1 / 7

Population Dynamics

This text explores the dynamics of a population model through the lens of linearization and eigenvalue analysis. It identifies critical points, specifically (0,0), (1,0), (0,2), and (0.5,0.5), highlighting their stability characteristics. The analysis reveals that (0,0) is an unstable node, while (1,0) and (0,2) are asymptotically stable nodes. The saddle point nature of (0.5,0.5) is also discussed. Phase portraits illustrate how trajectories interact with these equilibrium points, emphasizing the practicality of species survival in competitive dynamics.

nerice
Télécharger la présentation

Population Dynamics

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Population Dynamics Application of Eigenvalues & Eigenvectors

  2. Consider the system of equations • The critical points are (0,0), (1,0), (0,2) & (.5,.5). These critical points correspond to equilibrium solutions

  3. Linearization for critical point (0,0) • For this critical point the approximating linear system is • The eigenvalues and eigenvectors are • Thus (0,0) is an unstable node for both the linear and nonlinear systems

  4. Linearization for critical point (1,0) • For this critical point the approximating linear system is • The eigenvalues and eigenvectors are • Thus (1,0) is an asymptotically stable node of both the linear and nonlinear systems

  5. Linearization for critical point (0,2) • For this critical point the approximating linear system is • The eigenvalues and eigenvectors are • Thus (0,2) is an asymptotically stable node for both the linear and nonlinear systems

  6. Linearization for critical point (.5,.5) • For this critical point the approximating linear system is • The eigenvalues and eigenvectors are • Thus (0,2) is a unstable saddle node for both the linear and nonlinear systems

  7. Phase Portrait & Direction Field Trajectories starting above the separatrix approach the node at (0,2), while those below approach the node at (1,0). If initial state lies on separatrix, then the solution will approach the saddle point, but the slightest perturbation will send the trajectory to one of the nodes instead. Thus in practice, one species will survive the competition and the other species will not.

More Related