Differential Equations - Equilibrium Solutions

Introduction

• Logistic Growth: More realistic model compared to constant growth rate.
• Key Variables:
  • r: Intrinsic growth rate (unlimited conditions)
  • K: Carrying capacity or saturation level.

Logistic Growth Equation

• Formula: ( P = r \left(1 - \frac{P}{K}\right) P )
• Example with ( r = \frac{1}{2} ) and ( K = 10 ).
• Equilibrium solutions occur when derivative ( \frac{dP}{dt} = 0 ), i.e., ( P = 0 ) and ( P = 10 ).

Direction Fields

• Derivative zero at equilibrium points.
• Population does not grow at ( P = 0 ), stabilizes at ( P = 10 ).
• Population declines if ( P > 10 ).

Realistic Population Behavior

• Unlimited growth is unrealistic; resources limit population size.
• If initial population is above carrying capacity, it will reduce to ( K ).

Autonomous Differential Equations

• Form: ( \frac{dy}{dt} = f(y) ), independent of time variable.
• Equilibrium solutions: ( f(y_0) = 0 ) implies solutions are constant over time.

Classifying Equilibrium Solutions

• Stable: Solutions near equilibrium move towards it.
• Unstable: Solutions near equilibrium move away from it.

Examples

1. Example 1: Equation ( y = y^2 - 6 )
   • Find and classify equilibrium solutions.
2. Example 2: Equation ( y = (y^2 - 4)(y + 1)^2 )
   • Introduces the third classification of equilibrium solutions.

Conclusion

• Equilibrium solutions help predict long-term behavior of differential equations.
• Important to classify and understand behavior near equilibrium points.