Analysis of the Lotka Volterra Equations as a Technological Substitution Model

1. Analysis of the Lotka Volterra Equations as a Technological Substitution Model Steven Morris October 25, 2001

2. DISCONTINUOUS TECHNOLOGY SUBSTITUTION OLD DOMINANT TECHNOLOGY OBSOLESENCE COMPETITION AMONG MANY, NO STANDARD PRODUCT EMERGENCE OF STANDARD PRODUCT INNOVATION NEW DOMINANT TECHNOLOGY

3. Lotka-Volterra Competition Equations • Model interaction of between species competing for the same resources • Model short term “struggle for existence” • Original applied to biological systems

4. Historical Use of LVC model • Qualitative analysis using nullclines • Under arbitrary initial conditions, find stable solutions • No scaling of parameters • No quantitative analysis of dynamic response

5. Key Question: State the problem as commonly found in practice:Given a dominant competitor at equilibrium, what happens when a better suited competitor appears? We describe the resulting replacement curve as “LVC substitution”.

6. DISCONTINUOUS TECHNOLOGY SUBSTITUTION OLD DOMINANT TECHNOLOGY OBSOLESENCE COMPETITION AMONG MANY, NO STANDARD PRODUCT EMERGENCE OF STANDARD PRODUCT INNOVATION NEW DOMINANT TECHNOLOGY

7. Analysis of LVC Substitution • Step 1: Normalize LVC equations • Step 2:Find models parameters and initial conditions that result in LVC substitution • Step 3: Analyze dynamics of substitution • Step 4: Compare to other substitution models

8. LVC Equations

9. Some Definitions

10. Normalized LVC Equations

11. Nullcline Equations

12. Decreasing u1 Decreasing u2 Increasing u1 Increasing u2 ANALYSIS OF NULLCLINES

13. CO-EXISTENCE

14. LOCKOUT

15. Dominance of u1, extinction of u2

16. Dominance of u2, extinction of u1

17. LVC Substitution of u1by u2 • u1 is dominant competitor • u2 is invader with advantage • 2 > 1, 1 < 1

19. Final condition: u1 = 0, u2= 1 u2= 1 (b) 1-u2 (a) u1= 0 Initial condition: u1 = 1, u2 0 u2= 0 u1= 1 1-u1

20. u2 u1

21. u2 

22. e =0.8 1 e =1.5 2 g =1.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 u2

23. FISHER PRY PLOT e =0.8 1 e =1.5 2 g =1.0 

24. 4. Right asymmetric with  asymptote 1+ 3. Right asymmetric with 2-1 asymptote 2 2. Logistic 1. Left asymmetric 1 0 1 1

30. u Slope = 1 delay

31. 1  2 0.9880 1.0248 10.1466 1.946

32. a u BASS MODEL

33. inset ‘c’ u

34. NSRL MODEL  u

35. inset ‘c’ u

36. SHARIF-KABIR MODEL

37. 5 4 3 2 inset ‘c’ u 1 5 4 3 2 1

38. TOTAL CANS FOR VEGETABLES SUM OF LEAD-FREE AND SOLDERED CANS BILLION UNITS SOLDERED CANS LEAD-FREE CANS YEAR

39. 1 = 0.75  2 = 1.25 a1 = a2 = 2 yr -1 K1 = K2 = 1 t(0)=1975 u2(0)=1.52% SOLDERED CANS PERCENT OF MARKET LEAD-FREE CANS SOLDERED CANS LEAD-FREE CANS YEAR

40. SIGNIFICANT RESULTS • NORMALIZED LVC MODEL • ASYMPTOTIC BEHAVIOR DURING SUBSTITUTION • PREDICTION OF RESPONSE CLASSES BASED ON LVC PARAMETERS • REVERSION OF LVC TO LOGISTIC SUBSTITUTION • LOGISTIC SUBSTITUTION IN FIXED MARKET • GRAPHICAL ANALYSIS TECHNIQUE • NORMALIZED BASS MODEL • COMPARISON OF LVC MODEL TO OTHER SUBSTITUTION MODELS