Glossary

2.2 Systems of ODEs and phase plane analysis

3 min read•july 25, 2024

are powerful tools for modeling complex biological phenomena. They allow us to represent interrelated variables and their rates of change over time. By converting higher-order ODEs and analyzing , we can gain insights into system behavior.

helps us understand the long-term behavior of biological systems. By finding , calculating , and classifying stability, we can predict how populations or other variables will evolve under different conditions. This knowledge is crucial for making informed decisions in fields like ecology and epidemiology.

Systems of ODEs

Conversion of higher-order ODEs

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  • Identify ODE order
  • Introduce new variables for derivatives up to one less than order
  • Express highest-order derivative using lower-order derivatives and original variables
  • Write first-order ODE system with new variables
    • Each equation represents one variable's derivative in terms of others
  • Equation number matches original ODE order
  • Example: Convert d2ydt2+3dydt+2y=0\frac{d^2y}{dt^2} + 3\frac{dy}{dt} + 2y = 0 to first-order system
    1. Let x1=yx_1 = y and x2=dydtx_2 = \frac{dy}{dt}
    2. Rewrite as dx1dt=x2\frac{dx_1}{dt} = x_2 and dx2dt=−3x2−2x1\frac{dx_2}{dt} = -3x_2 - 2x_1

Phase portraits for ODE systems

  • Determine system type (linear or nonlinear)
  • Linear systems:
    • Calculate eigenvalues and eigenvectors
    • Identify equilibrium point type (node, saddle, focus, center)
    • Draw eigenvectors and flow direction
  • Nonlinear systems:
    • Locate equilibrium points solving dxdt=0\frac{dx}{dt} = 0 and dydt=0\frac{dy}{dt} = 0
    • Linearize system around equilibrium points
    • Analyze behavior near equilibrium points
  • Plot trajectories in phase plane
    • Use arrows for flow direction
    • Include nullclines (dxdt=0\frac{dx}{dt} = 0 and dydt=0\frac{dy}{dt} = 0)
  • Example: Sketch phase portrait for predator-prey system
    • dxdt=ax−bxy\frac{dx}{dt} = ax - bxy, dydt=−cy+dxy\frac{dy}{dt} = -cy + dxy ()

Stability Analysis

Equilibrium points and stability analysis

  • Locate equilibrium points solving dxdt=0\frac{dx}{dt} = 0 and dydt=0\frac{dy}{dt} = 0
  • Calculate at equilibrium points
  • Find Jacobian eigenvalues
    • 2x2 systems: use det⁡(J−λI)=0\det(J - \lambda I) = 0
  • Determine stability from eigenvalues:
    • Negative real parts: stable
    • Positive real parts: unstable
    • Pure imaginary: center (neutrally stable)
    • Zero real part (non-zero imaginary): further analysis needed
  • Calculate eigenvectors: (J−λI)v=0(J - \lambda I)v = 0
  • Classify equilibrium points:
    • Node: real, distinct eigenvalues, same sign
    • Saddle: real, distinct eigenvalues, opposite signs
    • Focus: complex conjugate eigenvalues, non-zero real parts
    • Center: pure imaginary eigenvalues
  • Example: Analyze stability of equilibrium points
    • dSdt=−βSI\frac{dS}{dt} = -\beta SI, dIdt=βSI−γI\frac{dI}{dt} = \beta SI - \gamma I, dRdt=γI\frac{dR}{dt} = \gamma I

Linearization near equilibrium points

  • Approximate nonlinear systems with linear systems near equilibrium
  • Compute Jacobian matrix at equilibrium point
  • Analyze linearized system:
    • Use Jacobian eigenvalues and eigenvectors
    • Determine local equilibrium point stability
  • Apply Hartman-Grobman theorem
    • Linearization represents local behavior for hyperbolic equilibria
  • Recognize linearization limitations
    • Not applicable for non-hyperbolic equilibria (zero real part eigenvalues)
  • Use higher-order terms for non-hyperbolic cases
    • Center manifold theory
    • Normal form theory
  • Example: Linearize Van der Pol oscillator near equilibrium
    • dxdt=y\frac{dx}{dt} = y, dydt=μ(1−x2)y−x\frac{dy}{dt} = \mu(1-x^2)y - x

Key Terms to Review (19)

