Competition, Persistence, Extinction in a Climax Population Model

1. Competition, Persistence, Extinction in a Climax Population Model Shurron Farmer Department of Mathematics Morgan State University Ph. D. Advisor: Dr. A. A. Yakubu, Howard University

2. MAIN QUESTION • What is the role of age-structure in the persistence of species?

3. Outline • What are climax species? • Mathematical Model • Theorems • Simulations • Conclusions • Further Study

4. What are Climax Species? • Species that may go extinct at small densities but have initial sets of densities that do not lead to extinction • Example: the oak tree Quercus floribunda • x(t+1)= x(t)g(x(t))

5. A Climax Growth Function

6. Example of x(t+1) = x(t)g(x(t))

7. MATHEMATICAL MODEL x(t+1) = y(t)g(ax(t) + y(t)) y(t+1) = x(t) where x(t) - population of juveniles at generation t y(t) - population of adults at generation t g - per capita growth function a - intra-specific competition coefficient

8. Reproduction Function F(x, y) = (yg(ax+y), x) where (x, y) = (x(t), y(t)) F(x, y) = (x(t+1), y(t+1)) Ft(x,y) is the population size after t generations. The domain of F is the nonnegative cone.

9. THEOREMS • Suppose the maximum value of the growth function g is less than one. Then all positive population sizes are attracted to the origin. • Suppose the maximum value of the growth function g is equal to one. Then all positive population sizes are attracted either to an equilibrium point or a 2-cycle.

10. Graph of Juvenile-adult phase plane; Maximum of g >1, a > 1

11. From one region to another

12. Maximum Value of g > 1, existence of fixed points and period 2-cycles • For any a,(0, 0),(c/(1+a), c/(1+a)), and (d/(1+a), d/(1+a)) are fixed points. • For a = 1, infinitely many 2-cycles of the form {(u, v), (v, u)} where u+v = c or u+v = d. • For a not equal to 1, if no interior 2-cycles exist, then {(0, c), (c, 0)}, {(d, 0), (0, d)}, are the only 2-cycles.

13. Theorem: Maximum Value of g > 1, no chaotic orbits • All positive population sizes are attracted either to a fixed point or a 2-cycle.

14. Sketch of Proof for I.C. In R1 • R1 is an F-invariant set. • By induction, the sequences of even and odd iterates for the juveniles (and hence for the adults) are bounded and decreasing. • Determine that the omega-limit set is the origin.

15. Ricker’s Model as Growth Function • Model (no age structure) is f(x) = x2er-x,r > 0. • The model(with or without age structure) undergoes period-doubling bifurcation route to chaos. • The model with age structure supports Hopf bifurcation and chaotic attractors.

16. Bif. Diagram (No age structure) r

17. Ricker’s Model as Growth Function (no age structure), r = 1.3

18. Ricker’s Model as growth function; r=1.3, a=2.

19. Ricker’s Model as growth function; r=1.3, a=0.1.

21. Sigmoidal Model • Growth function is g(x) = rx/(x2+s), where r, s > 0. • There are no chaotic dynamics (with or without age-structure). • Positive solutions converge to equilibrium points or 2-cycles.

22. Rep. Function for Sigmoidal Model (No Age Structure); r = 7, s = 9

23. Sigmoidal Model (Age Structure); r = 7, s = 9, a = 2.

24. CONCLUSIONS • Age structure makes it possible for a density that has extinction as its ultimate life history to have persistence as its ultimate fate with juvenile-adult competition. • Juvenile-adult competition is important in the diversity of a species.

25. Further Study • Model where juveniles and adults reproduce • Model where NOT ALL juveniles become adults • Effects of dispersion on juvenile-adult competition • Population models with some local dynamics under climax behavior and other local dynamics under pioneer behavior