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1. Presentation On Analysis of the 'Rabbits vs. Sheep' Model Presenter Shamima Akter M.S Student

2. Agenda introduce the "Rabbits vs. Sheep" model. Finding fixed Point Investigate their stability Drawing Nullclines Sketch Plausible Phase portraits Indicating basins of Attraction of any stable fixed points.

3. Rabbits vs. Sheep Model The Rabbit vs. Sheep model provides a concise illustration of Lotka -Volterra's competition dynamics between two species, featuring rabbits and sheep as the focal organisms. Considering Organisms competing for same food Resources are limited

4. Mathematical Model:   Were Population of Sheep’s Population of Rabbits  

5. Fixed Points of the system : S olving equation We find following fixed points   There are 4 different types of equilibrium points based on eigenvalues: Eigenvalues Types of Node 1 are real and same sign. ( Node 2 are real and non zero opposite sign. ( Saddle 3 complex and real part non zero equal Re( Spiral 4 are purely imaginary Centre Eigenvalues Types of Node 1 Node 2 Saddle 3 Spiral 4 Centre

6. Stability Analysis   Jacobi matrix for the system Steady State Eigenvalues Stability Unstable Node Saddle point Stable Node Steady State Eigenvalues Stability Unstable Node Saddle point Stable Node Solving Charactristicequation for each fixed point we get following eigenvalues.  

7. 7 Nullclines

8. 8 Trajectories are tangential to the slow eigendirection (i.e. smallest |λ|) at a node Phase portrait For Each fixed point  

9. 9 Phase Portrait  

10. 10 Phase portrait at  

11. 11 Putting these together, the phase portrait becomes

12. 12 Approaching Stable fixed point for different initial condition

13. 13 Rabbits vs sheep's dynamics respect to time

14. 14 On Going work Advance plot for basis of attraction using MATLAB.

15. 9/3/20XX Presentation Title 15 Biological interpretation In general, one species eventually drives the other to extinction; which species eventually dominates depends on initial populations   Basin of attraction of an attracting fixed point x ∗ defined as the set of initial conditions x0 such that x → x ∗ as t → ∞

16. Thank you