Competition, Persistence, Competition, Persistence, Extinction in a Climax Population Extinction in a Climax Population ModelModel

Shurron FarmerShurron FarmerDepartment of MathematicsDepartment of Mathematics

Morgan State UniversityMorgan State UniversityPh. D. Advisor: Dr. A. A. Yakubu, Ph. D. Advisor: Dr. A. A. Yakubu,

Howard UniversityHoward University

MAIN QUESTIONMAIN QUESTION

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

OutlineOutline

What are climax species? Mathematical ModelTheoremsSimulationsConclusionsFurther Study

What are Climax Species?What are Climax Species?

Species that may go extinct at small densities but have initial sets of densities that do not lead to extinctionExample: the oak tree Quercus floribundax(t+1)= x(t)g(x(t))

A Climax Growth FunctionA Climax Growth Function

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

MATHEMATICAL MODELMATHEMATICAL MODEL

x(t+1) = y(t)g(ax(t) + y(t))y(t+1) = x(t)

wherex(t) - population of juveniles at generation ty(t) - population of adults at generation tg - per capita growth functiona - intra-specific competition coefficient

Reproduction FunctionReproduction FunctionF(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.

THEOREMSTHEOREMS

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

From one region to From one region to anotheranother

R1

R1

R1

R1 R2 R4 R5

R3

R4

R1 R2 R5

R6

R5 R6

R4

R2

R1 R2 R3 R4 R5 R6 R7

R7

Maximum Value of Maximum Value of g > 1g > 1, , existence of fixed points and existence of fixed points and period 2-cyclesperiod 2-cycles

For any For any a,a, (0, 0),(0, 0), (c/(1+a), c/(1+a)),(c/(1+a), c/(1+a)), and and (d/(1+a), d/(1+a))(d/(1+a), d/(1+a)) are fixed points. are fixed points.

For a = 1, infinitely many 2-cycles of For a = 1, infinitely many 2-cycles of the form the form {(u, v), (v, u)}{(u, v), (v, u)} where where u+v = cu+v = c or or u+v = d.u+v = d.

For For aa not equal to 1, if no interior 2- not equal to 1, if no interior 2-cycles exist, then cycles exist, then {(0, c), (c, 0)}{(0, c), (c, 0)}, , {(d, {(d, 0), (0, d)},0), (0, d)}, are the only 2-cycles. are the only 2-cycles.

Theorem: Maximum Value Theorem: Maximum Value of of g > 1,g > 1, no chaotic orbits no chaotic orbits

All positive population sizes are All positive population sizes are attracted either to a fixed point or a 2-attracted either to a fixed point or a 2-cycle.cycle.

Sketch of Proof for I.C. In Sketch of Proof for I.C. In R1R1

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 Ricker’s Model as Growth FunctionFunction

Model (no age structure) is Model (no age structure) is f(x) = f(x) = xx22eer-xr-x,, r > 0.r > 0.

The modelThe model (with or without age (with or without age structure) undergoes period-structure) undergoes period-doubling bifurcation route to chaos.doubling bifurcation route to chaos.

The model with age structure The model with age structure supports Hopf bifurcation and supports Hopf bifurcation and chaotic attractors.chaotic attractors.

Bif. Diagram (No age Bif. Diagram (No age structure)structure)

r

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

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

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

Sigmoidal ModelSigmoidal Model

Growth function is Growth function is g(x) = rx/(xg(x) = rx/(x22+s), +s), wherewhere r, s > 0. r, s > 0.

There are no chaotic dynamics There are no chaotic dynamics (with or without age-structure).(with or without age-structure).

Positive solutions converge to Positive solutions converge to equilibrium points or 2-cycles.equilibrium points or 2-cycles.

Rep. Function for Sigmoidal Rep. Function for Sigmoidal Model (No Age Structure); Model (No Age Structure); r = r = 7, s = 97, s = 9

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

CONCLUSIONSCONCLUSIONS

Age structure makes it possible for Age structure makes it possible for a density that has extinction as its a density that has extinction as its ultimate life history to have ultimate life history to have persistence as its ultimate fate with persistence as its ultimate fate with juvenile-adult competition.juvenile-adult competition.

Juvenile-adult competition is Juvenile-adult competition is important in the diversity of a important in the diversity of a species.species.

Further StudyFurther Study

Model where juveniles and adults reproduceModel where NOT ALL juveniles become adultsEffects of dispersion on juvenile-adult competition Population models with some local dynamics under climax behavior and other local dynamics under pioneer behavior