This will be a research using mathematical concepts of a specific endangered species’ population over time. At least half of the paper has to be calculations and operations that show integration of the following concepts below. I will also upload a sample paper that is similar to the essay you will write. Use these mathematical concepts.
1. Differential Equations
- Modeling Population Growth: Differential equations are foundational in modeling changes in systems over time. The logistic growth model, with its concept of carrying capacity, is a classic example of how non-linear dynamics can lead to different types of behavior, from stable to chaotic, under varying conditions.
- Lotka-Volterra Equations: These represent predator-prey interactions, a classic example of non-linear systems in ecology. They can exhibit oscillatory or even chaotic dynamics under certain conditions, illustrating the complexity inherent in biological systems.
2. Stability Analysis
- Equilibrium Points and Stability: Understanding the stability of equilibrium points in a system is crucial in chaos theory. It helps in identifying how systems respond to small perturbations, which is essential for understanding chaotic dynamics.
- 3. Non-linear Dynamics and Chaos
- Logistic Map: This is a quintessential example of how simple non-linear equations can exhibit complex, chaotic behavior. By varying the rate parameter
, the logistic map can transition from stable to periodic and then to chaotic behavior, serving as a clear illustration of chaos theory in action. - Sensitivity to Initial Conditions: A hallmark of chaotic systems is their extreme sensitivity to initial conditions. This principle is directly explored through calculus and numerical simulations, demonstrating how small differences can lead to vastly divergent outcomes over time.