Competition, Persistence, Extinction in a Climax Population Model 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

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

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

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))

A Climax Growth Function

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

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

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)) F t (x,y) is the population size after t generations. The domain of F is the nonnegative cone.

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.

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

From one region to another

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.

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

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.

Ricker’s Model as Growth Function Model (no age structure) is f(x) = x 2 e r-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.

Bif. Diagram (No age structure) r

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

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

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

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

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

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

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.

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