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))
Ft(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.
plane; Maximum of g >1, a >
1
From one region to another
R1 R2 R7
R1 R1 R3 R4 R1 R2 R3 R4 R5 R6 R7
R1 R2 R4 R5 R4 R5 R6
R6
R1 R2 R5
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) = 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.
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/(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.
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