an initial value
  • The presence of the delay term enables DDEs to exhibit complex dynamics, such as oscillations, instabilities, and chaotic behavior due to the delayed feedback
  • Example: A DDE modeling population growth with a maturation delay: $N'(t) = rN(t)(1 - \frac{N(t-τ)}{K})$, where $N(t)$ is the population size at time $t$, $r$ is the growth rate, $K$ is the carrying capacity, and $τ$ is the maturation delay

Comparison to Ordinary Differential Equations

• DDEs differ from ordinary differential equations (ODEs) in that they involve the values of the unknown function at previous times
  • ODEs only depend on the current state of the system and do not incorporate time delays
  • The solution of an ODE is determined by an initial value, while the solution of a DDE requires an initial function specified over an interval
  • DDEs can capture more complex behaviors and dynamics compared to ODEs due to the presence of the delay term
  • Example: An ODE modeling exponential growth: $N'(t) = rN(t)$, where $N(t)$ is the population size at time $t$ and $r$ is the growth rate, does not account for any delays in the growth process

Applications of DDEs

Biological Systems

• Population dynamics: DDEs model the growth and interactions of populations with delayed feedback
  • Maturation time or gestation period can be incorporated as a delay term, affecting the population growth
  • Example: Nicholson's blowflies equation, which models the population of adult blowflies with a delay representing the time from egg to adult: $N'(t) = -δN(t) + pN(t-τ)e^{-aN(t-τ)}$, where $N(t)$ is the adult population, $δ$ is the death rate, $p$ is the maximum per capita daily egg production, $a$ is the size at which the population reproduces at its maximum rate, and $τ$ is the maturation delay
• Epidemiology: DDEs describe the spread of infectious diseases with incubation periods or delayed immune responses
  • The delay term represents the time between infection and the onset of symptoms or the time for the immune system to respond
  • Example: An SIR model with a fixed infectious period: $S'(t) = -βS(t)I(t)$, $I'(t) = βS(t)I(t) - βS(t-τ)I(t-τ)$, $R'(t) = βS(t-τ)I(t-τ)$, where $S(t)$, $I(t)$, and $R(t)$ are the susceptible, infected, and recovered populations, respectively, $β$ is the transmission rate, and $τ$ is the fixed infectious period
• Physiological processes: DDEs describe the delayed feedback in hormonal regulation, blood cell production, or circadian rhythms
  • The delay term captures the time required for the production, secretion, or transport of hormones or cells
  • Example: A DDE model for the regulation of blood cell production: $P'(t) = -γP(t) + β(M - P(t-τ))$, where $P(t)$ is the concentration of blood cells, $γ$ is the decay rate, $β$ is the production rate, $M$ is the target concentration, and $τ$ is the delay in the feedback loop

Physical and Engineering Systems

• Control systems: DDEs represent the time delays in feedback control loops
  • The delay term accounts for the time between the measurement of a system's state and the application of a control action
  • Example: A proportional-integral-derivative (PID) controller with a delay: $u(t) = K_p e(t) + K_i \int_0^t e(s) ds + K_d \frac{de(t-τ)}{dt}$, where $u(t)$ is the control signal, $e(t)$ is the error signal, $K_p$, $K_i$, and $K_d$ are the proportional, integral, and derivative gains, respectively, and $τ$ is the delay in the derivative term
• Neural networks: DDEs capture the time delays in signal transmission between neurons
  • The delay term represents the time required for a signal to propagate from one neuron to another, affecting the synchronization and stability of neural activity
  • Example: A DDE model for a single neuron with delayed feedback: $v'(t) = -v(t) + f(v(t-τ))$, where $v(t)$ is the membrane potential, $f$ is the activation function, and $τ$ is the feedback delay

Economic and Social Systems

• Economics: DDEs model the delayed impact of economic policies, investment decisions, or price adjustments on market dynamics
  • The delay term captures the time lag between the implementation of a policy or decision and its effect on the economy
  • Example: A DDE model for the price adjustment in a market with a production delay: $P'(t) = α(D(P(t)) - S(P(t-τ)))$, where $P(t)$ is the price at time $t$, $D(P)$ is the demand function, $S(P)$ is the supply function, $α$ is the price adjustment coefficient, and $τ$ is the production delay
• Social dynamics: DDEs describe the spread of information, opinions, or behaviors in social networks with communication delays
  • The delay term represents the time required for information to propagate through the network or for individuals to process and respond to the information
  • Example: A DDE model for the spread of a rumor in a social network: $S'(t) = -βS(t)I(t-τ)$, $I'(t) = βS(t)I(t-τ) - γI(t)$, $R'(t) = γI(t)$, where $S(t)$, $I(t)$, and $R(t)$ are the susceptible, infected (rumor-spreading), and recovered (non-spreading) populations, respectively, $β$ is the transmission rate, $γ$ is the recovery rate, and $τ$ is the communication delay

Classifying DDEs

Types of Delay

• Constant delay: The delay term $τ$ is a fixed value, independent of time or the system's state
  • The DDE takes the form $x'(t) = f(t, x(t), x(t-τ))$, where $τ$ is a constant
  • Example: A DDE with a constant delay: $x'(t) = -ax(t) + bx(t-τ)$, where $a$ and $b$ are constants and $τ$ is a fixed delay
• Time-dependent delay: The delay term $τ(t)$ varies with time
  • The DDE has the form $x'(t) = f(t, x(t), x(t-τ(t)))$
  • Example: A DDE with a time-dependent delay: $x'(t) = -ax(t) + bx(t-τ(t))$, where $a$ and $b$ are constants and $τ(t)$ is a function of time
• State-dependent delay: The delay term $τ(x(t))$ depends on the current state of the system
  • The DDE has the form $x'(t) = f(t, x(t), x(t-τ(x(t))))$
  • Example: A DDE with a state-dependent delay: $x'(t) = -ax(t) + bx(t-τ(x(t)))$, where $a$ and $b$ are constants and $τ(x(t))$ is a function of the current state $x(t)$