Asymptotic stability: Asymptotic stability refers to the property of a dynamical system where, after a disturbance, the system returns to a stable equilibrium point as time approaches infinity. In other words, not only does the solution converge to the equilibrium point, but it also does so in a way that ensures all nearby trajectories eventually follow this path back to stability. This concept is vital when analyzing systems of ordinary differential equations (ODEs) and understanding their long-term behavior in phase plane analysis.
Bifurcation Diagram: A bifurcation diagram is a visual representation that shows how the qualitative behavior of a dynamical system changes as a parameter is varied. It illustrates the points at which the system's equilibrium states transition, leading to sudden changes in stability or the emergence of new behaviors. This diagram is crucial in analyzing how small changes in parameters can result in significant shifts in the dynamics of systems governed by ordinary differential equations (ODEs).
Eigenvalues: Eigenvalues are special numbers associated with a linear transformation represented by a matrix that indicate how much a corresponding eigenvector is stretched or compressed during the transformation. They play a critical role in understanding the stability and dynamics of systems described by differential equations, particularly in relation to their phase portraits and equilibria. By analyzing eigenvalues, one can determine the nature of fixed points in models, including their stability and behavior over time.
Epidemiological models: Epidemiological models are mathematical frameworks used to describe the dynamics of infectious diseases within populations. These models help in understanding how diseases spread, the impact of interventions, and predicting future outbreaks by using systems of ordinary differential equations (ODEs) and analyzing their behavior through phase plane analysis.
Equilibrium Points: Equilibrium points are specific states in a dynamic system where the variables of interest do not change over time. At these points, the forces acting on the system are balanced, resulting in no net change, which makes them crucial for understanding stability and behavior in biological models and ecological systems.
Jacobian Matrix: The Jacobian matrix is a mathematical representation that describes the rates of change of a vector-valued function relative to its variables. In the context of systems of ordinary differential equations, it provides essential information about the system's behavior near equilibrium points and helps in analyzing stability and bifurcations. By applying the Jacobian matrix to the Lotka-Volterra model, we can better understand predator-prey dynamics and predict how small changes in population sizes affect overall system behavior.
Limit cycle: A limit cycle is a closed trajectory in phase space that represents periodic solutions of a dynamical system, particularly in the context of ordinary differential equations (ODEs). It indicates that regardless of the initial conditions nearby, the system will eventually settle into this repetitive behavior over time, creating stable oscillations. The presence of a limit cycle implies that the system has reached a steady state in terms of its oscillatory dynamics, which can be essential in understanding biological rhythms and population dynamics.
Linear system: A linear system refers to a set of linear equations or differential equations that can be expressed in the form of matrices. These systems are essential in analyzing and understanding the behavior of dynamic systems, as they can often be simplified and solved using established mathematical techniques. In the study of systems of ordinary differential equations (ODEs), linear systems are particularly useful for examining equilibrium points and stability through phase plane analysis.
Lotka-Volterra equations: The Lotka-Volterra equations are a pair of first-order, nonlinear differential equations used to describe the dynamics of biological systems in which two species interact, typically a predator and its prey. These equations model the population changes over time, capturing the cyclical nature of predator-prey interactions and demonstrating how the populations influence one another's growth rates.
Lyapunov Stability: Lyapunov stability refers to a property of dynamical systems where an equilibrium point is stable if small perturbations from that point lead to trajectories that remain close to the equilibrium over time. This concept is crucial in understanding the long-term behavior of systems described by ordinary differential equations (ODEs) and plays a key role in phase plane analysis, where the stability of various equilibrium points can be visualized and analyzed.
Nonlinear system: A nonlinear system is a type of mathematical system in which the output is not directly proportional to the input, meaning that small changes in initial conditions can lead to vastly different outcomes. In these systems, equations governing the relationships between variables contain nonlinear terms, such as products or powers, which complicate their behavior and analysis. Nonlinear systems are essential for understanding complex phenomena in various fields, including mathematical biology, where they can describe interactions among species or populations.
Perturbation Theory: Perturbation theory is a mathematical approach used to find an approximate solution to a problem that cannot be solved exactly, by introducing a small change, or 'perturbation,' to a known solution of a simpler problem. This technique helps analyze how slight modifications in parameters or conditions can influence the behavior of a system, making it especially useful in understanding dynamics in various mathematical models, including those related to population structures and systems of differential equations.
Phase Portraits: Phase portraits are graphical representations used to visualize the trajectories of dynamic systems in the phase space, typically derived from a set of ordinary differential equations (ODEs). They provide insight into the behavior of systems over time, illustrating how the state of the system evolves and revealing stable and unstable equilibria, periodic orbits, and other phenomena that characterize the dynamics.
Population dynamics: Population dynamics refers to the changes in population size, structure, and distribution over time, influenced by birth rates, death rates, immigration, and emigration. This concept helps in understanding how populations grow, shrink, or stabilize under various environmental pressures and interactions, such as competition and predation.
SIR Model: The SIR model is a fundamental mathematical framework used to describe the spread of infectious diseases within a population, categorizing individuals into three compartments: Susceptible, Infected, and Recovered. This model helps to analyze how diseases propagate over time and can be extended to incorporate more complex interactions and network structures in epidemiology, making it foundational for understanding disease dynamics.
Stability Analysis: Stability analysis is a mathematical method used to determine the behavior of solutions to differential equations, particularly in terms of their response to small perturbations. This concept is crucial for understanding whether a system will return to equilibrium after a disturbance or diverge away from it, impacting the long-term dynamics and predictions made by various biological models.
Steady-state solution: A steady-state solution refers to a condition in a dynamic system where the variables no longer change with time, indicating equilibrium. In systems described by ordinary differential equations (ODEs), it represents a point where the derivatives of the system's state variables are equal to zero, leading to stable behavior over time. This concept is crucial for understanding the long-term behavior of systems and is often analyzed through phase plane analysis, where trajectories may converge to these steady-state points.
Systems of ODEs: Systems of ordinary differential equations (ODEs) refer to a set of equations involving multiple dependent variables and their derivatives with respect to one independent variable, usually time. These systems are essential in modeling complex biological processes where several interacting components, such as populations or chemical concentrations, change over time. The behavior of these systems can be analyzed using various techniques, including phase plane analysis, to gain insights into stability and dynamic behavior.
Transient solution: A transient solution refers to a type of solution for a system of ordinary differential equations (ODEs) that describes the behavior of the system over time before it reaches a steady state. These solutions are typically temporary and highlight how the system evolves from an initial condition towards equilibrium, which can be crucial in understanding dynamic processes. They play a significant role in phase plane analysis, where the trajectory of the system in the phase space reveals how transient behavior influences long-term outcomes.
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