Notes on Systems of Nonlinear Differential Equations

Identify System: Typically in the form of
- ( \frac{dx}{dt} = f(x,y) )
- ( \frac{dy}{dt} = g(x,y) )
Key Terminology:
- Nullcline
- Jacobian
- Equilibria
Example: Lotka-Volterra Predator-Prey Model

Definition:
- x-nullclines: Curves where ( f(x,y) = 0 )
- y-nullclines: Curves where ( g(x,y) = 0 )
Example: For Lotka-Volterra equations:
- From model ( \frac{dx}{dt} = \frac{1}{2}x - 2xy )
- Set to zero to find nullclines:
- x-nullclines: ( y = \frac{\frac{1}{2} x}{2x} ) or simplified forms.
- y-nullclines: ( x = 0 ) or rearranged forms.

Definition:
- Pair ( (x^,y^) ) where both ( \frac{dx}{dt} = 0 ) and ( \frac{dy}{dt} = 0 )
Finding Equilibria: Solve
- From equations ( f(x^, y^) = 0 ) and ( g(x^, y^) = 0 )
- Example from Lotka-Volterra model:
- Equilibria identified by nullclines' intersections.

Stability in 1-D:
- Consider ( f'(x^*) ):
- ( f'(x^*) > 0 ) ⇒ unstable
- ( f'(x^*) < 0 ) ⇒ stable
Jacobian for 2-D Systems:
- Definition: Jacobian ( J ) is a ( 2 \times 2 ) matrix defined as:
- ( J = \begin{bmatrix} \frac{\partial f}{\partial x} & \frac{\partial f}{\partial y} \ \frac{\partial g}{\partial x} & \frac{\partial g}{\partial y} \end{bmatrix} )
- Evaluated at equilibrium points.
- Purpose: To determine the stability of equilibria using trace and determinant conditions.

Calculate Jacobian: Perform calculations for given equilibrium points.
Trace and Determinant Conditions:
- Evaluate to conclude whether equilibria are centers, nodes, or spirals based on eigenvalues derived from Jacobian.

Examined observed fixed points through graphical intersection of nullclines.
Stability evaluated through Jacobian analysis, enabling classification of behavior near equilibria (stable/unstable).

Direction Vectors:
- Can sketch vector fields to visualize system trajectories.
Qualitative Behavior:
- Understand qualitative dynamics around nullclines and equilibria to predict system behavior over time.

Identify System: Typically in the form of
( \frac{dx}{dt} = f(x,y) )
( \frac{dy}{dt} = g(x,y) )
This involves recognizing the dependent and independent variables, as well as understanding the nature of their interactions governed by the specific functions f and g.
Key Terminology:
- Nullcline: A curve in the phase plane where one of the rates of change (either \frac{dx}{dt} or \frac{dy}{dt}) is zero.
- Jacobian: A matrix of all first-order partial derivatives of a vector-valued function, essential for determining the local behavior of nonlinear systems near equilibria.
- Equilibria: Points in the system where the system remains at rest if disturbed slightly, i.e., both \frac{dx}{dt} and \frac{dy}{dt} equal zero.
Example: Lotka-Volterra Predator-Prey Model: This is a well-known application demonstrating interaction dynamics between two species, a predator and its prey, often used in ecological studies.

Definition:
- x-nullclines: Curves where ( f(x,y) = 0 ), indicating no change in x.
- y-nullclines: Curves where ( g(x,y) = 0 ), indicating no change in y.
  Understanding these curves helps visualize where populations could remain constant.
Example: For Lotka-Volterra equations:
From the model ( \frac{dx}{dt} = \frac{1}{2}x - 2xy ) set to zero to find nullclines:
- x-nullclines: Solve ( 0 = \frac{1}{2}x - 2xy ), yielding equations like ( y = \frac{\frac{1}{2} x}{2x} ), or simplifying further to express y in terms of x.
- y-nullclines: Solve ( 0 = g(x,y) ), e.g., ( x = 0 ) or other rearranged forms that help find intersections.

Definition:
A pair ( (x^,y^) ) representing conditions where both ( \frac{dx}{dt} = 0 ) and ( \frac{dy}{dt} = 0 ) are satisfied simultaneously, characterizing steady states of the system.
Finding Equilibria: Solve using the nullclines derived from both equations ( f(x^, y^) = 0 ) and ( g(x^, y^) = 0 ) to find intersection points.
For the Lotka-Volterra model, analysis of nullclines can reveal these equilibria, often corresponding to population sizes where neither species drives the other to extinction.

Stability in 1-D:
- For a one-dimensional system, check the derivative ( f'(x^\*) ) at the equilibrium point:
- ( f'(x^\*) > 0 ) indicates an unstable equilibrium as perturbations lead to divergence.
- ( f'(x^\*) < 0 ) indicates a stable equilibrium as perturbations lead to convergence back to equilibrium.
Jacobian for 2-D Systems:
- Definition: The Jacobian ( J ) is a ( 2 \times 2 ) matrix defined as:
  ( J = \begin{bmatrix} \frac{\partial f}{\partial x} & \frac{\partial f}{\partial y} \ \frac{\partial g}{\partial x} & \frac{\partial g}{\partial y} \end{bmatrix} )
- Evaluated at the equilibrium points to analyze local behavior and stability.
- Purpose: Used to classify the nature of equilibria—centers, nodes, foci—through analysis of their eigenvalues derived from this matrix, informing about local behavior.
  The trace and determinant conditions can inform whether equilibria are stable, unstable, or semi-stable based on their signs and relationships.

Calculate Jacobian: Obtain the Jacobian matrix by deriving the functions involved with respect to x and y at the determined equilibrium points.
This step often provides insight into the interaction strength between predator and prey at equal population levels.
Trace and Determinant Conditions:
- Evaluate these conditions (det(J), trace(J)) to classify equilibrium points accurately. For instance, positive trace and determinant can indicate spiral or center dynamics, whereas negative determinant with a positive trace may suggest a node.

Examined observed fixed points through graphical intersection of nullclines to identify equilibria.
- The state of each equilibrium was analyzed through Jacobian evaluation, allowing for predictions about the dynamic behavior near these points, describing them as stable or unstable based on local matrices.

Direction Vectors:
- Employ vector fields to represent the direction of movement in state space, illustrating possible trajectories of the system over time.
Qualitative Behavior:
- Develop an understanding of the system dynamics around nullclines and equilibria to anticipate shifts in behavior and population dynamics over time, illustrating potential outcomes and stabilities in realistic scenarios.