Discrete and Distributed Delays

• Discrete delay: The delay term takes on discrete values, typically integer multiples of a fixed delay
  • The DDE can be written as $x'(t) = f(t, x(t), x(t-τ_1), x(t-τ_2), ...)$, where $τ_1$, $τ_2$, ... are discrete delays
  • Example: A DDE with multiple discrete delays: $x'(t) = -ax(t) + bx(t-τ_1) + cx(t-τ_2)$, where $a$, $b$, and $c$ are constants and $τ_1$ and $τ_2$ are fixed discrete delays
• Distributed delay: The DDE involves an integral term that accounts for the cumulative effect of the past states over a continuous interval
  • The general form is $x'(t) = f(t, x(t), \int_{t-τ}^t g(t, s, x(s)) ds)$, where $g$ is a kernel function
  • Example: A DDE with a distributed delay: $x'(t) = -ax(t) + b\int_{t-τ}^t x(s) ds$, where $a$ and $b$ are constants and $τ$ is the maximum delay

Solutions for DDEs

Existence and Uniqueness

• Initial value problem: For a DDE $x'(t) = f(t, x(t), x(t-τ))$ with initial condition $x(t) = φ(t)$ for $t ∈ [-τ, 0]$, where $φ$ is a given function, the existence and uniqueness of solutions depend on the properties of $f$ and $φ$
  • If the function $f$ satisfies the Lipschitz condition with respect to $x(t)$ and $x(t-τ)$, and the initial function $φ$ is continuous, then the DDE has a unique solution on some interval $[0, T]$
  • Example: Consider the DDE $x'(t) = -ax(t) + bx(t-τ)$ with initial condition $x(t) = φ(t)$ for $t ∈ [-τ, 0]$. If $a$ and $b$ are constants and $φ$ is a continuous function, then the DDE has a unique solution on some interval $[0, T]$
• Continuation of solutions: If the solution of a DDE exists on an interval $[0, T]$, it can be extended to a larger interval by considering the value of the solution at $T$ as the new initial condition and applying the existence and uniqueness theorem
  • This process can be repeated to extend the solution to its maximal interval of existence
  • Example: If the solution of the DDE $x'(t) = -ax(t) + bx(t-τ)$ exists on the interval $[0, T]$, then it can be extended to a larger interval by using the value $x(T)$ as the new initial condition for the DDE on the interval $[T, T+τ]$

Methods for Solving DDEs

• Method of steps: For DDEs with constant delays, the method of steps can be used to construct the solution step by step on intervals of length $τ$
  • The solution from the previous interval is used as the initial condition for the next interval
  • The DDE is reduced to an ODE on each interval, which can be solved using standard techniques
  • Example: Consider the DDE $x'(t) = -ax(t) + bx(t-τ)$ with initial condition $x(t) = φ(t)$ for $t ∈ [-τ, 0]$. Using the method of steps, the solution on the interval $[0, τ]$ is obtained by solving the ODE $x'(t) = -ax(t) + bφ(t-τ)$ with initial condition $x(0) = φ(0)$. The solution on the interval $[τ, 2τ]$ is then obtained by solving the ODE $x'(t) = -ax(t) + bx(t-τ)$ with initial condition $x(τ)$ obtained from the previous step
• Numerical methods: Various numerical methods can be used to approximate the solutions of DDEs
  • Runge-Kutta methods: Adapted Runge-Kutta methods, such as the Runge-Kutta-Fehlberg method or the Dormand-Prince method, can be used to solve DDEs by incorporating the delay terms in the step-by-step computation
  • Predictor-corrector methods: Predictor-corrector methods, such as the Adams-Bashforth-Moulton method, can be extended to solve DDEs by using an interpolation scheme to approximate the delayed terms
  • Example: Consider the DDE $x'(t) = -ax(t) + bx(t-τ)$ with initial condition $x(t) = φ(t)$ for $t ∈ [-τ, 0]$. A Runge-Kutta method can be used to approximate the solution by discretizing the time interval and computing the solution step by step, using the delayed terms from the previous steps

Smoothness and Stability

• Smoothness of solutions: The smoothness of the solution depends on the smoothness of the function $f$ and the initial condition $φ$
  • If $f$ and $φ$ are continuously differentiable, then the solution will also be continuously differentiable
  • Discontinuities in $f$ or $φ$ can lead to discontinuities or non-smoothness in the solution
  • Example: If the function $f(t, x(t), x(t-τ))$ and the initial condition $φ(t)$ are both continuously differentiable, then the solution of the DDE $x'(t) = f(t, x(t), x(t-τ))$ with initial condition $x(t) = φ(t)$ for $t ∈ [-τ, 0]$ will be continuously differentiable
• Stability analysis: The stability of solutions of DDEs can be investigated using various techniques
  • Lyapunov-Krasovskii functionals: Lyapunov-Krasovskii functionals, which are extensions of Lyapunov functions for DDEs, can be used to study the stability of equilibrium points or periodic solutions
  • Characteristic equations: The stability of linear DDEs can be determined by analyzing the roots of the characteristic equation, which is obtained by substituting an exponential solution into the DDE
  • Example: Consider the linear DDE $x'(t) = -ax(t) + bx(t-τ)$. The characteristic equation is given by $λ + a - be^{-λτ} = 0$. If all the roots of the characteristic equation have negative real parts, then the zero solution of the DDE is asymptotically stable